 MAGMA  2.5.4 Matrix Algebra for GPU and Multicore Architectures
geqrf: QR factorization

## Functions

magma_int_t magma_cgeqrf (magma_int_t m, magma_int_t n, magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex *tau, magmaFloatComplex *work, magma_int_t lwork, magma_int_t *info)
CGEQRF computes a QR factorization of a COMPLEX M-by-N matrix A: A = Q * R. More...

magma_int_t magma_cgeqrf2_gpu (magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex *tau, magma_int_t *info)
CGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R. More...

magma_int_t magma_cgeqrf3_gpu (magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex *tau, magmaFloatComplex_ptr dT, magma_int_t *info)
CGEQRF3 computes a QR factorization of a complex M-by-N matrix A: A = Q * R. More...

magma_int_t magma_cgeqrf_gpu (magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex *tau, magmaFloatComplex_ptr dT, magma_int_t *info)
CGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R. More...

magma_int_t magma_cgeqrf_m (magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex *tau, magmaFloatComplex *work, magma_int_t lwork, magma_int_t *info)
CGEQRF computes a QR factorization of a COMPLEX M-by-N matrix A: A = Q * R using multiple GPUs. More...

magma_int_t magma_cgeqrf2_mgpu (magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dlA[], magma_int_t ldda, magmaFloatComplex *tau, magma_int_t *info)
CGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R. More...

magma_int_t magma_cgeqrf_ooc (magma_int_t m, magma_int_t n, magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex *tau, magmaFloatComplex *work, magma_int_t lwork, magma_int_t *info)
CGEQRF_OOC computes a QR factorization of a COMPLEX M-by-N matrix A: A = Q * R. More...

magma_int_t magma_dgeqrf (magma_int_t m, magma_int_t n, double *A, magma_int_t lda, double *tau, double *work, magma_int_t lwork, magma_int_t *info)
DGEQRF computes a QR factorization of a DOUBLE PRECISION M-by-N matrix A: A = Q * R. More...

magma_int_t magma_dgeqrf2_gpu (magma_int_t m, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, double *tau, magma_int_t *info)
DGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R. More...

magma_int_t magma_dgeqrf3_gpu (magma_int_t m, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, double *tau, magmaDouble_ptr dT, magma_int_t *info)
DGEQRF3 computes a QR factorization of a real M-by-N matrix A: A = Q * R. More...

magma_int_t magma_dgeqrf_gpu (magma_int_t m, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, double *tau, magmaDouble_ptr dT, magma_int_t *info)
DGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R. More...

magma_int_t magma_dgeqrf_m (magma_int_t ngpu, magma_int_t m, magma_int_t n, double *A, magma_int_t lda, double *tau, double *work, magma_int_t lwork, magma_int_t *info)
DGEQRF computes a QR factorization of a DOUBLE PRECISION M-by-N matrix A: A = Q * R using multiple GPUs. More...

magma_int_t magma_dgeqrf2_mgpu (magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaDouble_ptr dlA[], magma_int_t ldda, double *tau, magma_int_t *info)
DGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R. More...

magma_int_t magma_dgeqrf_ooc (magma_int_t m, magma_int_t n, double *A, magma_int_t lda, double *tau, double *work, magma_int_t lwork, magma_int_t *info)
DGEQRF_OOC computes a QR factorization of a DOUBLE PRECISION M-by-N matrix A: A = Q * R. More...

magma_int_t magma_sgeqrf (magma_int_t m, magma_int_t n, float *A, magma_int_t lda, float *tau, float *work, magma_int_t lwork, magma_int_t *info)
SGEQRF computes a QR factorization of a REAL M-by-N matrix A: A = Q * R. More...

magma_int_t magma_sgeqrf2_gpu (magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, float *tau, magma_int_t *info)
SGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R. More...

magma_int_t magma_sgeqrf3_gpu (magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, float *tau, magmaFloat_ptr dT, magma_int_t *info)
SGEQRF3 computes a QR factorization of a real M-by-N matrix A: A = Q * R. More...

magma_int_t magma_sgeqrf_gpu (magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, float *tau, magmaFloat_ptr dT, magma_int_t *info)
SGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R. More...

magma_int_t magma_sgeqrf_m (magma_int_t ngpu, magma_int_t m, magma_int_t n, float *A, magma_int_t lda, float *tau, float *work, magma_int_t lwork, magma_int_t *info)
SGEQRF computes a QR factorization of a REAL M-by-N matrix A: A = Q * R using multiple GPUs. More...

magma_int_t magma_sgeqrf2_mgpu (magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaFloat_ptr dlA[], magma_int_t ldda, float *tau, magma_int_t *info)
SGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R. More...

magma_int_t magma_sgeqrf_ooc (magma_int_t m, magma_int_t n, float *A, magma_int_t lda, float *tau, float *work, magma_int_t lwork, magma_int_t *info)
SGEQRF_OOC computes a QR factorization of a REAL M-by-N matrix A: A = Q * R. More...

magma_int_t magma_zgeqrf (magma_int_t m, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *tau, magmaDoubleComplex *work, magma_int_t lwork, magma_int_t *info)
ZGEQRF computes a QR factorization of a COMPLEX_16 M-by-N matrix A: A = Q * R. More...

magma_int_t magma_zgeqrf2_gpu (magma_int_t m, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex *tau, magma_int_t *info)
ZGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R. More...

magma_int_t magma_zgeqrf3_gpu (magma_int_t m, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex *tau, magmaDoubleComplex_ptr dT, magma_int_t *info)
ZGEQRF3 computes a QR factorization of a complex M-by-N matrix A: A = Q * R. More...

magma_int_t magma_zgeqrf_gpu (magma_int_t m, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex *tau, magmaDoubleComplex_ptr dT, magma_int_t *info)
ZGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R. More...

magma_int_t magma_zgeqrf_m (magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *tau, magmaDoubleComplex *work, magma_int_t lwork, magma_int_t *info)
ZGEQRF computes a QR factorization of a COMPLEX_16 M-by-N matrix A: A = Q * R using multiple GPUs. More...

magma_int_t magma_zgeqrf2_mgpu (magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaDoubleComplex_ptr dlA[], magma_int_t ldda, magmaDoubleComplex *tau, magma_int_t *info)
ZGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R. More...

magma_int_t magma_zgeqrf_ooc (magma_int_t m, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *tau, magmaDoubleComplex *work, magma_int_t lwork, magma_int_t *info)
ZGEQRF_OOC computes a QR factorization of a COMPLEX_16 M-by-N matrix A: A = Q * R. More...

## ◆ magma_cgeqrf()

 magma_int_t magma_cgeqrf ( magma_int_t m, magma_int_t n, magmaFloatComplex * A, magma_int_t lda, magmaFloatComplex * tau, magmaFloatComplex * work, magma_int_t lwork, magma_int_t * info )

CGEQRF computes a QR factorization of a COMPLEX M-by-N matrix A: A = Q * R.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine.

This uses 2 queues to overlap communication and computation.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A COMPLEX array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_cgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v'


where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_cgeqrf2_gpu()

 magma_int_t magma_cgeqrf2_gpu ( magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex * tau, magma_int_t * info )

CGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R.

This version has LAPACK-complaint arguments.

Other versions (magma_cgeqrf_gpu and magma_cgeqrf3_gpu) store the intermediate T matrices.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA COMPLEX array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_cgeqrf3_gpu()

 magma_int_t magma_cgeqrf3_gpu ( magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex * tau, magmaFloatComplex_ptr dT, magma_int_t * info )

CGEQRF3 computes a QR factorization of a complex M-by-N matrix A: A = Q * R.

This version stores the triangular dT matrices used in the block QR factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster. Also, the upper triangular matrices for V have 0s in them. The corresponding parts of the upper triangular R are stored separately in dT.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA COMPLEX array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT (workspace) COMPLEX array on the GPU, dimension (2*MIN(M, N) + ceil(N/32)*32 )*NB, where NB can be obtained through magma_get_cgeqrf_nb( M, N ). It starts with a MIN(M,N)*NB block that stores the triangular T matrices, followed by a MIN(M,N)*NB block that stores the diagonal blocks of the R matrix. The rest of the array is used as workspace. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_cgeqrf_gpu()

 magma_int_t magma_cgeqrf_gpu ( magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex * tau, magmaFloatComplex_ptr dT, magma_int_t * info )

CGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R.

This version stores the triangular dT matrices used in the block QR factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster. Also, the upper triangular matrices for V have 0s in them. The corresponding parts of the upper triangular R are inverted and stored separately in dT.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA COMPLEX array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT (workspace) COMPLEX array on the GPU, dimension (2*MIN(M, N) + ceil(N/32)*32 )*NB, where NB can be obtained through magma_get_cgeqrf_nb( M, N ). It starts with a MIN(M,N)*NB block that stores the triangular T matrices, followed by a MIN(M,N)*NB block that stores inverses of the diagonal blocks of the R matrix. The rest of the array is used as workspace. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_cgeqrf_m()

 magma_int_t magma_cgeqrf_m ( magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaFloatComplex * A, magma_int_t lda, magmaFloatComplex * tau, magmaFloatComplex * work, magma_int_t lwork, magma_int_t * info )

CGEQRF computes a QR factorization of a COMPLEX M-by-N matrix A: A = Q * R using multiple GPUs.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine.

Parameters
 [in] ngpu INTEGER Number of GPUs to use. ngpu > 0. [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A COMPLEX array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_cgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_cgeqrf2_mgpu()

 magma_int_t magma_cgeqrf2_mgpu ( magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dlA[], magma_int_t ldda, magmaFloatComplex * tau, magma_int_t * info )

CGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R.

This is a GPU interface of the routine.

Parameters
 [in] ngpu INTEGER Number of GPUs to use. ngpu > 0. [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dlA COMPLEX array of pointers on the GPU, dimension (ngpu). On entry, the M-by-N matrix A distributed over GPUs (d_lA[d] points to the local matrix on d-th GPU). It uses 1D block column cyclic format with the block size of nb, and each local matrix is stored by column. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_cgeqrf_ooc()

 magma_int_t magma_cgeqrf_ooc ( magma_int_t m, magma_int_t n, magmaFloatComplex * A, magma_int_t lda, magmaFloatComplex * tau, magmaFloatComplex * work, magma_int_t lwork, magma_int_t * info )

CGEQRF_OOC computes a QR factorization of a COMPLEX M-by-N matrix A: A = Q * R.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine. This is an out-of-core (ooc) version that is similar to magma_cgeqrf but the difference is that this version can use a GPU even if the matrix does not fit into the GPU memory at once.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A COMPLEX array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_cgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v'


where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_dgeqrf()

 magma_int_t magma_dgeqrf ( magma_int_t m, magma_int_t n, double * A, magma_int_t lda, double * tau, double * work, magma_int_t lwork, magma_int_t * info )

DGEQRF computes a QR factorization of a DOUBLE PRECISION M-by-N matrix A: A = Q * R.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine.

This uses 2 queues to overlap communication and computation.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_dgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v'


where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_dgeqrf2_gpu()

 magma_int_t magma_dgeqrf2_gpu ( magma_int_t m, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, double * tau, magma_int_t * info )

DGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R.

This version has LAPACK-complaint arguments.

Other versions (magma_dgeqrf_gpu and magma_dgeqrf3_gpu) store the intermediate T matrices.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA DOUBLE PRECISION array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_dgeqrf3_gpu()

 magma_int_t magma_dgeqrf3_gpu ( magma_int_t m, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, double * tau, magmaDouble_ptr dT, magma_int_t * info )

DGEQRF3 computes a QR factorization of a real M-by-N matrix A: A = Q * R.

This version stores the triangular dT matrices used in the block QR factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster. Also, the upper triangular matrices for V have 0s in them. The corresponding parts of the upper triangular R are stored separately in dT.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA DOUBLE PRECISION array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT (workspace) DOUBLE PRECISION array on the GPU, dimension (2*MIN(M, N) + ceil(N/32)*32 )*NB, where NB can be obtained through magma_get_dgeqrf_nb( M, N ). It starts with a MIN(M,N)*NB block that stores the triangular T matrices, followed by a MIN(M,N)*NB block that stores the diagonal blocks of the R matrix. The rest of the array is used as workspace. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_dgeqrf_gpu()

 magma_int_t magma_dgeqrf_gpu ( magma_int_t m, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, double * tau, magmaDouble_ptr dT, magma_int_t * info )

DGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R.

This version stores the triangular dT matrices used in the block QR factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster. Also, the upper triangular matrices for V have 0s in them. The corresponding parts of the upper triangular R are inverted and stored separately in dT.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA DOUBLE PRECISION array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT (workspace) DOUBLE PRECISION array on the GPU, dimension (2*MIN(M, N) + ceil(N/32)*32 )*NB, where NB can be obtained through magma_get_dgeqrf_nb( M, N ). It starts with a MIN(M,N)*NB block that stores the triangular T matrices, followed by a MIN(M,N)*NB block that stores inverses of the diagonal blocks of the R matrix. The rest of the array is used as workspace. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_dgeqrf_m()

 magma_int_t magma_dgeqrf_m ( magma_int_t ngpu, magma_int_t m, magma_int_t n, double * A, magma_int_t lda, double * tau, double * work, magma_int_t lwork, magma_int_t * info )

DGEQRF computes a QR factorization of a DOUBLE PRECISION M-by-N matrix A: A = Q * R using multiple GPUs.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine.

Parameters
 [in] ngpu INTEGER Number of GPUs to use. ngpu > 0. [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_dgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_dgeqrf2_mgpu()

 magma_int_t magma_dgeqrf2_mgpu ( magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaDouble_ptr dlA[], magma_int_t ldda, double * tau, magma_int_t * info )

DGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R.

This is a GPU interface of the routine.

Parameters
 [in] ngpu INTEGER Number of GPUs to use. ngpu > 0. [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dlA DOUBLE PRECISION array of pointers on the GPU, dimension (ngpu). On entry, the M-by-N matrix A distributed over GPUs (d_lA[d] points to the local matrix on d-th GPU). It uses 1D block column cyclic format with the block size of nb, and each local matrix is stored by column. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_dgeqrf_ooc()

 magma_int_t magma_dgeqrf_ooc ( magma_int_t m, magma_int_t n, double * A, magma_int_t lda, double * tau, double * work, magma_int_t lwork, magma_int_t * info )

DGEQRF_OOC computes a QR factorization of a DOUBLE PRECISION M-by-N matrix A: A = Q * R.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine. This is an out-of-core (ooc) version that is similar to magma_dgeqrf but the difference is that this version can use a GPU even if the matrix does not fit into the GPU memory at once.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_dgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v'


where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_sgeqrf()

 magma_int_t magma_sgeqrf ( magma_int_t m, magma_int_t n, float * A, magma_int_t lda, float * tau, float * work, magma_int_t lwork, magma_int_t * info )

SGEQRF computes a QR factorization of a REAL M-by-N matrix A: A = Q * R.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine.

This uses 2 queues to overlap communication and computation.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_sgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v'


where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_sgeqrf2_gpu()

 magma_int_t magma_sgeqrf2_gpu ( magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, float * tau, magma_int_t * info )

SGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R.

This version has LAPACK-complaint arguments.

Other versions (magma_sgeqrf_gpu and magma_sgeqrf3_gpu) store the intermediate T matrices.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA REAL array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_sgeqrf3_gpu()

 magma_int_t magma_sgeqrf3_gpu ( magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, float * tau, magmaFloat_ptr dT, magma_int_t * info )

SGEQRF3 computes a QR factorization of a real M-by-N matrix A: A = Q * R.

This version stores the triangular dT matrices used in the block QR factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster. Also, the upper triangular matrices for V have 0s in them. The corresponding parts of the upper triangular R are stored separately in dT.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA REAL array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT (workspace) REAL array on the GPU, dimension (2*MIN(M, N) + ceil(N/32)*32 )*NB, where NB can be obtained through magma_get_sgeqrf_nb( M, N ). It starts with a MIN(M,N)*NB block that stores the triangular T matrices, followed by a MIN(M,N)*NB block that stores the diagonal blocks of the R matrix. The rest of the array is used as workspace. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_sgeqrf_gpu()

 magma_int_t magma_sgeqrf_gpu ( magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, float * tau, magmaFloat_ptr dT, magma_int_t * info )

SGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R.

This version stores the triangular dT matrices used in the block QR factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster. Also, the upper triangular matrices for V have 0s in them. The corresponding parts of the upper triangular R are inverted and stored separately in dT.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA REAL array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT (workspace) REAL array on the GPU, dimension (2*MIN(M, N) + ceil(N/32)*32 )*NB, where NB can be obtained through magma_get_sgeqrf_nb( M, N ). It starts with a MIN(M,N)*NB block that stores the triangular T matrices, followed by a MIN(M,N)*NB block that stores inverses of the diagonal blocks of the R matrix. The rest of the array is used as workspace. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_sgeqrf_m()

 magma_int_t magma_sgeqrf_m ( magma_int_t ngpu, magma_int_t m, magma_int_t n, float * A, magma_int_t lda, float * tau, float * work, magma_int_t lwork, magma_int_t * info )

SGEQRF computes a QR factorization of a REAL M-by-N matrix A: A = Q * R using multiple GPUs.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine.

Parameters
 [in] ngpu INTEGER Number of GPUs to use. ngpu > 0. [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_sgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_sgeqrf2_mgpu()

 magma_int_t magma_sgeqrf2_mgpu ( magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaFloat_ptr dlA[], magma_int_t ldda, float * tau, magma_int_t * info )

SGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R.

This is a GPU interface of the routine.

Parameters
 [in] ngpu INTEGER Number of GPUs to use. ngpu > 0. [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dlA REAL array of pointers on the GPU, dimension (ngpu). On entry, the M-by-N matrix A distributed over GPUs (d_lA[d] points to the local matrix on d-th GPU). It uses 1D block column cyclic format with the block size of nb, and each local matrix is stored by column. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_sgeqrf_ooc()

 magma_int_t magma_sgeqrf_ooc ( magma_int_t m, magma_int_t n, float * A, magma_int_t lda, float * tau, float * work, magma_int_t lwork, magma_int_t * info )

SGEQRF_OOC computes a QR factorization of a REAL M-by-N matrix A: A = Q * R.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine. This is an out-of-core (ooc) version that is similar to magma_sgeqrf but the difference is that this version can use a GPU even if the matrix does not fit into the GPU memory at once.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_sgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v'


where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_zgeqrf()

 magma_int_t magma_zgeqrf ( magma_int_t m, magma_int_t n, magmaDoubleComplex * A, magma_int_t lda, magmaDoubleComplex * tau, magmaDoubleComplex * work, magma_int_t lwork, magma_int_t * info )

ZGEQRF computes a QR factorization of a COMPLEX_16 M-by-N matrix A: A = Q * R.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine.

This uses 2 queues to overlap communication and computation.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A COMPLEX_16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_zgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v'


where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_zgeqrf2_gpu()

 magma_int_t magma_zgeqrf2_gpu ( magma_int_t m, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex * tau, magma_int_t * info )

ZGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R.

This version has LAPACK-complaint arguments.

Other versions (magma_zgeqrf_gpu and magma_zgeqrf3_gpu) store the intermediate T matrices.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA COMPLEX_16 array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_zgeqrf3_gpu()

 magma_int_t magma_zgeqrf3_gpu ( magma_int_t m, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex * tau, magmaDoubleComplex_ptr dT, magma_int_t * info )

ZGEQRF3 computes a QR factorization of a complex M-by-N matrix A: A = Q * R.

This version stores the triangular dT matrices used in the block QR factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster. Also, the upper triangular matrices for V have 0s in them. The corresponding parts of the upper triangular R are stored separately in dT.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA COMPLEX_16 array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT (workspace) COMPLEX_16 array on the GPU, dimension (2*MIN(M, N) + ceil(N/32)*32 )*NB, where NB can be obtained through magma_get_zgeqrf_nb( M, N ). It starts with a MIN(M,N)*NB block that stores the triangular T matrices, followed by a MIN(M,N)*NB block that stores the diagonal blocks of the R matrix. The rest of the array is used as workspace. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_zgeqrf_gpu()

 magma_int_t magma_zgeqrf_gpu ( magma_int_t m, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex * tau, magmaDoubleComplex_ptr dT, magma_int_t * info )

ZGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R.

This version stores the triangular dT matrices used in the block QR factorization so that they can be applied directly (i.e., without being recomputed) later. As a result, the application of Q is much faster. Also, the upper triangular matrices for V have 0s in them. The corresponding parts of the upper triangular R are inverted and stored separately in dT.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dA COMPLEX_16 array on the GPU, dimension (LDDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT (workspace) COMPLEX_16 array on the GPU, dimension (2*MIN(M, N) + ceil(N/32)*32 )*NB, where NB can be obtained through magma_get_zgeqrf_nb( M, N ). It starts with a MIN(M,N)*NB block that stores the triangular T matrices, followed by a MIN(M,N)*NB block that stores inverses of the diagonal blocks of the R matrix. The rest of the array is used as workspace. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v^H


where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_zgeqrf_m()

 magma_int_t magma_zgeqrf_m ( magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaDoubleComplex * A, magma_int_t lda, magmaDoubleComplex * tau, magmaDoubleComplex * work, magma_int_t lwork, magma_int_t * info )

ZGEQRF computes a QR factorization of a COMPLEX_16 M-by-N matrix A: A = Q * R using multiple GPUs.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine.

Parameters
 [in] ngpu INTEGER Number of GPUs to use. ngpu > 0. [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A COMPLEX_16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_zgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_zgeqrf2_mgpu()

 magma_int_t magma_zgeqrf2_mgpu ( magma_int_t ngpu, magma_int_t m, magma_int_t n, magmaDoubleComplex_ptr dlA[], magma_int_t ldda, magmaDoubleComplex * tau, magma_int_t * info )

ZGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R.

This is a GPU interface of the routine.

Parameters
 [in] ngpu INTEGER Number of GPUs to use. ngpu > 0. [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] dlA COMPLEX_16 array of pointers on the GPU, dimension (ngpu). On entry, the M-by-N matrix A distributed over GPUs (d_lA[d] points to the local matrix on d-th GPU). It uses 1D block column cyclic format with the block size of nb, and each local matrix is stored by column. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). To benefit from coalescent memory accesses LDDA must be divisible by 16. [out] tau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).

## ◆ magma_zgeqrf_ooc()

 magma_int_t magma_zgeqrf_ooc ( magma_int_t m, magma_int_t n, magmaDoubleComplex * A, magma_int_t lda, magmaDoubleComplex * tau, magmaDoubleComplex * work, magma_int_t lwork, magma_int_t * info )

ZGEQRF_OOC computes a QR factorization of a COMPLEX_16 M-by-N matrix A: A = Q * R.

This version does not require work space on the GPU passed as input. GPU memory is allocated in the routine. This is an out-of-core (ooc) version that is similar to magma_zgeqrf but the difference is that this version can use a GPU even if the matrix does not fit into the GPU memory at once.

Parameters
 [in] m INTEGER The number of rows of the matrix A. M >= 0. [in] n INTEGER The number of columns of the matrix A. N >= 0. [in,out] A COMPLEX_16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). Higher performance is achieved if A is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] tau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] work (workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK returns the optimal LWORK. Higher performance is achieved if WORK is in pinned memory, e.g. allocated using magma_malloc_pinned. [in] lwork INTEGER The dimension of the array WORK. LWORK >= N*NB, where NB can be obtained through magma_get_zgeqrf_nb( M, N ). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value or another error occured, such as memory allocation failed.

## Further Details

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).


Each H(i) has the form

H(i) = I - tau * v * v'


where tau is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i).