MAGMA
2.3.0
Matrix Algebra for GPU and Multicore Architectures

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...  
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.
[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 offdiagonal 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 mbynb 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 nbynb 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. 
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:i1) = 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:i1) = 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 mbynb matrix V and the nbbyn 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.
[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 offdiagonal 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 mbynb 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 nbynb 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. 
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:i1) = 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:i1) = 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 mbynb matrix V and the nbbyn 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.
[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 offdiagonal 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 mbynb 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 nbynb 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. 
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:i1) = 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:i1) = 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 mbynb matrix V and the nbbyn 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.
[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 offdiagonal 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 mbynb 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 nbynb 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. 
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:i1) = 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:i1) = 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 mbynb matrix V and the nbbyn 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).