MAGMA  2.3.0
Matrix Algebra for GPU and Multicore Architectures
 All Classes Files Functions Friends Groups Pages
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...
 
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. More...
 

Detailed Description

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]mINTEGER The number of rows in the matrix A.
[in]nINTEGER The number of columns in the matrix A.
[in]nbINTEGER The number of leading rows and columns of A to be reduced.
[in,out]ACOMPLEX 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]ldaINTEGER The leading dimension of the array A. LDA >= max(1,M).
[in,out]dACOMPLEX array, dimension (LDDA,N) Copy of A on GPU.
[in]lddaINTEGER The leading dimension of the array dA. LDDA >= max(1,M).
[out]dCOMPLEX array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).
[out]eCOMPLEX array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix.
[out]tauqCOMPLEX array dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
[out]taupCOMPLEX array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
[out]XCOMPLEX array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A.
[in]ldxINTEGER The leading dimension of the array X. LDX >= M.
[out]dXCOMPLEX array, dimension (LDDX,NB) Copy of X on GPU.
[in]lddxINTEGER The leading dimension of the array dX. LDDX >= M.
[out]YCOMPLEX array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A.
[in]ldyINTEGER The leading dimension of the array Y. LDY >= N.
[out]dYCOMPLEX array, dimension (LDDY,NB) Copy of Y on GPU.
[in]lddyINTEGER The leading dimension of the array dY. LDDY >= N.
workCOMPLEX array, dimension (LWORK) Workspace.
[in]lworkINTEGER The dimension of the array WORK. LWORK >= max( M, N ).
[in]queuemagma_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]mINTEGER The number of rows in the matrix A.
[in]nINTEGER The number of columns in the matrix A.
[in]nbINTEGER The number of leading rows and columns of A to be reduced.
[in,out]ADOUBLE 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]ldaINTEGER The leading dimension of the array A. LDA >= max(1,M).
[in,out]dADOUBLE PRECISION array, dimension (LDDA,N) Copy of A on GPU.
[in]lddaINTEGER The leading dimension of the array dA. LDDA >= max(1,M).
[out]dDOUBLE PRECISION array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).
[out]eDOUBLE PRECISION array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix.
[out]tauqDOUBLE PRECISION array dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
[out]taupDOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
[out]XDOUBLE PRECISION array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A.
[in]ldxINTEGER The leading dimension of the array X. LDX >= M.
[out]dXDOUBLE PRECISION array, dimension (LDDX,NB) Copy of X on GPU.
[in]lddxINTEGER The leading dimension of the array dX. LDDX >= M.
[out]YDOUBLE PRECISION array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A.
[in]ldyINTEGER The leading dimension of the array Y. LDY >= N.
[out]dYDOUBLE PRECISION array, dimension (LDDY,NB) Copy of Y on GPU.
[in]lddyINTEGER The leading dimension of the array dY. LDDY >= N.
workDOUBLE PRECISION array, dimension (LWORK) Workspace.
[in]lworkINTEGER The dimension of the array WORK. LWORK >= max( M, N ).
[in]queuemagma_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]mINTEGER The number of rows in the matrix A.
[in]nINTEGER The number of columns in the matrix A.
[in]nbINTEGER The number of leading rows and columns of A to be reduced.
[in,out]AREAL 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]ldaINTEGER The leading dimension of the array A. LDA >= max(1,M).
[in,out]dAREAL array, dimension (LDDA,N) Copy of A on GPU.
[in]lddaINTEGER The leading dimension of the array dA. LDDA >= max(1,M).
[out]dREAL array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).
[out]eREAL array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix.
[out]tauqREAL array dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
[out]taupREAL array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
[out]XREAL array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A.
[in]ldxINTEGER The leading dimension of the array X. LDX >= M.
[out]dXREAL array, dimension (LDDX,NB) Copy of X on GPU.
[in]lddxINTEGER The leading dimension of the array dX. LDDX >= M.
[out]YREAL array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A.
[in]ldyINTEGER The leading dimension of the array Y. LDY >= N.
[out]dYREAL array, dimension (LDDY,NB) Copy of Y on GPU.
[in]lddyINTEGER The leading dimension of the array dY. LDDY >= N.
workREAL array, dimension (LWORK) Workspace.
[in]lworkINTEGER The dimension of the array WORK. LWORK >= max( M, N ).
[in]queuemagma_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]mINTEGER The number of rows in the matrix A.
[in]nINTEGER The number of columns in the matrix A.
[in]nbINTEGER The number of leading rows and columns of A to be reduced.
[in,out]ACOMPLEX_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]ldaINTEGER The leading dimension of the array A. LDA >= max(1,M).
[in,out]dACOMPLEX_16 array, dimension (LDDA,N) Copy of A on GPU.
[in]lddaINTEGER The leading dimension of the array dA. LDDA >= max(1,M).
[out]dCOMPLEX_16 array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).
[out]eCOMPLEX_16 array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix.
[out]tauqCOMPLEX_16 array dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
[out]taupCOMPLEX_16 array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
[out]XCOMPLEX_16 array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A.
[in]ldxINTEGER The leading dimension of the array X. LDX >= M.
[out]dXCOMPLEX_16 array, dimension (LDDX,NB) Copy of X on GPU.
[in]lddxINTEGER The leading dimension of the array dX. LDDX >= M.
[out]YCOMPLEX_16 array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A.
[in]ldyINTEGER The leading dimension of the array Y. LDY >= N.
[out]dYCOMPLEX_16 array, dimension (LDDY,NB) Copy of Y on GPU.
[in]lddyINTEGER The leading dimension of the array dY. LDDY >= N.
workCOMPLEX_16 array, dimension (LWORK) Workspace.
[in]lworkINTEGER The dimension of the array WORK. LWORK >= max( M, N ).
[in]queuemagma_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).