## cuSOLVER

The API reference guide for cuSOLVER, a GPU accelerated library for decompositions and linear system solutions for both dense and sparse matrices.

## 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:
1. The cuSolver API on a single GPU
2. 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.

### 1.1. cuSolverDN: Dense LAPACK

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

 $Ax=b$

where the coefficient matrix $A\in {R}^{\mathrm{nxn}}$ , 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.

### 1.2. cuSolverSP: Sparse LAPACK

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

 $Ax=b$

and the least-squares problem

 $x=\mathrm{argmin}\mathrm{||}A*z-b\mathrm{||}$

where sparse matrix $A\in {R}^{\mathrm{mxn}}$ , 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 $Ax=b$ , 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.

### 1.3. 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 ${A}_{i}\in {R}^{\mathrm{nxn}}$ , 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 provides two different APIs; legacy and generic.

The functions in the legacy API are available for data types float, double, cuComplex, and cuDoubleComplex. The naming convention for the legacy API is as follows:

 cusolverDn

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 functions in the generic API provide a single entry point for each routine and support for 64-bit integers to define matrix and vector dimensions. The naming convention for the generic API is data-agnostic and is as follows:

 cusolverDn

where <operation> can be Cholesky factorization (potrf), LU with partial pivoting (getrf) and QR factorization (geqrf).

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

 cusolverSp[Host][][]

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__[[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);


## 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.

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:
1. Create CUDA streams using the function cudaStreamCreate(), and
2. 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.

### 2.1.6. 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.

### 2.1.7. 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.

### 2.2.1. 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 A*x = lambda*B*x CUSOLVER_EIG_TYPE_2 A*B*x = lambda*x CUSOLVER_EIG_TYPE_3 B*A*x = 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, this value is what is set when creating the params structure. IRS solver will return error. CUSOLVER_IRS_REFINE_NONE No refinement solver, the IRS solver performs a factorisation followed by a solve without any refinement. For example if the IRS solver was cusolverDnIRSXgesv(), this is equivalent to a Xgesv routine without refinement and where the factorisation is carried out in the lowest precision. If for example the main precision was CUSOLVER_R_64F and the lowest was CUSOLVER_R_64F as well, then this is equivalent to a call to cusolverDnDgesv(). 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. based on our experimentation, we recommend this setting. 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 consists of 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. cusolverDnFunction_t

The cusolverDnFunction_t type indicates which routine needs to be configured by cusolverDnSetAdvOptions(). The value CUSOLVERDN_GETRF corresponds to the routine Getrf.

 Value Meaning CUSOLVERDN_GETRF corresponds to Getrf

### 2.2.1.10. cusolverAlgMode_t

The cusolverAlgMode_t type indicates which algorithm is selected by cusolverDnSetAdvOptions(). The set of algorithms supported for each routine is described in detail along with the routine documentation.

The default algorithm is CUSOLVER_ALG_0. The user can also provide NULL to use the default algorithm.

### 2.2.1.11. 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.

### 2.2.3.1. 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.

### 2.2.3.2. 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.

### 2.2.3.3. 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.

### 2.2.3.4. 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.

### 2.2.3.5. 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.

### 2.2.3.6. 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_ALG1 algorithm 1. (default) CUSOLVER_TRIANGULAR_SOLVE_ALG2 algorithm 2. Domino-based scheme. CUSOLVER_TRIANGULAR_SOLVE_ALG3 algorithm 3. Domino-based scheme.

### 2.2.3.7. 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.

### 2.2.3.8. 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.

### 2.3.1. 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

 $x=\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ ⋮\\ {x}_{n}\end{array}\right)$

is represented as

 $\left(\begin{array}{cccc}{x}_{1}& {x}_{2}& \dots & {x}_{n}\end{array}\right)$

### 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

 $\left(\begin{array}{ccc}{a}_{1,1}& \dots & {a}_{1,n}\\ {a}_{2,1}& \dots & {a}_{2,n}\\ ⋮\\ {a}_{m,1}& \dots & {a}_{m,n}\end{array}\right)$

is represented as

 $\left(\begin{array}{ccc}{a}_{1,1}& \dots & {a}_{1,n}\\ {a}_{2,1}& \dots & {a}_{2,n}\\ ⋮& \ddots & ⋮\\ {a}_{m,1}& \dots & {a}_{m,n}\\ ⋮& \ddots & ⋮\\ {a}_{\mathrm{lda},1}& \dots & {a}_{\mathrm{lda},n}\end{array}\right)$

with its elements arranged linearly in memory as

 $\left(\begin{array}{ccccccccccccc}{a}_{1,1}& {a}_{2,1}& \dots & {a}_{m,1}& \dots & {a}_{\mathrm{lda},1}& \dots & {a}_{1,n}& {a}_{2,n}& \dots & {a}_{m,n}& \dots & {a}_{\mathrm{lda},n}\end{array}\right)$

where ldam 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

 $A=\left(\begin{array}{cccc}\mathrm{1.0}& \mathrm{3.0}& \mathrm{0.0}& \mathrm{0.0}\\ \mathrm{0.0}& \mathrm{4.0}& \mathrm{6.0}& \mathrm{0.0}\\ \mathrm{2.0}& \mathrm{5.0}& \mathrm{7.0}& \mathrm{8.0}\\ \mathrm{0.0}& \mathrm{0.0}& \mathrm{0.0}& \mathrm{9.0}\end{array}\right)$

is represented as

 $\mathrm{csrRowPtr}=\left(\begin{array}{ccccc}0& 2& 4& 8& 9\end{array}\right)$
 $\mathrm{csrColInd}=\left(\begin{array}{ccccccccc}0& 1& 1& 2& 0& 1& 2& 3& 3\end{array}\right)$
 $\mathrm{csrVal}=\left(\begin{array}{ccccccccc}1.0& 3.0& 4.0& 6.0& 2.0& 5.0& 7.0& 8.0& 9.0\end{array}\right)$

### 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

 $A=\left(\begin{array}{cccc}\mathrm{1.0}& \mathrm{3.0}& \mathrm{0.0}& \mathrm{0.0}\\ \mathrm{0.0}& \mathrm{4.0}& \mathrm{6.0}& \mathrm{0.0}\\ \mathrm{2.0}& \mathrm{5.0}& \mathrm{7.0}& \mathrm{8.0}\\ \mathrm{0.0}& \mathrm{0.0}& \mathrm{0.0}& \mathrm{9.0}\end{array}\right)$

is represented as

 $\mathrm{cscColPtr}=\left(\begin{array}{ccccc}0& 2& 5& 7& 9\end{array}\right)$
 $\mathrm{cscRowInd}=\left(\begin{array}{ccccccccc}0& 2& 0& 1& 2& 1& 2& 2& 3\end{array}\right)$
 $\mathrm{cscVal}=\left(\begin{array}{ccccccccc}1.0& 2.0& 3.0& 4.0& 5.0& 6.0& 7.0& 8.0& 9.0\end{array}\right)$

### 2.4. cuSolverDN: dense LAPACK Function Reference

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

### 2.4.1. 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.

### 2.4.1.3. 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.

### 2.4.1.4. 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() or the cusolverDnIRSXgels() functions to default values. The params structure created by this function can be used by one or more call to the same or to a different IRS solver. Note that, in CUDA-10.2, the behavior was different and a new params structure was needed to be created per each call to an IRS solver. Also note that, the user can also change configurations of the params then call a new IRS instance, but be careful that the previous call was done because any change to the configuration before that the previous call was done could affect it.

 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.

 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. cusolverDnIRSParamsSetSolverPrecisions()

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


This function set both, the main and the lowest precision for the Iterative Refinement Solver (IRS). By main precision, we mean the precision of the Input and Output datatype. By lowest precision, we mean the solver is allowed to use as lowest computational precision during the LU factorization process. Note that, the user has to set both the main and lowest precision before a first call to the IRS solver because they are NOT set by default with the params structure creation, as it depends on the Input Output data type and user request. It is a wrapper to both cusolverDnIRSParamsSetSolverMainPrecision() and cusolverDnIRSParamsSetSolverLowestPrecision(). All possible combinations of main/lowest precision are described in the table below. Usually the lowest precision defines the speedup that can be achieved. The ratio of the performance of the lowest precision over the main precision (e.g., Inputs/Outputs datatype) define somehow the upper bound of the speedup that could be obtained. More precisely, it depends on many factors, but for large matrices sizes, it is the ratio of the matrix-matrix rank-k product (e.g., GEMM where K is 256 and M=N=size of the matrix) that define the possible speedup. For instance, if the inout precision is real double precision CUSOLVER_R_64F and the lowest precision is CUSOLVER_R_32F, then we can expect a speedup of at most 2X for large problem sizes. If the lowest precision was CUSOLVER_R_16F, then we can expect 3X-4X. A reasonable strategy should take the number of right-hand sides, 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 solver_main_precision host input Allowed Inputs/Outputs datatype (for example CUSOLVER_R_FP64 for a real double precision data). See the table below for the supported precisions. solver_lowest_precision host input Allowed lowest compute type (for example CUSOLVER_R_16F for half precision computation). See the table below for the supported precisions.
Status Returned
 CUSOLVER_STATUS_SUCCESS The operation completed successfully. CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED The Params structure was not created.
Supported Inputs/Outputs data type and lower precision for the IRS solver
 Inputs/Outputs Data Type (e.g., main precision) Supported values for the lowest precision CUSOLVER_C_64F CUSOLVER_C_64F, CUSOLVER_C_32F, CUSOLVER_C_16F, CUSOLVER_C_16BF, CUSOLVER_C_TF32 CUSOLVER_C_32F CUSOLVER_C_32F, CUSOLVER_C_16F, CUSOLVER_C_16BF, CUSOLVER_C_TF32 CUSOLVER_R_64F CUSOLVER_R_64F, CUSOLVER_R_32F, CUSOLVER_R_16F, CUSOLVER_R_16BF, CUSOLVER_R_TF32 CUSOLVER_R_32F CUSOLVER_R_32F, CUSOLVER_R_16F, CUSOLVER_R_16BF, CUSOLVER_R_TF32

### 2.4.1.22. cusolverDnIRSParamsSetSolverMainPrecision()

cusolverStatus_t
cusolverDnIRSParamsSetSolverMainPrecision(
cusolverDnIRSParams_t params,
cusolverPrecType_t solver_main_precision);


This function sets the main precision for the Iterative Refinement Solver (IRS). By main precision, we mean, the type of the Input and Output data. Note that, the user has to set both the main and lowest precision before a first call to the IRS solver because they are NOT set by default with the params structure creation, as it depends on the Input Output data type and user request. user can set it by either calling this function or by calling cusolverDnIRSParamsSetSolverPrecisions() which set both the main and the lowest precision together. All possible combinations of main/lowest precision are described in the table in the cusolverDnIRSParamsSetSolverPrecisions() section above.

 parameter Memory In/out Meaning params host in/out The cusolverDnIRSParams_t Params structure solver_main_precision host input Allowed Inputs/Outputs datatype (for example CUSOLVER_R_FP64 for a real double precision data). See the table in the cusolverDnIRSParamsSetSolverPrecisions() section above for the 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,
cusolverPrecType_t lowest_precision_type);


This function sets the lowest precision that will be used by Iterative Refinement Solver. By lowest precision, we mean the solver is allowed to use as lowest computational precision during the LU factorization process. Note that, the user has to set both the main and lowest precision before a first call to the IRS solver because they are NOT set by default with the params structure creation, as it depends on the Input Output data type and user request. Usually the lowest precision defines the speedup that can be achieved. The ratio of the performance of the lowest precision over the main precision (e.g., Inputs/Outputs datatype) define somehow the upper bound of the speedup that could be obtained. More precisely, it depends on many factors, but for large matrices sizes, it is the ratio of the matrix-matrix rank-k product (e.g., GEMM where K is 256 and M=N=size of the matrix) that define the possible speedup. For instance, if the inout precision is real double precision CUSOLVER_R_64F and the lowest precision is CUSOLVER_R_32F, then we can expect a speedup of at most 2X for large problem sizes. If the lowest precision was CUSOLVER_R_16F, then we can expect 3X-4X. A reasonable strategy should take the number of right-hand sides, 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 lowest compute type (for example CUSOLVER_R_16F for half precision computation). See the table in the cusolverDnIRSParamsSetSolverPrecisions() section above for the 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. 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() or the cusolverDnIRSXgels() functions. Note that, the user has to set the refinement algorithm before a first call to the IRS solver because it is NOT set by default with the creating of params. Details about values that can be set to and theirs meaning are described in the table below.

 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() or cusolverDnIRSXgels
Status Returned
 CUSOLVER_STATUS_SUCCESS The operation completed successfully. CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED The Params structure was not created.
 Value Meaning CUSOLVER_IRS_REFINE_NOT_SET Solver is not set, this value is what is set when creating the params structure. IRS solver will return error. CUSOLVER_IRS_REFINE_NONE No refinement solver, the IRS solver performs a factorization followed by a solve without any refinement. For example, if the IRS solver was cusolverDnIRSXgesv(), this is equivalent to a Xgesv routine without refinement and where the factorization is carried out in the lowest precision. If for example the main precision was CUSOLVER_R_64F and the lowest was CUSOLVER_R_64F as well, then this is equivalent to a call to cusolverDnDgesv(). 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. Based on our experimentation, we recommend this setting. 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 consists of 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 preconditioned system.

### 2.4.1.25. cusolverDnIRSParamsSetTol()

cusolverStatus_t
cusolverDnIRSParamsSetTol(
cusolverDnIRSParams_t params,
double val );


This function sets the tolerance for the refinement solver. By default it is 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 for the Inputs/Outputs datatype that matches LAPACK <X>LAMCH('Epsilon')
• BWDMAX, 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 more control such a way he can investigate and control every detail of the IRS solver. Note the, the tolerance value is always in real double precision whatever the Inputs/Outputs datatype is.

 parameter Memory In/out Meaning params host in/out The cusolverDnIRSParams_t Params structure val host input double precision real value to which the refinement tolerance will be set.
Status Returned
 CUSOLVER_STATUS_SUCCESS The operation completed successfully. CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED The Params structure was not created.

### 2.4.1.26. cusolverDnIRSParamsSetTolInner()

cusolverStatus_t
cusolverDnIRSParamsSetTolInner(
cusolverDnIRSParams_t params,
double val );


This function sets the tolerance for the inner refinement solver when the refinement solver consists of two-levels solver (e.g., CUSOLVER_IRS_REFINE_CLASSICAL_GMRES or CUSOLVER_IRS_REFINE_GMRES_GMRES cases). It is not referenced in case of one level refinement solver such as CUSOLVER_IRS_REFINE_CLASSICAL or CUSOLVER_IRS_REFINE_GMRES. It is set to 1e-4 by default. This function set the tolerance for the inner solver (e.g. the inner GMRES). For example, if the Refinement Solver 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. Our goal is to give the user more control such a way he can investigate and control every detail of the IRS solver. Note the, the tolerance value is always in real double precision whatever the Inputs/Outputs datatype is.

 parameter Memory In/out Meaning params host in/out The cusolverDnIRSParams_t Params structure val host input Double precision real value to which the tolerance of the inner refinement solver will be set.
Status Returned
 CUSOLVER_STATUS_SUCCESS The operation completed successfully. CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED The Params structure was not created.

### 2.4.1.27. cusolverDnIRSParamsSetMaxIters()

cusolverStatus_t
cusolverDnIRSParamsSetMaxIters(
cusolverDnIRSParams_t params,
int max_iters);


This function sets the total number of allowed refinement iterations after which the solver will stop. Total means any iteration which means the sum of the outer and the inner iterations (inner is meaningful when two-levels refinement solver is set). Default value is set to 50. Our goal is to give the user more control 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 50. Note that this value could not be larger than the MaxIters since MaxIters is the total number of allowed iterations. Note that, if the user call cusolverDnIRSParamsSetMaxIters after calling this function, the SetMaxIters has priority and will overwrite MaxItersInner to the minimum value of (MaxIters, MaxItersInner).

 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 inner 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. cusolverDnIRSParamsEnableFallback()

cusolverStatus_t
cusolverDnIRSParamsEnableFallback(
cusolverDnIRSParams_t params );



This function enable the fallback to the main precision in case the Iterative Refinement Solver (IRS) failed to converge. In other term, if the IRS solver failed to converge, the solver will return a no convergence code (e.g., niter < 0), but can either return the non-convergent solution as it is (e.g., disable fallback) or can fallback (e.g., enable fallback) to the main precision (which is the precision of the Inputs/Outputs data) and solve the problem from scratch returning the good solution. This is the behavior by default, and it will guarantee that the IRS solver always provide the good solution. this function is provided because we provided cusolverDnIRSParamsDisableFallback which allow the user to disable the fallback and thus this function allow him to re-enable it.

 parameter Memory In/out Meaning params host in/out The cusolverDnIRSParams_t Params structure
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. cusolverDnIRSParamsDisableFallback()

cusolverStatus_t
cusolverDnIRSParamsDisableFallback(
cusolverDnIRSParams_t params );



This function disables the fallback to the main precision in case the Iterative Refinement Solver (IRS) failed to converge. In other term, if the IRS solver failed to converge, the solver will return a no convergence code (e.g., niter < 0), but can either return the non-convergent solution as it is (e.g., disable fallback) or can fallback (e.g., enable fallback) to the main precision (which is the precision of the Inputs/Outputs data) and solve the problem from scratch returning the good solution. This function disables the fallback and the returned solution is whatever the refinement solver was able to reach before it returns. Disabling fallback does not guarantee that the solution is the good one. However, for some users, who want to keep getting the solution of the lower precision, in case the IRS did not converge after certain number of iterations, they need to disable the fallback. The user can re-enable it by calling cusolverDnIRSParamsEnableFallback.

 parameter Memory In/out Meaning params host in/out The cusolverDnIRSParams_t Params structure
Status Returned
 CUSOLVER_STATUS_SUCCESS The operation completed successfully. CUSOLVER_STATUS_IRS_PARAMS_NOT_INITIALIZED The Params structure was not created.

### 2.4.1.31. cusolverDnIRSParamsGetMaxIters()


cusolverStatus_t
cusolverDnIRSParamsGetMaxIters(
cusolverDnIRSParams_t params,
cusolver_int_t *maxiters );


This function returns the current setting in the params structure for the maximal allowed number of iterations (e.g., either the default MaxIters, or the one set by the user in case he set it using cusolverDnIRSParamsSetMaxIters). Note that, this function returns the current setting in the params configuration and not to be confused with the cusolverDnIRSInfosGetMaxIters which return the maximal allowed number of iterations for a particular call to an IRS solver. To be clearer, the params structure can be used for many calls to an IRS solver. A user can change the allowed MaxIters between calls while the Infos structure in cusolverDnIRSInfosGetMaxIters contains information about a particular call and cannot be reused for different call, and thus, cusolverDnIRSInfosGetMaxIters return the allowed MaxIters for that call.

 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.

### 2.4.1.32. cusolverDnIRSInfosCreate()

cusolverStatus_t
cusolverDnIRSInfosCreate(
cusolverDnIRSInfos_t* infos )


This function creates and initializes the Infos structure that will hold the refinement information of an Iterative Refinement Solver (IRS) call. Such information includes the total number of iterations that was needed to converge (Niters), the outer number of iterations (meaningful when two-levels preconditioner such as CUSOLVER_IRS_REFINE_CLASSICAL_GMRES is used ), the maximal number of iterations that was allowed for that call, and a pointer to the matrix of the convergence history residual norms. The Infos structure need to be created before a call to an IRS solver. The Infos structure is valid for only one call to an IRS solver, since it holds info about that solve and thus each solve will requires its own Infos structure.

 parameter Memory In/out Meaning 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.

### 2.4.1.33. cusolverDnIRSInfosDestroy()

cusolverStatus_t
cusolverDnIRSInfosDestroy(
cusolverDnIRSInfos_t infos );


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

### 2.4.1.34. cusolverDnIRSInfosGetMaxIters()

cusolverStatus_t
cusolverDnIRSInfosGetMaxIters(
cusolverDnIRSInfos_t infos,
cusolver_int_t *maxiters );


This function returns the maximal allowed number of iterations that was set for the corresponding call to the IRS solver. Note that, this function returns the setting that was set when that call happened and not to be confused with the cusolverDnIRSParamsGetMaxIters which return the current setting in the params configuration structure. To be clearer, the params structure can be used for many calls to an IRS solver. A user can change the allowed MaxIters between calls while the Infos structure in cusolverDnIRSInfosGetMaxIters contains information about a particular call and cannot be reused for different call, and thus cusolverDnIRSInfosGetMaxIters return the allowed MaxIters for that call.

 parameter Memory In/out Meaning 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_INFOS_NOT_INITIALIZED The Infos structure was not created.

### 2.4.1.35. cusolverDnIRSInfosGetNiters()

cusolverStatus_t cusolverDnIRSInfosGetNiters(
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 did not converge and if the user did not disable the fallback to full precision, then the fallback to a full precision solution happened and solution is good. Please refer to the description of negative niters values in the corresponding IRS linear solver functions such as cusolverDnXgesv() or cusolverDnXgels()

 parameter Memory In/out Meaning 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_INFOS_NOT_INITIALIZED The Infos structure was not created.

### 2.4.1.36. cusolverDnIRSInfosGetOuterNiters()

cusolverStatus_t
cusolverDnIRSInfosGetOuterNiters(
cusolverDnIRSInfos_t infos,
cusolver_int_t *outer_niters );


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

 parameter Memory In/out Meaning 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_INFOS_NOT_INITIALIZED The Infos structure was not created.

### 2.4.1.37. cusolverDnIRSInfosRequestResidual()

cusolverStatus_t cusolverDnIRSInfosRequestResidual(
cusolverDnIRSInfos_t infos );


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

 parameter Memory In/out Meaning infos host in The cusolverDnIRSInfos_t Infos structure
Status Returned
 CUSOLVER_STATUS_SUCCESS The operation completed successfully. CUSOLVER_STATUS_IRS_INFOS_NOT_INITIALIZED The Infos structure was not created.

### 2.4.1.38. cusolverDnIRSInfosGetResidualHistory()

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


If the user called cusolverDnIRSInfosRequestResidual() before the call to the IRS function, then the IRS solver will store the convergence history (residual norms) of the refinement phase in a matrix, that could be accessed via a pointer returned by this function. The datatype of the residual norms depends on the input and output data type. If the Inputs/Outputs datatype is double precision real or complex (CUSOLVER_R_FP64 or CUSOLVER_C_FP64), this residual will be of type real double precision (FP64) double, otherwise if the Inputs/Outputs datatype is single precision real or complex (CUSOLVER_R_FP32 or CUSOLVER_C_FP32), this residual will be real single precision FP32 float.

The residual history matrix consists of two columns (even for the multiple right-hand side case NRHS) of MaxIters+1 row, thus a matrix of size (MaxIters+1,2). Only the first OuterNiters+1 rows contains the residual norms the other (e.g., OuterNiters+2:Maxiters+1) are garbage. On the first column, each row "i" specify the total number of iterations happened till 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 rows are the residual obtained at each outer iteration of the refinement loop. Note, it only consists of the history of the outer loop.

If the refinement solver 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 refinement solver 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 till this step the second columns correspond to the residual norm at this step.

For example, let say the user specify CUSOLVER_IRS_REFINE_CLASSICAL_GMRES as a refinement solver 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. Row 0 correspond to the first residual before the refinement start, so it has 0 in its first column. 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 the norm of the residual at this outer iteration.

 parameter Memory In/out Meaning 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 datatype and the inout datatype.
Status Returned
 CUSOLVER_STATUS_SUCCESS The operation completed successfully. 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.

### 2.4.1.39. cusolverDnCreateParams()

cusolverStatus_t
cusolverDnCreateParams(
cusolverDnParams_t *params);


This function creates and initializes the structure of 64-bit API to default values.

 parameter Memory In/out Meaning params host output the pointer to the structure of 64-bit API.
Status Returned
 CUSOLVER_STATUS_SUCCESS the structure was initialized successfully. CUSOLVER_STATUS_ALLOC_FAILED the resources could not be allocated.

### 2.4.1.40. cusolverDnDestroyParams()

cusolverStatus_t
cusolverDnDestroyParams(
cusolverDnParams_t params);


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

 parameter Memory In/out Meaning params host input the structure of 64-bit API.
Status Returned
 CUSOLVER_STATUS_SUCCESS the resources are released successfully.

cusolverStatus_t
cusolverDnParams_t params,
cusolverDnFunction_t function,
cusolverAlgMode_t algo   );


This function configures algorithm algo of function, a 64-bit API routine.

 parameter Memory In/out Meaning params host in/out the pointer to the structure of 64-bit API. function host input which routine to be configured. algo host input which algorithm to be configured.
Status Returned
 CUSOLVER_STATUS_SUCCESS the operation completed successfully. CUSOLVER_STATUS_INVALID_VALUE wrong combination of function and algo

### 2.4.2. Dense Linear Solver Reference (legacy)

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

### 2.4.2.1. 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 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, 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

### 2.4.2.2. cusolverDnPotrf()[DEPRECATED]

[[DEPRECATED]] use cusolverDnXpotrf() instead. The routine will be removed in the next major release.

The helper functions below can calculate the sizes needed for pre-allocated buffer.
cusolverStatus_t
cusolverDnPotrf_bufferSize(
cusolverDnHandle_t handle,
cusolverDnParams_t params,
cublasFillMode_t uplo,
int64_t n,
const void *A,
int64_t lda,
size_t *workspaceInBytes )

The routine bellow
cusolverStatus_t
cusolverDnPotrf(
cusolverDnHandle_t handle,
cusolverDnParams_t params,
cublasFillMode_t uplo,
int64_t n,
void *A,
int64_t lda,
void *pBuffer,
size_t workspaceInBytes,
int *info )



computes the Cholesky factorization of a Hermitian positive-definite matrix using the generic API interfacte.

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 pBuffer. The input parameter workspaceInBytes is size in bytes of the working space, and it is returned by cusolverDnPotrf_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 info would indicate smallest leading minor of A which is not positive definite.

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

Currently, cusolverDnPotrf supports only the default algorithm.

Table of algorithms supported by cusolverDnPotrf
 CUSOLVER_ALG_0 or NULL Default algorithm.

List of input arguments for cusolverDnPotrf_bufferSize and cusolverDnPotrf:

API of potrf
 parameter Memory In/out Meaning handle host input handle to the cuSolverDN library context. params host input structure with information collected by cusolverDnSetAdvOptions. 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. dataTypeA host in data type of array A. A device in/out 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. computeType host in data type of computation. pBuffer device in/out Working space. Array of type void of size workspaceInBytes bytes. workspaceInBytes host input size in bytes of pBuffer, returned by cusolverDnPotrf_bufferSize. info device output if info = 0, the Cholesky factorization is successful. if info = -i, the i-th parameter is wrong (not counting handle). if info = i, the leading minor of order i is not positive definite.

The generic API has two different types, dataTypeA is data type of the matrix A, computeType is compute type of the operation. cusolverDnPotrf only supports the following four combinations.

Valid combination of data type and compute type
 DataTypeA ComputeType Meaning CUDA_R_32F CUDA_R_32F SPOTRF CUDA_R_64F CUDA_R_64F DPOTRF CUDA_C_32F CUDA_C_32F CPOTRF CUDA_C_64F CUDA_C_64F ZPOTRF
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

### 2.4.2.3. 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 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 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

### 2.4.2.4. cusolverDnPotrs()[DEPRECATED]

[[DEPRECATED]] use cusolverDnXpotrs() instead. The routine will be removed in the next major release.

cusolverStatus_t
cusolverDnPotrs(
cusolverDnHandle_t handle,
cusolverDnParams_t params,
cublasFillMode_t uplo,
int64_t n,
int64_t nrhs,
const void *A,
int64_t lda,
void *B,
int64_t ldb,
int *info)



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 using the generic API interface. The input parameter uplo indicates which part of the matrix is used. The function would leave other part untouched.

The user has to call cusolverDnPotrf 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 info = -i (less than zero), the i-th parameter is wrong (not counting handle).

Currently, cusolverDnPotrs supports only the default algorithm.

Table of algorithms supported by cusolverDnPotrs
 CUSOLVER_ALG_0 or NULL Default algorithm.

List of input arguments for cusolverDnPotrs:

API of potrs
 parameter Memory In/out Meaning handle host input handle to the cuSolveDN library context. params host input structure with information collected by cusolverDnSetAdvOptions. 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. dataTypeA host in data type of array A. A device input 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. dataTypeB host in data type of array B. B device in/out 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. info device output if info = 0, the Cholesky factorization is successful. if info = -i, the i-th parameter is wrong (not counting handle).

The generic API has two different types, dataTypeA is data type of the matrix A, dataTypeB is data type of the matrix B. cusolverDnPotrs only supports the following four combinations.

Valid combination of data type and compute type
 dataTypeA dataTypeB Meaning CUDA_R_32F CUDA_R_32F SPOTRS CUDA_R_64F CUDA_R_64F DPOTRS CUDA_C_32F CUDA_C_32F CPOTRS CUDA_C_64F CUDA_C_64F ZPOTRS
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

### 2.4.2.5. 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 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, 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

### 2.4.2.6. 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.

Remark: getrf uses fastest implementation with large workspace of size m*n. The user can choose the legacy implementation with minimal workspace by Getrf and cusolverDnSetAdvOptions(params, CUSOLVERDN_GETRF, CUSOLVER_ALG_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 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, 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

### 2.4.2.7. cusolverDnGetrf()[DEPRECATED]

[[DEPRECATED]] use cusolverDnXgetrf() instead. The routine will be removed in the next major release.

The helper function below can calculate the sizes needed for pre-allocated buffer.
cusolverStatus_t
cusolverDnGetrf_bufferSize(
cusolverDnHandle_t handle,
cusolverDnParams_t params,
int64_t m,
int64_t n,
const void *A,
int64_t lda,
size_t *workspaceInBytes )


The function bellow
cusolverStatus_t
cusolverDnGetrf(
cusolverDnHandle_t handle,
cusolverDnParams_t params,
int64_t m,
int64_t n,
void *A,
int64_t lda,
int64_t *ipiv,
void *pBuffer,
size_t workspaceInBytes,
int *info )



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 using the generic API interface.

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

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

If ipiv 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 ipiv contains pivoting sequence, row i is interchanged with row ipiv(i).

The user has to provide working space which is pointed by input parameter pBuffer. The input parameter workspaceInBytes is size in bytes of the working space, and it is returned by cusolverDnGetrf_bufferSize().

The user can combine cusolverDnGetrf and cusolverDnGetrs to complete a linear solver.

Currently, cusolverDnGetrf supports two algorithms. To select legacy implementation, the user has to call cusolverDnSetAdvOptions.

Table of algorithms supported by cusolverDnGetrf
 CUSOLVER_ALG_0 or NULL Default algorithm. The fastest, requires a large workspace of m*n elements. CUSOLVER_ALG_1 Legacy implementation

List of input arguments for cusolverDnGetrf_bufferSize and cusolverDnGetrf:

API of cusolverDnGetrf
 parameter Memory In/out Meaning handle host input handle to the cuSolverDN library context. params host input structure with information collected by cusolverDnSetAdvOptions. m host input number of rows of matrix A. n host input number of columns of matrix A. dataTypeA host in data type of array A. A device in/out 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. ipiv device output array of size at least min(m,n), containing pivot indices. computeType host in data type of computation. pBuffer device in/out Working space. Array of type void of size workspaceInBytes bytes. workspaceInBytes host input size in bytes of pBuffer, returned by cusolverDnGetrf_bufferSize. info device output if info = 0, the LU factorization is successful. if info = -i, the i-th parameter is wrong (not counting handle). if info = i, the U(i,i) = 0.

The generic API has two different types, dataTypeA is data type of the matrix A, computeType is compute type of the operation. cusolverDnGetrf only supports the following four combinations.

valid combination of data type and compute type
 DataTypeA ComputeType Meaning CUDA_R_32F CUDA_R_32F SGETRF CUDA_R_64F CUDA_R_64F DGETRF CUDA_C_32F CUDA_C_32F CGETRF CUDA_C_64F CUDA_C_64F ZGETRF
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

### 2.4.2.8. 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

 $\mathrm{op\left(A\right)}*X=B$

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

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 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 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

### 2.4.2.9. cusolverDnGetrs()[DEPRECATED]

[[DEPRECATED]] use cusolverDnXgetrs() instead. The routine will be removed in the next major release.

cusolverStatus_t
cusolverDnGetrs(
cusolverDnHandle_t handle,
cusolverDnParams_t params,
cublasOperation_t trans,
int64_t n,
int64_t nrhs,
const void *A,
int64_t lda,
const int64_t *ipiv,
void *B,
int64_t ldb,
int *info )


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

 $\mathrm{op\left(A\right)}*X=B$

where A is a n×n matrix, and was LU-factored by cusolverDnGetrf, 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 using the generic API interface.

The input parameter trans is defined by

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

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

The user can combine cusolverDnGetrf and cusolverDnGetrs to complete a linear solver.

Currently, cusolverDnGetrs supports only the default algorithm.

Table of algorithms supported by cusolverDnGetrs
 CUSOLVER_ALG_0 or NULL Default algorithm.

List of input arguments for cusolverDnGetrss:

 parameter Memory In/out Meaning handle host input handle to the cuSolverDN library context. params host input structure with information collected by cusolverDnSetAdvOptions. 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. dataTypeA host in data type of array A. A device input 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 input array of size at least n, containing pivot indices. dataTypeB host in data type of array B. B device output 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. info device output if info = 0, the operation is successful. if info = -i, the i-th parameter is wrong (not counting handle).

The generic API has two different types, dataTypeA is data type of the matrix A and dataTypeB is data type of the matrix B. cusolverDnGetrs only supports the following four combinations.

Valid combination of data type and compute type
 DataTypeA dataTypeB Meaning CUDA_R_32F CUDA_R_32F SGETRS CUDA_R_64F CUDA_R_64F DGETRS CUDA_C_32F CUDA_C_32F CGETRS CUDA_C_64F CUDA_C_64F ZGETRS
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

### 2.4.2.10. cusolverDn<t1><t2>gesv()

These functions are modelled after functions DSGESV and ZCGESV from LAPACK. They compute the solution of a system of linear equations with one or multiple right hand sides using mixed precision iterative refinement techniques based on the LU factorization Xgesv. These functions are similar in term of functionalities to the full precision LU solver (Xgesv, where X denotes Z,C,D,S) but it uses lower precision internally in order to provide faster time to solution, from here cames the name mixed precision. Mixed precision iterative refinement techniques means that the solver compute an LU factorization in lower precision and then iteratively refine the solution to achieve the accuracy of the Inputs/Outputs datatype precision. The <t1> corresponds to the Inputs/Outputs datatype precision while <t2> represent the internal lower precision at which the factorization will be carried on.

 $A×X=B$

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

Functions API are designed to be as close as possible to LAPACK API to be considered as a quick and easy drop-in replacement. Parameters and behavior are mostly the same as LAPACK counterparts. Description of these functions and differences from LAPACK is given below. <t1><t2>gesv() functions are designated by two floating point precisions The <t1> corresponds to the main precision (e.g., Inputs/Outputs datatype precision) and the <t2> represent the internal lower precision at which the factorization will be carried on. 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 the main precision <t1>. If the approach fails to converge, then the method fallback to the main precision factorization and solve (Xgesv) such a way that there is always a good solution at the output of these functions. If <t2> is equal to <t1>, then it is not a mixed precision process but rather a full one precision factorisation, solve and refinement within the same main precision.

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.

The function returns value describes the results of the solving process. A CUSOLVER_STATUS_SUCCESS indicates that the function finished with success otherwise, it indicates if one of the API arguments is incorrect, or if the function did not finish with success. More details about the error will be in the niters and the dinfo API parameters. See their description below for more details. User should provide the required workspace allocated on device memory. The amount of bytes required can be queried by calling the respective function <t1><t2>gesv_bufferSize().

Note that in addition to the two mixed precision functions available in LAPACK (e.g., dsgesv and zcgesv), we provide a large set of mixed precision functions that include half, bfloat and tensorfloat as a lower precision as well as same precision functions (e.g., main and lowest precision are equal <t2> is equal to <t1>). The following table specifies which precisions will be used for which interface function.

Tensor Float (TF32), introduced with NVIDIA Ampere Architecture GPUs, is the most robust tensor core accelerated compute mode for the iterative refinement solver. It is able to solve the widest range of problems in HPC arising from different applications and provides up to 4X and 5X speedup for real and complex systems, respectively. On Volta and Turing architecture GPUs, half precision tensor core acceleration is recommended. In cases where the iterative refinement solver fails to converge to the desired accuracy (main precision, INOUT data precision), it is recommended to use main precision as internal lowest precision (i.e., cusolverDn[DD,ZZ]gesv for the FP64 case).

Supported combinations of floating point precisions for cusolver <t1><t2>gesv() functions
 Interface function Main precision (matrix, rhs and solution datatype) Lowest precision allowed to be used internally cusolverDnZZgesv cuDoubleComplex double complex cusolverDnZCgesv *has LAPACK counterparts cuDoubleComplex single complex cusolverDnZKgesv cuDoubleComplex half complex cusolverDnZEgesv cuDoubleComplex bfloat complex cusolverDnZYgesv cuDoubleComplex tensorfloat complex cusolverDnCCgesv cuComplex single complex cusolverDnCKgesv cuComplex half complex cusolverDnCEgesv cuComplex bfloat complex cusolverDnCYgesv cuComplex tensorfloat complex cusolverDnDDgesv double double cusolverDnDSgesv *has LAPACK counterparts double single cusolverDnDHgesv double half cusolverDnDBgesv double bfloat cusolverDnDXgesv double tensorfloat cusolverDnSSgesv float single cusolverDnSHgesv float half cusolverDnSBgesv float bfloat cusolverDnSXgesv float tensorfloat
cusolverDn<t1><t2>gesv_bufferSize() functions will return workspace buffer size in bytes required for the corresponding cusolverDn<t1><t2>gesv() function.
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
cusolverDnZEgesv_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
cusolverDnZYgesv_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);

cusolverStatus_t
cusolverDnCEgesv_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
cusolverDnCYgesv_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
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
cusolverDnDBgesv_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
cusolverDnDXgesv_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
cusolverDnSBgesv_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
cusolverDnSXgesv_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);


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. dA device None 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 None 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 None 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 output 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                 *   niter,
int                 *   dinfo);

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                 *   niter,
int                 *   dinfo);

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                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnZEgesv(
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                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnZYgesv(
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                 *   niter,
int                 *   dinfo);

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                 *   niter,
int                 *   dinfo);

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                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnCEgesv(
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                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnCYgesv(
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                 *   niter,
int                 *   dinfo);

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                 *   niter,
int                 *   dinfo);

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                 *   niter,
int                 *   dinfo);

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                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnDBgesv(
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                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnDXgesv(
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                 *   niter,
int                 *   dinfo);

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                 *   niter,
int                 *   dinfo);

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                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnSBgesv(
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                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnSXgesv(
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                 *   niter,
int                 *   dinfo);


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. dA device in/out Matrix A with size n-by-n. Can't be NULL. On return - unchanged if the iterative refinement process converged. If not - will contains the factorization of the matrix A in the main precision (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 output Vector that defines permutation for the factorization - row i was interchanged with row ipiv[i] dB device input 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 output 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 input Pointer to an allocated workspace in device memory of size lwork_bytes. lwork_bytes host input Size of the allocated device workspace. Should be at least what was returned by cusolverDngesv_bufferSize() function niters host output If iter is <0 : iterative refinement has failed, main precision (Inputs/Outputs 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 main to lower precision -3 : failure during the factorization -5 : overflow occured during computation -50: 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 IRS solver on the return. If 0 - solve was successful. If dinfo = -i then i-th argument is not valid. If dinfo = i, then U(i,i) computed in main 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, for example: n<0 lda

### 2.4.2.11. 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. cusolverDnIRSXgesv() allows additional control of the solver parameters such as setting:
• the main precision (Inputs/Outputs precision) of the solver,
• the lowest precision to be used internally by the solver,
• the refinement solver type,
• the maximum allowed number of iterations in the refinement phase,
• the tolerance of the refinement solver,
• the fallback to main precision,
• and more other.
through the configuration parameters structure gesv_irs_params and its helper functions. For more details about what configuration can be set and its meaning please refer to all the functions in the cuSolverDN Helper Function Section that start with cusolverDnIRSParamsxxxx(). Moreover, cusolverDnIRSXgesv() provides additional informations on the output such as the convergence history (e.g., the residual norms) at each iteration and the number of iterations needed to converge. For more details about what informations can be retrieved and its meaning please refer to all the functions in the cuSolverDN Helper Function Section that start with cusolverDnIRSInfosxxxx()

The function returns value describes the results of the solving process. A CUSOLVER_STATUS_SUCCESS indicates that the function finished with success otherwise, it indicates if one of the API arguments is incorrect, or if the configurations of params/infos structure is incorrect or if the function did not finish with success. More details about the error can be found by checking the niters and the dinfo API parameters. See their description below for further details. User should provide the required workspace allocated on device for the cusolverDnIRSXgesv() function. The amount of bytes required for the function can be queried by calling the respective function cusolverDnIRSXgesv_bufferSize(). Note that, if the user would like a praticular configuration to be set via the params structure, it should be set before the call to cusolverDnIRSXgesv_bufferSize() to get the size of the required workspace.

Tensor Float (TF32), introduced with NVIDIA Ampere Architecture GPUs, is the most robust tensor core accelerated compute mode for the iterative refinement solver. It is able to solve the widest range of problems in HPC arising from different applications and provides up to 4X and 5X speedup for real and complex systems, respectively. On Volta and Turing architecture GPUs, half precision tensor core acceleration is recommended. In cases where the iterative refinement solver fails to converge to the desired accuracy (main precision, INOUT data precision), it is recommended to use main precision as internal lowest precision.

The following table provides all possible combinations values for the lowest precision corresponding to the Inputs/Outputs data type. Note that if the lowest precision matches the Inputs/Outputs datatype, then the main precision factorization will be used.

Supported Inputs/Outputs data type and lower precision for the IRS solver
 Inputs/Outputs Data Type (e.g., main precision) Supported values for the lowest precision CUSOLVER_C_64F CUSOLVER_C_64F, CUSOLVER_C_32F, CUSOLVER_C_16F, CUSOLVER_C_16BF, CUSOLVER_C_TF32 CUSOLVER_C_32F CUSOLVER_C_32F, CUSOLVER_C_16F, CUSOLVER_C_16BF, CUSOLVER_C_TF32 CUSOLVER_R_64F CUSOLVER_R_64F, CUSOLVER_R_32F, CUSOLVER_R_16F, CUSOLVER_R_16BF, CUSOLVER_R_TF32 CUSOLVER_R_32F CUSOLVER_R_32F, CUSOLVER_R_16F, CUSOLVER_R_16BF, CUSOLVER_R_TF32
The cusolverDnIRSXgesv_bufferSize() function returns the required workspace buffer size in bytes for the corresponding cusolverDnXgesv() call with the given gesv_irs_params configuration.
cusolverStatus_t
cusolverDnIRSXgesv_bufferSize(
cusolverDnHandle_t          handle,
cusolverDnIRSParams_t       gesv_irs_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 configuration 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,
int                         n,
int                         nrhs,
void                    *   dA,
int                         ldda,
void                    *   dB,
int                         lddb,
void                    *   dX,
int                         lddx,
void                    *   dWorkspace,
size_t                      lwork_bytes,
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 Configuration parameters structure, can serve one or more calls to any IRS solver gesv_irs_infos host in/out Info structure, where information about a particular solve will be stored. The gesv_irs_infos struture correspond to a particular call. Thus different calls requires different gesv_irs_infos structure otherwise, it will be overwritten. 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. dA device in/out Matrix A with size n-by-n. Can't be NULL. On return - will contain the factorization of the matrix A in the main precision (A = P * L * U, where P - permutation matrix defined by vector ipiv, L and U - lower and upper triangular matrices) if the iterative refinement solver was set to CUSOLVER_IRS_REFINE_NONE and the lowest precision is equal to the main precision (Inputs/Ouputs datatype), or if the iterative refinement solver did not converge and the fallback to main precision was enabled (fallback enabled is the default setting); - unchanged otherwise. ldda host input leading dimension of two-dimensional array used to store matrix A. lda >= n. dB device input 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 output 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 input Pointer to an allocated workspace in device memory of size lwork_bytes. lwork_bytes host input Size of the allocated device workspace. Should be at least what was returned by cusolverDnIRSXgesv_bufferSize() function niters host output If iter is <0 : iterative refinement has failed, main precision (Inputs/Outputs precision) factorization has been performed if fallback is enabled. -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 main to lower precision -3 : failure during the factorization -5 : overflow occured during computation -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 IRS solver on the return. If 0 - solve was successful. If dinfo = -i then i-th argument is not valid. If dinfo = i, then U(i,i) computed in main 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, for example: n<0 lda1, and refinement solver was set to CUSOLVER_IRS_REFINE_GMRES CUSOLVER_STATUS_IRS_INFOS_NOT_INITIALIZED The information structure gesv_irs_infos was not created. CUSOLVER_STATUS_ALLOC_FAILED CPU memory allocation failed, most likely during the allocation of the residual array that store the residual norms.

### 2.4.2.12. 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

 $I-2*v*{v}^{H}=I-\tau *q*{q}^{H}$

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 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 array of dimension at least min(m,n). Workspace device in/out working space, 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

### 2.4.2.13. cusolverDnGeqrf()[DEPRECATED]

[[DEPRECATED]] use cusolverDnXgeqrf() instead. The routine will be removed in the next major release.

The helper functions below can calculate the sizes needed for pre-allocated buffer.
cusolverStatus_t
cusolverDnGeqrf_bufferSize(
cusolverDnHandle_t handle,
cusolverDnParams_t params,
int64_t m,
int64_t n,
const void *A,
int64_t lda,
const void *tau,
size_t *workspaceInBytes )


The routine bellow
cusolverStatus_t
cusolverDnGeqrf(
cusolverDnHandle_t handle,
cusolverDnParams_t params,
int64_t m,
int64_t n,
void *A,
int64_t lda,
void *tau,
void *pBuffer,
size_t workspaceInBytes,
int *info )


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 using the generic API interface.

The user has to provide working space which is pointed by input parameter pBuffer. The input parameter workspaceInBytes is size in bytes of the working space, and it is returned by cusolverDnGeqrf_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

 $I-2*v*{v}^{H}=I-\tau *q*{q}^{H}$

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

Currently, cusolverDnGeqrf supports only the default algorithm.

Table of algorithms supported by cusolverDnGeqrf
 CUSOLVER_ALG_0 or NULL Default algorithm.

List of input arguments for cusolverDnGeqrf_bufferSize and cusolverDnGeqrf:

API of geqrf
 parameter Memory In/out Meaning handle host input handle to the cuSolverDN library context. params host input structure with information collected by cusolverDnSetAdvOptions. m host input number of rows of matrix A. n host input number of columns of matrix A. dataTypeA host in data type of array A. A device in/out 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 array of dimension at least min(m,n). computeType host in data type of computation. pBuffer device in/out Working space. Array of type void of size workspaceInBytes bytes. workspaceInBytes host input size in bytes of working array pBuffer. info device output if info = 0, the LU factorization is successful. if info = -i, the i-th parameter is wrong (not counting handle).

The generic API has two different types, dataTypeA is data type of the matrix A and array tau and computeType is compute type of the operation. cusolverDnGeqrf only supports the following four combinations.

Valid combination of data type and compute type
 DataTypeA ComputeType Meaning CUDA_R_32F CUDA_R_32F SGEQRF CUDA_R_64F CUDA_R_64F DGEQRF CUDA_C_32F CUDA_C_32F CGEQRF CUDA_C_64F CUDA_C_64F ZGEQRF
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

### 2.4.2.14. cusolverDn<t1><t2>gels()

These functions compute the solution of a system of linear equations with one or multiple right hand sides using mixed precision iterative refinement techniques based on the QR factorization Xgels. These functions are similar in term of functionalities to the full precision LAPACK QR (least squares) solver (Xgels, where X denotes Z,C,D,S) but it uses lower precision internally in order to provide faster time to solution, from here cames the name mixed precision. Mixed precision iterative refinement techniques means that the solver compute an QR factorization in lower precision and then iteratively refine the solution to achieve the accuracy of the Inputs/Outputs datatype precision. The <t1> corresponds to the Inputs/Outputs datatype precision while <t2> represent the internal lower precision at which the factorization will be carried on.

 $A×X=B$

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

Functions API are designed to be as close as possible to LAPACK API to be considered as a quick and easy drop-in replacement. Description of these functions is given below. <t1><t2>gels() functions are designated by two floating point precisions The <t1> corresponds to the main precision (e.g., Inputs/Outputs datatype precision) and the <t2> represent the internal lower precision at which the factorization will be carried on. cusolver<t1><t2>gels() 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 the main precision <t1>. If the approach fails to converge, then the method fallback to the main precision factorization and solve (Xgels) such a way that there is always a good solution at the output of these functions. If <t2> is equal to <t1>, then it is not a mixed precision process but rather a full one precision factorisation, solve and refinement within the same main precision.

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.

The function returns value describes the results of the solving process. A CUSOLVER_STATUS_SUCCESS indicates that the function finished with success otherwise, it indicates if one of the API arguments is incorrect, or if the function did not finish with success. More details about the error will be in the niters and the dinfo API parameters. See their description below for more details. User should provide the required workspace allocated on device memory. The amount of bytes required can be queried by calling the respective function <t1><t2>gels_bufferSize().

We provide a large set of mixed precision functions that include half, bfloat and tensorfloat as a lower precision as well as same precision functions (e.g., main and lowest precision are equal <t2> is equal to <t1>). The following table specifies which precisions will be used for which interface function:

Tensor Float (TF32), introduced with NVIDIA Ampere Architecture GPUs, is the most robust tensor core accelerated compute mode for the iterative refinement solver. It is able to solve the widest range of problems in HPC arising from different applications and provides up to 4X and 5X speedup for real and complex systems, respectively. On Volta and Turing architecture GPUs, half precision tensor core acceleration is recommended. In cases where the iterative refinement solver fails to converge to the desired accuracy (main precision, INOUT data precision), it is recommended to use main precision as internal lowest precision (i.e., cusolverDn[DD,ZZ]gels for the FP64 case).

Supported combinations of floating point precisions for cusolver <t1><t2>gels() functions
 Interface function Main precision (matrix, rhs and solution datatype) Lowest precision allowed to be used internally cusolverDnZZgels cuDoubleComplex double complex cusolverDnZCgels cuDoubleComplex single complex cusolverDnZKgels cuDoubleComplex half complex cusolverDnZEgels cuDoubleComplex bfloat complex cusolverDnZYgels cuDoubleComplex tensorfloat complex cusolverDnCCgels cuComplex single complex cusolverDnCKgels cuComplex half complex cusolverDnCEgels cuComplex bfloat complex cusolverDnCYgels cuComplex tensorfloat complex cusolverDnDDgels double double cusolverDnDSgels double single cusolverDnDHgels double half cusolverDnDBgels double bfloat cusolverDnDXgels double tensorfloat cusolverDnSSgels float single cusolverDnSHgels float half cusolverDnSBgels float bfloat cusolverDnSXgels float tensorfloat
cusolverDn<t1><t2>gels_bufferSize() functions will return workspace buffer size in bytes required for the corresponding cusolverDn<t1><t2>gels() function.
cusolverStatus_t
cusolverDnZZgels_bufferSize(
cusolverHandle_t                handle,
int                             m,
int                             n,
int                             nrhs,
cuDoubleComplex             *   dA,
int                             ldda,
cuDoubleComplex             *   dB,
int                             lddb,
cuDoubleComplex             *   dX,
int                             lddx,
void                        *   dwork,
size_t                      *   lwork_bytes);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


Parameters of cusolverDn<T1><T2>gels_bufferSize() functions
 parameter Memory In/out Meaning handle host input Handle to the cusolverDN library context. m host input Number of rows of the matrix A. Should be non-negative and n<=m n host input Number of columns of the matrix A. Should be non-negative and n<=m. nrhs host input Number of right hand sides to solve. Should be non-negative. dA device None Matrix A with size m-by-n. Can be NULL. ldda host input leading dimension of two-dimensional array used to store matrix A. ldda >= m. dB device None Set of right hand sides B of size m-by-nrhs. Can be NULL. lddb host input leading dimension of two-dimensional array used to store matrix of right hand sides B. lddb >= max(1,m). dX device None 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. lddx >= max(1,n). dwork device none Pointer to device workspace. Not used and can be NULL. lwork_bytes host output Pointer to a variable where required size of temporary workspace in bytes will be stored. Can't be NULL.
cusolverStatus_t cusolverDnZZgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
cuDoubleComplex     *   dA,
int                     ldda,
cuDoubleComplex     *   dB,
int                     lddb,
cuDoubleComplex     *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnZCgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
cuDoubleComplex     *   dA,
int                     ldda,
cuDoubleComplex     *   dB,
int                     lddb,
cuDoubleComplex     *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnZKgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
cuDoubleComplex     *   dA,
int                     ldda,
cuDoubleComplex     *   dB,
int                     lddb,
cuDoubleComplex     *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnZEgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
cuDoubleComplex     *   dA,
int                     ldda,
cuDoubleComplex     *   dB,
int                     lddb,
cuDoubleComplex     *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnZYgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
cuDoubleComplex     *   dA,
int                     ldda,
cuDoubleComplex     *   dB,
int                     lddb,
cuDoubleComplex     *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnCCgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
cuComplex           *   dA,
int                     ldda,
cuComplex           *   dB,
int                     lddb,
cuComplex           *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnCKgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
cuComplex           *   dA,
int                     ldda,
cuComplex           *   dB,
int                     lddb,
cuComplex           *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnCEgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
cuComplex           *   dA,
int                     ldda,
cuComplex           *   dB,
int                     lddb,
cuComplex           *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnCYgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
cuComplex           *   dA,
int                     ldda,
cuComplex           *   dB,
int                     lddb,
cuComplex           *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnDDgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
double              *   dA,
int                     ldda,
double              *   dB,
int                     lddb,
double              *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnDSgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
double              *   dA,
int                     ldda,
double              *   dB,
int                     lddb,
double              *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnDHgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
double              *   dA,
int                     ldda,
double              *   dB,
int                     lddb,
double              *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnDBgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
double              *   dA,
int                     ldda,
double              *   dB,
int                     lddb,
double              *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnDXgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
double              *   dA,
int                     ldda,
double              *   dB,
int                     lddb,
double              *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnSSgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
float               *   dA,
int                     ldda,
float               *   dB,
int                     lddb,
float               *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnSHgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
float               *   dA,
int                     ldda,
float               *   dB,
int                     lddb,
float               *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnSBgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
float               *   dA,
int                     ldda,
float               *   dB,
int                     lddb,
float               *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);

cusolverStatus_t cusolverDnSXgels(
cusolverDnHandle_t      handle,
int                     m,
int                     n,
int                     nrhs,
float               *   dA,
int                     ldda,
float               *   dB,
int                     lddb,
float               *   dX,
int                     lddx,
void                *   dWorkspace,
size_t                  lwork_bytes,
int                 *   niter,
int                 *   dinfo);


Parameters of cusolverDn<T1><T2>gels() functions
 parameter Memory In/out Meaning handle host input Handle to the cusolverDN library context. m host input Number of rows of the matrix A. Should be non-negative and n<=m n host input Number of columns of the matrix A. Should be non-negative and n<=m. nrhs host input Number of right hand sides to solve. Should be non-negative. dA device in/out Matrix A with size m-by-n. Can't be NULL. On return - unchanged if the lowest precision is not equal to the main precision and the iterative refinement solver converged, - garbage otherwise. ldda host input leading dimension of two-dimensional array used to store matrix A. ldda >= m. dB device input Set of right hand sides B of size m-by-nrhs . Can't be NULL. lddb host input leading dimension of two-dimensional array used to store matrix of right hand sides B. lddb >= max(1,m). dX device output 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. lddx >= max(1,n). dWorkspace device input Pointer to an allocated workspace in device memory of size lwork_bytes. lwork_bytes host input Size of the allocated device workspace. Should be at least what was returned by cusolverDngels_bufferSize() function niters host output If iter is <0 : iterative refinement has failed, main precision (Inputs/Outputs 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 main to lower precision -3 : failure during the factorization -5 : overflow occured during computation -50: 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 IRS solver on the return. If 0 - solve was successful. If dinfo = -i then i-th argument is not valid.
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, for example: n<0 ldda

### 2.4.2.15. cusolverDnIRSXgels()

This function is designed to perform same functionality as cusolverDn<T1><T2>gels() 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. cusolverDnIRSXgels() allows additional control of the solver parameters such as setting:
• the main precision (Inputs/Outputs precision) of the solver,
• the lowest precision to be used internally by the solver,
• the refinement solver type,
• the maximum allowed number of iterations in the refinement phase,
• the tolerance of the refinement solver,
• the fallback to main precision,
• and more other.
through the configuration parameters structure gels_irs_params and its helper functions. For more details about what configuration can be set and its meaning please refer to all the functions in the cuSolverDN Helper Function Section that start with cusolverDnIRSParamsxxxx(). Moreover, cusolverDnIRSXgels() provides additional informations on the output such as the convergence history (e.g., the residual norms) at each iteration and the number of iterations needed to converge. For more details about what informations can be retrieved and its meaning please refer to all the functions in the cuSolverDN Helper Function Section that start with cusolverDnIRSInfosxxxx()

The function returns value describes the results of the solving process. A CUSOLVER_STATUS_SUCCESS indicates that the function finished with success otherwise, it indicates if one of the API arguments is incorrect, or if the configurations of params/infos structure is incorrect or if the function did not finish with success. More details about the error can be found by checking the niters and the dinfo API parameters. See their description below for further details. User should provide the required workspace allocated on device for the cusolverDnIRSXgels() function. The amount of bytes required for the function can be queried by calling the respective function cusolverDnIRSXgels_bufferSize(). Note that, if the user would like a praticular configuration to be set via the params structure, it should be set before the call to cusolverDnIRSXgels_bufferSize() to get the size of the required workspace.

The following table provides all possible combinations values for the lowest precision corresponding to the Inputs/Outputs data type. Note that if the lowest precision matches the Inputs/Outputs datatype, then main precision factorization will be used

Tensor Float (TF32), introduced with NVIDIA Ampere Architecture GPUs, is the most robust tensor core accelerated compute mode for the iterative refinement solver. It is able to solve the widest range of problems in HPC arising from different applications and provides up to 4X and 5X speedup for real and complex systems, respectively. On Volta and Turing architecture GPUs, half precision tensor core acceleration is recommended. In cases where the iterative refinement solver fails to converge to the desired accuracy (main precision, INOUT data precision), it is recommended to use main precision as internal lowest precision.

Supported Inputs/Outputs data type and lower precision for the IRS solver
 Inputs/Outputs Data Type (e.g., main precision) Supported values for the lowest precision CUSOLVER_C_64F CUSOLVER_C_64F, CUSOLVER_C_32F, CUSOLVER_C_16F, CUSOLVER_C_16BF, CUSOLVER_C_TF32 CUSOLVER_C_32F CUSOLVER_C_32F, CUSOLVER_C_16F, CUSOLVER_C_16BF, CUSOLVER_C_TF32 CUSOLVER_R_64F CUSOLVER_R_64F, CUSOLVER_R_32F, CUSOLVER_R_16F, CUSOLVER_R_16BF, CUSOLVER_R_TF32 CUSOLVER_R_32F CUSOLVER_R_32F, CUSOLVER_R_16F, CUSOLVER_R_16BF, CUSOLVER_R_TF32
The cusolverDnIRSXgels_bufferSize() function return the required workspace buffer size in bytes for the corresponding cusolverDnXgels() call with given gels_irs_params configuration.
cusolverStatus_t
cusolverDnIRSXgels_bufferSize(
cusolverDnHandle_t          handle,
cusolverDnIRSParams_t       gels_irs_params,
cusolver_int_t              m,
cusolver_int_t              n,
cusolver_int_t              nrhs,
size_t                  *   lwork_bytes);


Parameters of cusolverDnIRSXgels_bufferSize() functions
 parameter Memory In/out Meaning handle host input Handle to the cusolverDn library context. params host input Xgels configuration parameters m host input Number of rows of the matrix A. Should be non-negative and n<=m n host input Number of columns of the matrix A. Should be non-negative and n<=m. 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 cusolverDnIRSXgels_bufferSize. Can't be NULL.
cusolverStatus_t cusolverDnIRSXgels(
cusolverDnHandle_t          handle,
cusolverDnIRSParams_t       gels_irs_params,
cusolverDnIRSInfos_t        gels_irs_infos,
int                         m,
int                         n,
int                         nrhs,
void                    *   dA,
int                         ldda,
void                    *   dB,
int                         lddb,
void                    *   dX,
int                         lddx,
void                    *   dWorkspace,
size_t                      lwork_bytes,
int                     *   dinfo);


Parameters of cusolverDnIRSXgels() functions
 parameter Memory In/out Meaning handle host input Handle to the cusolverDn library context. gels_irs_params host input Configuration parameters structure, can serve one or more calls to any IRS solver gels_irs_infos host in/out Info structure, where information about a particular solve will be stored. The gels_irs_infos struture correspond to a particular call. Thus different calls requires different gels_irs_infos structure otherwise, it will be overwritten. m host input Number of rows of the matrix A. Should be non-negative and n<=m n host input Number of columns of the matrix A. Should be non-negative and n<=m. 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. dA device in/out Matrix A with size m-by-n. Can't be NULL. On return - unchanged if the lowest precision is not equal to the main precision and the iterative refinement solver converged, - garbage otherwise. ldda host input leading dimension of two-dimensional array used to store matrix A. ldda >= m. dB device input Set of right hand sides B of size m-by-nrhs . Can't be NULL. lddb host input leading dimension of two-dimensional array used to store matrix of right hand sides B. lddb >= max(1,m). dX device output 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. lddx >= max(1,n). dWorkspace device input Pointer to an allocated workspace in device memory of size lwork_bytes. lwork_bytes host input Size of the allocated device workspace. Should be at least what was returned by cusolverDnIRSXgels_bufferSize() function niters host output If iter is <0 : iterative refinement has failed, main precision (Inputs/Outputs precision) factorization has been performed if fallback is enabled. -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 main to lower precision -3 : failure during the factorization -5 : overflow occured during computation -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 IRS solver on the return. If 0 - solve was successful. If dinfo = -i then i-th argument is not valid.
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, for example: n<0 ldda1, and refinement solver was set to CUSOLVER_IRS_REFINE_GMRES CUSOLVER_STATUS_IRS_INFOS_NOT_INITIALIZED The information structure gels_irs_infos was not created. CUSOLVER_STATUS_ALLOC_FAILED CPU memory allocation failed, most likely during the allocation of the residual array that store the residual norms.

### 2.4.2.16. 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

The operation of Q is defined by

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 rows of matrix C. n host input Number of columns of matrix C. k host input Number of elementary relfections whose product defines the matrix Q. A device in/out 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 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 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, 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.

### 2.4.2.17. 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

 $Q=\mathrm{H\left(1\right)}*\mathrm{H\left(2\right)}*\mathrm{...}*\mathrm{H\left(k\right)}$

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 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 array of dimension k. tau(i) is the scalar of i-th elementary reflection vector. work device in/out working space, 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

### 2.4.2.18. 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 array of dimension lda * n with lda is not less than max(1,n). lda host