ZGEBRD(1) LAPACK routine (version 3.2) ZGEBRD(1)NAME
ZGEBRD - reduces 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 )
INTEGER INFO, LDA, LWORK, M, N
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
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) INTEGER
The number of rows in the matrix A. M >= 0.
N (input) INTEGER
The number of columns in the matrix A. N >= 0.
A (input/output) COMPLEX*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 overwrit‐
ten 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) INTEGER The leading dimension of the array A.
LDA >= max(1,M).
D (output) DOUBLE PRECISION array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B: D(i) =
A(i,i).
E (output) 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.
TAUQ (output) COMPLEX*16 array dimension (min(M,N))
The scalar factors of the elementary reflectors which represent
the unitary matrix Q. See Further Details. TAUP (output)
COMPLEX*16 array, dimension (min(M,N)) The scalar factors of
the elementary reflectors which represent the unitary matrix P.
See Further Details. WORK (workspace/output) COMPLEX*16
array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1)
returns the optimal LWORK.
LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,M,N). For opti‐
mum 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) 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 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 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).
LAPACK routine (version 3.2) November 2008 ZGEBRD(1)
