MAGMA  1.6.3
Matrix Algebra for GPU and Multicore Architectures
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double-complex precision

Functions

magma_int_t magma_zgebrd (magma_int_t m, magma_int_t n, magmaDoubleComplex *A, magma_int_t lda, double *d, double *e, magmaDoubleComplex *tauq, magmaDoubleComplex *taup, magmaDoubleComplex *work, magma_int_t lwork, magma_int_t *info)
 ZGEBRD reduces a general complex 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_zunmbr (magma_vect_t vect, magma_side_t side, magma_trans_t trans, magma_int_t m, magma_int_t n, magma_int_t k, magmaDoubleComplex *A, magma_int_t lda, magmaDoubleComplex *tau, magmaDoubleComplex *C, magma_int_t ldc, magmaDoubleComplex *work, magma_int_t lwork, magma_int_t *info)
 ZUNMBR multiplies by Q or P as part of the SVD decomposition. More...
 

Detailed Description

Function Documentation

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

ZGEBRD reduces a general complex 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]ACOMPLEX_16 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]ddouble precision array, dimension (min(M,N)) The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
[out]edouble 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]tauqCOMPLEX_16 array dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
[out]taupCOMPLEX_16 array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. 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 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 complex scalars, and v and u are complex 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 complex scalars, and v and u are complex 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_zunmbr ( magma_vect_t  vect,
magma_side_t  side,
magma_trans_t  trans,
magma_int_t  m,
magma_int_t  n,
magma_int_t  k,
magmaDoubleComplex *  A,
magma_int_t  lda,
magmaDoubleComplex *  tau,
magmaDoubleComplex *  C,
magma_int_t  ldc,
magmaDoubleComplex *  work,
magma_int_t  lwork,
magma_int_t *  info 
)

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

If VECT = MagmaQ, ZUNMBR 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

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

Here Q and P**H are the unitary matrices determined by ZGEBRD when reducing A complex 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;
  • = Magma_ConjTrans: 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 ZGEBRD. If VECT = MagmaP, the number of rows in the original matrix reduced by ZGEBRD. K >= 0.
[in]ACOMPLEX_16 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 ZGEBRD.
[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]tauCOMPLEX_16 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 ZGEBRD in the array argument TAUQ or TAUP.
[in,out]CCOMPLEX_16 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) 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. 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