MAGMA  1.6.1
Matrix Algebra for GPU and Multicore Architectures
 All Classes Files Functions Friends Groups Pages
double-complex precision

Functions

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_ptr dA, magma_int_t ldda, magmaDoubleComplex_ptr dW, magma_int_t lddw)
 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_ptr dA, magma_int_t ldda, magmaDoubleComplex_ptr dW, magma_int_t lddw, magmaDoubleComplex_ptr dwork, magma_int_t ldwork)
 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_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_ptr  dA,
magma_int_t  ldda,
magmaDoubleComplex_ptr  dW,
magma_int_t  lddw 
)

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).
dA
ldda
dW
lddw

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_ptr  dA,
magma_int_t  ldda,
magmaDoubleComplex_ptr  dW,
magma_int_t  lddw,
magmaDoubleComplex_ptr  dwork,
magma_int_t  ldwork 
)

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).
dATODO: dimension (ldda, n) ??
lddaTODO: ldda >= n ??
dWTODO: dimension (lddw, 2*nb) ??
lddwTODO: lddw >= n ??
dworkTODO: dimension (ldwork) ??
ldworkTODO: ldwork >= ceil(n/64)*ldda ??

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]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).
dA
[in]ldda
[in]offset
dW
[in]lddw
hwork
[in]lhwork
dwork
[in]ldwork
[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).