MAGMA  2.3.0 Matrix Algebra for GPU and Multicore Architectures
labrd: Partial factorization; used by gebrd

## Functions

magma_int_t magma_clabrd_gpu (magma_int_t m, magma_int_t n, magma_int_t nb, magmaFloatComplex *A, magma_int_t lda, magmaFloatComplex_ptr dA, magma_int_t ldda, float *d, float *e, magmaFloatComplex *tauq, magmaFloatComplex *taup, magmaFloatComplex *X, magma_int_t ldx, magmaFloatComplex_ptr dX, magma_int_t lddx, magmaFloatComplex *Y, magma_int_t ldy, magmaFloatComplex_ptr dY, magma_int_t lddy, magmaFloatComplex *work, magma_int_t lwork, magma_queue_t queue)
CLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. More...

magma_int_t magma_dlabrd_gpu (magma_int_t m, magma_int_t n, magma_int_t nb, double *A, magma_int_t lda, magmaDouble_ptr dA, magma_int_t ldda, double *d, double *e, double *tauq, double *taup, double *X, magma_int_t ldx, magmaDouble_ptr dX, magma_int_t lddx, double *Y, magma_int_t ldy, magmaDouble_ptr dY, magma_int_t lddy, double *work, magma_int_t lwork, magma_queue_t queue)
DLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. More...

magma_int_t magma_slabrd_gpu (magma_int_t m, magma_int_t n, magma_int_t nb, float *A, magma_int_t lda, magmaFloat_ptr dA, magma_int_t ldda, float *d, float *e, float *tauq, float *taup, float *X, magma_int_t ldx, magmaFloat_ptr dX, magma_int_t lddx, float *Y, magma_int_t ldy, magmaFloat_ptr dY, magma_int_t lddy, float *work, magma_int_t lwork, magma_queue_t queue)
SLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. More...

ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. More...

## Function Documentation

 magma_int_t magma_clabrd_gpu ( magma_int_t m, magma_int_t n, magma_int_t nb, magmaFloatComplex * A, magma_int_t lda, magmaFloatComplex_ptr dA, magma_int_t ldda, float * d, float * e, magmaFloatComplex * tauq, magmaFloatComplex * taup, magmaFloatComplex * X, magma_int_t ldx, magmaFloatComplex_ptr dX, magma_int_t lddx, magmaFloatComplex * Y, magma_int_t ldy, magmaFloatComplex_ptr dY, magma_int_t lddy, magmaFloatComplex * work, magma_int_t lwork, magma_queue_t queue )

CLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A.

If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form.

This is an auxiliary routine called by CGEBRD.

Parameters
 [in] m INTEGER The number of rows in the matrix A. [in] n INTEGER The number of columns in the matrix A. [in] nb INTEGER The number of leading rows and columns of A to be reduced. [in,out] A COMPLEX array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [in,out] dA COMPLEX array, dimension (LDDA,N) Copy of A on GPU. [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). [out] d COMPLEX array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i). [out] e COMPLEX array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix. [out] tauq COMPLEX array dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details. [out] taup COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details. [out] X COMPLEX array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A. [in] ldx INTEGER The leading dimension of the array X. LDX >= M. [out] dX COMPLEX array, dimension (LDDX,NB) Copy of X on GPU. [in] lddx INTEGER The leading dimension of the array dX. LDDX >= M. [out] Y COMPLEX array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A. [in] ldy INTEGER The leading dimension of the array Y. LDY >= N. [out] dY COMPLEX array, dimension (LDDY,NB) Copy of Y on GPU. [in] lddy INTEGER The leading dimension of the array dY. LDDY >= N. work COMPLEX array, dimension (LWORK) Workspace. [in] lwork INTEGER The dimension of the array WORK. LWORK >= max( M, N ). [in] queue magma_queue_t Queue to execute in.

## Further Details

The matrices Q and P are represented as products of elementary reflectors:

Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'

where tauq and taup are complex scalars, and v and u are complex vectors.

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y' - X*U'.

The contents of A on exit are illustrated by the following examples with nb = 2:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

(  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
(  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
(  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )


where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

 magma_int_t magma_dlabrd_gpu ( magma_int_t m, magma_int_t n, magma_int_t nb, double * A, magma_int_t lda, magmaDouble_ptr dA, magma_int_t ldda, double * d, double * e, double * tauq, double * taup, double * X, magma_int_t ldx, magmaDouble_ptr dX, magma_int_t lddx, double * Y, magma_int_t ldy, magmaDouble_ptr dY, magma_int_t lddy, double * work, magma_int_t lwork, magma_queue_t queue )

DLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A.

If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form.

This is an auxiliary routine called by DGEBRD.

Parameters
 [in] m INTEGER The number of rows in the matrix A. [in] n INTEGER The number of columns in the matrix A. [in] nb INTEGER The number of leading rows and columns of A to be reduced. [in,out] A DOUBLE PRECISION array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [in,out] dA DOUBLE PRECISION array, dimension (LDDA,N) Copy of A on GPU. [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). [out] d DOUBLE PRECISION array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i). [out] e DOUBLE PRECISION array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix. [out] tauq DOUBLE PRECISION array dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details. [out] taup DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details. [out] X DOUBLE PRECISION array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A. [in] ldx INTEGER The leading dimension of the array X. LDX >= M. [out] dX DOUBLE PRECISION array, dimension (LDDX,NB) Copy of X on GPU. [in] lddx INTEGER The leading dimension of the array dX. LDDX >= M. [out] Y DOUBLE PRECISION array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A. [in] ldy INTEGER The leading dimension of the array Y. LDY >= N. [out] dY DOUBLE PRECISION array, dimension (LDDY,NB) Copy of Y on GPU. [in] lddy INTEGER The leading dimension of the array dY. LDDY >= N. work DOUBLE PRECISION array, dimension (LWORK) Workspace. [in] lwork INTEGER The dimension of the array WORK. LWORK >= max( M, N ). [in] queue magma_queue_t Queue to execute in.

## Further Details

The matrices Q and P are represented as products of elementary reflectors:

Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'

where tauq and taup are real scalars, and v and u are real vectors.

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y' - X*U'.

The contents of A on exit are illustrated by the following examples with nb = 2:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

(  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
(  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
(  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )


where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

 magma_int_t magma_slabrd_gpu ( magma_int_t m, magma_int_t n, magma_int_t nb, float * A, magma_int_t lda, magmaFloat_ptr dA, magma_int_t ldda, float * d, float * e, float * tauq, float * taup, float * X, magma_int_t ldx, magmaFloat_ptr dX, magma_int_t lddx, float * Y, magma_int_t ldy, magmaFloat_ptr dY, magma_int_t lddy, float * work, magma_int_t lwork, magma_queue_t queue )

SLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A.

If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form.

This is an auxiliary routine called by SGEBRD.

Parameters
 [in] m INTEGER The number of rows in the matrix A. [in] n INTEGER The number of columns in the matrix A. [in] nb INTEGER The number of leading rows and columns of A to be reduced. [in,out] A REAL array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [in,out] dA REAL array, dimension (LDDA,N) Copy of A on GPU. [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). [out] d REAL array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i). [out] e REAL array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix. [out] tauq REAL array dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details. [out] taup REAL array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details. [out] X REAL array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A. [in] ldx INTEGER The leading dimension of the array X. LDX >= M. [out] dX REAL array, dimension (LDDX,NB) Copy of X on GPU. [in] lddx INTEGER The leading dimension of the array dX. LDDX >= M. [out] Y REAL array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A. [in] ldy INTEGER The leading dimension of the array Y. LDY >= N. [out] dY REAL array, dimension (LDDY,NB) Copy of Y on GPU. [in] lddy INTEGER The leading dimension of the array dY. LDDY >= N. work REAL array, dimension (LWORK) Workspace. [in] lwork INTEGER The dimension of the array WORK. LWORK >= max( M, N ). [in] queue magma_queue_t Queue to execute in.

## Further Details

The matrices Q and P are represented as products of elementary reflectors:

Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'

where tauq and taup are real scalars, and v and u are real vectors.

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y' - X*U'.

The contents of A on exit are illustrated by the following examples with nb = 2:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

(  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
(  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
(  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )


where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

 magma_int_t magma_zlabrd_gpu ( magma_int_t m, magma_int_t n, magma_int_t nb, magmaDoubleComplex * A, magma_int_t lda, magmaDoubleComplex_ptr dA, magma_int_t ldda, double * d, double * e, magmaDoubleComplex * tauq, magmaDoubleComplex * taup, magmaDoubleComplex * X, magma_int_t ldx, magmaDoubleComplex_ptr dX, magma_int_t lddx, magmaDoubleComplex * Y, magma_int_t ldy, magmaDoubleComplex_ptr dY, magma_int_t lddy, magmaDoubleComplex * work, magma_int_t lwork, magma_queue_t queue )

ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A.

If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form.

This is an auxiliary routine called by ZGEBRD.

Parameters
 [in] m INTEGER The number of rows in the matrix A. [in] n INTEGER The number of columns in the matrix A. [in] nb INTEGER The number of leading rows and columns of A to be reduced. [in,out] A COMPLEX_16 array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. [in] lda INTEGER The leading dimension of the array A. LDA >= max(1,M). [in,out] dA COMPLEX_16 array, dimension (LDDA,N) Copy of A on GPU. [in] ldda INTEGER The leading dimension of the array dA. LDDA >= max(1,M). [out] d COMPLEX_16 array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i). [out] e COMPLEX_16 array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix. [out] tauq COMPLEX_16 array dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details. [out] taup COMPLEX_16 array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details. [out] X COMPLEX_16 array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A. [in] ldx INTEGER The leading dimension of the array X. LDX >= M. [out] dX COMPLEX_16 array, dimension (LDDX,NB) Copy of X on GPU. [in] lddx INTEGER The leading dimension of the array dX. LDDX >= M. [out] Y COMPLEX_16 array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A. [in] ldy INTEGER The leading dimension of the array Y. LDY >= N. [out] dY COMPLEX_16 array, dimension (LDDY,NB) Copy of Y on GPU. [in] lddy INTEGER The leading dimension of the array dY. LDDY >= N. work COMPLEX_16 array, dimension (LWORK) Workspace. [in] lwork INTEGER The dimension of the array WORK. LWORK >= max( M, N ). [in] queue magma_queue_t Queue to execute in.

## Further Details

The matrices Q and P are represented as products of elementary reflectors:

Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'

where tauq and taup are complex scalars, and v and u are complex vectors.

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y' - X*U'.

The contents of A on exit are illustrated by the following examples with nb = 2:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

(  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
(  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
(  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )


where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).