Contents
zgebrd - reduce a general complex M-by-N matrix A to upper
or lower bidiagonal form B by a unitary transformation
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, doub-
lecomplex *taup, long *info);
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 bidi-
agonal.
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 bidi-
agonal 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 uni-
tary matrix P as a product of elementary reflec-
tors. 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 ille-
gal value.
The matrices Q and P are represented as products of elemen-
tary 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).