## SYNOPSIS

- SUBROUTINE DLABRD(
- M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )

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

- DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), X( LDX, * ), Y( LDY, * )

## PURPOSE

DLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal 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 DGEBRD

## ARGUMENTS

- 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) DOUBLE PRECISION 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 orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal 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 orthogonal 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 orthogonal 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) DOUBLE PRECISION array dimension (NB)
- The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details. TAUP (output) DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details. X (output) DOUBLE PRECISION 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 >= M.
- Y (output) DOUBLE PRECISION array, dimension (LDY,NB)
- The n-by-nb matrix Y required to update the unreduced part of A.
- LDY (input) INTEGER
- The leading dimension of the array Y. LDY >= N.

## FURTHER DETAILS

The matrices Q and P are represented as products of elementary reflectors:Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) 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 real scalars, and v and u are real 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).