Multiplication of Q From QR Factorization#

UNMQR (UNitary matrix Multiplication after QR factorization) overwrites the general M x N matrix C with

  • \(op(Q) * C\) if side = side::left

  • \(C * op(Q)\) if side = side::right

where Q is an unitary matrix from the QR factorization. Operation op(A) indicates if Q is non_transposed, or transposed for real data type, or conj_transposed for complex data type.

Q is of order M if side = side::left, and of order N if side = side::right.

cuSolverDx unmqr device functions are (see Execution Methods):

__device__ void execute(const data_type* A, const data_type* tau, data_type* C);
// with runtime leading dimensions
__device__ void execute(const data_type* A, const unsigned int lda, const data_type* tau,
                        data_type* C);
__device__ void execute(const data_type* A, const data_type* tau,
                        data_type* C, const unsigned int ldc);
__device__ void execute(const data_type* A, const unsigned int lda, const data_type* tau,
                        data_type* C, const unsigned int ldc);

A is a batched matrix coming from the output of the QR factorization. If side = side::left, A is M x K with leading dimension lda >= M if matrix A is in column-major layout, or lda >= K if matrix A is row-major. If side = side::right, A is instead N x K with leading dimension lda >= N if matrix A is in column-major layout, or lda >= K if matrix A is row-major.

Array tau is an input array of size K for each batch, coming from the output of the QR factorization. If side = side::left, then K cannot be larger than M. If side = side::right, then K cannot be larger than N.

The elements below the diagonal of A, with the array tau, represent the matrix Q as a product of a sequence of K Householder vectors.

Matrix C is an input and output matrix, having dimension of M x N per batch. After the function, matrix C is overwritten with the result of the multiplication.

The functions support:

  1. A and C being either column- or row-major memory layout, see Arrangement Operator,

  2. \(op(Q)\) either being non_transposed, transposed for real data type, or conj_transposed for complex data type, see TransposeMode Operator, and

  3. Side of multiplication being either side::left or side::right, see Side Operator.