CLABRD - 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 CLABRD(

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

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

REAL D( * ), E( * )

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

CLABRD 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 CGEBRD

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 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) REAL array, dimension (NB)

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

E (output) REAL array, dimension (NB)

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

TAUQ (output) COMPLEX array dimension (NB)

The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details. TAUP (output) COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details. X (output) COMPLEX 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 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).