single precision
[Symmetric eigenvalue: computational, 2-stage]

Functions

magma_int_t magma_sorgqr_2stage_gpu (magma_int_t m, magma_int_t n, magma_int_t k, magmaFloat_ptr dA, magma_int_t ldda, float *tau, magmaFloat_ptr dT, magma_int_t nb, magma_int_t *info)
 SORGQR generates an M-by-N REAL matrix Q with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M.
magma_int_t magma_sormqr_gpu_2stages (magma_side_t side, magma_trans_t trans, magma_int_t m, magma_int_t n, magma_int_t k, magmaFloat_ptr dA, magma_int_t ldda, magmaFloat_ptr dC, magma_int_t lddc, magmaFloat_ptr dT, magma_int_t nb, magma_int_t *info)
 SORMQR_GPU overwrites the general real M-by-N matrix C with.
magma_int_t magma_ssytrd_sb2st (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, magma_int_t Vblksiz, float *A, magma_int_t lda, float *d, float *e, float *V, magma_int_t ldv, float *TAU, magma_int_t compT, float *T, magma_int_t ldt)
magma_int_t magma_ssytrd_sy2sb (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, float *A, magma_int_t lda, float *tau, float *work, magma_int_t lwork, magmaFloat_ptr dT, magma_int_t *info)
 SSYTRD_HE2HB reduces a real symmetric matrix A to real symmetric band-diagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.
magma_int_t magma_ssytrd_sy2sb_mgpu (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, float *A, magma_int_t lda, float *tau, float *work, magma_int_t lwork, magmaFloat_ptr dAmgpu[], magma_int_t ldda, magmaFloat_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)
 SSYTRD_HE2HB reduces a real symmetric matrix A to real symmetric band-diagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.
magma_int_t magma_ssytrd_sy2sb_mgpu_spec (magma_uplo_t uplo, magma_int_t n, magma_int_t nb, float *A, magma_int_t lda, float *tau, float *work, magma_int_t lwork, magmaFloat_ptr dAmgpu[], magma_int_t ldda, magmaFloat_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)
 SSYTRD_HE2HB reduces a real symmetric matrix A to real symmetric band-diagonal form T by an orthogonal similarity transformation: Q**H * A * Q = T.

Function Documentation

magma_int_t magma_sorgqr_2stage_gpu ( magma_int_t  m,
magma_int_t  n,
magma_int_t  k,
magmaFloat_ptr  dA,
magma_int_t  ldda,
float *  tau,
magmaFloat_ptr  dT,
magma_int_t  nb,
magma_int_t *  info 
)

SORGQR generates an M-by-N REAL 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 SGEQRF_GPU.

Parameters:
[in] m INTEGER The number of rows of the matrix Q. M >= 0.
[in] n INTEGER The number of columns of the matrix Q. M >= N >= 0.
[in] k INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0.
[in,out] dA REAL 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 SGEQRF_GPU in the first k columns of its array argument A. On exit, the M-by-N matrix Q.
[in] ldda INTEGER The first dimension of the array A. LDDA >= max(1,M).
[in] tau REAL array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by SGEQRF_GPU.
[in] dT REAL work space array on the GPU device, dimension (MIN(M, N) )*NB. This must be the 6th argument of magma_sgeqrf_gpu [ note that if N here is bigger than N in magma_sgeqrf_gpu, the workspace requirement DT in magma_sgeqrf_gpu must be as specified in this routine ].
[in] nb INTEGER This is the block size used in SGEQRF_GPU, and correspondingly the size of the T matrices, used in the factorization, and stored in DT.
[out] info INTEGER

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument has an illegal value
magma_int_t magma_sormqr_gpu_2stages ( magma_side_t  side,
magma_trans_t  trans,
magma_int_t  m,
magma_int_t  n,
magma_int_t  k,
magmaFloat_ptr  dA,
magma_int_t  ldda,
magmaFloat_ptr  dC,
magma_int_t  lddc,
magmaFloat_ptr  dT,
magma_int_t  nb,
magma_int_t *  info 
)

SORMQR_GPU overwrites the general real M-by-N matrix C with.

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

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

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

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

Parameters:
[in] side magma_side_t

  • = MagmaLeft: apply Q or Q**H from the Left;
  • = MagmaRight: apply Q or Q**H from the Right.
[in] trans magma_trans_t

  • = MagmaNoTrans: No transpose, apply Q;
  • = MagmaTrans: Conjugate transpose, apply Q**H.
[in] m INTEGER The number of rows of the matrix C. M >= 0.
[in] n INTEGER The number of columns of the matrix C. N >= 0.
[in] k INTEGER The number of elementary reflectors whose product defines the matrix Q. If SIDE = MagmaLeft, M >= K >= 0; if SIDE = MagmaRight, N >= K >= 0.
[in] dA REAL 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 SGEQRF in the first k columns of its array argument DA. DA is modified by the routine but restored on exit.
[in] ldda INTEGER The leading dimension of the array DA. If SIDE = MagmaLeft, LDDA >= max(1,M); if SIDE = MagmaRight, LDDA >= max(1,N).
[in,out] dC REAL 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] lddc INTEGER The leading dimension of the array DC. LDDC >= max(1,M).
[in] dT REAL array on the GPU that is the output (the 9th argument) of magma_sgeqrf_gpu.
[in] nb INTEGER This is the blocking size that was used in pre-computing DT, e.g., the blocking size used in magma_sgeqrf_gpu.
[out] info INTEGER

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
magma_int_t magma_ssytrd_sb2st ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
magma_int_t  Vblksiz,
float *  A,
magma_int_t  lda,
float *  d,
float *  e,
float *  V,
magma_int_t  ldv,
float *  TAU,
magma_int_t  compT,
float *  T,
magma_int_t  ldt 
)
Parameters:
[in] uplo magma_uplo_t

  • = MagmaUpper: Upper triangles of A is stored;
  • = MagmaLower: Lower triangles of A is stored.
[in] n INTEGER The order of the matrix A. N >= 0.
[in] nb INTEGER The order of the band matrix A. N >= NB >= 0.
[in] Vblksiz INTEGER The size of the block of householder vectors applied at once.
[in] A (workspace) REAL array, dimension (LDA, N) On entry the band matrix stored in the following way:
[in] lda INTEGER The leading dimension of the array A. LDA >= 2*NB.
[out] d DOUBLE array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
[out] e DOUBLE 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] V REAL 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] ldv INTEGER The leading dimension of V. LDV > NB + VBLKSIZ + 1
[out] TAU REAL dimension(BLKCNT, VBLKSIZ) ???
[in] compT INTEGER if COMPT = 0 T is not computed if COMPT = 1 T is computed
[out] T REAL dimension(LDT *) if COMPT = 1 on exit contains the matrices T needed for Q2 if COMPT = 0 T is not referenced
[in] ldt INTEGER The leading dimension of T. LDT > Vblksiz
magma_int_t magma_ssytrd_sy2sb ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
float *  A,
magma_int_t  lda,
float *  tau,
float *  work,
magma_int_t  lwork,
magmaFloat_ptr  dT,
magma_int_t *  info 
)

SSYTRD_HE2HB reduces a real symmetric 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] uplo magma_uplo_t

  • = MagmaUpper: Upper triangle of A is stored;
  • = MagmaLower: Lower triangle of A is stored.
[in] n INTEGER The order of the matrix A. N >= 0.
[in,out] A REAL 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 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] lda INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out] tau REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out] work (workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in] lwork INTEGER 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] dT REAL 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] info INTEGER

  • = 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 real scalar, and v is a real 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 real scalar, and v is a real 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_ssytrd_sy2sb_mgpu ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
float *  A,
magma_int_t  lda,
float *  tau,
float *  work,
magma_int_t  lwork,
magmaFloat_ptr  dAmgpu[],
magma_int_t  ldda,
magmaFloat_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 
)

SSYTRD_HE2HB reduces a real symmetric 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] uplo magma_uplo_t

  • = MagmaUpper: Upper triangle of A is stored;
  • = MagmaLower: Lower triangle of A is stored.
[in] n INTEGER The order of the matrix A. N >= 0.
[in,out] A REAL 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 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] lda INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out] tau REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out] work (workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in] lwork INTEGER 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] dT REAL 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] info INTEGER

  • = 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 real scalar, and v is a real 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 real scalar, and v is a real 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_ssytrd_sy2sb_mgpu_spec ( magma_uplo_t  uplo,
magma_int_t  n,
magma_int_t  nb,
float *  A,
magma_int_t  lda,
float *  tau,
float *  work,
magma_int_t  lwork,
magmaFloat_ptr  dAmgpu[],
magma_int_t  ldda,
magmaFloat_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 
)

SSYTRD_HE2HB reduces a real symmetric 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] uplo magma_uplo_t

  • = MagmaUpper: Upper triangle of A is stored;
  • = MagmaLower: Lower triangle of A is stored.
[in] n INTEGER The order of the matrix A. N >= 0.
[in,out] A REAL 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 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] lda INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out] tau REAL array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).
[out] work (workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in] lwork INTEGER 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] dT REAL 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] info INTEGER

  • = 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 real scalar, and v is a real 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 real scalar, and v is a real 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).


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