---
name: "zgebd2"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/zgebd2/3lapack"
---

# zgebd2(3lapack)

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

## NAME

ZGEBD2 - reduces 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).\