ZLABRD - reduce the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A

SUBROUTINE ZLABRD(

M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )

INTEGER LDA, LDX, LDY, M, N, NB

DOUBLE PRECISION D( * ), E( * )

COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ), Y( LDY, * )

ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form.

This is an auxiliary routine called by ZGEBRD

M (input) INTEGER

The number of rows in the matrix A.

N (input) INTEGER

The number of columns in the matrix A.

NB (input) INTEGER

The number of leading rows and columns of A to be reduced.

A (input/output) COMPLEX*16 array, dimension (LDA,N)

On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, 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 (NB)

The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).

E (output) DOUBLE PRECISION array, dimension (NB)

The off-diagonal elements of the first NB rows and columns of the reduced matrix.

TAUQ (output) COMPLEX*16 array dimension (NB)

The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details. TAUP (output) COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details. X (output) COMPLEX*16 array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A.

LDX (input) INTEGER

The leading dimension of the array X. LDX >= max(1,M).

Y (output) COMPLEX*16 array, dimension (LDY,NB)

The n-by-nb matrix Y required to update the unreduced part of A.

LDY (output) INTEGER

The leading dimension of the array Y. LDY >= max(1,N).

The matrices Q and P are represented as products of elementary reflectors:

Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n)

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.

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 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).

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U' which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y' - X*U'.

The contents of A on exit are illustrated by the following examples with nb = 2:

m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):

( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) ( v1 v2 a a a ) ( v1 1 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a )

where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).