NAME

zgebrd - reduce a general complex M-by-N matrix A to upper or lower bidiagonal form B by a unitary transformation

SYNOPSIS

  SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, 
 *      INFO)
  DOUBLE COMPLEX A(LDA,*), TAUQ(*), TAUP(*), WORK(*)
  INTEGER M, N, LDA, LWORK, INFO
  DOUBLE PRECISION D(*), E(*)

  SUBROUTINE ZGEBRD_64( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, 
 *      INFO)
  DOUBLE COMPLEX A(LDA,*), TAUQ(*), TAUP(*), WORK(*)
  INTEGER*8 M, N, LDA, LWORK, INFO
  DOUBLE PRECISION D(*), E(*)

F95 INTERFACE

  SUBROUTINE GEBRD( [M], [N], A, [LDA], D, E, TAUQ, TAUP, [WORK], 
 *       [LWORK], [INFO])
  COMPLEX(8), DIMENSION(:) :: TAUQ, TAUP, WORK
  COMPLEX(8), DIMENSION(:,:) :: A
  INTEGER :: M, N, LDA, LWORK, INFO
  REAL(8), DIMENSION(:) :: D, E

  SUBROUTINE GEBRD_64( [M], [N], A, [LDA], D, E, TAUQ, TAUP, [WORK], 
 *       [LWORK], [INFO])
  COMPLEX(8), DIMENSION(:) :: TAUQ, TAUP, WORK
  COMPLEX(8), DIMENSION(:,:) :: A
  INTEGER(8) :: M, N, LDA, LWORK, INFO
  REAL(8), DIMENSION(:) :: D, E

C INTERFACE

#include <sunperf.h>

void zgebrd(int m, int n, doublecomplex *a, int lda, double *d, double *e, doublecomplex *tauq, doublecomplex *taup, int *info);

void zgebrd_64(long m, long n, doublecomplex *a, long lda, double *d, double *e, doublecomplex *tauq, doublecomplex *taup, long *info);

PURPOSE

zgebrd reduces a general complex M-by-N matrix A to upper or lower bidiagonal form B by a unitary transformation: Q**H * A * P = B.

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

ARGUMENTS

M (input)
The number of rows in the matrix A. M > = 0.
N (input)
The number of columns in the matrix A. N > = 0.
A (input/output)
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 unitary matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the unitary 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 unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. See Further Details.
LDA (input)
The leading dimension of the array A. LDA > = max(1,M).
D (output)
The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
E (output)
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.
TAUQ (output)
The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.
TAUP (output)
The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The length of the array WORK. LWORK > = max(1,M,N). For optimum performance 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.

INFO (output)

 = 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).