cuSOLVER :: CUDA Toolkit Documentation

cuSOLVER

The API reference guide for cuSOLVER, the CUDA sparse matrix library.

1. Introduction

The cuSOLVER library is a high-level package based on the cuBLAS and cuSPARSE libraries. It consists of two modules corresponding to two sets of API:

The cuSolver API on a single GPU
The cuSolverMG API on a single node multiGPU

Each of which can be used independently or in concert with other toolkit libraries. To simplify the notation, cuSolver denotes single GPU API and cuSolverMg denotes multiGPU API.

The intent of cuSolver is to provide useful LAPACK-like features, such as common matrix factorization and triangular solve routines for dense matrices, a sparse least-squares solver and an eigenvalue solver. In addition cuSolver provides a new refactorization library useful for solving sequences of matrices with a shared sparsity pattern.

cuSolver combines three separate components under a single umbrella. The first part of cuSolver is called cuSolverDN, and deals with dense matrix factorization and solve routines such as LU, QR, SVD and LDLT, as well as useful utilities such as matrix and vector permutations.

Next, cuSolverSP provides a new set of sparse routines based on a sparse QR factorization. Not all matrices have a good sparsity pattern for parallelism in factorization, so the cuSolverSP library also provides a CPU path to handle those sequential-like matrices. For those matrices with abundant parallelism, the GPU path will deliver higher performance. The library is designed to be called from C and C++.

The final part is cuSolverRF, a sparse re-factorization package that can provide very good performance when solving a sequence of matrices where only the coefficients are changed but the sparsity pattern remains the same.

The GPU path of the cuSolver library assumes data is already in the device memory. It is the responsibility of the developer to allocate memory and to copy data between GPU memory and CPU memory using standard CUDA runtime API routines, such as cudaMalloc(), cudaFree(), cudaMemcpy(), and cudaMemcpyAsync().

cuSolverMg is GPU-accelerated ScaLAPACK. By now, cuSolverMg supports 1-D column block cyclic layout and provides symmetric eigenvalue solver.

Note: The cuSolver library requires hardware with a CUDA compute capability (CC) of at least 2.0 or higher. Please see the CUDA C++ Programming Guide for a list of the compute capabilities corresponding to all NVIDIA GPUs.

cuSolverDN: Dense LAPACK

The cuSolverDN library was designed to solve dense linear systems of the form

A x = b

where the coefficient matrix $A \in R^{nxn}$ , right-hand-side vector $b \in R^{n}$ and solution vector $x \in R^{n}$

The cuSolverDN library provides QR factorization and LU with partial pivoting to handle a general matrix A, which may be non-symmetric. Cholesky factorization is also provided for symmetric/Hermitian matrices. For symmetric indefinite matrices, we provide Bunch-Kaufman (LDL) factorization.

The cuSolverDN library also provides a helpful bidiagonalization routine and singular value decomposition (SVD).

The cuSolverDN library targets computationally-intensive and popular routines in LAPACK, and provides an API compatible with LAPACK. The user can accelerate these time-consuming routines with cuSolverDN and keep others in LAPACK without a major change to existing code.

cuSolverSP: Sparse LAPACK

The cuSolverSP library was mainly designed to a solve sparse linear system

A x = b

and the least-squares problem

x = argmin || A * z - b ||

where sparse matrix $A \in R^{mxn}$ , right-hand-side vector $b \in R^{m}$ and solution vector $x \in R^{n}$ . For a linear system, we require m=n.

The core algorithm is based on sparse QR factorization. The matrix A is accepted in CSR format. If matrix A is symmetric/Hermitian, the user has to provide a full matrix, ie fill missing lower or upper part.

If matrix A is symmetric positive definite and the user only needs to solve $A x = b$ , Cholesky factorization can work and the user only needs to provide the lower triangular part of A.

On top of the linear and least-squares solvers, the cuSolverSP library provides a simple eigenvalue solver based on shift-inverse power method, and a function to count the number of eigenvalues contained in a box in the complex plane.

cuSolverRF: Refactorization

The cuSolverRF library was designed to accelerate solution of sets of linear systems by fast re-factorization when given new coefficients in the same sparsity pattern

A_{i} x_{i} = f_{i}

where a sequence of coefficient matrices $A_{i} \in R^{nxn}$ , right-hand-sides $f_{i} \in R^{n}$ and solutions $x_{i} \in R^{n}$ are given for i=1,...,k.

The cuSolverRF library is applicable when the sparsity pattern of the coefficient matrices $A_{i}$ as well as the reordering to minimize fill-in and the pivoting used during the LU factorization remain the same across these linear systems. In that case, the first linear system (i=1) requires a full LU factorization, while the subsequent linear systems (i=2,...,k) require only the LU re-factorization. The later can be performed using the cuSolverRF library.

Notice that because the sparsity pattern of the coefficient matrices, the reordering and pivoting remain the same, the sparsity pattern of the resulting triangular factors $L_{i}$ and $U_{i}$ also remains the same. Therefore, the real difference between the full LU factorization and LU re-factorization is that the required memory is known ahead of time.

1.4. Naming Conventions

The cuSolverDN library functions are available for data types float, double, cuComplex, and cuDoubleComplex. The naming convention is as follows:

cusolverDn<t><operation>

where <t> can be S, D, C, Z, or X, corresponding to the data types float, double, cuComplex, cuDoubleComplex, and the generic type, respectively. <operation> can be Cholesky factorization (potrf), LU with partial pivoting (getrf), QR factorization (geqrf) and Bunch-Kaufman factorization (sytrf).

The cuSolverSP library functions are available for data types float, double, cuComplex, and cuDoubleComplex. The naming convention is as follows:

cusolverSp[Host]<t>[<matrix data format>]<operation>[<output matrix data format>]<based on>

where cuSolverSp is the GPU path and cusolverSpHost is the corresponding CPU path. <t> can be S, D, C, Z, or X, corresponding to the data types float, double, cuComplex, cuDoubleComplex, and the generic type, respectively.

The <matrix data format> is csr, compressed sparse row format.

The <operation> can be ls, lsq, eig, eigs, corresponding to linear solver, least-square solver, eigenvalue solver and number of eigenvalues in a box, respectively.

The <output matrix data format> can be v or m, corresponding to a vector or a matrix.

<based on> describes which algorithm is used. For example, qr (sparse QR factorization) is used in linear solver and least-square solver.

All of the functions have the return type cusolverStatus_t and are explained in more detail in the chapters that follow.

cuSolverSP API

`routine`	`data format`	`operation`	`output format`	`based on`
`csrlsvlu`	`csr`	`linear solver (ls)`	`vector (v)`	`LU (lu) with partial pivoting`
`csrlsvqr`	`csr`	`linear solver (ls)`	`vector (v)`	`QR factorization (qr)`
`csrlsvchol`	`csr`	`linear solver (ls)`	`vector (v)`	`Cholesky factorization (chol)`
`csrlsqvqr`	`csr`	`least-square solver (lsq)`	`vector (v)`	`QR factorization (qr)`
`csreigvsi`	`csr`	`eigenvalue solver (eig)`	`vector (v)`	`shift-inverse`
`csreigs`	`csr`	`number of eigenvalues in a box (eigs)`
`csrsymrcm`	`csr`	`Symmetric Reverse Cuthill-McKee (symrcm)`

The cuSolverRF library routines are available for data type double. Most of the routines follow the naming convention:

cusolverRf_<operation>_[[Host]](...)

where the trailing optional Host qualifier indicates the data is accessed on the host versus on the device, which is the default. The <operation> can be Setup, Analyze, Refactor, Solve, ResetValues, AccessBundledFactors and ExtractSplitFactors.

Finally, the return type of the cuSolverRF library routines is cusolverStatus_t.

1.5. Asynchronous Execution

The cuSolver library functions prefer to keep asynchronous execution as much as possible. Developers can always use the cudaDeviceSynchronize() function to ensure that the execution of a particular cuSolver library routine has completed.

A developer can also use the cudaMemcpy() routine to copy data from the device to the host and vice versa, using the cudaMemcpyDeviceToHost and cudaMemcpyHostToDevice parameters, respectively. In this case there is no need to add a call to cudaDeviceSynchronize() because the call to cudaMemcpy() with the above parameters is blocking and completes only when the results are ready on the host.

1.6. Library Property

The libraryPropertyType data type is an enumeration of library property types. (ie. CUDA version X.Y.Z would yield MAJOR_VERSION=X, MINOR_VERSION=Y, PATCH_LEVEL=Z)

typedef enum libraryPropertyType_t
{
        MAJOR_VERSION,
        MINOR_VERSION,
        PATCH_LEVEL
} libraryPropertyType;

The following code can show the version of cusolver library.

    int major=-1,minor=-1,patch=-1;
    cusolverGetProperty(MAJOR_VERSION, &major);
    cusolverGetProperty(MINOR_VERSION, &minor);
    cusolverGetProperty(PATCH_LEVEL, &patch);
    printf("CUSOLVER Version (Major,Minor,PatchLevel): %d.%d.%d\n", major,minor,patch);

1.7. high precision package

The cusolver library uses high precision for iterative refinement when necessary.

2. Using the CUSOLVER API

2.1. General description

This chapter describes how to use the cuSolver library API. It is not a reference for the cuSolver API data types and functions; that is provided in subsequent chapters.

2.1.1. Thread Safety

The library is thread-safe, and its functions can be called from multiple host threads.

2.1.2. Scalar Parameters

In the cuSolver API, the scalar parameters can be passed by reference on the host.

2.1.3. Parallelism with Streams

If the application performs several small independent computations, or if it makes data transfers in parallel with the computation, then CUDA streams can be used to overlap these tasks.

The application can conceptually associate a stream with each task. To achieve the overlap of computation between the tasks, the developer should:

Create CUDA streams using the function cudaStreamCreate(), and
Set the stream to be used by each individual cuSolver library routine by calling, for example, cusolverDnSetStream(), just prior to calling the actual cuSolverDN routine.

The computations performed in separate streams would then be overlapped automatically on the GPU, when possible. This approach is especially useful when the computation performed by a single task is relatively small, and is not enough to fill the GPU with work, or when there is a data transfer that can be performed in parallel with the computation.

2.1.4. Link Third-party LAPACK Library

Starting with CUDA 10.1 update 2, NVIDIA LAPACK library liblapack_static.a is a subset of LAPACK and only contains GPU accelerated stedc and bdsqr. The user has to link libcusolver_static.a with liblapack_static.a in order to build the application successfully. Prior to CUDA 10.1 update 2, the user can replace liblapack_static.a with a third-party LAPACK library, for example, MKL. In CUDA 10.1 update 2, the third-party LAPACK library no longer affects the behavior of cusolver library, neither functionality nor performance. Furthermore the user cannot use liblapack_static.a as a standalone LAPACK library because it is only a subset of LAPACK.

Note: Theliblapack_static.a library, which is the binary of CLAPACK-3.2.1, is a new feature of CUDA 10.0.

If you use libcusolver_static.a, then you must link with liblapack_static.a explicitly, otherwise the linker will report missing symbols. No conflict of symbols between liblapack_static.a and other third-party LAPACK library, you are free to link the latter to your application.
The liblapack_static.a is built inside libcusolver.so. Hence, if you use libcusolver.so, then you don't need to specify a LAPACK library. The libcusolver.so will not pick up any routines from the third-party LAPACK library even you link the application with it.

2.1.5. convention of info

Each LAPACK routine returns an info which indicates the position of invalid parameter. If info = -i, then i-th parameter is invalid. To be consistent with base-1 in LAPACK, cusolver does not report invalid handle into info. Instead, cusolver returns CUSOLVER_STATUS_NOT_INITIALIZED for invalid handle.

2.1.6. usage of _bufferSize

There is no cudaMalloc inside cuSolver library, the user must allocate the device workspace explicitly. The routine xyz_bufferSize is to query the size of workspace of the routine xyz, for example xyz = potrf. To make the API simple, xyz_bufferSize follows almost the same signature of xyz even it only depends on some parameters, for example, device pointer is not used to decide the size of workspace. In most cases, xyz_bufferSize is called in the beginning before actual device data (pointing by a device pointer) is prepared or before the device pointer is allocated. In such case, the user can pass null pointer to xyz_bufferSize without breaking the functionality.

2.2. cuSolver Types Reference

2.2.1. cuSolverDN Types

The float, double, cuComplex, and cuDoubleComplex data types are supported. The first two are standard C data types, while the last two are exported from cuComplex.h. In addition, cuSolverDN uses some familiar types from cuBlas.

2.2.1.1. cusolverDnHandle_t

This is a pointer type to an opaque cuSolverDN context, which the user must initialize by calling cusolverDnCreate() prior to calling any other library function. An un-initialized Handle object will lead to unexpected behavior, including crashes of cuSolverDN. The handle created and returned by cusolverDnCreate() must be passed to every cuSolverDN function.

2.2.1.2. cublasFillMode_t

The type indicates which part (lower or upper) of the dense matrix was filled and consequently should be used by the function. Its values correspond to Fortran characters ‘L’ or ‘l’ (lower) and ‘U’ or ‘u’ (upper) that are often used as parameters to legacy BLAS implementations.

`Value`	`Meaning`
`CUBLAS_FILL_MODE_LOWER`	the lower part of the matrix is filled
`CUBLAS_FILL_MODE_UPPER`	the upper part of the matrix is filled

2.2.1.3. cublasOperation_t

The cublasOperation_t type indicates which operation needs to be performed with the dense matrix. Its values correspond to Fortran characters ‘N’ or ‘n’ (non-transpose), ‘T’ or ‘t’ (transpose) and ‘C’ or ‘c’ (conjugate transpose) that are often used as parameters to legacy BLAS implementations.

`Value`	`Meaning`
`CUBLAS_OP_N`	the non-transpose operation is selected
`CUBLAS_OP_T`	the transpose operation is selected
`CUBLAS_OP_C`	the conjugate transpose operation is selected

2.2.1.4. cusolverEigType_t

The cusolverEigType_t type indicates which type of eigenvalue solver is. Its values correspond to Fortran integer 1 (A*x = lambda*B*x), 2 (A*B*x = lambda*x), 3 (B*A*x = lambda*x), used as parameters to legacy LAPACK implementations.

`Value`	`Meaning`
`CUSOLVER_EIG_TYPE_1`	Ax = lambdaB*x
`CUSOLVER_EIG_TYPE_2`	ABx = lambda*x
`CUSOLVER_EIG_TYPE_3`	BAx = lambda*x

2.2.1.5. cusolverEigMode_t

The cusolverEigMode_t type indicates whether or not eigenvectors are computed. Its values correspond to Fortran character 'N' (only eigenvalues are computed), 'V' (both eigenvalues and eigenvectors are computed) used as parameters to legacy LAPACK implementations.

`Value`	`Meaning`
`CUSOLVER_EIG_MODE_NOVECTOR`	only eigenvalues are computed
`CUSOLVER_EIG_MODE_VECTOR`	both eigenvalues and eigenvectors are computed

2.2.1.6. cusolverIRSRefinement_t

The cusolverIRSRefinement_t type indicates which solver type would be used for the specific cusolver function. Most of our experimentation shows that CUSOLVER_IRS_REFINE_GMRES is the best option.

More details about the refinement process can be found in Azzam Haidar, Stanimire Tomov, Jack Dongarra, and Nicholas J. Higham. 2018. Harnessing GPU tensor cores for fast FP16 arithmetic to speed up mixed-precision iterative refinement solvers. In Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis (SC '18). IEEE Press, Piscataway, NJ, USA, Article 47, 11 pages.

`Value`	`Meaning`
`CUSOLVER_IRS_REFINE_NOT_SET`	Solver is not set. Default value.
`CUSOLVER_IRS_REFINE_NONE`	No solver
`CUSOLVER_IRS_REFINE_CLASSICAL`	Classical iterative refinement solver. Similar to the one used in LAPACK routines.
`CUSOLVER_IRS_REFINE_GMRES`	GMRES (Generalized Minimal Residual) based iterative refinement solver. In recent study, the GMRES method has drawn the scientific community attention for its ability to be used as refinement solver that outperforms the classical iterative refinement method.
`CUSOLVER_IRS_REFINE_CLASSICAL_GMRES`	Classical iterative refinement solver that uses the GMRES (Generalized Minimal Residual) internally to solve the correction equation at each iteration. We call the classical refinement iteration the outer iteration while the GMRES is called inner iteration. Note that if the tolerance of the inner GMRES is set very low, let say to machine precision, then the outer classical refinement iteration will performs only one iteration and thus this option will behaves like CUSOLVER_IRS_REFINE_GMRES.
`CUSOLVER_IRS_REFINE_GMRES_GMRES`	Similar to CUSOLVER_IRS_REFINE_CLASSICAL_GMRES which is classical refinement process that uses GMRES to solve the inner correction system, here it is a GMRES (Generalized Minimal Residual) based iterative refinement solver that uses another GMRES internally to solve the preconditionned system.

2.2.1.7. cusolverDnIRSParams_t

This is a pointer type to an opaque cusolverDnIRSParams_t structure, which holds parameters for the iterative refinement linear solvers such as cusolverDnXgesv(). Use corresponding helper functions described below to either Create/Destroy this structure or Set/Get solver parameters.

2.2.1.8. cusolverDnIRSInfos_t

This is a pointer type to an opaque cusolverDnIRSInfos_t structure, which holds information about the performed call to an iterative refinement linear solver (e.g., cusolverDnXgesv()). Use corresponding helper functions described below to either Create/Destroy this structure or retrieve solve information.

2.2.1.9. cusolverStatus_t

This is the same as cusolverStatus_t in the sparse LAPACK section.

2.2.2. cuSolverSP Types

The float, double, cuComplex, and cuDoubleComplex data types are supported. The first two are standard C data types, while the last two are exported from cuComplex.h.

2.2.2.1. cusolverSpHandle_t

This is a pointer type to an opaque cuSolverSP context, which the user must initialize by calling cusolverSpCreate() prior to calling any other library function. An un-initialized Handle object will lead to unexpected behavior, including crashes of cuSolverSP. The handle created and returned by cusolverSpCreate() must be passed to every cuSolverSP function.

2.2.2.2. cusparseMatDescr_t

We have chosen to keep the same structure as exists in cuSparse to describe the shape and properties of a matrix. This enables calls to either cuSparse or cuSolver using the same matrix description.

typedef struct {
    cusparseMatrixType_t MatrixType;
    cusparseFillMode_t FillMode;
    cusparseDiagType_t DiagType;
    cusparseIndexBase_t IndexBase;
} cusparseMatDescr_t;

Please read documenation of CUSPARSE Library to understand each field of cusparseMatDescr_t.

2.2.2.3. cusolverStatus_t

This is a status type returned by the library functions and it can have the following values.

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	The cuSolver library was not initialized. This is usually caused by the lack of a prior call, an error in the CUDA Runtime API called by the cuSolver routine, or an error in the hardware setup. To correct: call `cusolverCreate()` prior to the function call; and check that the hardware, an appropriate version of the driver, and the cuSolver library are correctly installed.
`CUSOLVER_STATUS_ALLOC_FAILED`	Resource allocation failed inside the cuSolver library. This is usually caused by a `cudaMalloc()` failure. To correct: prior to the function call, deallocate previously allocated memory as much as possible.
`CUSOLVER_STATUS_INVALID_VALUE`	An unsupported value or parameter was passed to the function (a negative vector size, for example). To correct: ensure that all the parameters being passed have valid values.
`CUSOLVER_STATUS_ARCH_MISMATCH`	The function requires a feature absent from the device architecture; usually caused by the lack of support for atomic operations or double precision. To correct: compile and run the application on a device with compute capability 2.0 or above.
`CUSOLVER_STATUS_EXECUTION_FAILED`	The GPU program failed to execute. This is often caused by a launch failure of the kernel on the GPU, which can be caused by multiple reasons. To correct: check that the hardware, an appropriate version of the driver, and the cuSolver library are correctly installed.
`CUSOLVER_STATUS_INTERNAL_ERROR`	An internal cuSolver operation failed. This error is usually caused by a `cudaMemcpyAsync()` failure. To correct: check that the hardware, an appropriate version of the driver, and the cuSolver library are correctly installed. Also, check that the memory passed as a parameter to the routine is not being deallocated prior to the routine’s completion.
`CUSOLVER_STATUS_MATRIX_TYPE_NOT_SUPPORTED`	The matrix type is not supported by this function. This is usually caused by passing an invalid matrix descriptor to the function. To correct: check that the fields in `descrA` were set correctly.

2.2.3. cuSolverRF Types

cuSolverRF only supports double.

cusolverRfHandle_t

The cusolverRfHandle_t is a pointer to an opaque data structure that contains the cuSolverRF library handle. The user must initialize the handle by calling cusolverRfCreate() prior to any other cuSolverRF library calls. The handle is passed to all other cuSolverRF library calls.

cusolverRfMatrixFormat_t

The cusolverRfMatrixFormat_t is an enum that indicates the input/output matrix format assumed by the cusolverRfSetupDevice(), cusolverRfSetupHost(), cusolverRfResetValues(), cusolveRfExtractBundledFactorsHost() and cusolverRfExtractSplitFactorsHost() routines.

`Value`	`Meaning`
`CUSOLVER_MATRIX_FORMAT_CSR`	matrix format CSR is assumed. (default)
`CUSOLVER_MATRIX_FORMAT_CSC`	matrix format CSC is assumed.

cusolverRfNumericBoostReport_t

The cusolverRfNumericBoostReport_t is an enum that indicates whether numeric boosting (of the pivot) was used during the cusolverRfRefactor() and cusolverRfSolve() routines. The numeric boosting is disabled by default.

`Value`	`Meaning`
`CUSOLVER_NUMERIC_BOOST_NOT_USED`	numeric boosting not used. (default)
`CUSOLVER_NUMERIC_BOOST_USED`	numeric boosting used.

cusolverRfResetValuesFastMode_t

The cusolverRfResetValuesFastMode_t is an enum that indicates the mode used for the cusolverRfResetValues() routine. The fast mode requires extra memory and is recommended only if very fast calls to cusolverRfResetValues() are needed.

`Value`	`Meaning`
`CUSOLVER_RESET_VALUES_FAST_MODE_OFF`	fast mode disabled. (default)
`CUSOLVER_RESET_VALUES_FAST_MODE_ON`	fast mode enabled.

cusolverRfFactorization_t

The cusolverRfFactorization_t is an enum that indicates which (internal) algorithm is used for refactorization in the cusolverRfRefactor() routine.

`Value`	`Meaning`
`CUSOLVER_FACTORIZATION_ALG0`	algorithm 0. (default)
`CUSOLVER_FACTORIZATION_ALG1`	algorithm 1.
`CUSOLVER_FACTORIZATION_ALG2`	algorithm 2. Domino-based scheme.

cusolverRfTriangularSolve_t

The cusolverRfTriangularSolve_t is an enum that indicates which (internal) algorithm is used for triangular solve in the cusolverRfSolve() routine.

`Value`	`Meaning`
`CUSOLVER_TRIANGULAR_SOLVE_ALG0`	algorithm 0.
`CUSOLVER_TRIANGULAR_SOLVE_ALG1`	algorithm 1. (default)
`CUSOLVER_TRIANGULAR_SOLVE_ALG2`	algorithm 2. Domino-based scheme.
`CUSOLVER_TRIANGULAR_SOLVE_ALG3`	algorithm 3. Domino-based scheme.

cusolverRfUnitDiagonal_t

The cusolverRfUnitDiagonal_t is an enum that indicates whether and where the unit diagonal is stored in the input/output triangular factors in the cusolverRfSetupDevice(), cusolverRfSetupHost() and cusolverRfExtractSplitFactorsHost() routines.

`Value`	`Meaning`
`CUSOLVER_UNIT_DIAGONAL_STORED_L`	unit diagonal is stored in lower triangular factor. (default)
`CUSOLVER_UNIT_DIAGONAL_STORED_U`	unit diagonal is stored in upper triangular factor.
`CUSOLVER_UNIT_DIAGONAL_ASSUMED_L`	unit diagonal is assumed in lower triangular factor.
`CUSOLVER_UNIT_DIAGONAL_ASSUMED_U`	unit diagonal is assumed in upper triangular factor.

cusolverStatus_t

The cusolverStatus_t is an enum that indicates success or failure of the cuSolverRF library call. It is returned by all the cuSolver library routines, and it uses the same enumerated values as the sparse and dense Lapack routines.

2.3. cuSolver Formats Reference

2.3.1. Index Base Format

The CSR or CSC format requires either zero-based or one-based index for a sparse matrix A. The GLU library supports only zero-based indexing. Otherwise, both one-based and zero-based indexing are supported in cuSolver.

2.3.2. Vector (Dense) Format

The vectors are assumed to be stored linearly in memory. For example, the vector

$x = (\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix})$

is represented as

$(\begin{matrix} x_{1} & x_{2} & \dots & x_{n} \end{matrix})$

2.3.3. Matrix (Dense) Format

The dense matrices are assumed to be stored in column-major order in memory. The sub-matrix can be accessed using the leading dimension of the original matrix. For examle, the m*n (sub-)matrix

$(\begin{matrix} a_{1, 1} & \dots & a_{1, n} \\ a_{2, 1} & \dots & a_{2, n} \\ ⋮ \\ a_{m, 1} & \dots & a_{m, n} \end{matrix})$

is represented as

$(\begin{matrix} a_{1, 1} & \dots & a_{1, n} \\ a_{2, 1} & \dots & a_{2, n} \\ ⋮ & ⋱ & ⋮ \\ a_{m, 1} & \dots & a_{m, n} \\ ⋮ & ⋱ & ⋮ \\ a_{lda, 1} & \dots & a_{lda, n} \end{matrix})$

with its elements arranged linearly in memory as

$(\begin{matrix} a_{1, 1} & a_{2, 1} & \dots & a_{m, 1} & \dots & a_{lda, 1} & \dots & a_{1, n} & a_{2, n} & \dots & a_{m, n} & \dots & a_{lda, n} \end{matrix})$

where lda ≥ m is the leading dimension of A.

2.3.4. Matrix (CSR) Format

In CSR format the matrix is represented by the following parameters

`parameter`	`type`	`size`	`Meaning`
`n`	`(int)`		the number of rows (and columns) in the matrix.
`nnz`	`(int)`		the number of non-zero elements in the matrix.
`csrRowPtr`	`(int *)`	`n+1`	the array of offsets corresponding to the start of each row in the arrays `csrColInd` and `csrVal`. This array has also an extra entry at the end that stores the number of non-zero elements in the matrix.
`csrColInd`	`(int *)`	`nnz`	the array of column indices corresponding to the non-zero elements in the matrix. It is assumed that this array is sorted by row and by column within each row.
`csrVal`	`(S\|D\|C\|Z)*`	`nnz`	the array of values corresponding to the non-zero elements in the matrix. It is assumed that this array is sorted by row and by column within each row.

Note that in our CSR format sparse matrices are assumed to be stored in row-major order, in other words, the index arrays are first sorted by row indices and then within each row by column indices. Also it is assumed that each pair of row and column indices appears only once.

For example, the 4x4 matrix

$A = (\begin{matrix} 1.0 & 3.0 & 0.0 & 0.0 \\ 0.0 & 4.0 & 6.0 & 0.0 \\ 2.0 & 5.0 & 7.0 & 8.0 \\ 0.0 & 0.0 & 0.0 & 9.0 \end{matrix})$

is represented as

$csrRowPtr = (\begin{matrix} 0 & 2 & 4 & 8 & 9 \end{matrix})$

$csrColInd = (\begin{matrix} 0 & 1 & 1 & 2 & 0 & 1 & 2 & 3 & 3 \end{matrix})$

$csrVal = (\begin{matrix} 1.0 & 3.0 & 4.0 & 6.0 & 2.0 & 5.0 & 7.0 & 8.0 & 9.0 \end{matrix})$

2.3.5. Matrix (CSC) Format

In CSC format the matrix is represented by the following parameters

`parameter`	`type`	`size`	`Meaning`
`n`	`(int)`		the number of rows (and columns) in the matrix.
`nnz`	`(int)`		the number of non-zero elements in the matrix.
`cscColPtr`	`(int *)`	`n+1`	the array of offsets corresponding to the start of each column in the arrays `cscRowInd` and `cscVal`. This array has also an extra entry at the end that stores the number of non-zero elements in the matrix.
`cscRowInd`	`(int *)`	`nnz`	the array of row indices corresponding to the non-zero elements in the matrix. It is assumed that this array is sorted by column and by row within each column.
`cscVal`	`(S\|D\|C\|Z)*`	`nnz`	the array of values corresponding to the non-zero elements in the matrix. It is assumed that this array is sorted by column and by row within each column.

Note that in our CSC format sparse matrices are assumed to be stored in column-major order, in other words, the index arrays are first sorted by column indices and then within each column by row indices. Also it is assumed that each pair of row and column indices appears only once.

For example, the 4x4 matrix

$A = (\begin{matrix} 1.0 & 3.0 & 0.0 & 0.0 \\ 0.0 & 4.0 & 6.0 & 0.0 \\ 2.0 & 5.0 & 7.0 & 8.0 \\ 0.0 & 0.0 & 0.0 & 9.0 \end{matrix})$

is represented as

$cscColPtr = (\begin{matrix} 0 & 2 & 5 & 7 & 9 \end{matrix})$

$cscRowInd = (\begin{matrix} 0 & 2 & 0 & 1 & 2 & 1 & 2 & 2 & 3 \end{matrix})$

$cscVal = (\begin{matrix} 1.0 & 2.0 & 3.0 & 4.0 & 5.0 & 6.0 & 7.0 & 8.0 & 9.0 \end{matrix})$

cuSolverDN: dense LAPACK Function Reference

This chapter describes the API of cuSolverDN, which provides a subset of dense LAPACK functions.

cuSolverDN Helper Function Reference

The cuSolverDN helper functions are described in this section.

2.4.1.1. cusolverDnCreate()

cusolverStatus_t 
cusolverDnCreate(cusolverDnHandle_t *handle);

This function initializes the cuSolverDN library and creates a handle on the cuSolverDN context. It must be called before any other cuSolverDN API function is invoked. It allocates hardware resources necessary for accessing the GPU.

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`output`	the pointer to the handle to the cuSolverDN context.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the initialization succeeded.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the CUDA Runtime initialization failed.
`CUSOLVER_STATUS_ALLOC_FAILED`	the resources could not be allocated.
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.

2.4.1.2. cusolverDnDestroy()

cusolverStatus_t 
cusolverDnDestroy(cusolverDnHandle_t handle);

This function releases CPU-side resources used by the cuSolverDN library.

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the shutdown succeeded.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.

cusolverDnSetStream()

cusolverStatus_t 
cusolverDnSetStream(cusolverDnHandle_t handle, cudaStream_t streamId)

This function sets the stream to be used by the cuSolverDN library to execute its routines.

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`streamId`	`host`	`input`	the stream to be used by the library.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the stream was set successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.

cusolverDnGetStream()

cusolverStatus_t 
cusolverDnGetStream(cusolverDnHandle_t handle, cudaStream_t *streamId)

This function sets the stream to be used by the cuSolverDN library to execute its routines.

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`streamId`	`host`	`output`	the stream to be used by the library.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the stream was set successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.

2.4.1.5. cusolverDnCreateSyevjInfo()

cusolverStatus_t 
cusolverDnCreateSyevjInfo(
    syevjInfo_t *info);

This function creates and initializes the structure of syevj, syevjBatched and sygvj to default values.

`parameter`	`Memory`	`In/out`	`Meaning`
`info`	`host`	`output`	the pointer to the structure of syevj.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the structure was initialized successfully.
`CUSOLVER_STATUS_ALLOC_FAILED`	the resources could not be allocated.

2.4.1.6. cusolverDnDestroySyevjInfo()

cusolverStatus_t 
cusolverDnDestroySyevjInfo(
    syevjInfo_t info);

This function destroys and releases any memory required by the structure.

`parameter`	`Memory`	`In/out`	`Meaning`
`info`	`host`	`input`	the structure of syevj.

Status Returned

CUSOLVER_STATUS_SUCCESS

the resources are released successfully.

2.4.1.7. cusolverDnXsyevjSetTolerance()

cusolverStatus_t 
cusolverDnXsyevjSetTolerance(
    syevjInfo_t info,
    double tolerance)

This function configures tolerance of syevj.

`parameter`	`Memory`	`In/out`	`Meaning`
`info`	`host`	`in/out`	the pointer to the structure of syevj.
`tolerance`	`host`	`input`	accuracy of numerical eigenvalues.

Status Returned

CUSOLVER_STATUS_SUCCESS

the operation completed successfully.

2.4.1.8. cusolverDnXsyevjSetMaxSweeps()

cusolverStatus_t 
cusolverDnXsyevjSetMaxSweeps(
    syevjInfo_t info,
    int max_sweeps)

This function configures maximum number of sweeps in syevj. The default value is 100.

`parameter`	`Memory`	`In/out`	`Meaning`
`info`	`host`	`in/out`	the pointer to the structure of syevj.
`max_sweeps`	`host`	`input`	maximum number of sweeps.

Status Returned

CUSOLVER_STATUS_SUCCESS

the operation completed successfully.

2.4.1.9. cusolverDnXsyevjSetSortEig()

cusolverStatus_t 
cusolverDnXsyevjSetSortEig(
    syevjInfo_t info,
    int sort_eig)

if sort_eig is zero, the eigenvalues are not sorted. This function only works for syevjBatched. syevj and sygvj always sort eigenvalues in ascending order. By default, eigenvalues are always sorted in ascending order.

`parameter`	`Memory`	`In/out`	`Meaning`
`info`	`host`	`in/out`	the pointer to the structure of syevj.
`sort_eig`	`host`	`input`	if `sort_eig` is zero, the eigenvalues are not sorted.

Status Returned

CUSOLVER_STATUS_SUCCESS

the operation completed successfully.

2.4.1.10. cusolverDnXsyevjGetResidual()

cusolverStatus_t 
cusolverDnXsyevjGetResidual(
    cusolverDnHandle_t handle,
    syevjInfo_t info,
    double *residual)

This function reports residual of syevj or sygvj. It does not support syevjBatched. If the user calls this function after syevjBatched, the error CUSOLVER_STATUS_NOT_SUPPORTED is returned.

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`info`	`host`	`input`	the pointer to the structure of syevj.
`residual`	`host`	`output`	residual of syevj.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_SUPPORTED`	does not support batched version

2.4.1.11. cusolverDnXsyevjGetSweeps()

cusolverStatus_t 
cusolverDnXsyevjGetSweeps(
    cusolverDnHandle_t handle,
    syevjInfo_t info,
    int *executed_sweeps)

This function reports number of executed sweeps of syevj or sygvj. It does not support syevjBatched. If the user calls this function after syevjBatched, the error CUSOLVER_STATUS_NOT_SUPPORTED is returned.

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`info`	`host`	`input`	the pointer to the structure of syevj.
`executed_sweeps`	`host`	`output`	number of executed sweeps.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_SUPPORTED`	does not support batched version

2.4.1.12. cusolverDnCreateGesvdjInfo()

cusolverStatus_t 
cusolverDnCreateGesvdjInfo(
    gesvdjInfo_t *info);

This function creates and initializes the structure of gesvdj and gesvdjBatched to default values.

`parameter`	`Memory`	`In/out`	`Meaning`
`info`	`host`	`output`	the pointer to the structure of gesvdj.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the structure was initialized successfully.
`CUSOLVER_STATUS_ALLOC_FAILED`	the resources could not be allocated.

2.4.1.13. cusolverDnDestroyGesvdjInfo()

cusolverStatus_t 
cusolverDnDestroyGesvdjInfo(
    gesvdjInfo_t info);

This function destroys and releases any memory required by the structure.

`parameter`	`Memory`	`In/out`	`Meaning`
`info`	`host`	`input`	the structure of gesvdj.

Status Returned

CUSOLVER_STATUS_SUCCESS

the resources are released successfully.

2.4.1.14. cusolverDnXgesvdjSetTolerance()

cusolverStatus_t 
cusolverDnXgesvdjSetTolerance(
    gesvdjInfo_t info,
    double tolerance)

This function configures tolerance of gesvdj.

`parameter`	`Memory`	`In/out`	`Meaning`
`info`	`host`	`in/out`	the pointer to the structure of gesvdj.
`tolerance`	`host`	`input`	accuracy of numerical singular values.

Status Returned

CUSOLVER_STATUS_SUCCESS

the operation completed successfully.

2.4.1.15. cusolverDnXgesvdjSetMaxSweeps()

cusolverStatus_t 
cusolverDnXgesvdjSetMaxSweeps(
    gesvdjInfo_t info,
    int max_sweeps)

This function configures maximum number of sweeps in gesvdj. The default value is 100.

`parameter`	`Memory`	`In/out`	`Meaning`
`info`	`host`	`in/out`	the pointer to the structure of gesvdj.
`max_sweeps`	`host`	`input`	maximum number of sweeps.

Status Returned

CUSOLVER_STATUS_SUCCESS

the operation completed successfully.

2.4.1.16. cusolverDnXgesvdjSetSortEig()

cusolverStatus_t 
cusolverDnXgesvdjSetSortEig(
    gesvdjInfo_t info,
    int sort_svd)

if sort_svd is zero, the singular values are not sorted. This function only works for gesvdjBatched. gesvdj always sorts singular values in descending order. By default, singular values are always sorted in descending order.

`parameter`	`Memory`	`In/out`	`Meaning`
`info`	`host`	`in/out`	the pointer to the structure of gesvdj.
`sort_svd`	`host`	`input`	if `sort_svd` is zero, the singular values are not sorted.

Status Returned

CUSOLVER_STATUS_SUCCESS

the operation completed successfully.

2.4.1.17. cusolverDnXgesvdjGetResidual()

cusolverStatus_t 
cusolverDnXgesvdjGetResidual(
    cusolverDnHandle_t handle,
    gesvdjInfo_t info,
    double *residual)

This function reports residual of gesvdj. It does not support gesvdjBatched. If the user calls this function after gesvdjBatched, the error CUSOLVER_STATUS_NOT_SUPPORTED is returned.

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`info`	`host`	`input`	the pointer to the structure of gesvdj.
`residual`	`host`	`output`	residual of gesvdj.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_SUPPORTED`	does not support batched version

2.4.1.18. cusolverDnXgesvdjGetSweeps()

cusolverStatus_t 
cusolverDnXgesvdjGetSweeps(
    cusolverDnHandle_t handle,
    gesvdjInfo_t info,
    int *executed_sweeps)

This function reports number of executed sweeps of gesvdj. It does not support gesvdjBatched. If the user calls this function after gesvdjBatched, the error CUSOLVER_STATUS_NOT_SUPPORTED is returned.

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`info`	`host`	`input`	the pointer to the structure of gesvdj.
`executed_sweeps`	`host`	`output`	number of executed sweeps.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_SUPPORTED`	does not support batched version

2.4.1.19. cusolverDnIRSParamsCreate()

cusolverStatus_t 
cusolverDnIRSParamsCreate(cusolverDnIRSParams_t *params);

This function creates and initializes the structure of parameters for an IRS solver such as the cusolverDnIRSXgesv() function to default values. For this release, this function is valid for only one call to an IRS solver, thus each new call to the IRS solver will requires its own Params structure. This restriction is going to be removed in future release and then if user want to reuse the same configuration to many call to an IRS solver it will allow him.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`output`	Pointer to the cusolverDnIRSParams_t Params structure

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The structure was created and initialized successfully.
`CUSOLVER_STATUS_ALLOC_FAILED`	The resources could not be allocated.

2.4.1.20. cusolverDnIRSParamsDestroy()

cusolverStatus_t 
cusolverDnIRSParamsDestroy(cusolverDnIRSParams_t params);

This function destroys and releases any memory required by the Params structure. Since the Infos structure (see cusolverDnIRSInfosCreate() for more details) depends on the Params structure, this function cannot be called to destroy the Params structure if the Infos structure was not destroyed. For this release, this function is valid for only one call to an IRS solver, thus each call to an IRS solver should have its own Params and Infos structure. This restriction is going to be removed in future release and then if user want to reuse the same configuration to many call to an IRS solver it will allow him.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`input`	The cusolverDnIRSParams_t Params structure

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The resources are released successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.
`CUSOLVER_STATUS_IRS_INFOS_NOT_DESTROYED`	Not all the Infos structure associated with this Params structure have been destroyed yet.

2.4.1.21. cusolverDnIRSParamsSetRefinementSolver()

cusolverStatus_t 
cusolverDnIRSParamsSetRefinementSolver(
    cusolverDnIRSParams_t params, 
    cusolverIRSRefinement_t solver);

This function sets the refinement solver to be used in the Iterative Refinement Solver functions such as the cusolverDnIRSXgesv() function. Details about values that can be set to and their meaning can be found in the cusolverIRSRefinement_t type section.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in/out`	The cusolverDnIRSParams_t Params structure
`solver`	`host`	`input`	Type of the refinement solver to be used by the IRS solver such as cusolverDnIRSXgesv()

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.

2.4.1.22. cusolverDnIRSParamsSetSolverMainPrecision()

cusolverStatus_t 
cusolverDnIRSParamsSetSolverMainPrecision(
    cusolverDnIRSParams_t params, 
    cudaDataType solver_main_precision);

This function sets the main precision (e.g., the INOUT data type) of the IRS solver. The value set here should be the same cuda datatype as the third argument on the call to the IRS solver. Note that, the user has to set the main precision before a first call to the IRS solver because it is NOT set by default with the Params creation. He can set it by either calling this function or cusolverDnIRSParamsSetSolverPrecisions(). Possible values are described in the table of the corresponding IRS solver for example, see the description of the third argument of the cusolverDnIRSXgesv() IRS function.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in/out`	The cusolverDnIRSParams_t Params structure
`solver_main_precision`	`host`	`input`	Allowed cuda datatype (for example CUDA_R_FP64). See the corresponding IRS solver for the table of supported precisions.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.

2.4.1.23. cusolverDnIRSParamsSetSolverLowestPrecision()

cusolverStatus_t 
cusolverDnIRSParamsSetSolverLowestPrecision(
    cusolverDnIRSParams_t params, 
    cudaDataType lowest_precision_type);

This function sets the lowest precision that will be used by Iterative Refinement Solver. Note that, the user has to set the lowest precision before a first call to the IRS solver because it is NOT set by default with the Params creation. He can set it by either calling this function or cusolverDnIRSParamsSetSolverPrecisions(). Usually this precision define the speedup that can be achieved. The ratio of the performance of the lowest precision over the inout_data_type precision define somehow the upper bound of the speedup that could be obtained. More precisely, it depends of many factors, but for large matrices sizes, it is the ratio of the matrix-matrix rank-k product (e.g., rank-k GEMM where k is around 256) that define the possible speedup. For instance, if the inout precision is real double precision FP64 and the lowest precision is FP32, then we can expect a speedup of at most 2X for large problem sizes. If the lowest precision was FP16, then we can expect 3X-4X. A reasonable strategy should take the number of right-hand sides and the size of the matrix as well as the convergence rate into account.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in/out`	The cusolverDnIRSParams_t Params structure
`lowest_precision_type`	`host`	`input`	Allowed cuda datatype (for example CUDA_R_FP16). See the corresponding IRS solver for the table of supported precisions.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.

2.4.1.24. cusolverDnIRSParamsSetSolverPrecisions()

cusolverStatus_t
    cusolverDnIRSParamsSetSolverPrecisions(
            cusolverDnIRSParams_t params,
            cudaDataType solver_main_precision,
            cudaDataType solver_lowest_precision );

This function set both, the main and the lowest precision of the IRS solver. It is a wrappers to both cusolverDnIRSParamsSetSolverMainPrecision() and cusolverDnIRSParamsSetSolverLowestPrecision(). Note that, the user has to set both the main and the lowest precision before a first call to the IRS solver because they are NOT set by default with the Params creation. He can set it by either calling this function or both functions cusolverDnIRSParamsSetSolverLowestPrecisions() and cusolverDnIRSParamsSetSolverMainPrecisions().

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in/out`	The cusolverDnIRSParams_t Params structure
`solver_main_precision`	`host`	`input`	Allowed cuda datatype (for example CUDA_R_FP64). See the corresponding IRS solver for the table of supported precisions.
`solver_lowest_precision`	`host`	`input`	Allowed cuda datatype (for example CUDA_R_FP16). See the corresponding IRS solver for the table of supported precisions.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.

2.4.1.25. cusolverDnIRSParamsSetTol()

cusolverStatus_t 
cusolverDnIRSParamsSetTol(
            cusolverDnIRSParams_t params,
            cudaDataType data_type,
            double val );

This function sets the tolerance for the refinement solver. By default it is set such that, all the RHS satisfy:

RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX where

RNRM is the infinity-norm of the residual
XNRM is the infinity-norm of the solution
ANRM is the infinity-operator-norm of the matrix A
EPS is the machine epsilon that matches LAPACK <t1>LAMCH('Epsilon')

The value BWDMAX is fixed to 1.0.

The user can use this function to change the tolerance to a lower or higher value. Our goal is to give the user as much control as we can such a way he can investigate and control every detail of the IRS solver.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in/out`	The cusolverDnIRSParams_t Params structure
`data_type`	`host`	`input`	cuda datatype of the inout_data_type
`val`	`host`	`input`	double precision value to which the refinement tolerance will be set internally.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.

2.4.1.26. cusolverDnIRSParamsSetTolInner()

cusolverStatus_t
    cusolverDnIRSParamsSetTolInner(
            cusolverDnIRSParams_t params,
            cudaDataType data_type,
            double val );

This function sets the tolerance for the inner refinement solver when the refinement solver consists of two level solvers (e.g., CUSOLVER_IRS_REFINE_CLASSICAL_GMRES or CUSOLVER_IRS_REFINE_GMRES_GMRES). It is not referenced in case of one level refinement solver such as CUSOLVER_IRS_REFINE_CLASSICAL or CUSOLVER_IRS_REFINE_GMRES. This function set the tolerance for the inner solver (e.g. the inner GMRES). For example, if the RefinementSolver was set to CUSOLVER_IRS_REFINE_CLASSICAL_GMRES setting this tolerance mean that the inner GMRES solver will converge to that tolerance at each outer iteration of the classical refinement solver. It is set to 1e-4 by default. Our goal is to give the user as much control as we can such a way he can investigate and control every detail of the IRS solver.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in/out`	The cusolverDnIRSParams_t Params structure
`data_type`	`host`	`input`	The cuda datatype (for example CUDA_R_FP64) of the inout data
`val`	`host`	`input`	Double precision real value to which the refinement tolerance will be set internally.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.

2.4.1.27. cusolverDnIRSParamsSetMaxIters()

cusolverStatus_t 
cusolverDnIRSParamsSetMaxIters(
    cusolverDnIRSParams_t params, 
    int max_iters);

This function sets the total number of allowed refinement iterations after which solver will stop. Total means the maximum number of iterations allowed (e.g., outer and inner iterations when two level refinement solver is set). Default value is set to 50. Our goal is to give the user as much control as we can such a way he can investigate and control every detail of the IRS solver.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in/out`	The cusolverDnIRSParams_t Params structure
`max_iters`	`host`	`input`	Maximum total number of iterations allowed for the refinement solver

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.

2.4.1.28. cusolverDnIRSParamsSetMaxItersInner()

cusolverStatus_t
    cusolverDnIRSParamsSetMaxItersInner(
            cusolverDnIRSParams_t params,
            cusolver_int_t maxiters_inner );

This function sets the maximal number of iterations allowed for the inner refinement solver. It is not referenced in case of one level refinement solver such as CUSOLVER_IRS_REFINE_CLASSICAL or CUSOLVER_IRS_REFINE_GMRES. The inner refinement solver will stop after reaching either the inner tolerance or the MaxItersInner value. By default, it is set to MaxIters (e.g., 50). Note that this value could not be larger than MaxIters since MaxIters is the total number of allowed iterations. Note that, if user call to set MaxIters after calling this function, the MaxIters has priority and will overwrite MaxItersInner to the minimum value of (MaxIters, MaxItersInner). Our goal is to give the user as much control as we can such a way he can investigate and control every detail of the IRS solver.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in/out`	The cusolverDnIRSParams_t Params structure
`maxiters_inner`	`host`	`input`	Maximum number of allowed inner iterations for the refinement solver. Meaningful when the refinement solver is a two levels solver such as CUSOLVER_IRS_REFINE_CLASSICAL_GMRES or CUSOLVER_IRS_REFINE_GMRES_GMRES. Value should be less or equal to MaxIters.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.
`CUSOLVER_STATUS_IRS_PARAMS_INVALID`	if the value was larger than MaxIters.

2.4.1.29. cusolverDnIRSParamsGetMaxIters()


cusolverStatus_t
    cusolverDnIRSParamsGetMaxIters(
            cusolverDnIRSParams_t params,
            cusolver_int_t *maxiters );

This function returns the maximal number of iterations MaxIters that is currently set within the current Params structure. Thus, it returns the value of the MaxIters parameter.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in`	The cusolverDnIRSParams_t Params structure
`maxiters`	`host`	`output`	The maximal number of iterations that is currently set

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.

2.4.1.30. cusolverDnIRSInfosCreate()

cusolverStatus_t 
cusolverDnIRSInfosCreate(
    cusolverDnIRSParams_t params,
    cusolverDnIRSInfos_t* infos )

This function creates and initializes the Infos structure that will hold the refinement informations of an IRS solver. Such information includes the total number of iterations needed to converge (Niters), the outer number of iterations, and a pointer to the matrix of the convergence history residual norms. This function need to be called after the Params structure (see cusolverDnIRSParamsCreate()) has been created and before the call to an IRS solver such as cusolverDnIRSXgesv(). This function is valid for only one call to an IRS solver, since it hold info about that solve and thus each solve will requires its own Infos structure.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in`	The cusolverDnIRSParams_t Params structure
`info`	`host`	`output`	Pointer to the cusolverDnIRSInfos_t Infos structure

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The structure was initialized successfully.
`CUSOLVER_STATUS_ALLOC_FAILED`	The resources could not be allocated.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.

2.4.1.31. cusolverDnIRSInfosDestroy()

cusolverStatus_t 
cusolverDnIRSInfosDestroy(
    cusolverDnIRSParams_t params,
    cusolverDnIRSInfos_t infos );

This function destroys and releases any memory required by the Infos structure. This function destroy all the informations (e.g., Niters performed, OuterNiters performed, residual history etc) about a solver call, thus a user is supposed to call it once he is done from the informations he need.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in/out`	The cusolverDnIRSParams_t Params structure
`info`	`host`	`in/out`	The cusolverDnIRSInfos_t Infos structure

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the resources are released successfully.
`CUSOLVER_STATUS_IRS_INFOS_NOT_INITIALIZED`	The Infos structure was not created.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.

2.4.1.32. cusolverDnIRSInfosGetMaxIters()

cusolverStatus_t
    cusolverDnIRSInfosGetMaxIters(
            cusolverDnIRSParams_t params,
            cusolverDnIRSInfos_t infos,
            cusolver_int_t *maxiters );

This function returns the maximal number of iterations that is currently set within the current Params structure. This function is a duplicate of cusolverDnIRSParamsGetMaxIters().

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in`	The cusolverDnIRSParams_t Params structure
`infos`	`host`	`in`	The cusolverDnIRSInfos_t Infos structure
`maxiters`	`host`	`output`	The maximal number of iterations that is currently set

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.
`CUSOLVER_STATUS_IRS_INFOS_NOT_INITIALIZED`	The Infos structure was not created.

2.4.1.33. cusolverDnIRSInfosGetNiters()

cusolverStatus_t cusolverDnIRSInfosGetNiters(
            cusolverDnIRSParams_t params,
            cusolverDnIRSInfos_t infos,
            cusolver_int_t *niters );

This function returns the total number of iterations performed by the IRS solver. If it was negative it means that the IRS solver had numerical issues a fall back to a full precision solution most like happened. Please refer to the description of negative niters values in the corresponding IRS linear solver functions such as cusolverDnXgesv().

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in`	The cusolverDnIRSParams_t Params structure
`infos`	`host`	`in`	The cusolverDnIRSInfos_t Infos structure
`niters`	`host`	`output`	The total number of iterations performed by the IRS solver

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.
`CUSOLVER_STATUS_IRS_INFOS_NOT_INITIALIZED`	The Infos structure was not created.

2.4.1.34. cusolverDnIRSInfosGetOuterNiters()

cusolverStatus_t
    cusolverDnIRSInfosGetOuterNiters(
            cusolverDnIRSParams_t params,
            cusolverDnIRSInfos_t infos,
            cusolver_int_t *outer_niters );

This function returns the number of iterations performed by outer refinement loop of the IRS solver. When RefinementSolver consists of one level solver such as CUSOLVER_IRS_REFINE_CLASSICAL or CUSOLVER_IRS_REFINE_GMRES, it is the same as Niters. When RefinementSolver consists of two levels solver such as CUSOLVER_IRS_REFINE_CLASSICAL_GMRES or CUSOLVER_IRS_REFINE_GMRES_GMRES, it is the number of the outer iteration. See description of cusolverIRSRefinementSolver_t type section for more details.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in`	The cusolverDnIRSParams_t Params structure
`infos`	`host`	`in`	The cusolverDnIRSInfos_t Infos structure
`outer_niters`	`host`	`output`	The number of iterations of the outer refinement loop of the IRS solver

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.
`CUSOLVER_STATUS_IRS_INFOS_NOT_INITIALIZED`	The Infos structure was not created.

2.4.1.35. cusolverDnIRSInfosRequestResidual()

cusolverStatus_t cusolverDnIRSInfosRequestResidual(
        cusolverDnIRSParams_t params,
        cusolverDnIRSInfos_t infos );

This function, once called, tell the IRS solver to store the convergence history of the refinement phase in a matrix, that could be accessed via a pointer returned by the cusolverDnIRSInfosGetResidualHistory() function.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in`	The cusolverDnIRSParams_t Params structure
`infos`	`host`	`in`	The cusolverDnIRSInfos_t Infos structure

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.
`CUSOLVER_STATUS_IRS_INFOS_NOT_INITIALIZED`	The Infos structure was not created.

2.4.1.36. cusolverDnIRSInfosGetResidualHistory()

cusolverStatus_t 
cusolverDnIRSInfosGetResidualHistory(
    cusolverDnIRSParams_t params,
    cusolverDnIRSInfos_t infos,
    void **residual_history );

If the user called cusolverDnIRSInfosRequestResidual() before the call to the IRS solve function, this function return a pointer to the matrix of the convergence history residual norms. Precision of residual norms depends on the IRS input data type. If inout datatype has double precision (CUDA_R_FP64 or CUDA_C_FP64 inout datatype), this residual will be real double precision. Otherwise (CUDA_R_FP32 or CUDA_C_FP32 inout datatype) - residual will be with real single precision.

The residual history matrix consists of two columns (even for NRHS case) of MaxIters+1 rows, thus a matrix of size (MaxIters+1,2). Only the first OuterNiters+1 rows contains the needed informations the other (e.g., OuterNiters+2:Maxiters+1) are garbage. On the first column, each row "i" specify the total number of iterations happened at this outer iteration "i" and on the second columns the residual norm corresponding to this outer iteration "i". Thus, the first row (e.g., outer iteration "0") consists of the initial residual (e.g., the residual before the refinement loop start) then the consecutive OuterNiters rows are the residual obtained at each outer iteration of the refinement loop. Note, it only consists of the history of the outer loop.

Thus, if the Refinementsolver was CUSOLVER_IRS_REFINE_CLASSICAL or CUSOLVER_IRS_REFINE_GMRES, then OuterNiters=Niters (Niters is the total number of iterations performed) and there is Niters+1 rows of norms that correspond to the Niters outer iterations.

If the Refinementsolver was CUSOLVER_IRS_REFINE_CLASSICAL_GMRES or CUSOLVER_IRS_REFINE_GMRES_GMRES, then OuterNiters <= Niters corresponds to the outer iterations performed by the outer refinement loop. Thus there is OuterNiters+1 residual norms where row "i" correspond to the outer iteration "i" and the first column specify the total number of iterations (outer and inner) that were performed while the second columns correspond to the residual norm at this stage.

For example, let say the user specify CUSOLVER_IRS_REFINE_CLASSICAL_GMRES as a Refinementsolver and let say it needed 3 outer iterations to converge and 4,3,3 inner iterations at each outer respectively. This consists of 10 total iterations. Thus on row 0 is for the first residual before the refinement start, so it has 0 iteration. On row 1 which correspond to the outer iteration 1, it will be shown 4 (4 is the total number of iterations that were performed till now) on row 2, it will be 7 and on row 3 it will be 10.

As summary, let define ldh=Maxiters+1, the leading dimension of the residual matrix. then residual_history[i] shows the total number of iterations performed at the outer iteration "i" and residual_history[i+ldh] correspond to its norm of the residual at this stage.

`parameter`	`Memory`	`In/out`	`Meaning`
`params`	`host`	`in`	The cusolverDnIRSParams_t Params structure
`infos`	`host`	`in`	The cusolverDnIRSInfos_t Infos structure
`residual_history`	`host`	`output`	Returns a void pointer to the matrix of the convergence history residual norms. See the description above for the relation between the residual norm types and the inout datatype.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED`	The Params structure was not created.
`CUSOLVER_STATUS_IRS_INFOS_NOT_INITIALIZED`	The Infos structure was not created.
`CUSOLVER_STATUS_INVALID_VALUE`	This function was called without calling `cusolverDnIRSInfosRequestResidual()` in advance.

Dense Linear Solver Reference

This chapter describes linear solver API of cuSolverDN, including Cholesky factorization, LU with partial pivoting, QR factorization and Bunch-Kaufman (LDLT) factorization.

cusolverDn<t>potrf()

These helper functions calculate the necessary size of work buffers.

cusolverStatus_t
cusolverDnSpotrf_bufferSize(cusolverDnHandle_t handle,
                 cublasFillMode_t uplo,
                 int n,
                 float *A,
                 int lda,
                 int *Lwork );

cusolverStatus_t
cusolverDnDpotrf_bufferSize(cusolveDnHandle_t handle,
                 cublasFillMode_t uplo,
                 int n,
                 double *A,
                 int lda,
                 int *Lwork );

cusolverStatus_t 
cusolverDnCpotrf_bufferSize(cusolverDnHandle_t handle,
                 cublasFillMode_t uplo,
                 int n,
                 cuComplex *A,
                 int lda,
                 int *Lwork );

cusolverStatus_t 
cusolverDnZpotrf_bufferSize(cusolverDnHandle_t handle,
                 cublasFillMode_t uplo,
                 int n,
                 cuDoubleComplex *A,
                 int lda,
                 int *Lwork);

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSpotrf(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           float *A,
           int lda,
           float *Workspace,
           int Lwork,
           int *devInfo );

cusolverStatus_t 
cusolverDnDpotrf(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           double *A,
           int lda,
           double *Workspace,
           int Lwork,
           int *devInfo );

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCpotrf(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           cuComplex *A,
           int lda,
           cuComplex *Workspace,
           int Lwork,
           int *devInfo );

cusolverStatus_t 
cusolverDnZpotrf(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           cuDoubleComplex *A,
           int lda,
           cuDoubleComplex *Workspace,
           int Lwork,
           int *devInfo );

This function computes the Cholesky factorization of a Hermitian positive-definite matrix.

A is a n×n Hermitian matrix, only lower or upper part is meaningful. The input parameter uplo indicates which part of the matrix is used. The function would leave other part untouched.

If input parameter uplo is CUBLAS_FILL_MODE_LOWER, only lower triangular part of A is processed, and replaced by lower triangular Cholesky factor L.

A = L * L^{H}

If input parameter uplo is CUBLAS_FILL_MODE_UPPER, only upper triangular part of A is processed, and replaced by upper triangular Cholesky factor U.

A = U^{H} * U

The user has to provide working space which is pointed by input parameter Workspace. The input parameter Lwork is size of the working space, and it is returned by potrf_bufferSize().

If Cholesky factorization failed, i.e. some leading minor of A is not positive definite, or equivalently some diagonal elements of L or U is not a real number. The output parameter devInfo would indicate smallest leading minor of A which is not positive definite.

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

API of potrf

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`uplo`	`host`	`input`	indicates if matrix `A` lower or upper part is stored, the other part is not referenced.
`n`	`host`	`input`	number of rows and columns of matrix `A`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,n)`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`.
`Workspace`	`device`	`in/out`	working space, <type> array of size `Lwork`.
`Lwork`	`host`	`input`	size of `Workspace`, returned by `potrf_bufferSize`.
`devInfo`	`device`	`output`	if `devInfo = 0`, the Cholesky factorization is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle). if `devInfo = i`, the leading minor of order i is not positive definite.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`n<0` or `lda<max(1,n)`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>potrs()

cusolverStatus_t 
cusolverDnSpotrs(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           int nrhs,
           const float *A,
           int lda,
           float *B,
           int ldb,
           int *devInfo);

cusolverStatus_t 
cusolverDnDpotrs(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           int nrhs,
           const double *A,
           int lda,
           double *B,
           int ldb,
           int *devInfo);

cusolverStatus_t 
cusolverDnCpotrs(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           int nrhs,
           const cuComplex *A,
           int lda,
           cuComplex *B,
           int ldb,
           int *devInfo);

cusolverStatus_t 
cusolverDnZpotrs(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           int nrhs,
           const cuDoubleComplex *A,
           int lda,
           cuDoubleComplex *B,
           int ldb,
           int *devInfo);

This function solves a system of linear equations

A * X = B

where A is a n×n Hermitian matrix, only lower or upper part is meaningful. The input parameter uplo indicates which part of the matrix is used. The function would leave other part untouched.

The user has to call potrf first to factorize matrix A. If input parameter uplo is CUBLAS_FILL_MODE_LOWER, A is lower triangular Cholesky factor L correspoding to $A = L * L^{H}$ . If input parameter uplo is CUBLAS_FILL_MODE_UPPER, A is upper triangular Cholesky factor U corresponding to $A = U^{H} * U$ .

The operation is in-place, i.e. matrix X overwrites matrix B with the same leading dimension ldb.

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

API of potrs

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolveDN library context.
`uplo`	`host`	`input`	indicates if matrix `A` lower or upper part is stored, the other part is not referenced.
`n`	`host`	`input`	number of rows and columns of matrix `A`.
`nrhs`	`host`	`input`	number of columns of matrix `X` and `B`.
`A`	`device`	`input`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,n)`. `A` is either lower cholesky factor `L` or upper Cholesky factor `U`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`.
`B`	`device`	`in/out`	<type> array of dimension `ldb * nrhs`. `ldb` is not less than `max(1,n)`. As an input, `B` is right hand side matrix. As an output, `B` is the solution matrix.
`devInfo`	`device`	`output`	if `devInfo = 0`, the Cholesky factorization is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle).

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`n<0`, `nrhs<0`, `lda<max(1,n)` or `ldb<max(1,n)`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>potri()

These helper functions calculate the necessary size of work buffers.

cusolverStatus_t
cusolverDnSpotri_bufferSize(cusolverDnHandle_t handle,
                 cublasFillMode_t uplo,
                 int n,
                 float *A,
                 int lda,
                 int *Lwork );

cusolverStatus_t
cusolverDnDpotri_bufferSize(cusolveDnHandle_t handle,
                 cublasFillMode_t uplo,
                 int n,
                 double *A,
                 int lda,
                 int *Lwork );

cusolverStatus_t 
cusolverDnCpotri_bufferSize(cusolverDnHandle_t handle,
                 cublasFillMode_t uplo,
                 int n,
                 cuComplex *A,
                 int lda,
                 int *Lwork );

cusolverStatus_t 
cusolverDnZpotri_bufferSize(cusolverDnHandle_t handle,
                 cublasFillMode_t uplo,
                 int n,
                 cuDoubleComplex *A,
                 int lda,
                 int *Lwork);

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSpotri(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           float *A,
           int lda,
           float *Workspace,
           int Lwork,
           int *devInfo );

cusolverStatus_t 
cusolverDnDpotri(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           double *A,
           int lda,
           double *Workspace,
           int Lwork,
           int *devInfo );

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCpotri(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           cuComplex *A,
           int lda,
           cuComplex *Workspace,
           int Lwork,
           int *devInfo );

cusolverStatus_t 
cusolverDnZpotri(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           cuDoubleComplex *A,
           int lda,
           cuDoubleComplex *Workspace,
           int Lwork,
           int *devInfo );

This function computes the inverse of a positive-definite matrix A using the Cholesky factorization

A = L * L^{H} = U^{H} * U

computed by potrf().

A is a n×n matrix containing the triangular factor L or U computed by the Cholesky factorization. Only lower or upper part is meaningful and the input parameter uplo indicates which part of the matrix is used. The function would leave the other part untouched.

If the input parameter uplo is CUBLAS_FILL_MODE_LOWER, only lower triangular part of A is processed, and replaced the by lower triangular part of the inverse of A.

If the input parameter uplo is CUBLAS_FILL_MODE_UPPER, only upper triangular part of A is processed, and replaced by the upper triangular part of the inverse of A.

The user has to provide the working space which is pointed to by input parameter Workspace. The input parameter Lwork is the size of the working space, returned by potri_bufferSize().

If the computation of the inverse fails, i.e. some leading minor of L or U, is null, the output parameter devInfo would indicate the smallest leading minor of L or U which is not positive definite.

If the output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting the handle).

API of potri

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`uplo`	`host`	`input`	indicates if matrix `A` lower or upper part is stored, the other part is not referenced.
`n`	`host`	`input`	number of rows and columns of matrix `A`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` where `lda` is not less than `max(1,n)`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`.
`Workspace`	`device`	`in/out`	working space, <type> array of size `Lwork`.
`Lwork`	`host`	`input`	size of `Workspace`, returned by `potri_bufferSize`.
`devInfo`	`device`	`output`	if `devInfo = 0`, the computation of the inverse is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle). if `devInfo = i`, the leading minor of order i is zero.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`n<0` or `lda<max(1,n)`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>getrf()

These helper functions calculate the size of work buffers needed.

cusolverStatus_t 
cusolverDnSgetrf_bufferSize(cusolverDnHandle_t handle,
                      int m,
                      int n,
                      float *A,
                      int lda,
                      int *Lwork );

cusolverStatus_t 
cusolverDnDgetrf_bufferSize(cusolverDnHandle_t handle,
                      int m,
                      int n,
                      double *A,
                      int lda,
                      int *Lwork );

cusolverStatus_t 
cusolverDnCgetrf_bufferSize(cusolverDnHandle_t handle,
                      int m,
                      int n,
                      cuComplex *A,
                      int lda,
                      int *Lwork );

cusolverStatus_t 
cusolverDnZgetrf_bufferSize(cusolverDnHandle_t handle,
                      int m,
                      int n,
                      cuDoubleComplex *A,
                      int lda,
                      int *Lwork );

The S and D data types are real single and double precision, respectively.

cusolverStatus_t 
cusolverDnSgetrf(cusolverDnHandle_t handle,
           int m,
           int n,
           float *A,
           int lda,
           float *Workspace,
           int *devIpiv,
           int *devInfo );

cusolverStatus_t 
cusolverDnDgetrf(cusolverDnHandle_t handle,
           int m,
           int n,
           double *A,
           int lda,
           double *Workspace,
           int *devIpiv,
           int *devInfo );

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCgetrf(cusolverDnHandle_t handle,
           int m,
           int n,
           cuComplex *A,
           int lda,
           cuComplex *Workspace,
           int *devIpiv,
           int *devInfo );

cusolverStatus_t 
cusolverDnZgetrf(cusolverDnHandle_t handle,
           int m,
           int n,
           cuDoubleComplex *A,
           int lda,
           cuDoubleComplex *Workspace,
           int *devIpiv,
           int *devInfo );

This function computes the LU factorization of a m×n matrix

P * A = L * U

where A is a m×n matrix, P is a permutation matrix, L is a lower triangular matrix with unit diagonal, and U is an upper triangular matrix.

The user has to provide working space which is pointed by input parameter Workspace. The input parameter Lwork is size of the working space, and it is returned by getrf_bufferSize().

If LU factorization failed, i.e. matrix A (U) is singular, The output parameter devInfo=i indicates U(i,i) = 0.

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

If devIpiv is null, no pivoting is performed. The factorization is A=L*U, which is not numerically stable.

No matter LU factorization failed or not, the output parameter devIpiv contains pivoting sequence, row i is interchanged with row devIpiv(i).

The user can combine getrf and getrs to complete a linear solver. Please refer to appendix D.1.

API of getrf

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`m`	`host`	`input`	number of rows of matrix `A`.
`n`	`host`	`input`	number of columns of matrix `A`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,m)`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`.
`Workspace`	`device`	`in/out`	working space, <type> array of size `Lwork`.
`devIpiv`	`device`	`output`	array of size at least `min(m,n)`, containing pivot indices.
`devInfo`	`device`	`output`	if `devInfo = 0`, the LU factorization is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle). if `devInfo = i`, the `U(i,i) = 0`.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`m,n<0` or `lda<max(1,m)`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>getrs()

cusolverStatus_t 
cusolverDnSgetrs(cusolverDnHandle_t handle,
           cublasOperation_t trans,
           int n,
           int nrhs,
           const float *A,
           int lda,
           const int *devIpiv,
           float *B,
           int ldb,
           int *devInfo );

cusolverStatus_t 
cusolverDnDgetrs(cusolverDnHandle_t handle,
           cublasOperation_t trans,
           int n,
           int nrhs,
           const double *A,
           int lda,
           const int *devIpiv,
           double *B,
           int ldb,
           int *devInfo );

cusolverStatus_t 
cusolverDnCgetrs(cusolverDnHandle_t handle,
           cublasOperation_t trans,
           int n,
           int nrhs,
           const cuComplex *A,
           int lda,
           const int *devIpiv,
           cuComplex *B,
           int ldb,
           int *devInfo );

cusolverStatus_t 
cusolverDnZgetrs(cusolverDnHandle_t handle,
           cublasOperation_t trans,
           int n,
           int nrhs,
           const cuDoubleComplex *A,
           int lda,
           const int *devIpiv,
           cuDoubleComplex *B,
           int ldb,
           int *devInfo );

This function solves a linear system of multiple right-hand sides

op(A) * X = B

where A is a n×n matrix, and was LU-factored by getrf, that is, lower trianular part of A is L, and upper triangular part (including diagonal elements) of A is U. B is a n×nrhs right-hand side matrix.

The input parameter trans is defined by

$op (A) = \{\begin{cases} A & if trans == CUBLAS_OP_N \\ A^{T} & if trans == CUBLAS_OP_T \\ A^{H} & if trans == CUBLAS_OP_C \end{cases}$

The input parameter devIpiv is an output of getrf. It contains pivot indices, which are used to permutate right-hand sides.

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

The user can combine getrf and getrs to complete a linear solver. Please refer to appendix D.1.

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`trans`	`host`	`input`	operation `op(A)` that is non- or (conj.) transpose.
`n`	`host`	`input`	number of rows and columns of matrix `A`.
`nrhs`	`host`	`input`	number of right-hand sides.
`A`	`device`	`input`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,n)`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`.
`devIpiv`	`device`	`input`	array of size at least `n`, containing pivot indices.
`B`	`device`	`output`	<type> array of dimension `ldb * nrhs` with `ldb` is not less than `max(1,n)`.
`ldb`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `B`.
`devInfo`	`device`	`output`	if `devInfo = 0`, the operation is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle).

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`n<0` or `lda<max(1,n)` or `ldb<max(1,n)`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>geqrf()

These helper functions calculate the size of work buffers needed.

cusolverStatus_t 
cusolverDnSgeqrf_bufferSize(cusolverDnHandle_t handle,
                      int m,
                      int n,
                      float *A,
                      int lda,
                      int *Lwork );

cusolverStatus_t 
cusolverDnDgeqrf_bufferSize(cusolverDnHandle_t handle,
                      int m,
                      int n,
                      double *A,
                      int lda,
                      int *Lwork );

cusolverStatus_t 
cusolverDnCgeqrf_bufferSize(cusolverDnHandle_t handle,
                      int m,
                      int n,
                      cuComplex *A,
                      int lda,
                      int *Lwork );

cusolverStatus_t 
cusolverDnZgeqrf_bufferSize(cusolverDnHandle_t handle,
                      int m,
                      int n,
                      cuDoubleComplex *A,
                      int lda,
                      int *Lwork );

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSgeqrf(cusolverDnHandle_t handle,
           int m,
           int n,
           float *A,
           int lda,
           float *TAU,
           float *Workspace,
           int Lwork,
           int *devInfo );

cusolverStatus_t 
cusolverDnDgeqrf(cusolverDnHandle_t handle,
           int m,
           int n,
           double *A,
           int lda,
           double *TAU,
           double *Workspace,
           int Lwork,
           int *devInfo );

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCgeqrf(cusolverDnHandle_t handle,
           int m,
           int n,
           cuComplex *A,
           int lda,
           cuComplex *TAU,
           cuComplex *Workspace,
           int Lwork,
           int *devInfo );

cusolverStatus_t 
cusolverDnZgeqrf(cusolverDnHandle_t handle,
           int m,
           int n,
           cuDoubleComplex *A,
           int lda,
           cuDoubleComplex *TAU,
           cuDoubleComplex *Workspace,
           int Lwork,
           int *devInfo );

This function computes the QR factorization of a m×n matrix

A = Q * R

where A is a m×n matrix, Q is a m×n matrix, and R is a n×n upper triangular matrix.

The user has to provide working space which is pointed by input parameter Workspace. The input parameter Lwork is size of the working space, and it is returned by geqrf_bufferSize().

The matrix R is overwritten in upper triangular part of A, including diagonal elements.

The matrix Q is not formed explicitly, instead, a sequence of householder vectors are stored in lower triangular part of A. The leading nonzero element of householder vector is assumed to be 1 such that output parameter TAU contains the scaling factor τ. If v is original householder vector, q is the new householder vector corresponding to τ, satisying the following relation

I - 2 * v * v^{H} = I - τ * q * q^{H}

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

API of geqrf

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`m`	`host`	`input`	number of rows of matrix `A`.
`n`	`host`	`input`	number of columns of matrix `A`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,m)`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`.
`TAU`	`device`	`output`	<type> array of dimension at least `min(m,n)`.
`Workspace`	`device`	`in/out`	working space, <type> array of size `Lwork`.
`Lwork`	`host`	`input`	size of working array `Workspace`.
`devInfo`	`device`	`output`	if `devInfo = 0`, the LU factorization is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle).

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`m,n<0` or `lda<max(1,m)`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>ormqr()

These helper functions calculate the size of work buffers needed.

cusolverStatus_t 
cusolverDnSormqr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasOperation_t trans,
    int m,
    int n,
    int k,
    const float *A,
    int lda,
    const float *tau,
    const float *C,
    int ldc,
    int *lwork);

cusolverStatus_t 
cusolverDnDormqr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasOperation_t trans,
    int m,
    int n,
    int k,
    const double *A,
    int lda,
    const double *tau,
    const double *C,
    int ldc,
    int *lwork);

cusolverStatus_t
cusolverDnCunmqr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasOperation_t trans,
    int m,
    int n,
    int k,
    const cuComplex *A,
    int lda,
    const cuComplex *tau,
    const cuComplex *C,
    int ldc,
    int *lwork);

cusolverStatus_t
cusolverDnZunmqr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasOperation_t trans,
    int m,
    int n,
    int k,
    const cuDoubleComplex *A,
    int lda,
    const cuDoubleComplex *tau,
    const cuDoubleComplex *C,
    int ldc,
    int *lwork);

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSormqr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasOperation_t trans,
    int m,
    int n,
    int k,
    const float *A,
    int lda,
    const float *tau,
    float *C,
    int ldc,
    float *work,
    int lwork,
    int *devInfo);

cusolverStatus_t
cusolverDnDormqr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasOperation_t trans,
    int m,
    int n,
    int k,
    const double *A,
    int lda,
    const double *tau,
    double *C,
    int ldc,
    double *work,
    int lwork,
    int *devInfo);

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCunmqr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasOperation_t trans,
    int m,
    int n,
    int k,
    const cuComplex *A,
    int lda,
    const cuComplex *tau,
    cuComplex *C,
    int ldc,
    cuComplex *work,
    int lwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnZunmqr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasOperation_t trans,
    int m,
    int n,
    int k,
    const cuDoubleComplex *A,
    int lda,
    const cuDoubleComplex *tau,
    cuDoubleComplex *C,
    int ldc,
    cuDoubleComplex *work,
    int lwork,
    int *devInfo);

This function overwrites m×n matrix C by

C = \{\begin{cases} op (Q) * C & if side == CUBLAS_SIDE_LEFT \\ C * op (Q) & if side == CUBLAS_SIDE_RIGHT \end{cases}

The operation of Q is defined by

op (Q) = \{\begin{cases} Q & if transa == CUBLAS_OP_N \\ Q^{T} & if transa == CUBLAS_OP_T \\ Q^{H} & if transa == CUBLAS_OP_C \end{cases}

Q is a unitary matrix formed by a sequence of elementary reflection vectors from QR factorization (geqrf) of A.

Q=H(1)H(2) ... H(k)

Q is of order m if side = CUBLAS_SIDE_LEFT and of order n if side = CUBLAS_SIDE_RIGHT.

The user has to provide working space which is pointed by input parameter work. The input parameter lwork is size of the working space, and it is returned by geqrf_bufferSize() or ormqr_bufferSize().

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

The user can combine geqrf, ormqr and trsm to complete a linear solver or a least-square solver. Please refer to appendix C.1.

API of ormqr

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	Handle to the cuSolverDn library context.
`side`	`host`	`input`	Indicates if matrix `Q` is on the left or right of `C`.
`trans`	`host`	`input`	Operation `op(Q)` that is non- or (conj.) transpose.
`m`	`host`	`input`	Number of columns of matrix `C`.
`n`	`host`	`input`	Number of rows of matrix `C`.
`k`	`host`	`input`	Number of elementary relfections whose product defines the matrix Q.
`A`	`device`	`in/out`	<type> array of dimension `lda * k` with `lda` is not less than `max(1,m)`. The matrix `A` is from `geqrf`, so i-th column contains elementary reflection vector.
`lda`	`host`	`input`	Leading dimension of two-dimensional array used to store matrix `A`. if `side` is `CUBLAS_SIDE_LEFT`, lda >= max(1,m); if `side` is `CUBLAS_SIDE_RIGHT`, lda >= max(1,n).
`tau`	`device`	`output`	<type> array of dimension at least `min(m,n)`. The vector `tau` is from `geqrf`, so `tau(i)` is the scalar of i-th elementary reflection vector.
`C`	`device`	`in/out`	<type> array of size `ldc * n`. On exit, `C` is overwritten by `op(Q)*C`.
`ldc`	`host`	`input`	Leading dimension of two-dimensional array of matrix `C`. ldc >= max(1,m).
`work`	`device`	`in/out`	Working space, <type> array of size `lwork`.
`lwork`	`host`	`input`	Size of working array `work`.
`devInfo`	`device`	`output`	If `devInfo = 0`, the ormqr is successful. If `devInfo = -i`, the `i-th` parameter is wrong (not counting handle).

Status Returned

`CUSOLVER_STATUS_SUCCESS`	The operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	The library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	Invalid parameters were passed (`m,n<0` or wrong `lda` or `ldc`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	The device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	An internal operation failed.

cusolverDn<t>orgqr()

These helper functions calculate the size of work buffers needed.

cusolverStatus_t 
cusolverDnSorgqr_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int k,
    const float *A,
    int lda,
    const float *tau,
    int *lwork);

cusolverStatus_t 
cusolverDnDorgqr_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int k,
    const double *A,
    int lda,
    const double *tau,
    int *lwork);

cusolverStatus_t 
cusolverDnCungqr_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int k,
    const cuComplex *A,
    int lda,
    const cuComplex *tau,
    int *lwork);

cusolverStatus_t 
cusolverDnZungqr_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int k,
    const cuDoubleComplex *A,
    int lda,
    const cuDoubleComplex *tau,
    int *lwork);

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSorgqr(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int k,
    float *A,
    int lda,
    const float *tau,
    float *work,
    int lwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnDorgqr(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int k,
    double *A,
    int lda,
    const double *tau,
    double *work,
    int lwork,
    int *devInfo);

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCungqr(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int k,
    cuComplex *A,
    int lda,
    const cuComplex *tau,
    cuComplex *work,
    int lwork,
    int *devInfo);

cusolverStatus_t
cusolverDnZungqr(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int k,
    cuDoubleComplex *A,
    int lda,
    const cuDoubleComplex *tau,
    cuDoubleComplex *work,
    int lwork,
    int *devInfo);

This function overwrites m×n matrix A by

Q = H(1) * H(2) * ... * H(k)

where Q is a unitary matrix formed by a sequence of elementary reflection vectors stored in A.

The user has to provide working space which is pointed by input parameter work. The input parameter lwork is size of the working space, and it is returned by orgqr_bufferSize().

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

The user can combine geqrf, orgqr to complete orthogonalization. Please refer to appendix C.2.

API of ormqr

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`m`	`host`	`input`	number of rows of matrix `Q`. m >= 0;
`n`	`host`	`input`	number of columns of matrix `Q`. m >= n >= 0;
`k`	`host`	`input`	number of elementary relfections whose product defines the matrix `Q`. n >= k >= 0;
`A`	`device`	`in/out`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,m)`. i-th column of `A` contains elementary reflection vector.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`. lda >= max(1,m).
`tau`	`device`	`output`	<type> array of dimension `k`. `tau(i)` is the scalar of i-th elementary reflection vector.
`work`	`device`	`in/out`	working space, <type> array of size `lwork`.
`lwork`	`host`	`input`	size of working array `work`.
`devInfo`	`device`	`output`	if `info = 0`, the orgqr is successful. if `info = -i`, the `i-th` parameter is wrong (not counting handle).

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`m,n,k<0`, `n>m`, `k>n` or `lda<m`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>sytrf()

These helper functions calculate the size of the needed buffers.

cusolverStatus_t 
cusolverDnSsytrf_bufferSize(cusolverDnHandle_t handle,
                      int n,
                      float *A,
                      int lda,
                      int *Lwork );

cusolverStatus_t 
cusolverDnDsytrf_bufferSize(cusolverDnHandle_t handle,
                      int n,
                      double *A,
                      int lda,
                      int *Lwork );

cusolverStatus_t 
cusolverDnCsytrf_bufferSize(cusolverDnHandle_t handle,
                      int n,
                      cuComplex *A,
                      int lda,
                      int *Lwork );

cusolverStatus_t 
cusolverDnZsytrf_bufferSize(cusolverDnHandle_t handle,
                      int n,
                      cuDoubleComplex *A,
                      int lda,
                      int *Lwork );

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSsytrf(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           float *A,
           int lda,
           int *ipiv,
           float *work,
           int lwork,
           int *devInfo );

cusolverStatus_t 
cusolverDnDsytrf(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           double *A,
           int lda,
           int *ipiv,
           double *work,
           int lwork,
           int *devInfo );

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCsytrf(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           cuComplex *A,
           int lda,
           int *ipiv,
           cuComplex *work,
           int lwork,
           int *devInfo );

cusolverStatus_t 
cusolverDnZsytrf(cusolverDnHandle_t handle,
           cublasFillMode_t uplo,
           int n,
           cuDoubleComplex *A,
           int lda,
           int *ipiv,
           cuDoubleComplex *work,
           int lwork,
           int *devInfo );

This function computes the Bunch-Kaufman factorization of a n×n symmetric indefinite matrix

A is a n×n symmetric matrix, only lower or upper part is meaningful. The input parameter uplo which part of the matrix is used. The function would leave other part untouched.

If input parameter uplo is CUBLAS_FILL_MODE_LOWER, only lower triangular part of A is processed, and replaced by lower triangular factor L and block diagonal matrix D. Each block of D is either 1x1 or 2x2 block, depending on pivoting.

P * A * P^{T} = L * D * L^{T}

If input parameter uplo is CUBLAS_FILL_MODE_UPPER, only upper triangular part of A is processed, and replaced by upper triangular factor U and block diagonal matrix D.

P * A * P^{T} = U * D * U^{T}

The user has to provide working space which is pointed by input parameter work. The input parameter lwork is size of the working space, and it is returned by sytrf_bufferSize().

If Bunch-Kaufman factorization failed, i.e. A is singular. The output parameter devInfo = i would indicate D(i,i)=0.

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

The output parameter devIpiv contains pivoting sequence. If devIpiv(i) = k > 0, D(i,i) is 1x1 block, and i-th row/column of A is interchanged with k-th row/column of A. If uplo is CUBLAS_FILL_MODE_UPPER and devIpiv(i-1) = devIpiv(i) = -m < 0, D(i-1:i,i-1:i) is a 2x2 block, and (i-1)-th row/column is interchanged with m-th row/column. If uplo is CUBLAS_FILL_MODE_LOWER and devIpiv(i+1) = devIpiv(i) = -m < 0, D(i:i+1,i:i+1) is a 2x2 block, and (i+1)-th row/column is interchanged with m-th row/column.

API of sytrf

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`uplo`	`host`	`input`	indicates if matrix `A` lower or upper part is stored, the other part is not referenced.
`n`	`host`	`input`	number of rows and columns of matrix `A`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,n)`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`.
`ipiv`	`device`	`output`	array of size at least `n`, containing pivot indices.
`work`	`device`	`in/out`	working space, <type> array of size `lwork`.
`lwork`	`host`	`input`	size of working space `work`.
`devInfo`	`device`	`output`	if `devInfo = 0`, the LU factorization is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle). if `devInfo = i`, the `D(i,i) = 0`.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`n<0` or `lda<max(1,n)`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>potrfBatched()

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t
cusolverDnSpotrfBatched(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    float *Aarray[],
    int lda,
    int *infoArray,
    int batchSize);

cusolverStatus_t 
cusolverDnDpotrfBatched(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    double *Aarray[],
    int lda,
    int *infoArray,
    int batchSize);

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCpotrfBatched(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    cuComplex *Aarray[],
    int lda,
    int *infoArray,
    int batchSize);

cusolverStatus_t 
cusolverDnZpotrfBatched(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    cuDoubleComplex *Aarray[],
    int lda,
    int *infoArray,
    int batchSize);

This function computes the Cholesky factorization of a squence of Hermitian positive-definite matrices.

Each Aarray[i] for i=0,1,..., batchSize-1 is a n×n Hermitian matrix, only lower or upper part is meaningful. The input parameter uplo indicates which part of the matrix is used.

If input parameter uplo is CUBLAS_FILL_MODE_LOWER, only lower triangular part of A is processed, and replaced by lower triangular Cholesky factor L.

A = L * L^{H}

If input parameter uplo is CUBLAS_FILL_MODE_UPPER, only upper triangular part of A is processed, and replaced by upper triangular Cholesky factor U.

A = U^{H} * U

If Cholesky factorization failed, i.e. some leading minor of A is not positive definite, or equivalently some diagonal elements of L or U is not a real number. The output parameter infoArray would indicate smallest leading minor of A which is not positive definite.

infoArray is an integer array of size batchsize. If potrfBatched returns CUSOLVER_STATUS_INVALID_VALUE, infoArray[0] = -i (less than zero), meaning that the i-th parameter is wrong (not counting handle). If potrfBatched returns CUSOLVER_STATUS_SUCCESS but infoArray[i] = k is positive, then i-th matrix is not positive definite and the Cholesky factorization failed at row k.

Remark: the other part of A is used as a workspace. For example, if uplo is CUBLAS_FILL_MODE_UPPER, upper triangle of A contains cholesky factor U and lower triangle of A is destroyed after potrfBatched.

API of potrfBatched

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`uplo`	`host`	`input`	indicates if lower or upper part is stored, the other part is used as a workspace.
`n`	`host`	`input`	number of rows and columns of matrix `A`.
`Aarray`	`device`	`in/out`	array of pointers to <type> array of dimension `lda * n` with `lda` is not less than `max(1,n)`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store each matrix `Aarray[i]`.
`infoArray`	`device`	`output`	array of size `batchSize`. `infoArray[i]` contains information of factorization of `Aarray[i]`. if `potrfBatched` returns `CUSOLVER_STATUS_INVALID_VALUE`, `infoArray[0] = -i` (less than zero) means the `i-th` parameter is wrong (not counting handle). if `potrfBatched` returns `CUSOLVER_STATUS_SUCCESS`, `infoArray[i] = 0` means the Cholesky factorization of `i-th` matrix is successful, and `infoArray[i] = k` means the leading submatrix of order k of `i-th` matrix is not positive definite.
`batchSize`	`host`	`input`	number of pointers in `Aarray`.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`n<0` or `lda<max(1,n)` or `batchSize<1`).
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>potrsBatched()

cusolverStatus_t 
cusolverDnSpotrsBatched(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    int nrhs,
    float *Aarray[],
    int lda,
    float *Barray[],
    int ldb,
    int *info,
    int batchSize);

cusolverStatus_t 
cusolverDnDpotrsBatched(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    int nrhs, 
    double *Aarray[],
    int lda,
    double *Barray[],
    int ldb,
    int *info,
    int batchSize);

cusolverStatus_t
cusolverDnCpotrsBatched(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    int nrhs,
    cuComplex *Aarray[],
    int lda,
    cuComplex *Barray[],
    int ldb,
    int *info,
    int batchSize);

cusolverStatus_t
cusolverDnZpotrsBatched(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    int nrhs,
    cuDoubleComplex *Aarray[],
    int lda,
    cuDoubleComplex *Barray[],
    int ldb,
    int *info,
    int batchSize);

This function solves a squence of linear systems

A[i] * X[i] = B[i]

where each Aarray[i] for i=0,1,..., batchSize-1 is a n×n Hermitian matrix, only lower or upper part is meaningful. The input parameter uplo indicates which part of the matrix is used.

The user has to call potrfBatched first to factorize matrix Aarray[i]. If input parameter uplo is CUBLAS_FILL_MODE_LOWER, A is lower triangular Cholesky factor L correspoding to $A = L * L^{H}$ . If input parameter uplo is CUBLAS_FILL_MODE_UPPER, A is upper triangular Cholesky factor U corresponding to $A = U^{H} * U$ .

The operation is in-place, i.e. matrix X overwrites matrix B with the same leading dimension ldb.

The output parameter info is a scalar. If info = -i (less than zero), the i-th parameter is wrong (not counting handle).

Remark 1: only nrhs=1 is supported.

Remark 2: infoArray from potrfBatched indicates if the matrix is positive definite. info from potrsBatched only shows which input parameter is wrong (not counting handle).

Remark 3: the other part of A is used as a workspace. For example, if uplo is CUBLAS_FILL_MODE_UPPER, upper triangle of A contains cholesky factor U and lower triangle of A is destroyed after potrsBatched.

API of potrsBatched

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolveDN library context.
`uplo`	`host`	`input`	indicates if matrix `A` lower or upper part is stored.
`n`	`host`	`input`	number of rows and columns of matrix `A`.
`nrhs`	`host`	`input`	number of columns of matrix `X` and `B`.
`Aarray`	`device`	`in/out`	array of pointers to <type> array of dimension `lda * n` with `lda` is not less than `max(1,n)`. `Aarray[i]` is either lower cholesky factor `L` or upper Cholesky factor `U`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store each matrix `Aarray[i]`.
`Barray`	`device`	`in/out`	array of pointers to <type> array of dimension `ldb * nrhs`. `ldb` is not less than `max(1,n)`. As an input, `Barray[i]` is right hand side matrix. As an output, `Barray[i]` is the solution matrix.
`ldb`	`host`	`input`	leading dimension of two-dimensional array used to store each matrix `Barray[i]`.
`info`	`device`	`output`	if `info = 0`, all parameters are correct. if `info = -i`, the `i-th` parameter is wrong (not counting handle).
`batchSize`	`host`	`input`	number of pointers in `Aarray`.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`n<0`, `nrhs<0`, `lda<max(1,n)`, `ldb<max(1,n)` or `batchSize<0`).
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t1><t2>gesv()

These functions are modelled after functions DSGESV and ZCGESV from LAPACK and compute the solution to a system of linear equations with multiple right hand sides using mixed precision iterative refinement

A \times X = B

Where A is n-by-n matrix and X and B are n-by-nrhs matrices.

Functions are designed to be as close to LAPACK drop-in replacements as possible. Parameters and behaviour are mostly the same as LAPACK counterparts. Description of LAPACK functions and differences from them are below.

<t1><t2>gesv() functions are designated by two floating point precisions - data type (full) precision and internal lower precision. cusolver<t1><t2>gesv() first attempts to factorize the matrix in lower precision and use this factorization within an iterative refinement procedure to obtain a solution with same normwise backward error as full precision. If the approach fails to converge, then the method switches to a full precision factorization and solve.

Note that in addition to the data type / lower floating point precision functions available in LAPACK - also functions with half precision as a lower precision are present. The following table specifies which precisions will be used for which interface function:

Supported combinations of floating point precisions for cusolver <t1><t2>gesv() functions

`Interface function`	`Data type (matrix, rhs and solution)`	`Data floating point precision`	`Internal (compute) floating point precision`
`cusolverDnDDgesv`	`double`	`double`	`double`
`cusolverDnDSgesv`	`double`	`double`	`single`
`cusolverDnDHgesv`	`double`	`double`	`half`
`cusolverDnSSgesv`	`float`	`single`	`single`
`cusolverDnSHgesv`	`float`	`single`	`half`
`cusolverDnZZgesv`	`cuDoubleComplex`	`double`	`double`
`cusolverDnZCgesv`	`cuDoubleComplex`	`double`	`single`
`cusolverDnZKgesv`	`cuDoubleComplex`	`double`	`half`
`cusolverDnCCgesv`	`cuComplex`	`single`	`single`
`cusolverDnCKgesv`	`cuComplex`	`single`	`half`

The iterative refinement process is stopped if
ITER > ITERMAX
or for all the RHS we have:
RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
where

ITER is the number of the current iteration in the iterative refinement process
RNRM is the infinity-norm of the residual
XNRM is the infinity-norm of the solution
ANRM is the infinity-operator-norm of the matrix A
EPS is the machine epsilon that matches LAPACK <t1>LAMCH('Epsilon')

The value ITERMAX and BWDMAX are fixed to 50 and 1.0 respectively.

Solve process results will be indicated by output parameter info , see parameter description.

User should provide a large enough workspace allocated on the device for the <t1><t2>gesv() functions. The amount of bytes required can be queried by the respective <t1><t2>gesv_bufferSize() functions.

cusolverDn<t1><t2>gesv_bufferSize() functions will return workspace buffer size in bytes required for corresponding cusolverDn<t1><t2>gesv() function.

cusolverStatus_t
cusolverDnDDgesv_bufferSize(
    cusolverHandle_t                handle,
    int                             n, 
    int                             nrhs,
    double                      *   dA, 
    int                             ldda,
    int                         *   dipiv,
    double                      *   dB, 
    int                             lddb,
    double                      *   dX, 
    int                             lddx,
    void                        *   dwork, 
    size_t                      *   lwork_bytes);

cusolverStatus_t
cusolverDnDSgesv_bufferSize(
    cusolverHandle_t                handle,
    int                             n, 
    int                             nrhs,
    double                      *   dA, 
    int                             ldda,
    int                         *   dipiv,
    double                      *   dB, 
    int                             lddb,
    double                      *   dX, 
    int                             lddx,
    void                        *   dwork, 
    size_t                      *   lwork_bytes);
 
cusolverStatus_t
cusolverDnDHgesv_bufferSize(
    cusolverHandle_t                handle,
    int                             n, 
    int                             nrhs,
    double                      *   dA, 
    int                             ldda,
    int                         *   dipiv,
    double                      *   dB, 
    int                             lddb,
    double                      *   dX, 
    int                             lddx,
    void                        *   dwork, 
    size_t                      *   lwork_bytes);

cusolverStatus_t
cusolverDnSSgesv_bufferSize(
    cusolverHandle_t                handle,
    int                             n, 
    int                             nrhs,
    float                       *   dA, 
    int                             ldda,
    int                         *   dipiv,
    float                       *   dB, 
    int                             lddb,
    float                       *   dX, 
    int                             lddx,
    void                        *   dwork, 
    size_t                      *   lwork_bytes);

cusolverStatus_t
cusolverDnSHgesv_bufferSize(
    cusolverHandle_t                handle,
    int                             n, 
    int                             nrhs,
    float                       *   dA, 
    int                             ldda,
    int                         *   dipiv,
    float                       *   dB, 
    int                             lddb,
    float                       *   dX, 
    int                             lddx,
    void                        *   dwork, 
    size_t                      *   lwork_bytes);

cusolverStatus_t
cusolverDnZZgesv_bufferSize(
    cusolverHandle_t                handle,
    int                             n, 
    int                             nrhs,
    cuDoubleComplex             *   dA, 
    int                             ldda,
    int                         *   dipiv,
    cuDoubleComplex             *   dB, 
    int                             lddb,
    cuDoubleComplex             *   dX, 
    int                             lddx,
    void                        *   dwork, 
    size_t                      *   lwork_bytes);

cusolverStatus_t
cusolverDnZCgesv_bufferSize(
    cusolverHandle_t                handle,
    int                             n, 
    int                             nrhs,
    cuDoubleComplex             *   dA, 
    int                             ldda,
    int                         *   dipiv,
    cuDoubleComplex             *   dB, 
    int                             lddb,
    cuDoubleComplex             *   dX, 
    int                             lddx,
    void                        *   dwork, 
    size_t                      *   lwork_bytes);
 
 
cusolverStatus_t
cusolverDnZKgesv_bufferSize(
    cusolverHandle_t                handle,
    int                             n, 
    int                             nrhs,
    cuDoubleComplex             *   dA, 
    int                             ldda,
    int                         *   dipiv,
    cuDoubleComplex             *   dB, 
    int                             lddb,
    cuDoubleComplex             *   dX, 
    int                             lddx,
    void                        *   dwork, 
    size_t                      *   lwork_bytes);

cusolverStatus_t
cusolverDnCCgesv_bufferSize(
    cusolverHandle_t                handle,
    int                             n, 
    int                             nrhs,
    cuComplex                   *   dA, 
    int                             ldda,
    int                         *   dipiv,
    cuComplex                   *   dB, 
    int                             lddb,
    cuComplex                   *   dX, 
    int                             lddx,
    void                        *   dwork, 
    size_t                      *   lwork_bytes);
 
cusolverStatus_t
cusolverDnCKgesv_bufferSize(
    cusolverHandle_t                handle,
    int                             n, 
    int                             nrhs,
    cuComplex                   *   dA, 
    int                             ldda,
    int                         *   dipiv,
    cuComplex                   *   dB, 
    int                             lddb,
    cuComplex                   *   dX, 
    int                             lddx,
    void                        *   dwork, 
    size_t                      *   lwork_bytes);

Parameters of cusolverDn<T1><T2>gesv_bufferSize() functions

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	Handle to the cusolverDN library context.
`n`	`host`	`input`	Number of rows and columns of square matrix `A`. Should be non-negative.
`nrhs`	`host`	`input`	Number of right hand sides to solve. Should be non-negative. `nrhs` is limited to 1 if selected IRS solver is CUSOLVER_IRS_GMRES.
`dA`	`device`	`in`	Matrix `A` with size `n-by-n`. Can be `NULL`.
`ldda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`. `lda >= n`.
`dipiv`	`device`	`None`	Pivoting sequence. Not used and can be `NULL`.
`dB`	`device`	`in`	Set of right hand sides `B` of size `n-by-nrhs`. Can be `NULL`.
`lddb`	`host`	`input`	leading dimension of two-dimensional array used to store matrix of right hand sides `B`. `ldb >= n`.
`dX`	`device`	`in`	Set of soultion vectors `X` of size `n-by-nrhs`. Can be `NULL`.
`lddx`	`host`	`input`	leading dimension of two-dimensional array used to store matrix of solution vectors `X`. `ldx >= n`.
`dwork`	`device`	`none`	Pointer to device workspace. Not used and can be `NULL`.
`lwork_bytes`	`host`	`out`	Pointer to a variable where required size of temporary workspace in bytes will be stored. Can't be NULL.

cusolverStatus_t cusolverDnZZgesv(
        cusolverDnHandle_t      handle,
        int                     n, 
        int                     nrhs,
        cuDoubleComplex     *   dA, 
        int                     ldda,
        int                 *   dipiv,
        cuDoubleComplex     *   dB, 
        int                     lddb,
        cuDoubleComplex     *   dX, 
        int                     lddx,
        void                *   dWorkspace, 
        size_t                  lwork_bytes,
        int                 *   iter,
        int                 *   d_info);

cusolverStatus_t cusolverDnZCgesv(
        cusolverDnHandle_t      handle,
        int                     n, 
        int                     nrhs,
        cuDoubleComplex     *   dA, 
        int                     ldda,
        int                 *   dipiv,
        cuDoubleComplex     *   dB, 
        int                     lddb,
        cuDoubleComplex     *   dX, 
        int                     lddx,
        void                *   dWorkspace, 
        size_t                  lwork_bytes,
        int                 *   iter,
        int                 *   d_info);

cusolverStatus_t cusolverDnZKgesv(
        cusolverDnHandle_t      handle,
        int                     n, 
        int                     nrhs,
        cuDoubleComplex     *   dA, 
        int                     ldda,
        int                 *   dipiv,
        cuDoubleComplex     *   dB, 
        int                     lddb,
        cuDoubleComplex     *   dX, 
        int                     lddx,
        void                *   dWorkspace, 
        size_t                  lwork_bytes,
        int                 *   iter,
        int                 *   d_info);

cusolverStatus_t cusolverDnCCgesv(
        cusolverDnHandle_t      handle,
        int                     n, 
        int                     nrhs,
        cuComplex           *   dA, 
        int                     ldda,
        int                 *   dipiv,
        cuComplex           *   dB, 
        int                     lddb,
        cuComplex           *   dX, 
        int                     lddx,
        void                *   dWorkspace, 
        size_t                  lwork_bytes,
        int                 *   iter,
        int                 *   d_info);

cusolverStatus_t cusolverDnCKgesv(
        cusolverDnHandle_t      handle,
        int                     n, 
        int                     nrhs,
        cuComplex           *   dA, 
        int                     ldda,
        int                 *   dipiv,
        cuComplex           *   dB, 
        int                     lddb,
        cuComplex           *   dX, 
        int                     lddx,
        void                *   dWorkspace, 
        size_t                  lwork_bytes,
        int                 *   iter,
        int                 *   d_info);

cusolverStatus_t cusolverDnDDgesv(
        cusolverDnHandle_t      handle,
        int                     n, 
        int                     nrhs,
        double              *   dA, 
        int                     ldda,
        int                 *   dipiv,
        double              *   dB, 
        int                     lddb,
        double              *   dX, 
        int                     lddx,
        void                *   dWorkspace, 
        size_t                  lwork_bytes,
        int                 *   iter,
        int                 *   d_info);

cusolverStatus_t cusolverDnDSgesv(
        cusolverDnHandle_t      handle,
        int                     n, 
        int                     nrhs,
        double              *   dA, 
        int                     ldda,
        int                 *   dipiv,
        double              *   dB, 
        int                     lddb,
        double              *   dX, 
        int                     lddx,
        void                *   dWorkspace, 
        size_t                  lwork_bytes,
        int                 *   iter,
        int                 *   d_info);

cusolverStatus_t cusolverDnDHgesv(
        cusolverDnHandle_t      handle,
        int                     n, 
        int                     nrhs,
        double              *   dA, 
        int                     ldda,
        int                 *   dipiv,
        double              *   dB, 
        int                     lddb,
        double              *   dX, 
        int                     lddx,
        void                *   dWorkspace, 
        size_t                  lwork_bytes,
        int                 *   iter,
        int                 *   d_info);

cusolverStatus_t cusolverDnSSgesv(
        cusolverDnHandle_t      handle,
        int                     n, 
        int                     nrhs,
        float               *   dA, 
        int                     ldda,
        int                 *   dipiv,
        float               *   dB, 
        int                     lddb,
        float               *   dX, 
        int                     lddx,
        void                *   dWorkspace, 
        size_t                  lwork_bytes,
        int                 *   iter,
        int                 *   d_info);

cusolverStatus_t cusolverDnSHgesv(
        cusolverDnHandle_t      handle,
        int                     n, 
        int                     nrhs,
        float               *   dA, 
        int                     ldda,
        int                 *   dipiv,
        float               *   dB, 
        int                     lddb,
        float               *   dX, 
        int                     lddx,
        void                *   dWorkspace, 
        size_t                  lwork_bytes,
        int                 *   iter,
        int                 *   d_info);

Parameters of cusolverDn<T1><T2>gesv() functions

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	Handle to the cusolverDN library context.
`n`	`host`	`input`	Number of rows and columns of square matrix `A`. Should be non-negative.
`nrhs`	`host`	`input`	Number of right hand sides to solve. Should be non-negative. `nrhs` is limited to 1 if selected IRS solver is CUSOLVER_IRS_GMRES.
`dA`	`device`	`in/out`	Matrix `A` with size `n-by-n`. Can't be `NULL`. On return - unchanged if solve process iterative refinement converged. If not - will contain full precision factorization of matrix A : `A = P * L * U`, where P - permutation matrix defined by vector ipiv, L and U - lower and upper triangular matrices.
`ldda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`. `lda >= n`.
`dipiv`	`device`	`in/out`	Vector that defines permutation matrix for factorization - row `i` was interchanged with row `ipiv[i]` If `NULL` then no pivoting is performed.
`dB`	`device`	`in/out`	Set of right hand sides `B` of size `n-by-nrhs` . Can't be `NULL`.
`lddb`	`host`	`input`	leading dimension of two-dimensional array used to store matrix of right hand sides `B`. `ldb >= n`.
`dX`	`device`	`in/out`	Set of soultion vectors `X` of size `n-by-nrhs`. Can't be `NULL`.
`lddx`	`host`	`input`	leading dimension of two-dimensional array used to store matrix of solution vectors `X`. `ldx >= n`.
`dWorkspace`	`device`	`in/out`	Pointer to a workspace in device memory of size `lwork_bytes`.
`lwork_bytes`	`host`	`in`	Size of provided device workspace in bytes. Should be at least what was returned by `cusolverDn<T1><T2>gesv_bufferSize()` function
`iter`	`host`	`output`	If iter is <0 : iterative refinement has failed, full precision factorization has been performed. -1 : taking into account machine parameters, n, nrhs, it is determined a priori it is not worth working in lower precision -2 : overflow of an entry when moving from double to lower precision -3 : failure of gesv function -31: solver stopped the iterative refinement after reaching maximum allowed iterations >0 : iter is a number of iterations solver perfromed to reach convergence criteria
`info`	`device`	`output`	Status of the iterative refinement on the return. If 0 - solve was successful. If info = -`i` then `i`-th argument has is not valid. If info = `i`, then `U(i,i)` computed in full precision is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed: `n<0` `ldda<max(1,n)` `lddb<max(1,n)` `lddx<max(1,n)` `NULL` where it's not allowed `nrhs` is larger than allowed
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 7.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

2.4.2.13. cusolverDnIRSXgesv()

This function is designed to perform same functionality as cusolverDn<T1><T2>gesv() functions, but wrapped in a more generic and expert interface that gives user more control to parametrize the function as well as it provides more informations on output. See cusolverDn<T1><T2>gesv() description for detailed explanation of the algorithm functionality and behaviour. cusolverDnIRSXgesv() allows additional control of the solver parameters such as setting the - lowest floating point precision authorized to be used by the solver, refinement solver type, maximum allowed number of iterative solver iterations, tolerance of the refinement solver - through Xgesv parameters structure and helper functions. cusolverDnIRSXgesv() provides additional informations on the output such as the convergence history of the residual array and the number of iterations needed to converge.

Following table provides authorized values for lowest precision parameter for specified full data type. Note that if lowest precision matches full datatype, then full precision factorization will be used

Supported lower floating point precisions for factorization for provided full datatype

`Data Type`	`Supported values for lowest precision in Xgesv parameters structure`
`CUDA_R_32F`	`CUDA_R_32F, CUDA_R_16F`
`CUDA_R_64F`	`CUDA_R_64F, CUDA_R_32F, CUDA_R_16F`
`CUDA_C_32F`	`CUDA_C_32F, CUDA_C_16F`
`CUDA_C_64F`	`CUDA_C_64F, CUDA_C_32F, CUDA_C_16F`

Solve process results will be indicated by output parameter info, see parameter description.

User should provide large enough workspace allocated on device for the cusolverDnIRSXgesv() function. Amount of bytes required for the function can be retrieved by respective function cusolverDnIRSXgesv_bufferSize()

cusolverDnIRSXgesv_bufferSize() functions will return workspace buffer size in bytes required for corresponding cusolverDnXgesv() function with given parameters.

cusolverStatus_t
cusolverDnIRSXgesv_bufferSize(
        cusolverDnHandle_t          handle,
        cusolverDnIRSParams_t       params,
        cusolver_int_t              n, 
        cusolver_int_t              nrhs,
        size_t                  *   lwork_bytes);

Parameters of cusolverDnIRSXgesv_bufferSize() functions

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	Handle to the cusolverDn library context.
`params`	`host`	`input`	Xgesv solve parameters
`n`	`host`	`input`	Number of rows and columns of the square matrix `A`. Should be non-negative.
`nrhs`	`host`	`input`	Number of right hand sides to solve. Should be non-negative. Note that, `nrhs` is limited to 1 if the selected IRS refinement solver is CUSOLVER_IRS_REFINE_GMRES, CUSOLVER_IRS_REFINE_GMRES_GMRES, CUSOLVER_IRS_REFINE_CLASSICAL_GMRES.
`lwork_bytes`	`host`	`out`	Pointer to a variable, where the required size in bytes, of the workspace will be stored after a call to cusolverDnIRSXgesv_bufferSize. Can't be NULL.

cusolverStatus_t cusolverDnIRSXgesv(
        cusolverDnHandle_t          handle,
        cusolverDnIRSParams_t       gesv_irs_params,
        cusolverDnIRSInfos_t        gesv_irs_infos,
        cudaDataType                inout_data_type,
        int                         n, 
        int                         nrhs,
        void                    *   dA, 
        int                         ldda,
        int                     *   dipiv,
        void                    *   dB,     
        int                         lddb,
        void                    *   dX, 
        int                         lddx,
        void                    *   dWorkspace, 
        size_t                      lwork_bytes,
        int                     *   niters,
        int                     *   dinfo);

Parameters of cusolverDnIRSXgesv() functions

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	Handle to the cusolverDn library context.
`gesv_irs_params`	`host`	`input`	Solve parameters handle
`gesv_irs_infos`	`host`	`in/out`	Info structure parameter handle where information about performed solve will be stored.
`inout_data_type`	`host`	`input`	Datatype of Matrix, right hand side and solution
`n`	`host`	`input`	Number of rows and columns of square matrix `A`. Should be non-negative.
`nrhs`	`host`	`input`	Number of right hand sides to solve. Should be non-negative. Note that, `nrhs` is limited to 1 if the selected IRS refinement solver is CUSOLVER_IRS_REFINE_GMRES, CUSOLVER_IRS_REFINE_GMRES_GMRES, CUSOLVER_IRS_REFINE_CLASSICAL_GMRES. Number of right hand sides to solve. Should be non-negative.
`dA`	`device`	`in`	Matrix `A` with size `n-by-n`. Can't be `NULL`. On return - unchanged if the iterative refinement solver converged. If not - will contain full precision factorization of matrix A : `A = P * L * U`, where P - permutation matrix defined by vector ipiv, L and U - lower and upper triangular matrices.
`ldda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`. `lda >= n`..
`dipiv`	`device`	`in/out`	Vector that defines permutation matrix for factorization - row `i` was interchanged with row `ipiv[i]` If NULL then no pivoting is performed.
`dB`	`device`	`in`	Set of right hand sides `B` of size `n-by-nrhs` . Can be `NULL`.
`lddb`	`host`	`input`	leading dimension of two-dimensional array used to store matrix of right hand sides `B`. `ldb >= n`..
`dX`	`device`	`in`	Set of soultion vectors `X` of size `n-by-nrhs` . Can be `NULL`.
`lddx`	`host`	`input`	leading dimension of two-dimensional array used to store matrix of solution vectors `X`. `ldx >= n`..
`dWorkspace`	`device`	`in/out`	Pointer to a workspace in device memory of size lwork_bytes.
`lwork_bytes`	`host`	`in`	Size of device workspace. Should be at least what was returned by `cusolverDnIRSXgesv_bufferSize()` function
`niters`	`host`	`output`	If iter is <0 : iterative refinement has failed, full precision factorization has been performed. -1 : taking into account machine parameters, n, nrhs, it is a priori not worth working in lower precision -2 : overflow of an entry when moving from double to lower precision -3 : failure of gesv function -maxiter: solver stopped the iterative refinement after reaching maximum allowed iterations >0 : iter is a number of iterations solver perfromed to reach convergence criteria
`dinfo`	`device`	`output`	Status of the iterative refinement on the return. If 0 - solve was successful. If dinfo = -`i` then `i`-th argument has is not valid. If dinfo = `i`, then `U(i,i)` computed in full precision is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed: `n<0` `lda<max(1,n)` `ldb<max(1,n)` `ldx<max(1,n)` `NULL` where it's not allowed `nrhs` is larger than allowed
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 7.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

Dense Eigenvalue Solver Reference

This chapter describes eigenvalue solver API of cuSolverDN, including bidiagonalization and SVD.

cusolverDn<t>gebrd()

These helper functions calculate the size of work buffers needed.

cusolverStatus_t 
cusolverDnSgebrd_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int *Lwork );

cusolverStatus_t 
cusolverDnDgebrd_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int *Lwork );

cusolverStatus_t
cusolverDnCgebrd_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int *Lwork );

cusolverStatus_t 
cusolverDnZgebrd_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int *Lwork );

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSgebrd(cusolverDnHandle_t handle,
           int m,
           int n,
           float *A,
           int lda,
           float *D,
           float *E,
           float *TAUQ,
           float *TAUP,
           float *Work,
           int Lwork,
           int *devInfo );

cusolverStatus_t 
cusolverDnDgebrd(cusolverDnHandle_t handle,
           int m,
           int n,
           double *A,
           int lda,
           double *D,
           double *E,
           double *TAUQ,
           double *TAUP,
           double *Work,
           int Lwork,
           int *devInfo );

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCgebrd(cusolverDnHandle_t handle,
           int m,
           int n,
           cuComplex *A,
           int lda,
           float *D,
           float *E,
           cuComplex *TAUQ,
           cuComplex *TAUP,
           cuComplex *Work,
           int Lwork,
           int *devInfo );

cusolverStatus_t 
cusolverDnZgebrd(cusolverDnHandle_t handle,
           int m,
           int n,
           cuDoubleComplex *A,
           int lda,
           double *D,
           double *E,
           cuDoubleComplex *TAUQ,
           cuDoubleComplex *TAUP,
           cuDoubleComplex *Work,
           int Lwork,
           int *devInfo );

This function reduces a general m×n matrix A to a real upper or lower bidiagonal form B by an orthogonal transformation: $Q^{H} * A * P = B$

If m>=n, B is upper bidiagonal; if m<n, B is lower bidiagonal.

The matrix Q and P are overwritten into matrix A in the following sense:

if m>=n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors.

if m<n, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors.

The user has to provide working space which is pointed by input parameter Work. The input parameter Lwork is size of the working space, and it is returned by gebrd_bufferSize().

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

Remark: gebrd only supports m>=n.

API of gebrd

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`m`	`host`	`input`	number of rows of matrix `A`.
`n`	`host`	`input`	number of columns of matrix `A`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,n)`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`.
`D`	`device`	`output`	real array of dimension `min(m,n)`. The diagonal elements of the bidiagonal matrix `B`: `D(i) = A(i,i)`.
`E`	`device`	`output`	real array of dimension `min(m,n)`. The off-diagonal elements of the bidiagonal matrix `B`: if `m>=n`, `E(i) = A(i,i+1)` for `i = 1,2,...,n-1`; if `m<n`, `E(i) = A(i+1,i)` for `i = 1,2,...,m-1`.
`TAUQ`	`device`	`output`	<type> array of dimension `min(m,n)`. The scalar factors of the elementary reflectors which represent the orthogonal matrix `Q`.
`TAUP`	`device`	`output`	<type> array of dimension `min(m,n)`. The scalar factors of the elementary reflectors which represent the orthogonal matrix `P`.
`Work`	`device`	`in/out`	working space, <type> array of size `Lwork`.
`Lwork`	`host`	`input`	size of `Work`, returned by `gebrd_bufferSize`.
`devInfo`	`device`	`output`	if `devInfo = 0`, the operation is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle).

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`m,n<0`, or `lda<max(1,m)`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>orgbr()

These helper functions calculate the size of work buffers needed.

cusolverStatus_t 
cusolverDnSorgbr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    int m,
    int n,
    int k,
    const float *A,
    int lda,
    const float *tau,
    int *lwork);

cusolverStatus_t
cusolverDnDorgbr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    int m,
    int n,
    int k,
    const double *A,
    int lda,
    const double *tau,
    int *lwork);

cusolverStatus_t 
cusolverDnCungbr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    int m,
    int n,
    int k,
    const cuComplex *A,
    int lda,
    const cuComplex *tau,
    int *lwork);

cusolverStatus_t 
cusolverDnZungbr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    int m,
    int n,
    int k,
    const cuDoubleComplex *A,
    int lda,
    const cuDoubleComplex *tau,
    int *lwork);

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSorgbr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    int m,
    int n,
    int k,
    float *A,
    int lda,
    const float *tau,
    float *work,
    int lwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnDorgbr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    int m,
    int n,
    int k,
    double *A,
    int lda,
    const double *tau,
    double *work,
    int lwork,
    int *devInfo);

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCungbr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    int m,
    int n,
    int k,
    cuComplex *A,
    int lda,
    const cuComplex *tau,
    cuComplex *work,
    int lwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnZungbr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    int m,
    int n,
    int k,
    cuDoubleComplex *A,
    int lda,
    const cuDoubleComplex *tau,
    cuDoubleComplex *work,
    int lwork,
    int *devInfo);

This function generates one of the unitary matrices Q or P**H determined by gebrd when reducing a matrix A to bidiagonal form: $Q^{H} * A * P = B$

Q and P**H are defined as products of elementary reflectors H(i) or G(i) respectively.

The user has to provide working space which is pointed by input parameter work. The input parameter lwork is size of the working space, and it is returned by orgbr_bufferSize().

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

API of orgbr

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`side`	`host`	`input`	if `side` = `CUBLAS_SIDE_LEFT`, generate Q. if `side` = `CUBLAS_SIDE_RIGHT`, generate P**T.
`m`	`host`	`input`	number of rows of matrix `Q` or `P**T`.
`n`	`host`	`input`	if `side` = `CUBLAS_SIDE_LEFT`, m>= n>= min(m,k). if `side` = `CUBLAS_SIDE_RIGHT`, n>= m>= min(n,k).
`k`	`host`	`input`	if `side` = `CUBLAS_SIDE_LEFT`, the number of columns in the original m-by-k matrix reduced by `gebrd`. if `side` = `CUBLAS_SIDE_RIGHT`, the number of rows in the original k-by-n matrix reduced by `gebrd`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` On entry, the vectors which define the elementary reflectors, as returned by `gebrd`. On exit, the m-by-n matrix `Q` or `P**T`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`. lda >= max(1,m);
`tau`	`device`	`output`	<type> array of dimension `min(m,k)` if `side` is `CUBLAS_SIDE_LEFT`; of dimension `min(n,k)` if `side` is `CUBLAS_SIDE_RIGHT`; tau(i) must contain the scalar factor of the elementary reflector H(i) or G(i), which determines Q or P**T, as returned by `gebrd` in its array argument `TAUQ` or `TAUP`.
`work`	`device`	`in/out`	working space, <type> array of size `lwork`.
`lwork`	`host`	`input`	size of working array `work`.
`devInfo`	`device`	`output`	if `info = 0`, the ormqr is successful. if `info = -i`, the `i-th` parameter is wrong (not counting handle).

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`m,n<0` or wrong `lda` ).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>sytrd()

These helper functions calculate the size of work buffers needed.

cusolverStatus_t 
cusolverDnSsytrd_bufferSize(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    const float *A,
    int lda,
    const float *d,
    const float *e,
    const float *tau,
    int *lwork);

cusolverStatus_t 
cusolverDnDsytrd_bufferSize(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    const double *A,
    int lda,
    const double *d,
    const double *e,
    const double *tau,
    int *lwork);

cusolverStatus_t
cusolverDnChetrd_bufferSize(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    const cuComplex *A,
    int lda,
    const float *d,
    const float *e,
    const cuComplex *tau,
    int *lwork);

cusolverStatus_t 
cusolverDnZhetrd_bufferSize(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    const cuDoubleComplex *A,
    int lda,
    const double *d,
    const double *e,
    const cuDoubleComplex *tau,
    int *lwork);

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSsytrd(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    float *A,
    int lda,
    float *d,
    float *e,
    float *tau,
    float *work,
    int lwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnDsytrd(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    double *A,
    int lda,
    double *d,
    double *e,
    double *tau,
    double *work,
    int lwork,
    int *devInfo);

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnChetrd(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    cuComplex *A,
    int lda,
    float *d,
    float *e,
    cuComplex *tau,
    cuComplex *work,
    int lwork,
    int *devInfo);

cusolverStatus_t CUDENSEAPI cusolverDnZhetrd(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    cuDoubleComplex *A,
    int lda,
    double *d,
    double *e,
    cuDoubleComplex *tau,
    cuDoubleComplex *work,
    int lwork,
    int *devInfo);

This function reduces a general symmetric (Hermitian) n×n matrix A to real symmetric tridiagonal form T by an orthogonal transformation: $Q^{H} * A * Q = T$

As an output, A contains T and householder reflection vectors. If uplo = CUBLAS_FILL_MODE_UPPER, the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors; If uplo = CUBLAS_FILL_MODE_LOWER, the diagonal and first subdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors.

The user has to provide working space which is pointed by input parameter work. The input parameter lwork is size of the working space, and it is returned by sytrd_bufferSize().

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

API of sytrd

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`uplo`	`host`	`input`	specifies which part of `A` is stored. `uplo` = `CUBLAS_FILL_MODE_LOWER`: Lower triangle of `A` is stored. `uplo` = `CUBLAS_FILL_MODE_UPPER`: Upper triangle of `A` is stored.
`n`	`host`	`input`	number of rows (columns) of matrix `A`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,n)`. If `uplo` = `CUBLAS_FILL_MODE_UPPER`, the leading n-by-n upper triangular part of `A` contains the upper triangular part of the matrix `A`, and the strictly lower triangular part of `A` is not referenced. If `uplo` = `CUBLAS_FILL_MODE_LOWER`, the leading n-by-n lower triangular part of `A` contains the lower triangular part of the matrix `A`, and the strictly upper triangular part of `A` is not referenced. On exit, `A` is overwritten by `T` and householder reflection vectors.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`. `lda >= max(1,n)`.
`D`	`device`	`output`	real array of dimension `n`. The diagonal elements of the tridiagonal matrix `T`: `D(i) = A(i,i)`.
`E`	`device`	`output`	real array of dimension `(n-1)`. The off-diagonal elements of the tridiagonal matrix `T`: if `uplo` = `CUBLAS_FILL_MODE_UPPER`, `E(i) = A(i,i+1)`. if `uplo` = `CUBLAS_FILL_MODE_LOWERE(i) = A(i+1,i)`.
`tau`	`device`	`output`	<type> array of dimension `(n-1)`. The scalar factors of the elementary reflectors which represent the orthogonal matrix `Q`.
`work`	`device`	`in/out`	working space, <type> array of size `lwork`.
`lwork`	`host`	`input`	size of `work`, returned by `sytrd_bufferSize`.
`devInfo`	`device`	`output`	if `devInfo = 0`, the operation is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle).

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`n<0`, or `lda<max(1,n)`, or `uplo` is not `CUBLAS_FILL_MODE_LOWER` or `CUBLAS_FILL_MODE_UPPER`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>ormtr()

These helper functions calculate the size of work buffers needed.

cusolverStatus_t 
cusolverDnSormtr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasFillMode_t uplo,
    cublasOperation_t trans,
    int m,
    int n,
    const float *A,
    int lda,
    const float *tau,
    const float *C,
    int ldc,
    int *lwork);

cusolverStatus_t 
cusolverDnDormtr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasFillMode_t uplo,
    cublasOperation_t trans,
    int m,
    int n,
    const double *A,
    int lda,
    const double *tau,
    const double *C,
    int ldc,
    int *lwork);

cusolverStatus_t 
cusolverDnCunmtr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasFillMode_t uplo,
    cublasOperation_t trans,
    int m,
    int n,
    const cuComplex *A,
    int lda,
    const cuComplex *tau,
    const cuComplex *C,
    int ldc,
    int *lwork);

cusolverStatus_t 
cusolverDnZunmtr_bufferSize(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasFillMode_t uplo,
    cublasOperation_t trans,
    int m,
    int n,
    const cuDoubleComplex *A,
    int lda,
    const cuDoubleComplex *tau,
    const cuDoubleComplex *C,
    int ldc,
    int *lwork);

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSormtr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasFillMode_t uplo,
    cublasOperation_t trans,
    int m,
    int n,
    float *A,
    int lda,
    float *tau,
    float *C,
    int ldc,
    float *work,
    int lwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnDormtr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasFillMode_t uplo,
    cublasOperation_t trans,
    int m,
    int n,
    double *A,
    int lda,
    double *tau,
    double *C,
    int ldc,
    double *work,
    int lwork,
    int *devInfo);

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCunmtr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasFillMode_t uplo,
    cublasOperation_t trans,
    int m,
    int n,
    cuComplex *A,
    int lda,
    cuComplex *tau,
    cuComplex *C,
    int ldc,
    cuComplex *work,
    int lwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnZunmtr(
    cusolverDnHandle_t handle,
    cublasSideMode_t side,
    cublasFillMode_t uplo,
    cublasOperation_t trans,
    int m,
    int n,
    cuDoubleComplex *A,
    int lda,
    cuDoubleComplex *tau,
    cuDoubleComplex *C,
    int ldc,
    cuDoubleComplex *work,
    int lwork,
    int *devInfo);

This function overwrites m×n matrix C by

C = \{\begin{cases} op (Q) * C & if side == CUBLAS_SIDE_LEFT \\ C * op (Q) & if side == CUBLAS_SIDE_RIGHT \end{cases}

where Q is a unitary matrix formed by a sequence of elementary reflection vectors from sytrd.

The operation on Q is defined by

op (Q) = \{\begin{cases} Q & if transa == CUBLAS_OP_N \\ Q^{T} & if transa == CUBLAS_OP_T \\ Q^{H} & if transa == CUBLAS_OP_C \end{cases}

The user has to provide working space which is pointed by input parameter work. The input parameter lwork is size of the working space, and it is returned by ormtr_bufferSize().

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

API of ormtr

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`side`	`host`	`input`	`side` = `CUBLAS_SIDE_LEFT`, apply Q or QT from the Left; `side` = `CUBLAS_SIDE_RIGHT`, apply Q or QT from the Right.
`uplo`	`host`	`input`	`uplo` = `CUBLAS_FILL_MODE_LOWER`: Lower triangle of `A` contains elementary reflectors from `sytrd`. `uplo` = `CUBLAS_FILL_MODE_UPPER`: Upper triangle of `A` contains elementary reflectors from `sytrd`.
`trans`	`host`	`input`	operation `op(Q)` that is non- or (conj.) transpose.
`m`	`host`	`input`	number of rows of matrix `C`.
`n`	`host`	`input`	number of columns of matrix `C`.
`A`	`device`	`in/out`	<type> array of dimension `lda * m` if `side` = `CUBLAS_SIDE_LEFT`; `lda * n` if `side` = `CUBLAS_SIDE_RIGHT`. The matrix `A` from `sytrd` contains the elementary reflectors.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`. if `side` is `CUBLAS_SIDE_LEFT`, lda >= max(1,m); if `side` is `CUBLAS_SIDE_RIGHT`, lda >= max(1,n).
`tau`	`device`	`output`	<type> array of dimension `(m-1)` if `side` is `CUBLAS_SIDE_LEFT`; of dimension `(n-1)` if `side` is `CUBLAS_SIDE_RIGHT`; The vector `tau` is from `sytrd`, so `tau(i)` is the scalar of i-th elementary reflection vector.
`C`	`device`	`in/out`	<type> array of size `ldc * n`. On exit, `C` is overwritten by `op(Q)C` or `Cop(Q)`.
`ldc`	`host`	`input`	leading dimension of two-dimensional array of matrix `C`. ldc >= max(1,m).
`work`	`device`	`in/out`	working space, <type> array of size `lwork`.
`lwork`	`host`	`input`	size of working array `work`.
`devInfo`	`device`	`output`	if `devInfo = 0`, the ormqr is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle).

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`m,n<0` or wrong `lda` or `ldc`).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>orgtr()

These helper functions calculate the size of work buffers needed.

cusolverStatus_t 
cusolverDnSorgtr_bufferSize(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    const float *A,
    int lda,
    const float *tau,
    int *lwork);

cusolverStatus_t 
cusolverDnDorgtr_bufferSize(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    const double *A,
    int lda,
    const double *tau,
    int *lwork);

cusolverStatus_t 
cusolverDnCungtr_bufferSize(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    const cuComplex *A,
    int lda,
    const cuComplex *tau,
    int *lwork);

cusolverStatus_t 
cusolverDnZungtr_bufferSize(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    const cuDoubleComplex *A,
    int lda,
    const cuDoubleComplex *tau,
    int *lwork);

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSorgtr(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    float *A,
    int lda,
    const float *tau,
    float *work,
    int lwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnDorgtr(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    double *A,
    int lda,
    const double *tau,
    double *work,
    int lwork,
    int *devInfo);

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCungtr(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    cuComplex *A,
    int lda,
    const cuComplex *tau,
    cuComplex *work,
    int lwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnZungtr(
    cusolverDnHandle_t handle,
    cublasFillMode_t uplo,
    int n,
    cuDoubleComplex *A,
    int lda,
    const cuDoubleComplex *tau,
    cuDoubleComplex *work,
    int lwork,
    int *devInfo);

This function generates a unitary matrix Q which is defined as the product of n-1 elementary reflectors of order n, as returned by sytrd:

The user has to provide working space which is pointed by input parameter work. The input parameter lwork is size of the working space, and it is returned by orgtr_bufferSize().

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle).

API of orgtr

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`uplo`	`host`	`input`	`uplo` = `CUBLAS_FILL_MODE_LOWER`: Lower triangle of `A` contains elementary reflectors from `sytrd`. `uplo` = `CUBLAS_FILL_MODE_UPPER`: Upper triangle of `A` contains elementary reflectors from `sytrd`.
`n`	`host`	`input`	number of rows (columns) of matrix `Q`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` On entry, matrix `A` from `sytrd` contains the elementary reflectors. On exit, matrix `A` contains the n-by-n orthogonal matrix `Q`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`. lda >= max(1,n).
`tau`	`device`	`output`	<type> array of dimension `(n-1)tau(i)` is the scalar of i-th elementary reflection vector.
`work`	`device`	`in/out`	working space, <type> array of size `lwork`.
`lwork`	`host`	`input`	size of working array `work`.
`devInfo`	`device`	`output`	if `devInfo = 0`, the orgtr is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle).

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`n<0` or wrong `lda` ).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>gesvd()

The helper functions below can calculate the sizes needed for pre-allocated buffer.

cusolverStatus_t 
cusolverDnSgesvd_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int *lwork );

cusolverStatus_t 
cusolverDnDgesvd_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int *lwork );

cusolverStatus_t 
cusolverDnCgesvd_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int *lwork );

cusolverStatus_t
cusolverDnZgesvd_bufferSize(
    cusolverDnHandle_t handle,
    int m,
    int n,
    int *lwork );

The S and D data types are real valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnSgesvd (
    cusolverDnHandle_t handle,
    signed char jobu,
    signed char jobvt,
    int m,
    int n,
    float *A,
    int lda,
    float *S,
    float *U,
    int ldu,
    float *VT,
    int ldvt,
    float *work,
    int lwork,
    float *rwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnDgesvd (
    cusolverDnHandle_t handle,
    signed char jobu,
    signed char jobvt,
    int m,
    int n,
    double *A,
    int lda,
    double *S,
    double *U,
    int ldu,
    double *VT,
    int ldvt,
    double *work,
    int lwork,
    double *rwork,
    int *devInfo);

The C and Z data types are complex valued single and double precision, respectively.

cusolverStatus_t 
cusolverDnCgesvd (
    cusolverDnHandle_t handle,
    signed char jobu,
    signed char jobvt,
    int m,
    int n,
    cuComplex *A,
    int lda,
    float *S,
    cuComplex *U,
    int ldu,
    cuComplex *VT,
    int ldvt,
    cuComplex *work,
    int lwork,
    float *rwork,
    int *devInfo);

cusolverStatus_t 
cusolverDnZgesvd (
    cusolverDnHandle_t handle,
    signed char jobu,
    signed char jobvt,
    int m,
    int n,
    cuDoubleComplex *A,
    int lda,
    double *S,
    cuDoubleComplex *U,
    int ldu,
    cuDoubleComplex *VT,
    int ldvt,
    cuDoubleComplex *work,
    int lwork,
    double *rwork,
    int *devInfo);

This function computes the singular value decomposition (SVD) of a m×n matrix A and corresponding the left and/or right singular vectors. The SVD is written

A = U * Σ * V^{H}

where Σ is an m×n matrix which is zero except for its min(m,n) diagonal elements, U is an m×m unitary matrix, and V is an n×n unitary matrix. The diagonal elements of Σ are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A.

The user has to provide working space which is pointed by input parameter work. The input parameter lwork is size of the working space, and it is returned by gesvd_bufferSize().

If output parameter devInfo = -i (less than zero), the i-th parameter is wrong (not counting handle). if bdsqr did not converge, devInfo specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.

The rwork is real array of dimension (min(m,n)-1). If devInfo>0 and rwork is not nil, rwork contains the unconverged superdiagonal elements of an upper bidiagonal matrix. This is slightly different from LAPACK which puts unconverged superdiagonal elements in work if type is real; in rwork if type is complex. rwork can be a NULL pointer if the user does not want the information from supperdiagonal.

Appendix G.1 provides a simple example of gesvd.

Remark 1: gesvd only supports m>=n.

Remark 2: the routine returns $V^{H}$ , not V.

API of gesvd

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`jobu`	`host`	`input`	specifies options for computing all or part of the matrix `U`: = 'A': all m columns of U are returned in array U: = 'S': the first min(m,n) columns of U (the left singular vectors) are returned in the array U; = 'O': the first min(m,n) columns of U (the left singular vectors) are overwritten on the array A; = 'N': no columns of U (no left singular vectors) are computed.
`jobvt`	`host`	`input`	specifies options for computing all or part of the matrix VT: = 'A': all N rows of VT are returned in the array VT; = 'S': the first min(m,n) rows of VT (the right singular vectors) are returned in the array VT; = 'O': the first min(m,n) rows of VT (the right singular vectors) are overwritten on the array A; = 'N': no rows of V**T (no right singular vectors) are computed.
`m`	`host`	`input`	number of rows of matrix `A`.
`n`	`host`	`input`	number of columns of matrix `A`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,m)`. On exit, the contents of `A` are destroyed.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`.
`S`	`device`	`output`	real array of dimension `min(m,n)`. The singular values of A, sorted so that `S(i) >= S(i+1)`.
`U`	`device`	`output`	<type> array of dimension `ldu * m` with `ldu` is not less than `max(1,m)`. `U` contains the `m×m` unitary matrix `U`.
`ldu`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `U`.
`VT`	`device`	`output`	<type> array of dimension `ldvt * n` with `ldvt` is not less than `max(1,n)`. `VT` contains the `n×n` unitary matrix V**T.
`ldvt`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `Vt`.
`work`	`device`	`in/out`	working space, <type> array of size `lwork`.
`lwork`	`host`	`input`	size of `work`, returned by `gesvd_bufferSize`.
`rwork`	`device`	`input`	real array of dimension min(m,n)-1. It contains the unconverged superdiagonal elements of an upper bidiagonal matrix if `devInfo > 0`.
`devInfo`	`device`	`output`	if `devInfo = 0`, the operation is successful. if `devInfo = -i`, the `i-th` parameter is wrong (not counting handle). if `devInfo > 0`, `devInfo` indicates how many superdiagonals of an intermediate bidiagonal form did not converge to zero.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`m,n<0` or `lda<max(1,m)` or `ldu<max(1,m)` or `ldvt<max(1,n)` ).
`CUSOLVER_STATUS_ARCH_MISMATCH`	the device only supports compute capability 2.0 and above.
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>gesvdj()

The helper functions below can calculate the sizes needed for pre-allocated buffer.

cusolverStatus_t
cusolverDnSgesvdj_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int econ,
    int m,
    int n, 
    const float *A,
    int lda,
    const float *S,
    const float *U,
    int ldu,
    const float *V,
    int ldv, 
    int *lwork,
    gesvdjInfo_t params);

cusolverStatus_t
cusolverDnDgesvdj_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz, 
    int econ,             
    int m,                
    int n,                
    const double *A,      
    int lda,             
    const double *S, 
    const double *U,      
    int ldu,              
    const double *V,      
    int ldv,              
    int *lwork,
    gesvdjInfo_t params);

cusolverStatus_t
cusolverDnCgesvdj_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz, 
    int econ,             
    int m,                
    int n,                
    const cuComplex *A,      
    int lda,             
    const float *S, 
    const cuComplex *U,      
    int ldu,              
    const cuComplex *V,      
    int ldv,              
    int *lwork,
    gesvdjInfo_t params);

cusolverStatus_t
cusolverDnZgesvdj_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz, 
    int econ,             
    int m,                
    int n,                
    const cuDoubleComplex *A,      
    int lda,             
    const double *S, 
    const cuDoubleComplex *U,      
    int ldu,              
    const cuDoubleComplex *V,      
    int ldv,              
    int *lwork,
    gesvdjInfo_t params);

The S and D data types are real valued single and double precision, respectively.


cusolverStatus_t 
cusolverDnSgesvdj(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int econ,
    int m,
    int n,
    float *A,
    int lda,
    float *S,
    float *U,
    int ldu,
    float *V,
    int ldv,
    float *work,
    int lwork,
    int *info,
    gesvdjInfo_t params);

cusolverStatus_t 
cusolverDnDgesvdj(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz, 
    int econ,             
    int m,                
    int n,                
    double *A,            
    int lda,              
    double *S,       
    double *U,            
    int ldu,              
    double *V,            
    int ldv,              
    double *work,
    int lwork,
    int *info,
    gesvdjInfo_t params);

The C and Z data types are complex valued single and double precision, respectively.


cusolverStatus_t 
cusolverDnCgesvdj(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz, 
    int econ,             
    int m,                
    int n,                
    cuComplex *A,            
    int lda,              
    float *S,       
    cuComplex *U,            
    int ldu,              
    cuComplex *V,            
    int ldv,              
    cuComplex *work,
    int lwork,
    int *info,
    gesvdjInfo_t params);

cusolverStatus_t 
cusolverDnZgesvdj(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz, 
    int econ,             
    int m,                
    int n,                
    cuDoubleComplex *A,            
    int lda,              
    double *S,       
    cuDoubleComplex *U,            
    int ldu,              
    cuDoubleComplex *V,            
    int ldv,              
    cuDoubleComplex *work,
    int lwork,
    int *info,
    gesvdjInfo_t params);

This function computes the singular value decomposition (SVD) of a m×n matrix A and corresponding the left and/or right singular vectors. The SVD is written

A = U * Σ * V^{H}

gesvdj has the same functionality as gesvd. The difference is that gesvd uses QR algorithm and gesvdj uses Jacobi method. The parallelism of Jacobi method gives GPU better performance on small and medium size matrices. Moreover the user can configure gesvdj to perform approximation up to certain accuracy.

gesvdj iteratively generates a sequence of unitary matrices to transform matrix A to the following form

U^{H} * A * V = S + E

where S is diagonal and diagonal of E is zero.

During the iterations, the Frobenius norm of E decreases monotonically. As E goes down to zero, S is the set of singular values. In practice, Jacobi method stops if

{||E||}_{F} \leq eps * {||A||}_{F}

where eps is given tolerance.

gesvdj has two parameters to control the accuracy. First parameter is tolerance (eps). The default value is machine accuracy but The user can use function cusolverDnXgesvdjSetTolerance to set a priori tolerance. The second parameter is maximum number of sweeps which controls number of iterations of Jacobi method. The default value is 100 but the user can use function cusolverDnXgesvdjSetMaxSweeps to set a proper bound. The experimentis show 15 sweeps are good enough to converge to machine accuracy. gesvdj stops either tolerance is met or maximum number of sweeps is met.

Jacobi method has quadratic convergence, so the accuracy is not proportional to number of sweeps. To guarantee certain accuracy, the user should configure tolerance only.

The user has to provide working space which is pointed by input parameter work. The input parameter lwork is the size of the working space, and it is returned by gesvdj_bufferSize().

If output parameter info = -i (less than zero), the i-th parameter is wrong (not counting handle). If info = min(m,n)+1, gesvdj does not converge under given tolerance and maximum sweeps.

If the user sets an improper tolerance, gesvdj may not converge. For example, tolerance should not be smaller than machine accuracy.

Appendix G.2 provides a simple example of gesvdj.

Remark 1: gesvdj supports any combination of m and n.

Remark 2: the routine returns V, not $V^{H}$ . This is different from gesvd.

API of gesvdj

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`jobz`	`host`	`input`	specifies options to either compute singular value only or singular vectors as well: `jobz` = `CUSOLVER_EIG_MODE_NOVECTOR` : Compute singular values only; `jobz` = `CUSOLVER_EIG_MODE_VECTOR` : Compute singular values and singular vectors.
`econ`	`host`	`input`	econ = 1 for economy size for `U` and `V`.
`m`	`host`	`input`	number of rows of matrix `A`.
`n`	`host`	`input`	number of columns of matrix `A`.
`A`	`device`	`in/out`	<type> array of dimension `lda * n` with `lda` is not less than `max(1,m)`. On exit, the contents of `A` are destroyed.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `A`.
`S`	`device`	`output`	real array of dimension `min(m,n)`. The singular values of A, sorted so that `S(i) >= S(i+1)`.
`U`	`device`	`output`	<type> array of dimension `ldu * m` if `econ` is zero. If `econ` is nonzero, the dimension is `ldu * min(m,n)`. `U` contains the left singular vectors.
`ldu`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `U`. `ldu` is not less than `max(1,m)`.
`V`	`device`	`output`	<type> array of dimension `ldv * n` if `econ` is zero. If `econ` is nonzero, the dimension is `ldv * min(m,n)`. `V` contains the right singular vectors.
`ldv`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `V`. `ldv` is not less than `max(1,n)`.
`work`	`device`	`in/out`	<type> array of size `lwork`, working space.
`lwork`	`host`	`input`	size of `work`, returned by `gesvdj_bufferSize`.
`info`	`device`	`output`	if `info = 0`, the operation is successful. if `info = -i`, the `i-th` parameter is wrong (not counting handle). if `info = min(m,n)+1`, `gesvdj` dose not converge under given tolerance and maximum sweeps.
`params`	`host`	`in/out`	structure filled with parameters of Jacobi algorithm and results of gesvdj.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`m,n<0` or `lda<max(1,m)` or `ldu<max(1,m)` or `ldv<max(1,n)` or `jobz` is not `CUSOLVER_EIG_MODE_NOVECTOR` or `CUSOLVER_EIG_MODE_VECTOR` ).
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>gesvdjBatched()

The helper functions below can calculate the sizes needed for pre-allocated buffer.


cusolverStatus_t 
cusolverDnSgesvdjBatched_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int m,
    int n,
    const float *A,
    int lda,
    const float *S,
    const float *U,
    int ldu,
    const float *V,
    int ldv,
    int *lwork,
    gesvdjInfo_t params,
    int batchSize);

cusolverStatus_t 
cusolverDnDgesvdjBatched_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz, 
    int m,                
    int n,                
    const double *A,      
    int lda,              
    const double *S, 
    const double *U,      
    int ldu,              
    const double *V,     
    int ldv,              
    int *lwork,
    gesvdjInfo_t params,
    int batchSize);

cusolverStatus_t 
cusolverDnCgesvdjBatched_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz, 
    int m,                
    int n,                
    const cuComplex *A,      
    int lda,              
    const float *S, 
    const cuComplex *U,      
    int ldu,              
    const cuComplex *V,     
    int ldv,              
    int *lwork,
    gesvdjInfo_t params,
    int batchSize);

cusolverStatus_t 
cusolverDnZgesvdjBatched_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz, 
    int m,                
    int n,                
    const cuDoubleComplex *A,      
    int lda,              
    const double *S, 
    const cuDoubleComplex *U,      
    int ldu,              
    const cuDoubleComplex *V,     
    int ldv,              
    int *lwork,
    gesvdjInfo_t params,
    int batchSize);

The S and D data types are real valued single and double precision, respectively.


cusolverStatus_t
cusolverDnSgesvdjBatched(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int m,
    int n,
    float *A,
    int lda,
    float *S,
    float *U,
    int ldu,
    float *V,
    int ldv,
    float *work,
    int lwork,
    int *info,
    gesvdjInfo_t params,
    int batchSize);

cusolverStatus_t
cusolverDnDgesvdjBatched(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz, 
    int m,           
    int n,           
    double *A,       
    int lda,         
    double *S,  
    double *U,       
    int ldu,         
    double *V,       
    int ldv,         
    double *work,
    int lwork,
    int *info,       
    gesvdjInfo_t params,
    int batchSize);

The C and Z data types are complex valued single and double precision, respectively.


cusolverStatus_t
cusolverDnCgesvdjBatched(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int m,
    int n,
    cuComplex *A,
    int lda,
    float *S,
    cuComplex *U,
    int ldu,
    cuComplex *V,
    int ldv,
    cuComplex *work,
    int lwork,
    int *info,
    gesvdjInfo_t params,
    int batchSize);

cusolverStatus_t
cusolverDnZgesvdjBatched(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int m,
    int n,
    cuDoubleComplex *A,
    int lda,
    double *S,
    cuDoubleComplex *U,
    int ldu,
    cuDoubleComplex *V,
    int ldv,
    cuDoubleComplex *work,
    int lwork,
    int *info,
    gesvdjInfo_t params,
    int batchSize);

This function computes singular values and singular vectors of a squence of general m×n matrices

A_{j} = U_{j} * Σ_{j} * V_{j}^{H}

where $Σ_{j}$ is a real m×n diagonal matrix which is zero except for its min(m,n) diagonal elements. $U_{j}$ (left singular vectors) is a m×m unitary matrix and $V_{j}$ (right singular vectors) is a n×n unitary matrix. The diagonal elements of $Σ_{j}$ are the singular values of $A_{j}$ in either descending order or non-sorting order.

gesvdjBatched performs gesvdj on each matrix. It requires that all matrices are of the same size m,n no greater than 32 and are packed in contiguous way,

$A = (\begin{matrix} A0 & A1 & \dots \end{matrix})$

Each matrix is column-major with leading dimension lda, so the formula for random access is $A_{k} (i,j) = A[ i + lda*j + lda*n*k]$ .

The parameter S also contains singular values of each matrix in contiguous way,

$S = (\begin{matrix} S0 & S1 & \dots \end{matrix})$

The formula for random access of S is $S_{k} (j) = S[ j + min(m,n)*k]$ .

Except for tolerance and maximum sweeps, gesvdjBatched can either sort the singular values in descending order (default) or chose as-is (without sorting) by the function cusolverDnXgesvdjSetSortEig. If the user packs several tiny matrices into diagonal blocks of one matrix, non-sorting option can separate singular values of those tiny matrices.

gesvdjBatched cannot report residual and executed sweeps by function cusolverDnXgesvdjGetResidual and cusolverDnXgesvdjGetSweeps. Any call of the above two returns CUSOLVER_STATUS_NOT_SUPPORTED. The user needs to compute residual explicitly.

The user has to provide working space pointed by input parameter work. The input parameter lwork is the size of the working space, and it is returned by gesvdjBatched_bufferSize().

The output parameter info is an integer array of size batchSize. If the function returns CUSOLVER_STATUS_INVALID_VALUE, the first element info[0] = -i (less than zero) indicates i-th parameter is wrong (not counting handle). Otherwise, if info[i] = min(m,n)+1, gesvdjBatched does not converge on i-th matrix under given tolerance and maximum sweeps.

Appendix G.3 provides a simple example of gesvdjBatched.

API of syevjBatched

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`jobz`	`host`	`input`	specifies options to either compute singular value only or singular vectors as well: `jobz` = `CUSOLVER_EIG_MODE_NOVECTOR` : Compute singular values only; `jobz` = `CUSOLVER_EIG_MODE_VECTOR` : Compute singular values and singular vectors.
`m`	`host`	`input`	number of rows of matrix `Aj`. `m` is no greater than 32.
`n`	`host`	`input`	number of columns of matrix `Aj`. `n` is no greater than 32.
`A`	`device`	`in/out`	<type> array of dimension `lda * n * batchSize` with `lda` is not less than `max(1,n)`. on Exit: the contents of `Aj` are destroyed.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `Aj`.
`S`	`device`	`output`	a real array of dimension `min(m,n)*batchSize`. It stores the singular values of `Aj` in descending order or non-sorting order.
`U`	`device`	`output`	<type> array of dimension `ldu * m * batchSize`. `Uj` contains the left singular vectors of `Aj`.
`ldu`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `Uj`. `ldu` is not less than `max(1,m)`.
`V`	`device`	`output`	<type> array of dimension `ldv * n * batchSize`. `Vj` contains the right singular vectors of `Aj`.
`ldv`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `Vj`. `ldv` is not less than `max(1,n)`.
`work`	`device`	`in/out`	<type> array of size `lwork`, working space.
`lwork`	`host`	`input`	size of `work`, returned by `gesvdjBatched_bufferSize`.
`info`	`device`	`output`	an integer array of dimension `batchSize`. If `CUSOLVER_STATUS_INVALID_VALUE` is returned, `info[0] = -i` (less than zero) indicates `i-th` parameter is wrong (not counting handle). Otherwise, if `info[i] = 0`, the operation is successful. if `info[i] = min(m,n)+1`, `gesvdjBatched` dose not converge on `i-th` matrix under given tolerance and maximum sweeps.
`params`	`host`	`in/out`	structure filled with parameters of Jacobi algorithm.
`batchSize`	`host`	`input`	number of matrices.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`m,n<0` or `lda<max(1,m)` or `ldu<max(1,m)` or `ldv<max(1,n)` or `jobz` is not `CUSOLVER_EIG_MODE_NOVECTOR` or `CUSOLVER_EIG_MODE_VECTOR` , or `batchSize<0` ).
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>gesvdaStridedBatched()

The helper functions below can calculate the sizes needed for pre-allocated buffer.


cusolverStatus_t 
cusolverDnSgesvdaStridedBatched_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int rank,
    int m,
    int n,
    const float *A,
    int lda,
    long long int strideA,
    const float *S,
    long long int strideS,
    const float *U,
    int ldu,
    long long int strideU,
    const float *V,
    int ldv,
    long long int strideV,
    int *lwork,
    int batchSize);

cusolverStatus_t 
cusolverDnDgesvdaStridedBatched_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int rank,
    int m,
    int n,
    const double *A,
    int lda,
    long long int strideA,
    const double *S,
    long long int strideS,
    const double *U,
    int ldu,
    long long int strideU,
    const double *V,
    int ldv,
    long long int strideV,
    int *lwork,
    int batchSize);

cusolverStatus_t 
cusolverDnCgesvdaStridedBatched_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int rank,
    int m,
    int n,
    const cuComplex *A,
    int lda,
    long long int strideA,
    const float *S,
    long long int strideS,
    const cuComplex *U,
    int ldu,
    long long int strideU,
    const cuComplex *V,
    int ldv,
    long long int strideV,
    int *lwork,
    int batchSize);

cusolverStatus_t 
cusolverDnZgesvdaStridedBatched_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int rank,
    int m,
    int n,
    const cuDoubleComplex *A,
    int lda,
    long long int strideA,
    const double *S,
    long long int strideS,
    const cuDoubleComplex *U,
    int ldu,
    long long int strideU,
    const cuDoubleComplex *V,
    int ldv,
    long long int strideV,
    int *lwork,
    int batchSize);

The S and D data types are real valued single and double precision, respectively.


cusolverStatus_t 
cusolverDnSgesvdaStridedBatched(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int rank,
    int m,
    int n,
    const float *A,
    int lda,
    long long int strideA,
    float *S,
    long long int strideS,
    float *U,
    int ldu,
    long long int strideU,
    float *V,
    int ldv,
    long long int strideV,
    float *work,
    int lwork,
    int *info,
    double *h_R_nrmF,
    int batchSize);

cusolverStatus_t 
cusolverDnDgesvdaStridedBatched(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int rank,
    int m,
    int n,
    const double *A,
    int lda,
    long long int strideA,
    double *S,
    long long int strideS,
    double *U,
    int ldu,
    long long int strideU,
    double *V,
    int ldv,
    long long int strideV,
    double *work,
    int lwork,
    int *info,
    double *h_R_nrmF,
    int batchSize);

The C and Z data types are complex valued single and double precision, respectively.


cusolverStatus_t 
cusolverDnCgesvdaStridedBatched(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int rank,
    int m,
    int n,
    const cuComplex *A,
    int lda,
    long long int strideA,
    float *S,
    long long int strideS,
    cuComplex *U,
    int ldu,
    long long int strideU,
    cuComplex *V,
    int ldv,
    long long int strideV,
    cuComplex *work,
    int lwork,
    int *info,
    double *h_R_nrmF,
    int batchSize);

cusolverStatus_t 
cusolverDnZgesvdaStridedBatched(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    int rank,
    int m,
    int n,
    const cuDoubleComplex *A,
    int lda,
    long long int strideA,
    double *S,
    long long int strideS,
    cuDoubleComplex *U,
    int ldu,
    long long int strideU,
    cuDoubleComplex *V,
    int ldv,
    long long int strideV,
    cuDoubleComplex *work,
    int lwork,
    int *info,
    double *h_R_nrmF,
    int batchSize);

This function gesvda (a stands for approximate) approximates the singular value decomposition of a tall skinny m×n matrix A and corresponding the left and right singular vectors. The economy form of SVD is written by

A = U * Σ * V^{H}

where Σ is an n×n matrix. U is an m×n unitary matrix, and V is an n×n unitary matrix. The diagonal elements of Σ are the singular values of A; they are real and non-negative, and are returned in descending order. U and V are the left and right singular vectors of A.

gesvda computes eigenvalues of A**T*A to approximate singular values and singular vectors. It generates matrices U and V and transforms the matrix A to the following form

U^{H} * A * V = S + E

where S is diagonal and E depends on rounding errors. To certain conditions, U, V and S approximate singular values and singular vectors up to machine zero of single precision. In general, V is unitary, S is more accurate than U. If singular value is far from zero, then left singular vector U is accurate. In other words, the accuracy of singular values and left singular vectors depend on the distance between singular value and zero.

The input parameter rank decides the number of singualr values and singular vectors are computed in parameter S, U and V.

The output parameter h_RnrmF computes Frobenius norm of residual.

A - U * S * V^{H}

if the paramter rank is equal n. Otherwise, h_RnrmF reports

|| U * S * V^{H} || - ||S||

in Frobenius norm sense. That is, how far U is from unitary.

gesvdaStridedBatched performs gesvda on each matrix. It requires that all matrices are of the same size m,n and are packed in contiguous way,

$A = (\begin{matrix} A0 & A1 & \dots \end{matrix})$

Each matrix is column-major with leading dimension lda, so the formula for random access is $A_{k} (i,j) = A[ i + lda*j + strideA*k]$ . Similarly, the formula for random access of S is $S_{k} (j) = S[ j + StrideS*k]$ , the formula for random access of U is $U_{k} (i,j) = U[ i + ldu*j + strideU*k]$ and the formula for random access of V is $V_{k} (i,j) = V[ i + ldv*j + strideV*k]$ .

The user has to provide working space which is pointed by input parameter work. The input parameter lwork is the size of the working space, and it is returned by gesvdaStridedBatched_bufferSize().

The output parameter info is an integer array of size batchSize. If the function returns CUSOLVER_STATUS_INVALID_VALUE, the first element info[0] = -i (less than zero) indicates i-th parameter is wrong (not counting handle). Otherwise, if info[i] = min(m,n)+1, gesvdaStridedBatched does not converge on i-th matrix under given tolerance.

Appendix G.4 provides a simple example of gesvda.

Remark 1: the routine returns V, not $V^{H}$ . This is different from gesvd.

Remark 2: if the user is confident on the accuracy of singular values and singular vectors, for example, certain conditions hold (required singular value is far from zero), then the performance can be improved by passing null pointer to h_RnrmF, i.e. no computation of residual norm.

API of gesvda

`parameter`	`Memory`	`In/out`	`Meaning`
`handle`	`host`	`input`	handle to the cuSolverDN library context.
`jobz`	`host`	`input`	specifies options to either compute singular value only or singular vectors as well: `jobz` = `CUSOLVER_EIG_MODE_NOVECTOR` : Compute singular values only; `jobz` = `CUSOLVER_EIG_MODE_VECTOR` : Compute singular values and singular vectors.
`rank`	`host`	`input`	number of singular values (from largest to smallest).
`m`	`host`	`input`	number of rows of matrix `Aj`.
`n`	`host`	`input`	number of columns of matrix `Aj`.
`A`	`device`	`input`	<type> array of dimension `strideA * batchSize` with `lda` is not less than `max(1,m)`. `Aj` is of dimension `m * n`.
`lda`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `Aj`.
`strideA`	`host`	`input`	value of type long long int that gives the address offset between `A[i]` and `A[i+1]`. `strideA` is not less than `lda*n`.
`S`	`device`	`output`	a real array of dimension `strideSbatchSize`. It stores the singular values of `Aj` in descending order. `Sj` is of dimension `rank 1`
`strideS`	`host`	`input`	value of type long long int that gives the address offset between `S[i]` and `S[i+1]`. `strideS` is not less than `rank`.
`U`	`device`	`output`	<type> array of dimension `strideU * batchSize`. `Uj` contains the left singular vectors of `Aj`. `Uj` is of dimension `m * rank`.
`ldu`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `Uj`. `ldu` is not less than `max(1,m)`.
`strideU`	`host`	`input`	value of type long long int that gives the address offset between `U[i]` and `U[i+1]`. `strideU` is not less than `ldu*rank`.
`V`	`device`	`output`	<type> array of dimension `strideV * batchSize`. `Vj` contains the right singular vectors of `Aj`. `Vj` is of dimension `n * rank`.
`ldv`	`host`	`input`	leading dimension of two-dimensional array used to store matrix `Vj`. `ldv` is not less than `max(1,n)`.
`strideV`	`host`	`input`	value of type long long int that gives the address offset between `V[i]` and `V[i+1]`. `strideV` is not less than `ldv*rank`.
`work`	`device`	`in/out`	<type> array of size `lwork`, working space.
`lwork`	`host`	`input`	size of `work`, returned by `gesvdaStridedBatched_bufferSize`.
`info`	`device`	`output`	an integer array of dimension `batchSize`. If `CUSOLVER_STATUS_INVALID_VALUE` is returned, `info[0] = -i` (less than zero) indicates `i-th` parameter is wrong (not counting handle). Otherwise, if `info[i] = 0`, the operation is successful. if `info[i] = min(m,n)+1`, `gesvdaStridedBatched` dose not converge on `i-th` matrix.
`h_RnrmF`	`host`	`output`	<double> array of size `batchSize`. `h_RnrmF[i]` is norm of residual of `i-th` matrix.
`batchSize`	`host`	`input`	number of matrices. `batchSize` is not less than 1.

Status Returned

`CUSOLVER_STATUS_SUCCESS`	the operation completed successfully.
`CUSOLVER_STATUS_NOT_INITIALIZED`	the library was not initialized.
`CUSOLVER_STATUS_INVALID_VALUE`	invalid parameters were passed (`m,n<0` or `lda<max(1,m)` or `ldu<max(1,m)` or `ldv<max(1,n)` or `strideA<ldan` or `strideS<rank` or `strideU<ldurank` or `strideV<ldv*rank` or `batchSize<1` or `jobz` is not `CUSOLVER_EIG_MODE_NOVECTOR` or `CUSOLVER_EIG_MODE_VECTOR` ).
`CUSOLVER_STATUS_INTERNAL_ERROR`	an internal operation failed.

cusolverDn<t>syevd()

The helper functions below can calculate the sizes needed for pre-allocated buffer.

cusolverStatus_t 
cusolverDnSsyevd_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    cublasFillMode_t uplo,
    int n,
    const float *A,
    int lda,
    const float *W,
    int *lwork);

cusolverStatus_t 
cusolverDnDsyevd_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    cublasFillMode_t uplo,
    int n,
    const double *A,
    int lda,
    const double *W,
    int *lwork);

cusolverStatus_t 
cusolverDnCheevd_bufferSize(
    cusolverDnHandle_t handle,
    cusolverEigMode_t jobz,
    cublasFillMode_t uplo,
    int n,
    const cuComplex *A,
    int lda,
    const float *W,
    int *lwork);

cusolverStatus_t 
cusolverDnZheevd_bufferSize(
    cusolver