MAGMA  2.0.0
Matrix Algebra for GPU and Multicore Architectures

Functions

magma_int_t magma_zhetrd_hb2st (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magma_int_t Vblksiz, magmaDoubleComplex *A, magma_int_t lda, double *d, double *e, magmaDoubleComplex *V, magma_int_t ldv, magmaDoubleComplex *TAU, magma_int_t wantz, magmaDoubleComplex *T, magma_int_t ldt)
 
magma_int_t magma_zhetrd_he2hb (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *tau, magmaDoubleComplex *work, magma_int_t lwork, magmaDoubleComplex_ptr dT, magma_int_t *info)
 ZHETRD_HE2HB reduces a complex Hermitian matrix A to real symmetric band-diagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
 
magma_int_t magma_zhetrd_he2hb_mgpu (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *tau, magmaDoubleComplex *work, magma_int_t lwork, magmaDoubleComplex_ptr dAmgpu[], magma_int_t ldda, magmaDoubleComplex_ptr dTmgpu[], magma_int_t lddt, magma_int_t ngpu, magma_int_t distblk, magma_queue_t queues[][20], magma_int_t nqueue, magma_int_t *info)
 ZHETRD_HE2HB reduces a complex Hermitian matrix A to real symmetric band-diagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T. More...
 
magma_int_t magma_zungqr_2stage_gpu (magma_int_t m, magma_int_t n, magma_int_t k, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex *tau, magmaDoubleComplex_ptr dT, magma_int_t nb, magma_int_t *info)
 ZUNGQR generates an M-by-N COMPLEX_16 matrix Q with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M. More...
 
magma_int_t magma_zunmqr_gpu_2stages (magma_side_t side, magma_trans_t trans, magma_int_t m, magma_int_t n, magma_int_t k, magmaDoubleComplex_ptr dA, magma_int_t ldda, magmaDoubleComplex_ptr dC, magma_int_t lddc, magmaDoubleComplex_ptr dT, magma_int_t nb, magma_int_t *info)
 ZUNMQR_GPU overwrites the general complex M-by-N matrix C with. More...
 

Detailed Description

Function Documentation

magma_int_t magma_zhetrd_hb2st ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magma_int_t  Vblksiz,
magmaDoubleComplex *  A,
magma_int_t  lda,
double *  d,
double *  e,
magmaDoubleComplex *  V,
magma_int_t  ldv,
magmaDoubleComplex *  TAU,
magma_int_t  wantz,
magmaDoubleComplex *  T,
magma_int_t  ldt 
)
Parameters
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangles of A is stored;
  • = MagmaLower: Lower triangles of A is stored.
[in]nINTEGER The order of the matrix A. n >= 0.
[in]nbINTEGER The order of the band matrix A. n >= nb >= 0.
[in]VblksizINTEGER The size of the block of householder vectors applied at once.
[in]A(workspace) COMPLEX_16 array, dimension (lda, n) On entry the band matrix stored in the following way:
[in]ldaINTEGER The leading dimension of the array A. lda >= 2*nb.
[out]dDOUBLE array, dimension (n) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out]eDOUBLE array, dimension (n-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = MagmaUpper, E(i) = A(i+1,i) if UPLO = MagmaLower.
[out]VCOMPLEX_16 array, dimension (BLKCNT, LDV, VBLKSIZ) On exit it contains the blocks of householder reflectors BLKCNT is the number of block and it is returned by the funtion MAGMA_BULGE_GET_BLKCNT.
[in]ldvINTEGER The leading dimension of V. LDV > nb + VBLKSIZ + 1
[out]TAUCOMPLEX_16 dimension(BLKCNT, VBLKSIZ) ???
[in]wantzINTEGER if COMPT = 0 T is not computed if COMPT = 1 T is computed
[out]TCOMPLEX_16 dimension(LDT *) if COMPT = 1 on exit contains the matrices T needed for Q2 if COMPT = 0 T is not referenced
[in]ldtINTEGER The leading dimension of T. LDT > Vblksiz
magma_int_t magma_zhetrd_he2hb ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  tau,
magmaDoubleComplex *  work,
magma_int_t  lwork,
magmaDoubleComplex_ptr  dT,
magma_int_t *  info 
)

ZHETRD_HE2HB reduces a complex Hermitian matrix A to real symmetric band-diagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

This version stores the triangular matrices T used in the accumulated Householder transformations (I - V T V').

Parameters
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangle of A is stored;
  • = MagmaLower: Lower triangle of A is stored.
[in]nINTEGER The order of the matrix A. n >= 0.
[in]nbINTEGER The inner blocking. nb >= 0.
[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 Upper band-diagonal of A is overwritten by the corresponding elements of the band-diagonal matrix T, and the elements above the band diagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = MagmaLower, the the Lower band-diagonal of A is overwritten by the corresponding elements of the band-diagonal matrix T, and the elements below the band-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 >= max(1,N).
[out]tauCOMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]work(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >= N*NB, where NB is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
[out]dTCOMPLEX_16 array on the GPU, dimension N*NB, where NB is the optimal blocksize. On exit dT holds the upper triangular matrices T from the accumulated Householder transformations (I - V T V') used in the factorization. The nb x nb matrices T are ordered consecutively in memory one after another.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value

Further Details

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

Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i).

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

Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i).

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

if UPLO = MagmaUpper: if UPLO = MagmaLower:

( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).

magma_int_t magma_zhetrd_he2hb_mgpu ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  tau,
magmaDoubleComplex *  work,
magma_int_t  lwork,
magmaDoubleComplex_ptr  dAmgpu[],
magma_int_t  ldda,
magmaDoubleComplex_ptr  dTmgpu[],
magma_int_t  lddt,
magma_int_t  ngpu,
magma_int_t  distblk,
magma_queue_t  queues[][20],
magma_int_t  nqueue,
magma_int_t *  info 
)

ZHETRD_HE2HB reduces a complex Hermitian matrix A to real symmetric band-diagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

This version stores the triangular matrices T used in the accumulated Householder transformations (I - V T V').

Parameters
[in]uplomagma_uplo_t
  • = MagmaUpper: Upper triangle of A is stored;
  • = MagmaLower: Lower triangle of A is stored.
[in]nINTEGER The order of the matrix A. N >= 0.
[in]nbINTEGER The inner blocking. nb >= 0.
[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 Upper band-diagonal of A is overwritten by the corresponding elements of the band-diagonal matrix T, and the elements above the band diagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = MagmaLower, the the Lower band-diagonal of A is overwritten by the corresponding elements of the band-diagonal matrix T, and the elements below the band-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 >= max(1,N).
[out]tauCOMPLEX_16 array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out]work(workspace) COMPLEX_16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >= N*NB, where NB is the optimal blocksize.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
[in,out]dAmgpuCOMPLEX_16 array of pointer, dimension (ngpu) Each point to a COMPLEX_16 array, dimension (LDDA, nlocal) which hold the local matrix on each GPU.
[in]lddaINTEGER The leading dimension of the array dAmgpu. ldda >= max(1,n).
[in,out]dTmgpuCOMPLEX_16 array of pointer, dimension (ngpu) Each point to a COMPLEX_16 array on the GPU, dimension n*nb, where nb is the optimal blocksize. On exit dT holds the upper triangular matrices T from the accumulated Householder transformations (I - V T V') used in the factorization. The nb x nb matrices T are ordered consecutively in memory one after another.
[in]lddtINTEGER The leading dimension of each array dT. lddt >= max(1,nb).
[in]ngpuINTEGER The number of GPUs.
[in]distblkINTEGER Internal parameter for performance tuning. The size of the distribution/computation.
[in]queuesArray of magma_queue_t that point to the queues to be used in execution/communications. Dimension >= max(3, ngpu+1) Queue to execute in.
[in]nqueueINTEGER The number of queues should be >= max(3, ngpu+1).
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value

Further Details

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

Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i).

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

Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i).

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

if UPLO = MagmaUpper: if UPLO = MagmaLower:

(  d   e   v2  v3  v4 )              (  d                  )
(      d   e   v3  v4 )              (  e   d              )
(          d   e   v4 )              (  v1  e   d          )
(              d   e  )              (  v1  v2  e   d      )
(                  d  )              (  v1  v2  v3  e   d  )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).

magma_int_t magma_zungqr_2stage_gpu ( magma_int_t  m,
magma_int_t  n,
magma_int_t  k,
magmaDoubleComplex_ptr  dA,
magma_int_t  ldda,
magmaDoubleComplex *  tau,
magmaDoubleComplex_ptr  dT,
magma_int_t  nb,
magma_int_t *  info 
)

ZUNGQR generates an M-by-N COMPLEX_16 matrix Q with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M.

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

as returned by ZGEQRF_GPU.

Parameters
[in]mINTEGER The number of rows of the matrix Q. M >= 0.
[in]nINTEGER The number of columns of the matrix Q. M >= N >= 0.
[in]kINTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0.
[in,out]dACOMPLEX_16 array A on the GPU device, dimension (LDDA,N). On entry, the i-th column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGEQRF_GPU in the first k columns of its array argument A. On exit, the M-by-N matrix Q.
[in]lddaINTEGER The first dimension of the array A. LDDA >= max(1,M).
[in]tauCOMPLEX_16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by ZGEQRF_GPU.
[in]dTCOMPLEX_16 work space array on the GPU device, dimension (MIN(M, N) )*NB. This must be the 6th argument of magma_zgeqrf_gpu [ note that if N here is bigger than N in magma_zgeqrf_gpu, the workspace requirement DT in magma_zgeqrf_gpu must be as specified in this routine ].
[in]nbINTEGER This is the block size used in ZGEQRF_GPU, and correspondingly the size of the T matrices, used in the factorization, and stored in DT.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
magma_int_t magma_zunmqr_gpu_2stages ( magma_side_t  side,
magma_trans_t  trans,
magma_int_t  m,
magma_int_t  n,
magma_int_t  k,
magmaDoubleComplex_ptr  dA,
magma_int_t  ldda,
magmaDoubleComplex_ptr  dC,
magma_int_t  lddc,
magmaDoubleComplex_ptr  dT,
magma_int_t  nb,
magma_int_t *  info 
)

ZUNMQR_GPU overwrites the general complex M-by-N matrix C with.

                           SIDE = MagmaLeft    SIDE = MagmaRight
TRANS = MagmaNoTrans:      Q * C               C * Q
TRANS = Magma_ConjTrans:   Q**H * C            C * Q**H

where Q is a complex unitary matrix defined as the product of k elementary reflectors

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

as returned by ZGEQRF. Q is of order M if SIDE = MagmaLeft and of order N if SIDE = MagmaRight.

Parameters
[in]sidemagma_side_t
  • = MagmaLeft: apply Q or Q**H from the Left;
  • = MagmaRight: apply Q or Q**H from the Right.
[in]transmagma_trans_t
  • = MagmaNoTrans: No transpose, apply Q;
  • = Magma_ConjTrans: Conjugate transpose, apply Q**H.
[in]mINTEGER The number of rows of the matrix C. M >= 0.
[in]nINTEGER The number of columns of the matrix C. N >= 0.
[in]kINTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = MagmaLeft, M >= K >= 0; if SIDE = MagmaRight, N >= K >= 0.
[in]dACOMPLEX_16 array on the GPU, dimension (LDDA,K) The i-th column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZGEQRF in the first k columns of its array argument DA. DA is modified by the routine but restored on exit.
[in]lddaINTEGER The leading dimension of the array DA. If SIDE = MagmaLeft, LDDA >= max(1,M); if SIDE = MagmaRight, LDDA >= max(1,N).
[in,out]dCCOMPLEX_16 array on the GPU, dimension (LDDC,N) On entry, the M-by-N matrix C. On exit, C is overwritten by Q*C or Q**H * C or C * Q**H or C*Q.
[in]lddcINTEGER The leading dimension of the array DC. LDDC >= max(1,M).
[in]dTCOMPLEX_16 array on the GPU that is the output (the 9th argument) of magma_zgeqrf_gpu.
[in]nbINTEGER This is the blocking size that was used in pre-computing DT, e.g., the blocking size used in magma_zgeqrf_gpu.
[out]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value