SGEBRD(l) ) SGEBRD(l)NAME
SGEBRD - reduce a general real M-by-N matrix A to upper or lower bidi‐
agonal 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 bidiag‐
onal 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 overwrit‐
ten 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 subdiagonal, with the array TAUQ, represent the orthogo‐
nal 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 factors of the ele‐
mentary reflectors which represent the orthogonal matrix P. See
Further Details. WORK (workspace/output) REAL array, dimen‐
sion (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 opti‐
mum 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 illegal value.
FURTHER DETAILS
The matrices Q and P are represented as products of elementary reflec‐
tors:
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(l)
