MAGMA  1.7.0
Matrix Algebra for GPU and Multicore Architectures
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single precision

Functions

magma_int_t magma_sgebrd (magma_int_t m, magma_int_t n, float *A, magma_int_t lda, float *d, float *e, float *tauq, float *taup, float *work, magma_int_t lwork, magma_int_t *info)
 SGEBRD reduces a general real M-by-N matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**H * A * P = B. More...
 
magma_int_t magma_sormbr (magma_vect_t vect, magma_side_t side, magma_trans_t trans, magma_int_t m, magma_int_t n, magma_int_t k, float *A, magma_int_t lda, float *tau, float *C, magma_int_t ldc, float *work, magma_int_t lwork, magma_int_t *info)
 SORMBR multiplies by Q or P as part of the SVD decomposition. More...
 

Detailed Description

Function Documentation

magma_int_t magma_sgebrd ( magma_int_t  m,
magma_int_t  n,
float *  A,
magma_int_t  lda,
float *  d,
float *  e,
float *  tauq,
float *  taup,
float *  work,
magma_int_t  lwork,
magma_int_t *  info 
)

SGEBRD reduces a general real M-by-N matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**H * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

Parameters
[in]mINTEGER The number of rows in the matrix A. M >= 0.
[in]nINTEGER The number of columns in the matrix A. N >= 0.
[in,out]AREAL array, dimension (LDA,N) On entry, the M-by-N general matrix to be reduced. On exit, if m >= n, the diagonal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details.
[in]ldaINTEGER The leading dimension of the array A. LDA >= max(1,M).
[out]dreal array, dimension (min(M,N)) The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
[out]ereal array, dimension (min(M,N)-1) The off-diagonal elements of the bidiagonal matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
[out]tauqREAL array dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
[out]taupREAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
[out]work(workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The length of the array WORK. LWORK >= (M+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]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value.

Further Details

The matrices Q and P are represented as products of elementary reflectors:

If m >= n,

Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'

where tauq and taup are real scalars, and v and u are real vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n,

Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'

where tauq and taup are real scalars, and v and u are real vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The contents of A on exit are illustrated by the following examples:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

  (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
  (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
  (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
  (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
  (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
  (  v1  v2  v3  v4  v5 )

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

magma_int_t magma_sormbr ( magma_vect_t  vect,
magma_side_t  side,
magma_trans_t  trans,
magma_int_t  m,
magma_int_t  n,
magma_int_t  k,
float *  A,
magma_int_t  lda,
float *  tau,
float *  C,
magma_int_t  ldc,
float *  work,
magma_int_t  lwork,
magma_int_t *  info 
)

SORMBR multiplies by Q or P as part of the SVD decomposition.

If VECT = MagmaQ, SORMBR 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

If VECT = MagmaP, SORMBR overwrites the general real M-by-N matrix C with SIDE = MagmaLeft SIDE = MagmaRight TRANS = MagmaNoTrans: P*C C*P TRANS = MagmaTrans: P**H*C C*P**H

Here Q and P**H are the unitary matrices determined by SGEBRD when reducing A real matrix A to bidiagonal form: A = Q*B * P**H. Q and P**H are defined as products of elementary reflectors H(i) and G(i) respectively.

Let nq = m if SIDE = MagmaLeft and nq = n if SIDE = MagmaRight. Thus nq is the order of the unitary matrix Q or P**H that is applied.

If VECT = MagmaQ, A is assumed to have been an NQ-by-K matrix: if nq >= k, Q = H(1) H(2) . . . H(k); if nq < k, Q = H(1) H(2) . . . H(nq-1).

If VECT = MagmaP, A is assumed to have been A K-by-NQ matrix: if k < nq, P = G(1) G(2) . . . G(k); if k >= nq, P = G(1) G(2) . . . G(nq-1).

Parameters
[in]vectmagma_vect_t
  • = MagmaQ: apply Q or Q**H;
  • = MagmaP: apply P or P**H.
[in]sidemagma_side_t
  • = MagmaLeft: apply Q, Q**H, P or P**H from the Left;
  • = MagmaRight: apply Q, Q**H, P or P**H from the Right.
[in]transmagma_trans_t
  • = MagmaNoTrans: No transpose, apply Q or P;
  • = MagmaTrans: Conjugate transpose, apply Q**H or P**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 If VECT = MagmaQ, the number of columns in the original matrix reduced by SGEBRD. If VECT = MagmaP, the number of rows in the original matrix reduced by SGEBRD. K >= 0.
[in]AREAL array, dimension (LDA,min(nq,K)) if VECT = MagmaQ (LDA,nq) if VECT = MagmaP The vectors which define the elementary reflectors H(i) and G(i), whose products determine the matrices Q and P, as returned by SGEBRD.
[in]ldaINTEGER The leading dimension of the array A. If VECT = MagmaQ, LDA >= max(1,nq); if VECT = MagmaP, LDA >= max(1,min(nq,K)).
[in]tauREAL array, dimension (min(nq,K)) TAU(i) must contain the scalar factor of the elementary reflector H(i) or G(i) which determines Q or P, as returned by SGEBRD in the array argument TAUQ or TAUP.
[in,out]CREAL array, dimension (LDC,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 or P*C or P**H*C or C*P or C*P**H.
[in]ldcINTEGER The leading dimension of the array C. LDC >= max(1,M).
[out]work(workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in]lworkINTEGER The dimension of the array WORK. If SIDE = MagmaLeft, LWORK >= max(1,N); if SIDE = MagmaRight, LWORK >= max(1,M); if N = 0 or M = 0, LWORK >= 1. For optimum performance if SIDE = MagmaLeft, LWORK >= max(1,N*NB); if SIDE = MagmaRight, LWORK >= max(1,M*NB), where NB is the optimal blocksize. (NB = 0 if M = 0 or N = 0.)
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]infoINTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value