sgebrd(3)

NAME

SGEBRD - reduce a general real M-by-N matrix A to upper or

lower bidiagonal form B by an orthogonal transformation

SYNOPSIS

SUBROUTINE  SGEBRD(  M, N, A, LDA, D, E, TAUQ, TAUP, WORK,
LWORK, INFO )
    INTEGER        INFO, LDA, LWORK, M, N
    REAL           A( LDA, * ), D( * ), E( * ), TAUP( * ),
TAUQ( * ), WORK( * )

PURPOSE

SGEBRD reduces a general real M-by-N matrix A to upper or

lower bidiagonal form B by an orthogonal transformation: Q**T * 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) REAL 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 be

low the diagonal, with the array TAUQ, represent the orthogonal

matrix Q as a product of elementary reflectors, and the elements

above the first superdiagonal, with the array TAUP, represent the

orthogonal 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 sub

diagonal, with the array TAUQ, represent the orthogonal matrix Q

as a product of elementary reflectors, and the elements above the

diagonal, 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) REAL array, dimension (min(M,N))

The diagonal elements of the bidiagonal matrix B:

D(i) = A(i,i).

E (output) REAL 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) REAL array dimension (min(M,N))

The scalar factors of the elementary reflectors

which represent the orthogonal matrix Q. See Further Details.

TAUP (output) REAL array, dimension (min(M,N)) The scalar fac

tors of the elementary reflectors which represent the orthogonal

matrix P. See Further Details. WORK (workspace/output) REAL

array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns

the optimal LWORK.

LWORK (input) INTEGER

The length of the array WORK. LWORK >=

max(1,M,N). For optimum performance LWORK >= (M+N)*NB, where NB

is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed;

the routine only calculates the optimal size of the WORK array,

returns this value as the first entry of the WORK array, and no

error message related to LWORK is issued by XERBLA.

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value.

FURTHER DETAILS

The matrices Q and P are represented as products of ele

mentary 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 real scalars, and v and u are real

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 real scalars, and v and u are real

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).

LAPACK version 3.0 15 June 2000

SGEBRD(3)