double precision
[SVD: computational]

Functions

magma_int_t magma_dgebrd (magma_int_t m, magma_int_t n, double *A, magma_int_t lda, double *d, double *e, double *tauq, double *taup, double *work, magma_int_t lwork, magma_int_t *info)
 DGEBRD 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.
magma_int_t magma_dormbr (magma_vect_t vect, magma_side_t side, magma_trans_t trans, magma_int_t m, magma_int_t n, magma_int_t k, double *A, magma_int_t lda, double *tau, double *C, magma_int_t ldc, double *work, magma_int_t lwork, magma_int_t *info)
 If VECT = MagmaQ, DORMBR 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.

Function Documentation

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

DGEBRD 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] m INTEGER The number of rows in the matrix A. M >= 0.
[in] n INTEGER The number of columns in the matrix A. N >= 0.
[in,out] A DOUBLE_PRECISION 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] lda INTEGER The leading dimension of the array A. LDA >= max(1,M).
[out] d double precision array, dimension (min(M,N)) The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
[out] e double precision 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] tauq DOUBLE_PRECISION array dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
[out] taup DOUBLE_PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
[out] work (workspace) DOUBLE_PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK[0] returns the optimal LWORK.
[in] lwork INTEGER 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] info INTEGER

  • = 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_dormbr ( magma_vect_t  vect,
magma_side_t  side,
magma_trans_t  trans,
magma_int_t  m,
magma_int_t  n,
magma_int_t  k,
double *  A,
magma_int_t  lda,
double *  tau,
double *  C,
magma_int_t  ldc,
double *  work,
magma_int_t  lwork,
magma_int_t *  info 
)

If VECT = MagmaQ, DORMBR 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, DORMBR 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 DGEBRD 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] vect magma_vect_t

  • = MagmaQ: apply Q or Q**H;
  • = MagmaP: apply P or P**H.
[in] side magma_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] trans magma_trans_t

  • = MagmaNoTrans: No transpose, apply Q or P;
  • = MagmaTrans: Conjugate transpose, apply Q**H or P**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 If VECT = MagmaQ, the number of columns in the original matrix reduced by DGEBRD. If VECT = MagmaP, the number of rows in the original matrix reduced by DGEBRD. K >= 0.
[in] A DOUBLE_PRECISION 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 DGEBRD.
[in] lda INTEGER The leading dimension of the array A. If VECT = MagmaQ, LDA >= max(1,nq); if VECT = MagmaP, LDA >= max(1,min(nq,K)).
[in] tau DOUBLE_PRECISION 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 DGEBRD in the array argument TAUQ or TAUP.
[in,out] C DOUBLE_PRECISION 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] ldc INTEGER The leading dimension of the array C. LDC >= max(1,M).
[out] work (workspace) DOUBLE_PRECISION 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. 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] info INTEGER

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value

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