MAGMA  2.3.0 Matrix Algebra for GPU and Multicore Architectures
geqr2: QR panel factorization

## Functions

magma_int_t magma_cgeqr2x2_gpu (magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dtau, magmaFloatComplex_ptr dT, magmaFloatComplex_ptr ddA, magmaFloat_ptr dwork, magma_int_t *info)
CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R. More...

magma_int_t magma_cgeqr2x3_gpu (magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dtau, magmaFloatComplex_ptr dT, magmaFloatComplex_ptr ddA, magmaFloat_ptr dwork, magma_int_t *info)
CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R. More...

magma_int_t magma_cgeqr2x_gpu (magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dtau, magmaFloatComplex_ptr dT, magmaFloatComplex_ptr ddA, magmaFloat_ptr dwork, magma_int_t *info)
CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R. More...

DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R. More...

DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R. More...

DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R. More...

magma_int_t magma_sgeqr2x2_gpu (magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dtau, magmaFloat_ptr dT, magmaFloat_ptr ddA, magmaFloat_ptr dwork, magma_int_t *info)
SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R. More...

magma_int_t magma_sgeqr2x3_gpu (magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dtau, magmaFloat_ptr dT, magmaFloat_ptr ddA, magmaFloat_ptr dwork, magma_int_t *info)
SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R. More...

magma_int_t magma_sgeqr2x_gpu (magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dtau, magmaFloat_ptr dT, magmaFloat_ptr ddA, magmaFloat_ptr dwork, magma_int_t *info)
SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R. More...

ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R. More...

ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R. More...

ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R. More...

magma_int_t magma_cgeqr2_gpu (magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dtau, magmaFloat_ptr dwork, magma_queue_t queue, magma_int_t *info)
CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R using the non-blocking Householder QR. More...

magma_int_t magma_cgeqr2x4_gpu (magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dtau, magmaFloatComplex_ptr dT, magmaFloatComplex_ptr ddA, magmaFloat_ptr dwork, magma_queue_t queue, magma_int_t *info)
CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R. More...

magma_int_t magma_dgeqr2_gpu (magma_int_t m, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, magmaDouble_ptr dtau, magmaDouble_ptr dwork, magma_queue_t queue, magma_int_t *info)
DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R using the non-blocking Householder QR. More...

DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R. More...

magma_int_t magma_sgeqr2_gpu (magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dtau, magmaFloat_ptr dwork, magma_queue_t queue, magma_int_t *info)
SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R using the non-blocking Householder QR. More...

magma_int_t magma_sgeqr2x4_gpu (magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dtau, magmaFloat_ptr dT, magmaFloat_ptr ddA, magmaFloat_ptr dwork, magma_queue_t queue, magma_int_t *info)
SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R. More...

magma_int_t magma_zgeqr2_gpu (magma_int_t m, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex_ptr dtau, magmaDouble_ptr dwork, magma_queue_t queue, magma_int_t *info)
ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R using the non-blocking Householder QR. More...

ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R. More...

## Function Documentation

 magma_int_t magma_cgeqr2x2_gpu ( magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dtau, magmaFloatComplex_ptr dT, magmaFloatComplex_ptr ddA, magmaFloat_ptr dwork, magma_int_t * info )

CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard cgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's cgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the unitary matrix Q as a product of elementary reflectors (see Further Details). 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 unitary matrix Q as a product of elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT COMPLEX array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA COMPLEX array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) REAL array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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_int_t magma_cgeqr2x3_gpu ( magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dtau, magmaFloatComplex_ptr dT, magmaFloatComplex_ptr ddA, magmaFloat_ptr dwork, magma_int_t * info )

CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard cgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's cgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the unitary matrix Q as a product of elementary reflectors (see Further Details). 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 unitary matrix Q as a product of elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT COMPLEX array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA COMPLEX array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) REAL array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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_int_t magma_cgeqr2x_gpu ( magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dtau, magmaFloatComplex_ptr dT, magmaFloatComplex_ptr ddA, magmaFloat_ptr dwork, magma_int_t * info )

CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard cgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's cgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R.

This version implements the right-looking QR. A hard-coded requirement for N is to be <= min(M, 128). For larger N one should use a blocking QR version.

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. 0 <= N <= min(M, 128). [in,out] dA COMPLEX array, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the unitary matrix Q as a product of elementary reflectors (see Further Details). 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 unitary matrix Q as a product of elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT COMPLEX array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA COMPLEX array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) COMPLEX array, dimension (N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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).

DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard dgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's dgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the orthogonal matrix Q as a product of elementary reflectors (see Further Details). 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 elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT DOUBLE PRECISION array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA DOUBLE PRECISION array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) DOUBLE PRECISION array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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).

DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard dgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's dgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the orthogonal matrix Q as a product of elementary reflectors (see Further Details). 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 elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT DOUBLE PRECISION array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA DOUBLE PRECISION array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) DOUBLE PRECISION array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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).

DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard dgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's dgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R.

This version implements the right-looking QR. A hard-coded requirement for N is to be <= min(M, 128). For larger N one should use a blocking QR version.

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. 0 <= N <= min(M, 128). [in,out] dA DOUBLE PRECISION array, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the orthogonal matrix Q as a product of elementary reflectors (see Further Details). 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 elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT DOUBLE PRECISION array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA DOUBLE PRECISION array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) DOUBLE PRECISION array, dimension (N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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_int_t magma_sgeqr2x2_gpu ( magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dtau, magmaFloat_ptr dT, magmaFloat_ptr ddA, magmaFloat_ptr dwork, magma_int_t * info )

SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard sgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's sgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the orthogonal matrix Q as a product of elementary reflectors (see Further Details). 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 elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT REAL array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA REAL array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) REAL array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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_int_t magma_sgeqr2x3_gpu ( magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dtau, magmaFloat_ptr dT, magmaFloat_ptr ddA, magmaFloat_ptr dwork, magma_int_t * info )

SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard sgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's sgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the orthogonal matrix Q as a product of elementary reflectors (see Further Details). 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 elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT REAL array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA REAL array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) REAL array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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_int_t magma_sgeqr2x_gpu ( magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dtau, magmaFloat_ptr dT, magmaFloat_ptr ddA, magmaFloat_ptr dwork, magma_int_t * info )

SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard sgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's sgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R.

This version implements the right-looking QR. A hard-coded requirement for N is to be <= min(M, 128). For larger N one should use a blocking QR version.

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. 0 <= N <= min(M, 128). [in,out] dA REAL array, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the orthogonal matrix Q as a product of elementary reflectors (see Further Details). 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 elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT REAL array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA REAL array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) REAL array, dimension (N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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).

ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard zgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's zgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the unitary matrix Q as a product of elementary reflectors (see Further Details). 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 unitary matrix Q as a product of elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT COMPLEX_16 array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA COMPLEX_16 array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) DOUBLE PRECISION array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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).

ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard zgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's zgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the unitary matrix Q as a product of elementary reflectors (see Further Details). 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 unitary matrix Q as a product of elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT COMPLEX_16 array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA COMPLEX_16 array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) DOUBLE PRECISION array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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).

ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard zgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's zgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R.

This version implements the right-looking QR. A hard-coded requirement for N is to be <= min(M, 128). For larger N one should use a blocking QR version.

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. 0 <= N <= min(M, 128). [in,out] dA COMPLEX_16 array, dimension (LDDA,N) On entry, the m by n matrix A. On exit, the unitary matrix Q as a product of elementary reflectors (see Further Details). 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 unitary matrix Q as a product of elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDDA >= max(1,M). [out] dtau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT COMPLEX_16 array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA COMPLEX_16 array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) COMPLEX_16 array, dimension (N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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_int_t magma_cgeqr2_gpu ( magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dtau, magmaFloat_ptr dwork, magma_queue_t queue, magma_int_t * info )

CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R using the non-blocking Householder QR.

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, 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 unitary matrix Q as a product of elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] dtau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). dwork (workspace) REAL array, dimension (N) [in] queue magma_queue_t Queue to execute in. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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_int_t magma_cgeqr2x4_gpu ( magma_int_t m, magma_int_t n, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dtau, magmaFloatComplex_ptr dT, magmaFloatComplex_ptr ddA, magmaFloat_ptr dwork, magma_queue_t queue, magma_int_t * info )

CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard cgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's cgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDA,N) On entry, the m by n matrix A. On exit, the unitary matrix Q as a product of elementary reflectors (see Further Details). 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 unitary matrix Q as a product of elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] dtau COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT COMPLEX array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA COMPLEX array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) REAL array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value [in] queue magma_queue_t Queue to execute in.

## 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_int_t magma_dgeqr2_gpu ( magma_int_t m, magma_int_t n, magmaDouble_ptr dA, magma_int_t ldda, magmaDouble_ptr dtau, magmaDouble_ptr dwork, magma_queue_t queue, magma_int_t * info )

DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R using the non-blocking Householder QR.

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, 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 elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] dtau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). dwork (workspace) DOUBLE PRECISION array, dimension (N) [in] queue magma_queue_t Queue to execute in. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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).

DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard dgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's dgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDA,N) On entry, the m by n matrix A. On exit, the orthogonal matrix Q as a product of elementary reflectors (see Further Details). 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 elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] dtau DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT DOUBLE PRECISION array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA DOUBLE PRECISION array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) DOUBLE PRECISION array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value [in] queue magma_queue_t Queue to execute in.

## 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_int_t magma_sgeqr2_gpu ( magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dtau, magmaFloat_ptr dwork, magma_queue_t queue, magma_int_t * info )

SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R using the non-blocking Householder QR.

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, 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 elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] dtau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). dwork (workspace) REAL array, dimension (N) [in] queue magma_queue_t Queue to execute in. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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_int_t magma_sgeqr2x4_gpu ( magma_int_t m, magma_int_t n, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dtau, magmaFloat_ptr dT, magmaFloat_ptr ddA, magmaFloat_ptr dwork, magma_queue_t queue, magma_int_t * info )

SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard sgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's sgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDA,N) On entry, the m by n matrix A. On exit, the orthogonal matrix Q as a product of elementary reflectors (see Further Details). 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 elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] dtau REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT REAL array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA REAL array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) REAL array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value [in] queue magma_queue_t Queue to execute in.

## 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_int_t magma_zgeqr2_gpu ( magma_int_t m, magma_int_t n, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex_ptr dtau, magmaDouble_ptr dwork, magma_queue_t queue, magma_int_t * info )

ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R using the non-blocking Householder QR.

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, 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 unitary matrix Q as a product of elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] dtau COMPLEX*16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). dwork (workspace) DOUBLE PRECISION array, dimension (N) [in] queue magma_queue_t Queue to execute in. [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

## 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).

ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R.

This expert routine requires two more arguments than the standard zgeqr2, namely, dT and ddA, explained below. The storage for A is also not as in the LAPACK's zgeqr2 routine (see below).

The first is used to output the triangular n x n factor T of the block reflector used in the factorization. The second holds the diagonal nxn blocks of A, i.e., the diagonal submatrices of R. This routine implements the left looking QR.

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, dimension (LDA,N) On entry, the m by n matrix A. On exit, the unitary matrix Q as a product of elementary reflectors (see Further Details). 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 unitary matrix Q as a product of elementary reflectors (see Further Details). [in] ldda INTEGER The leading dimension of the array A. LDA >= max(1,M). [out] dtau COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). [out] dT COMPLEX_16 array, dimension N x N. Stores the triangular N x N factor T of the block reflector used in the factorization. The lower triangular part is 0. [out] ddA COMPLEX_16 array, dimension N x N. Stores the elements of the upper N x N diagonal block of A. LAPACK stores this array in A. There are 0s below the diagonal. dwork (workspace) DOUBLE PRECISION array, dimension (3 N) [out] info INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value [in] queue magma_queue_t Queue to execute in.

## 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).