NAG Library Routine Document
F08KEF (DGEBRD)
1 Purpose
F08KEF (DGEBRD) reduces a real m by n matrix to bidiagonal form.
2 Specification
| SUBROUTINE F08KEF ( |
M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) |
| INTEGER |
M, N, LDA, LWORK, INFO |
| REAL (KIND=nag_wp) |
A(LDA,*), D(*), E(*), TAUQ(*), TAUP(*), WORK(max(1,LWORK)) |
|
The routine may be called by its
LAPACK
name dgebrd.
3 Description
F08KEF (DGEBRD) reduces a real m by n matrix A to bidiagonal form B by an orthogonal transformation: A=QBPT, where Q and PT are orthogonal matrices of order m and n respectively.
If
m≥n, the reduction is given by:
where
B1 is an
n by
n upper bidiagonal matrix and
Q1 consists of the first
n columns of
Q.
If
m<n, the reduction is given by
where
B1 is an
m by
m lower bidiagonal matrix and
P1T consists of the first
m rows of
PT.
The orthogonal matrices
Q and
P are not formed explicitly but are represented as products of elementary reflectors (see the
F08 Chapter Introduction for details). Routines are provided to work with
Q and
P in this representation (see
Section 8).
4 References
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: M – INTEGERInput
-
On entry:
m, the number of rows of the matrix A.
Constraint:
M≥0.
- 2: N – INTEGERInput
-
On entry:
n, the number of columns of the matrix A.
Constraint:
N≥0.
- 3: A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the m by n matrix A.
On exit: if
m≥n, the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix
B, elements below the diagonal are overwritten by details of the
orthogonal
matrix
Q and elements above the first superdiagonal are overwritten by details of the
orthogonal
matrix
P.
If m<n, the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix B, elements below the first subdiagonal are overwritten by details of the
orthogonal
matrix Q and elements above the diagonal are overwritten by details of the
orthogonal
matrix P.
- 4: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which F08KEF (DGEBRD) is called.
Constraint:
LDA≥max1,M.
- 5: D(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
D
must be at least
max1,minM,N.
On exit: the diagonal elements of the bidiagonal matrix B.
- 6: E(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
E
must be at least
max1,minM,N-1.
On exit: the off-diagonal elements of the bidiagonal matrix B.
- 7: TAUQ(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
TAUQ
must be at least
max1,minM,N.
On exit: further details of the matrix Q.
- 8: TAUP(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
TAUP
must be at least
max1,minM,N.
On exit: further details of the matrix P.
- 9: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 10: LWORK – INTEGERInput
-
On entry:
the dimension of the array
WORK as declared in the (sub)program from which F08KEF (DGEBRD) is called.
If
LWORK=-1, 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.
Suggested value:
for optimal performance, LWORK≥M+N×nb, where nb is the optimal block size.
Constraint:
LWORK≥max1,M,N or LWORK=-1.
- 11: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
7 Accuracy
The computed bidiagonal form
B satisfies
QBPT=A+E, where
cn is a modestly increasing function of
n, and
ε is the
machine precision.
The elements of B themselves may be sensitive to small perturbations in A or to rounding errors in the computation, but this does not affect the stability of the singular values and vectors.
8 Further Comments
The total number of floating point operations is approximately 43n23m-n if m≥n or 43m23n-m if m<n.
If
m≫n, it can be more efficient to first call
F08AEF (DGEQRF) to perform a
QR factorization of
A, and then to call F08KEF (DGEBRD) to reduce the factor
R to bidiagonal form. This requires approximately
2n2m+n floating point operations.
If
m≪n, it can be more efficient to first call
F08AHF (DGELQF) to perform an
LQ factorization of
A, and then to call F08KEF (DGEBRD) to reduce the factor
L to bidiagonal form. This requires approximately
2m2m+n operations.
To form the orthogonal matrices
PT and/or
Q F08KEF (DGEBRD) may be followed by calls to
F08KFF (DORGBR):
to form the
m by
m orthogonal matrix
Q
CALL DORGBR('Q',M,M,N,A,LDA,TAUQ,WORK,LWORK,INFO)
but note that the second dimension of the array
A must be at least
M, which may be larger than was required by F08KEF (DGEBRD);
to form the
n by
n orthogonal matrix
PT
CALL DORGBR('P',N,N,M,A,LDA,TAUP,WORK,LWORK,INFO)
but note that the first dimension of the array
A, specified by the parameter
LDA, must be at least
N, which may be larger than was required by F08KEF (DGEBRD).
To apply
Q or
P to a real rectangular matrix
C, F08KEF (DGEBRD) may be followed by a call to
F08KGF (DORMBR).
The complex analogue of this routine is
F08KSF (ZGEBRD).
9 Example
This example reduces the matrix
A to bidiagonal form, where
9.1 Program Text
Program Text (f08kefe.f90)
9.2 Program Data
Program Data (f08kefe.d)
9.3 Program Results
Program Results (f08kefe.r)