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]	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).

Functions

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Further Details

Further Details

Further Details

Further Details