zgebrd(3P) Sun Performance Library zgebrd(3P)NAMEzgebrd - 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);
PURPOSEzgebrd 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 over‐
written with the upper bidiagonal matrix B; the elements
below the diagonal, with the array TAUQ, represent the uni‐
tary 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 ele‐
ments below the first subdiagonal, with the array TAUQ, rep‐
resent 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 repre‐
sent the unitary matrix Q. See Further Details.
TAUP (output)
The scalar factors of the elementary reflectors which repre‐
sent 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 opti‐
mal 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 reflec‐
tors:
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 vec‐
tors; 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 vec‐
tors; 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).
6 Mar 2009 zgebrd(3P)
