Functions
magma_int_t	magma_cggrqf (magma_int_t m, magma_int_t p, magma_int_t n, magmaFloatComplex A, magma_int_t lda, magmaFloatComplex taua, magmaFloatComplex B, magma_int_t ldb, magmaFloatComplex taub, magmaFloatComplex work, magma_int_t lwork, magma_int_t info)
	CGGRQF computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B:

magma_int_t	magma_dggrqf (magma_int_t m, magma_int_t p, magma_int_t n, double A, magma_int_t lda, double taua, double B, magma_int_t ldb, double taub, double work, magma_int_t lwork, magma_int_t info)
	DGGRQF computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B:

magma_int_t	magma_sggrqf (magma_int_t m, magma_int_t p, magma_int_t n, float A, magma_int_t lda, float taua, float B, magma_int_t ldb, float taub, float work, magma_int_t lwork, magma_int_t info)
	SGGRQF computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B:

magma_int_t	magma_zggrqf (magma_int_t m, magma_int_t p, magma_int_t n, magmaDoubleComplex A, magma_int_t lda, magmaDoubleComplex taua, magmaDoubleComplex B, magma_int_t ldb, magmaDoubleComplex taub, magmaDoubleComplex work, magma_int_t lwork, magma_int_t info)
	ZGGRQF computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B:

Detailed Description

Function Documentation

◆ magma_cggrqf()

magma_int_t magma_cggrqf	(	magma_int_t	m,
		magma_int_t	p,
		magma_int_t	n,
		magmaFloatComplex *	A,
		magma_int_t	lda,
		magmaFloatComplex *	taua,
		magmaFloatComplex *	B,
		magma_int_t	ldb,
		magmaFloatComplex *	taub,
		magmaFloatComplex *	work,
		magma_int_t	lwork,
		magma_int_t *	info )

CGGRQF computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B:

        A = R*Q,        B = Z*T*Q,

where Q is an N-by-N unitary matrix, Z is a P-by-P unitary
matrix, and R and T assume one of the forms:

if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
N-M M ( R21 ) N
N

where R12 or R21 is upper triangular, and

if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
( 0 ) P-N P N-P
N

where T11 is upper triangular.

In particular, if B is square and nonsingular, the GRQ factorization
of A and B implicitly gives the RQ factorization of A*inv(B):

         A*inv(B) = (R*inv(T))*Z'

where inv(B) denotes the inverse of the matrix B, and Z' denotes the
conjugate transpose of the matrix Z.

Parameters

[in]	m	INTEGER The number of rows of the matrix A. M >= 0.
[in]	p	INTEGER The number of rows of the matrix B. P >= 0.
[in]	n	INTEGER The number of columns of the matrices A and B. N >= 0.
[in,out]	A	COMPLEX array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, if M <= N, the upper triangle of the subarray A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; if M > N, the elements on and above the (M-N)-th subdiagonal contain the M-by-N upper trapezoidal matrix R; the remaining elements, with the array TAUA, represent the unitary matrix Q as a product of elementary reflectors (see Further Details).
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,M).
[out]	taua	COMPLEX array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the unitary matrix Q (see Further Details).
[in,out]	B	COMPLEX array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, the elements on and above the diagonal of the array contain the min(P,N)-by-N upper trapezoidal matrix T (T is upper triangular if P >= N); the elements below the diagonal, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details).
[in]	ldb	INTEGER The leading dimension of the array B. LDB >= max(1,P).
[out]	taub	COMPLEX array, dimension (min(P,N)) The scalar factors of the elementary reflectors which represent the unitary matrix Z (see Further Details).
[out]	work	(workspace) COMPLEX array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= max(1,N,M,P). For optimum performance LWORK >= max(N,M,P)*NB, where NB is the optimal blocksize for the QR factorization of a P-by-N matrix. 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 by XERBLA.
[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 - taua * v * v'

where taua is a complex scalar, and v is a complex vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine ZUNGRQ.
To use Q to update another matrix, use LAPACK subroutine CUNMRQ.

The matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(p,n).

Each H(i) has the form

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

where taub is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine CUNGQR.
To use Z to update another matrix, use LAPACK subroutine CUNMQR.

◆ magma_dggrqf()

magma_int_t magma_dggrqf	(	magma_int_t	m,
		magma_int_t	p,
		magma_int_t	n,
		double *	A,
		magma_int_t	lda,
		double *	taua,
		double *	B,
		magma_int_t	ldb,
		double *	taub,
		double *	work,
		magma_int_t	lwork,
		magma_int_t *	info )

DGGRQF computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B:

        A = R*Q,        B = Z*T*Q,

where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
matrix, and R and T assume one of the forms:

if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
N-M M ( R21 ) N
N

where R12 or R21 is upper triangular, and

if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
( 0 ) P-N P N-P
N

where T11 is upper triangular.

In particular, if B is square and nonsingular, the GRQ factorization
of A and B implicitly gives the RQ factorization of A*inv(B):

         A*inv(B) = (R*inv(T))*Z'

where inv(B) denotes the inverse of the matrix B, and Z' denotes the
conjugate transpose of the matrix Z.

Parameters

[in]	m	INTEGER The number of rows of the matrix A. M >= 0.
[in]	p	INTEGER The number of rows of the matrix B. P >= 0.
[in]	n	INTEGER The number of columns of the matrices A and B. N >= 0.
[in,out]	A	DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, if M <= N, the upper triangle of the subarray A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; if M > N, the elements on and above the (M-N)-th subdiagonal contain the M-by-N upper trapezoidal matrix R; the remaining elements, with the array TAUA, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details).
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,M).
[out]	taua	DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q (see Further Details).
[in,out]	B	DOUBLE PRECISION array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, the elements on and above the diagonal of the array contain the min(P,N)-by-N upper trapezoidal matrix T (T is upper triangular if P >= N); the elements below the diagonal, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details).
[in]	ldb	INTEGER The leading dimension of the array B. LDB >= max(1,P).
[out]	taub	DOUBLE PRECISION array, dimension (min(P,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Z (see Further Details).
[out]	work	(workspace) DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= max(1,N,M,P). For optimum performance LWORK >= max(N,M,P)*NB, where NB is the optimal blocksize for the QR factorization of a P-by-N matrix. 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 by XERBLA.
[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 - taua * v * v'

where taua is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine ZUNGRQ.
To use Q to update another matrix, use LAPACK subroutine DORMRQ.

The matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(p,n).

Each H(i) has the form

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

where taub is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine DORGQR.
To use Z to update another matrix, use LAPACK subroutine DORMQR.

◆ magma_sggrqf()

magma_int_t magma_sggrqf	(	magma_int_t	m,
		magma_int_t	p,
		magma_int_t	n,
		float *	A,
		magma_int_t	lda,
		float *	taua,
		float *	B,
		magma_int_t	ldb,
		float *	taub,
		float *	work,
		magma_int_t	lwork,
		magma_int_t *	info )

SGGRQF computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B:

        A = R*Q,        B = Z*T*Q,

where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
matrix, and R and T assume one of the forms:

if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
N-M M ( R21 ) N
N

where R12 or R21 is upper triangular, and

if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
( 0 ) P-N P N-P
N

where T11 is upper triangular.

In particular, if B is square and nonsingular, the GRQ factorization
of A and B implicitly gives the RQ factorization of A*inv(B):

         A*inv(B) = (R*inv(T))*Z'

where inv(B) denotes the inverse of the matrix B, and Z' denotes the
conjugate transpose of the matrix Z.

Parameters

[in]	m	INTEGER The number of rows of the matrix A. M >= 0.
[in]	p	INTEGER The number of rows of the matrix B. P >= 0.
[in]	n	INTEGER The number of columns of the matrices A and B. N >= 0.
[in,out]	A	REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, if M <= N, the upper triangle of the subarray A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; if M > N, the elements on and above the (M-N)-th subdiagonal contain the M-by-N upper trapezoidal matrix R; the remaining elements, with the array TAUA, represent the orthogonal matrix Q as a product of elementary reflectors (see Further Details).
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,M).
[out]	taua	REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q (see Further Details).
[in,out]	B	REAL array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, the elements on and above the diagonal of the array contain the min(P,N)-by-N upper trapezoidal matrix T (T is upper triangular if P >= N); the elements below the diagonal, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details).
[in]	ldb	INTEGER The leading dimension of the array B. LDB >= max(1,P).
[out]	taub	REAL array, dimension (min(P,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Z (see Further Details).
[out]	work	(workspace) REAL array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= max(1,N,M,P). For optimum performance LWORK >= max(N,M,P)*NB, where NB is the optimal blocksize for the QR factorization of a P-by-N matrix. 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 by XERBLA.
[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 - taua * v * v'

where taua is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine ZUNGRQ.
To use Q to update another matrix, use LAPACK subroutine SORMRQ.

The matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(p,n).

Each H(i) has the form

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

where taub is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine SORGQR.
To use Z to update another matrix, use LAPACK subroutine SORMQR.

◆ magma_zggrqf()

magma_int_t magma_zggrqf	(	magma_int_t	m,
		magma_int_t	p,
		magma_int_t	n,
		magmaDoubleComplex *	A,
		magma_int_t	lda,
		magmaDoubleComplex *	taua,
		magmaDoubleComplex *	B,
		magma_int_t	ldb,
		magmaDoubleComplex *	taub,
		magmaDoubleComplex *	work,
		magma_int_t	lwork,
		magma_int_t *	info )

ZGGRQF computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B:

        A = R*Q,        B = Z*T*Q,

where Q is an N-by-N unitary matrix, Z is a P-by-P unitary
matrix, and R and T assume one of the forms:

if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
N-M M ( R21 ) N
N

where R12 or R21 is upper triangular, and

if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
( 0 ) P-N P N-P
N

where T11 is upper triangular.

In particular, if B is square and nonsingular, the GRQ factorization
of A and B implicitly gives the RQ factorization of A*inv(B):

         A*inv(B) = (R*inv(T))*Z'

where inv(B) denotes the inverse of the matrix B, and Z' denotes the
conjugate transpose of the matrix Z.

Parameters

[in]	m	INTEGER The number of rows of the matrix A. M >= 0.
[in]	p	INTEGER The number of rows of the matrix B. P >= 0.
[in]	n	INTEGER The number of columns of the matrices A and B. N >= 0.
[in,out]	A	COMPLEX_16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, if M <= N, the upper triangle of the subarray A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; if M > N, the elements on and above the (M-N)-th subdiagonal contain the M-by-N upper trapezoidal matrix R; the remaining elements, with the array TAUA, represent the unitary matrix Q as a product of elementary reflectors (see Further Details).
[in]	lda	INTEGER The leading dimension of the array A. LDA >= max(1,M).
[out]	taua	COMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the unitary matrix Q (see Further Details).
[in,out]	B	COMPLEX_16 array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, the elements on and above the diagonal of the array contain the min(P,N)-by-N upper trapezoidal matrix T (T is upper triangular if P >= N); the elements below the diagonal, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details).
[in]	ldb	INTEGER The leading dimension of the array B. LDB >= max(1,P).
[out]	taub	COMPLEX_16 array, dimension (min(P,N)) The scalar factors of the elementary reflectors which represent the unitary matrix Z (see Further Details).
[out]	work	(workspace) COMPLEX_16 array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]	lwork	INTEGER The dimension of the array WORK. LWORK >= max(1,N,M,P). For optimum performance LWORK >= max(N,M,P)*NB, where NB is the optimal blocksize for the QR factorization of a P-by-N matrix. 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 by XERBLA.
[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 - taua * v * v'

where taua is a complex scalar, and v is a complex vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine ZUNGRQ.
To use Q to update another matrix, use LAPACK subroutine ZUNMRQ.

The matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(p,n).

Each H(i) has the form

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

where taub is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine ZUNGQR.
To use Z to update another matrix, use LAPACK subroutine ZUNMQR.

Functions

Detailed Description

Function Documentation

◆ magma_cggrqf()

Further Details

◆ magma_dggrqf()

Further Details

◆ magma_sggrqf()

Further Details

◆ magma_zggrqf()

Further Details