DGEBRD(1) LAPACK routine (version 3.2) DGEBRD(1)NAME
DGEBRD - reduces a general real M-by-N matrix A to upper or lower bidi‐
agonal form B by an orthogonal transformation
SYNOPSIS
SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )
INTEGER INFO, LDA, LWORK, M, N
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
TAUQ( * ), WORK( * )
PURPOSE
DGEBRD 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) DOUBLE PRECISION 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) 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) DOUBLE PRECISION array dimension (min(M,N))
The scalar factors of the elementary reflectors which represent
the orthogonal matrix Q. See Further Details. TAUP (output)
DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors
of the elementary reflectors which represent the orthogonal
matrix P. See Further Details. WORK (workspace/output) DOU‐
BLE PRECISION array, dimension (MAX(1,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 routine (version 3.2) November 2008 DGEBRD(1)
