---
name: "zlabrd"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/zlabrd/3lapack"
---

# zlabrd(3lapack)

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

## NAME

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

## SYNOPSIS

- 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, \* )

## PURPOSE

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\

## 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) 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 (input) INTEGER

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