ZGEBD2(l) — LAPACK routine (version 2.0)

NAME

ZGEBD2 - reduce a complex general m by n matrix A to upper or lower real bidiagonal form B by a unitary transformation

SYNOPSIS

SUBROUTINE ZGEBD2(
M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )

INTEGER INFO, LDA, M, N

DOUBLE PRECISION D( ∗ ), E( ∗ )

COMPLEX∗16 A( LDA, ∗ ), TAUP( ∗ ), TAUQ( ∗ ), WORK( ∗ )

PURPOSE

ZGEBD2 reduces a complex general m by n matrix A to upper or lower real bidiagonal form B by a unitary transformation: Q’ ∗ 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 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) 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) COMPLEX∗16 array, dimension (max(M,N))

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 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, 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).