MAGMA  2.3.0
Matrix Algebra for GPU and Multicore Architectures
 All Classes Files Functions Friends Groups Pages
latrd: Partial factorization; used by hetrd

Functions

magma_int_t magma_clatrd (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magmaFloatComplex *A, magma_int_t lda, float *e, magmaFloatComplex *tau, magmaFloatComplex *W, magma_int_t ldw, magmaFloatComplex *work, magma_int_t lwork, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dW, magma_int_t lddw, magma_queue_t queue)
 CLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_clatrd2 (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magmaFloatComplex *A, magma_int_t lda, float *e, magmaFloatComplex *tau, magmaFloatComplex *W, magma_int_t ldw, magmaFloatComplex *work, magma_int_t lwork, magmaFloatComplex_ptr dA, magma_int_t ldda, magmaFloatComplex_ptr dW, magma_int_t lddw, magmaFloatComplex_ptr dwork, magma_int_t ldwork, magma_queue_t queue)
 CLATRD2 reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_clatrd_mgpu (magma_int_t ngpu, magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magma_int_t nb0, magmaFloatComplex *A, magma_int_t lda, float *e, magmaFloatComplex *tau, magmaFloatComplex *W, magma_int_t ldw, magmaFloatComplex_ptr dA[], magma_int_t ldda, magma_int_t offset, magmaFloatComplex_ptr dW[], magma_int_t lddw, magmaFloatComplex *hwork, magma_int_t lhwork, magmaFloatComplex_ptr dwork[], magma_int_t ldwork, magma_queue_t queues[])
 CLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_dlatrd (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, double *A, magma_int_t lda, double *e, double *tau, double *W, magma_int_t ldw, double *work, magma_int_t lwork, magmaDouble_ptr dA, magma_int_t ldda, magmaDouble_ptr dW, magma_int_t lddw, magma_queue_t queue)
 DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_dlatrd2 (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, double *A, magma_int_t lda, double *e, double *tau, double *W, magma_int_t ldw, double *work, magma_int_t lwork, magmaDouble_ptr dA, magma_int_t ldda, magmaDouble_ptr dW, magma_int_t lddw, magmaDouble_ptr dwork, magma_int_t ldwork, magma_queue_t queue)
 DLATRD2 reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_dlatrd_mgpu (magma_int_t ngpu, magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magma_int_t nb0, double *A, magma_int_t lda, double *e, double *tau, double *W, magma_int_t ldw, magmaDouble_ptr dA[], magma_int_t ldda, magma_int_t offset, magmaDouble_ptr dW[], magma_int_t lddw, double *hwork, magma_int_t lhwork, magmaDouble_ptr dwork[], magma_int_t ldwork, magma_queue_t queues[])
 DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_slatrd (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, float *A, magma_int_t lda, float *e, float *tau, float *W, magma_int_t ldw, float *work, magma_int_t lwork, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dW, magma_int_t lddw, magma_queue_t queue)
 SLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_slatrd2 (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, float *A, magma_int_t lda, float *e, float *tau, float *W, magma_int_t ldw, float *work, magma_int_t lwork, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dW, magma_int_t lddw, magmaFloat_ptr dwork, magma_int_t ldwork, magma_queue_t queue)
 SLATRD2 reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_slatrd_mgpu (magma_int_t ngpu, magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magma_int_t nb0, float *A, magma_int_t lda, float *e, float *tau, float *W, magma_int_t ldw, magmaFloat_ptr dA[], magma_int_t ldda, magma_int_t offset, magmaFloat_ptr dW[], magma_int_t lddw, float *hwork, magma_int_t lhwork, magmaFloat_ptr dwork[], magma_int_t ldwork, magma_queue_t queues[])
 SLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_zlatrd (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magmaDoubleComplex *A, magma_int_t lda, double *e, magmaDoubleComplex *tau, magmaDoubleComplex *W, magma_int_t ldw, magmaDoubleComplex *work, magma_int_t lwork, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex_ptr dW, magma_int_t lddw, magma_queue_t queue)
 ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_zlatrd2 (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magmaDoubleComplex *A, magma_int_t lda, double *e, magmaDoubleComplex *tau, magmaDoubleComplex *W, magma_int_t ldw, magmaDoubleComplex *work, magma_int_t lwork, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex_ptr dW, magma_int_t lddw, magmaDoubleComplex_ptr dwork, magma_int_t ldwork, magma_queue_t queue)
 ZLATRD2 reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 
magma_int_t magma_zlatrd_mgpu (magma_int_t ngpu, magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magma_int_t nb0, magmaDoubleComplex *A, magma_int_t lda, double *e, magmaDoubleComplex *tau, magmaDoubleComplex *W, magma_int_t ldw, magmaDoubleComplex_ptr dA[], magma_int_t ldda, magma_int_t offset, magmaDoubleComplex_ptr dW[], magma_int_t lddw, magmaDoubleComplex *hwork, magma_int_t lhwork, magmaDoubleComplex_ptr dwork[], magma_int_t ldwork, magma_queue_t queues[])
 ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. More...
 

Detailed Description

Function Documentation

magma_int_t magma_clatrd ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magmaFloatComplex *  A,
magma_int_t  lda,
float *  e,
magmaFloatComplex *  tau,
magmaFloatComplex *  W,
magma_int_t  ldw,
magmaFloatComplex *  work,
magma_int_t  lwork,
magmaFloatComplex_ptr  dA,
magma_int_t  ldda,
magmaFloatComplex_ptr  dW,
magma_int_t  lddw,
magma_queue_t  queue 
)

CLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, CLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, CLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by CHETRD.

Parameters
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in,out]ACOMPLEX array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eCOMPLEX array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauCOMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WCOMPLEX array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
workTODO: dimension (lwork)?
lworkTODO: size??
dATODO: dimension (ldda, n)?
lddaTODO: ldda >= n?
dWTODO: dimension (lddw, ??)
lddwTODO: lddw >= n ??
[in]queuemagma_queue_t Queue to execute in.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_clatrd2 ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magmaFloatComplex *  A,
magma_int_t  lda,
float *  e,
magmaFloatComplex *  tau,
magmaFloatComplex *  W,
magma_int_t  ldw,
magmaFloatComplex *  work,
magma_int_t  lwork,
magmaFloatComplex_ptr  dA,
magma_int_t  ldda,
magmaFloatComplex_ptr  dW,
magma_int_t  lddw,
magmaFloatComplex_ptr  dwork,
magma_int_t  ldwork,
magma_queue_t  queue 
)

CLATRD2 reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, CLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, CLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by CHETRD2_GPU. It uses an accelerated HEMV that needs extra memory.

Parameters
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in,out]ACOMPLEX array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eCOMPLEX array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauCOMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WCOMPLEX array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
workTODO: dimension (lwork)?
lworkTODO: size??
dATODO: dimension (ldda, n) ??
lddaTODO: ldda >= n ??
dWTODO: dimension (lddw, 2*nb) ??
lddwTODO: lddw >= n ??
dworkTODO: dimension (ldwork) ??
ldworkTODO: ldwork >= ceil(n/64)*ldda ??
[in]queuemagma_queue_t Queue to execute in.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_clatrd_mgpu ( magma_int_t  ngpu,
magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magma_int_t  nb0,
magmaFloatComplex *  A,
magma_int_t  lda,
float *  e,
magmaFloatComplex *  tau,
magmaFloatComplex *  W,
magma_int_t  ldw,
magmaFloatComplex_ptr  dA[],
magma_int_t  ldda,
magma_int_t  offset,
magmaFloatComplex_ptr  dW[],
magma_int_t  lddw,
magmaFloatComplex *  hwork,
magma_int_t  lhwork,
magmaFloatComplex_ptr  dwork[],
magma_int_t  ldwork,
magma_queue_t  queues[] 
)

CLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, CLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, CLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by CHETRD.

Parameters
[in]ngpuINTEGER Number of GPUs to use. ngpu > 0.
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in]nb0INTEGER The block size used for the matrix distribution. nb and nb0 can be different for the final step of chetrd.
[in,out]ACOMPLEX array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eCOMPLEX array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauCOMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WCOMPLEX array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
dACOMPLEX array of pointers on the GPU, dimension (ngpu) On entry, the Hermitian matrix dA distributed over GPUs (d_lA[d] points to the local matrix on the d-th GPU). It is distributed in 1D block column or row cyclic (with the block size of nb) if UPLO = MagmaUpper or MagmaLower, respectively. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of dA contains the upper triangular part of the matrix dA, and the strictly lower triangular part of dA is not referenced. If UPLO = MagmaLower, the leading N-by-N lower triangular part of dA contains the lower triangular part of the matrix dA, and the strictly upper triangular part of dA is not referenced.
On exit, the current panel is factorized.
[in]lddaINTEGER The leading dimension of the array dA[d]. LDDA >= max(1,M).
[in]offsetINTEGER The offset to the current panel.
dWCOMPLEX array on the GPU, dimension (ngpu*lddw*(3*nb)). On exit, it stores two block columns/rows needed for the trailing submatrix update, V and W.
[in]lddwINTEGER The leading dimension of the array dW[d]. LDDW >= max(1,M).
hwork(workspace) COMPLEX array on the CPU, dimension (lhwork)
[in]lhworkINTEGER The dimension of the array HWORK.
dwork(workspace) COMPLEX array on the GPU, dimension (ldwork)
[in]ldworkINTEGER The dimension of the array DORK.
[in]queuesmagma_queue_t array of dimension (ngpu). queues[dev] is an execution queue on GPU dev.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_dlatrd ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
double *  A,
magma_int_t  lda,
double *  e,
double *  tau,
double *  W,
magma_int_t  ldw,
double *  work,
magma_int_t  lwork,
magmaDouble_ptr  dA,
magma_int_t  ldda,
magmaDouble_ptr  dW,
magma_int_t  lddw,
magma_queue_t  queue 
)

DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, DLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, DLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by DSYTRD.

Parameters
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in,out]ADOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eDOUBLE PRECISION array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauDOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WDOUBLE PRECISION array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
workTODO: dimension (lwork)?
lworkTODO: size??
dATODO: dimension (ldda, n)?
lddaTODO: ldda >= n?
dWTODO: dimension (lddw, ??)
lddwTODO: lddw >= n ??
[in]queuemagma_queue_t Queue to execute in.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_dlatrd2 ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
double *  A,
magma_int_t  lda,
double *  e,
double *  tau,
double *  W,
magma_int_t  ldw,
double *  work,
magma_int_t  lwork,
magmaDouble_ptr  dA,
magma_int_t  ldda,
magmaDouble_ptr  dW,
magma_int_t  lddw,
magmaDouble_ptr  dwork,
magma_int_t  ldwork,
magma_queue_t  queue 
)

DLATRD2 reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, DLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, DLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by DSYTRD2_GPU. It uses an accelerated HEMV that needs extra memory.

Parameters
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in,out]ADOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eDOUBLE PRECISION array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauDOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WDOUBLE PRECISION array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
workTODO: dimension (lwork)?
lworkTODO: size??
dATODO: dimension (ldda, n) ??
lddaTODO: ldda >= n ??
dWTODO: dimension (lddw, 2*nb) ??
lddwTODO: lddw >= n ??
dworkTODO: dimension (ldwork) ??
ldworkTODO: ldwork >= ceil(n/64)*ldda ??
[in]queuemagma_queue_t Queue to execute in.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_dlatrd_mgpu ( magma_int_t  ngpu,
magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magma_int_t  nb0,
double *  A,
magma_int_t  lda,
double *  e,
double *  tau,
double *  W,
magma_int_t  ldw,
magmaDouble_ptr  dA[],
magma_int_t  ldda,
magma_int_t  offset,
magmaDouble_ptr  dW[],
magma_int_t  lddw,
double *  hwork,
magma_int_t  lhwork,
magmaDouble_ptr  dwork[],
magma_int_t  ldwork,
magma_queue_t  queues[] 
)

DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, DLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, DLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by DSYTRD.

Parameters
[in]ngpuINTEGER Number of GPUs to use. ngpu > 0.
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in]nb0INTEGER The block size used for the matrix distribution. nb and nb0 can be different for the final step of dsytrd.
[in,out]ADOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eDOUBLE PRECISION array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauDOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WDOUBLE PRECISION array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
dADOUBLE PRECISION array of pointers on the GPU, dimension (ngpu) On entry, the symmetric matrix dA distributed over GPUs (d_lA[d] points to the local matrix on the d-th GPU). It is distributed in 1D block column or row cyclic (with the block size of nb) if UPLO = MagmaUpper or MagmaLower, respectively. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of dA contains the upper triangular part of the matrix dA, and the strictly lower triangular part of dA is not referenced. If UPLO = MagmaLower, the leading N-by-N lower triangular part of dA contains the lower triangular part of the matrix dA, and the strictly upper triangular part of dA is not referenced.
On exit, the current panel is factorized.
[in]lddaINTEGER The leading dimension of the array dA[d]. LDDA >= max(1,M).
[in]offsetINTEGER The offset to the current panel.
dWDOUBLE PRECISION array on the GPU, dimension (ngpu*lddw*(3*nb)). On exit, it stores two block columns/rows needed for the trailing submatrix update, V and W.
[in]lddwINTEGER The leading dimension of the array dW[d]. LDDW >= max(1,M).
hwork(workspace) DOUBLE PRECISION array on the CPU, dimension (lhwork)
[in]lhworkINTEGER The dimension of the array HWORK.
dwork(workspace) DOUBLE PRECISION array on the GPU, dimension (ldwork)
[in]ldworkINTEGER The dimension of the array DORK.
[in]queuesmagma_queue_t array of dimension (ngpu). queues[dev] is an execution queue on GPU dev.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_slatrd ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
float *  A,
magma_int_t  lda,
float *  e,
float *  tau,
float *  W,
magma_int_t  ldw,
float *  work,
magma_int_t  lwork,
magmaFloat_ptr  dA,
magma_int_t  ldda,
magmaFloat_ptr  dW,
magma_int_t  lddw,
magma_queue_t  queue 
)

SLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, SLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, SLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by SSYTRD.

Parameters
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in,out]AREAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eREAL array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauREAL array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WREAL array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
workTODO: dimension (lwork)?
lworkTODO: size??
dATODO: dimension (ldda, n)?
lddaTODO: ldda >= n?
dWTODO: dimension (lddw, ??)
lddwTODO: lddw >= n ??
[in]queuemagma_queue_t Queue to execute in.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_slatrd2 ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
float *  A,
magma_int_t  lda,
float *  e,
float *  tau,
float *  W,
magma_int_t  ldw,
float *  work,
magma_int_t  lwork,
magmaFloat_ptr  dA,
magma_int_t  ldda,
magmaFloat_ptr  dW,
magma_int_t  lddw,
magmaFloat_ptr  dwork,
magma_int_t  ldwork,
magma_queue_t  queue 
)

SLATRD2 reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, SLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, SLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by SSYTRD2_GPU. It uses an accelerated HEMV that needs extra memory.

Parameters
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in,out]AREAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eREAL array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauREAL array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WREAL array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
workTODO: dimension (lwork)?
lworkTODO: size??
dATODO: dimension (ldda, n) ??
lddaTODO: ldda >= n ??
dWTODO: dimension (lddw, 2*nb) ??
lddwTODO: lddw >= n ??
dworkTODO: dimension (ldwork) ??
ldworkTODO: ldwork >= ceil(n/64)*ldda ??
[in]queuemagma_queue_t Queue to execute in.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_slatrd_mgpu ( magma_int_t  ngpu,
magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magma_int_t  nb0,
float *  A,
magma_int_t  lda,
float *  e,
float *  tau,
float *  W,
magma_int_t  ldw,
magmaFloat_ptr  dA[],
magma_int_t  ldda,
magma_int_t  offset,
magmaFloat_ptr  dW[],
magma_int_t  lddw,
float *  hwork,
magma_int_t  lhwork,
magmaFloat_ptr  dwork[],
magma_int_t  ldwork,
magma_queue_t  queues[] 
)

SLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, SLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, SLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by SSYTRD.

Parameters
[in]ngpuINTEGER Number of GPUs to use. ngpu > 0.
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in]nb0INTEGER The block size used for the matrix distribution. nb and nb0 can be different for the final step of ssytrd.
[in,out]AREAL array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eREAL array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauREAL array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WREAL array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
dAREAL array of pointers on the GPU, dimension (ngpu) On entry, the symmetric matrix dA distributed over GPUs (d_lA[d] points to the local matrix on the d-th GPU). It is distributed in 1D block column or row cyclic (with the block size of nb) if UPLO = MagmaUpper or MagmaLower, respectively. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of dA contains the upper triangular part of the matrix dA, and the strictly lower triangular part of dA is not referenced. If UPLO = MagmaLower, the leading N-by-N lower triangular part of dA contains the lower triangular part of the matrix dA, and the strictly upper triangular part of dA is not referenced.
On exit, the current panel is factorized.
[in]lddaINTEGER The leading dimension of the array dA[d]. LDDA >= max(1,M).
[in]offsetINTEGER The offset to the current panel.
dWREAL array on the GPU, dimension (ngpu*lddw*(3*nb)). On exit, it stores two block columns/rows needed for the trailing submatrix update, V and W.
[in]lddwINTEGER The leading dimension of the array dW[d]. LDDW >= max(1,M).
hwork(workspace) REAL array on the CPU, dimension (lhwork)
[in]lhworkINTEGER The dimension of the array HWORK.
dwork(workspace) REAL array on the GPU, dimension (ldwork)
[in]ldworkINTEGER The dimension of the array DORK.
[in]queuesmagma_queue_t array of dimension (ngpu). queues[dev] is an execution queue on GPU dev.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_zlatrd ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magmaDoubleComplex *  A,
magma_int_t  lda,
double *  e,
magmaDoubleComplex *  tau,
magmaDoubleComplex *  W,
magma_int_t  ldw,
magmaDoubleComplex *  work,
magma_int_t  lwork,
magmaDoubleComplex_ptr  dA,
magma_int_t  ldda,
magmaDoubleComplex_ptr  dW,
magma_int_t  lddw,
magma_queue_t  queue 
)

ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, ZLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, ZLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by ZHETRD.

Parameters
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in,out]ACOMPLEX_16 array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eCOMPLEX_16 array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauCOMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WCOMPLEX_16 array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
workTODO: dimension (lwork)?
lworkTODO: size??
dATODO: dimension (ldda, n)?
lddaTODO: ldda >= n?
dWTODO: dimension (lddw, ??)
lddwTODO: lddw >= n ??
[in]queuemagma_queue_t Queue to execute in.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_zlatrd2 ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magmaDoubleComplex *  A,
magma_int_t  lda,
double *  e,
magmaDoubleComplex *  tau,
magmaDoubleComplex *  W,
magma_int_t  ldw,
magmaDoubleComplex *  work,
magma_int_t  lwork,
magmaDoubleComplex_ptr  dA,
magma_int_t  ldda,
magmaDoubleComplex_ptr  dW,
magma_int_t  lddw,
magmaDoubleComplex_ptr  dwork,
magma_int_t  ldwork,
magma_queue_t  queue 
)

ZLATRD2 reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, ZLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, ZLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by ZHETRD2_GPU. It uses an accelerated HEMV that needs extra memory.

Parameters
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in,out]ACOMPLEX_16 array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eCOMPLEX_16 array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauCOMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WCOMPLEX_16 array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
workTODO: dimension (lwork)?
lworkTODO: size??
dATODO: dimension (ldda, n) ??
lddaTODO: ldda >= n ??
dWTODO: dimension (lddw, 2*nb) ??
lddwTODO: lddw >= n ??
dworkTODO: dimension (ldwork) ??
ldworkTODO: ldwork >= ceil(n/64)*ldda ??
[in]queuemagma_queue_t Queue to execute in.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

(  a   a   a   v4  v5 )              (  d                  )
(      a   a   v4  v5 )              (  1   d              )
(          a   1   v5 )              (  v1  1   a          )
(              d   1  )              (  v1  v2  a   a      )
(                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

magma_int_t magma_zlatrd_mgpu ( magma_int_t  ngpu,
magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magma_int_t  nb0,
magmaDoubleComplex *  A,
magma_int_t  lda,
double *  e,
magmaDoubleComplex *  tau,
magmaDoubleComplex *  W,
magma_int_t  ldw,
magmaDoubleComplex_ptr  dA[],
magma_int_t  ldda,
magma_int_t  offset,
magmaDoubleComplex_ptr  dW[],
magma_int_t  lddw,
magmaDoubleComplex *  hwork,
magma_int_t  lhwork,
magmaDoubleComplex_ptr  dwork[],
magma_int_t  ldwork,
magma_queue_t  queues[] 
)

ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = MagmaUpper, ZLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = MagmaLower, ZLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by ZHETRD.

Parameters
[in]ngpuINTEGER Number of GPUs to use. ngpu > 0.
[in]uplomagma_uplo_t Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
  • = MagmaUpper: Upper triangular
  • = MagmaLower: Lower triangular
[in]nINTEGER The order of the matrix A.
[in]nbINTEGER The number of rows and columns to be reduced.
[in]nb0INTEGER The block size used for the matrix distribution. nb and nb0 can be different for the final step of zhetrd.
[in,out]ACOMPLEX_16 array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = MagmaUpper, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = MagmaLower, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit:
  • if UPLO = MagmaUpper, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors;
  • if UPLO = MagmaLower, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= (1,N).
[out]eCOMPLEX_16 array, dimension (N-1) If UPLO = MagmaUpper, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = MagmaLower, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
[out]tauCOMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = MagmaUpper, and in TAU(1:nb) if UPLO = MagmaLower. See Further Details.
[out]WCOMPLEX_16 array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]ldwINTEGER The leading dimension of the array W. LDW >= max(1,N).
dACOMPLEX_16 array of pointers on the GPU, dimension (ngpu) On entry, the Hermitian matrix dA distributed over GPUs (d_lA[d] points to the local matrix on the d-th GPU). It is distributed in 1D block column or row cyclic (with the block size of nb) if UPLO = MagmaUpper or MagmaLower, respectively. If UPLO = MagmaUpper, the leading N-by-N upper triangular part of dA contains the upper triangular part of the matrix dA, and the strictly lower triangular part of dA is not referenced. If UPLO = MagmaLower, the leading N-by-N lower triangular part of dA contains the lower triangular part of the matrix dA, and the strictly upper triangular part of dA is not referenced.
On exit, the current panel is factorized.
[in]lddaINTEGER The leading dimension of the array dA[d]. LDDA >= max(1,M).
[in]offsetINTEGER The offset to the current panel.
dWCOMPLEX_16 array on the GPU, dimension (ngpu*lddw*(3*nb)). On exit, it stores two block columns/rows needed for the trailing submatrix update, V and W.
[in]lddwINTEGER The leading dimension of the array dW[d]. LDDW >= max(1,M).
hwork(workspace) COMPLEX_16 array on the CPU, dimension (lhwork)
[in]lhworkINTEGER The dimension of the array HWORK.
dwork(workspace) COMPLEX_16 array on the GPU, dimension (ldwork)
[in]ldworkINTEGER The dimension of the array DORK.
[in]queuesmagma_queue_t array of dimension (ngpu). queues[dev] is an execution queue on GPU dev.

Further Details

If UPLO = MagmaUpper, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = MagmaLower, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

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

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: A := A - V*W' - W*V'.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = MagmaUpper: if UPLO = MagmaLower:

( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).