
SGEBRD(l) ) SGEBRD(l)
NAME
SGEBRD  reduce a general real MbyN 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 MbyN 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 MbyN general matrix to be reduced. On exit, if m >= n, the diago
nal and the first superdiagonal are overwritten with the upper bidiagonal matrix
B; the elements below 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 Fur
ther 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 offdiagonal elements of the bidiagonal matrix B: if m >= n, E(i) = A(i,i+1)
for i = 1,2,...,n1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m1.
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 factors 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 ith 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(n1)
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:i1) = 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(m1) 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:i1) = 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 offdiagonal 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(l) 
