NAG Library Routine Document
F08KSF (ZGEBRD)
1 Purpose
F08KSF (ZGEBRD) reduces a complex m by n matrix to bidiagonal form.
2 Specification
SUBROUTINE F08KSF ( |
M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) |
INTEGER |
M, N, LDA, LWORK, INFO |
REAL (KIND=nag_wp) |
D(*), E(*) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), TAUQ(*), TAUP(*), WORK(max(1,LWORK)) |
|
The routine may be called by its
LAPACK
name zgebrd.
3 Description
F08KSF (ZGEBRD) reduces a complex m by n matrix A to real bidiagonal form B by a unitary transformation: A=QBPH, where Q and PH are unitary matrices of order m and n respectively.
If
m≥n, the reduction is given by:
where
B1 is a real
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 a real
m by
m lower bidiagonal matrix and
P1H consists of the first
m rows of
PH.
The unitary 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,*) – COMPLEX (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 unitary matrix
Q and elements above the first superdiagonal are overwritten by details of the unitary 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 unitary matrix Q and elements above the diagonal are overwritten by details of the unitary matrix P.
- 4: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08KSF (ZGEBRD) 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(*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
TAUQ
must be at least
max1,minM,N.
On exit: further details of the unitary matrix Q.
- 8: TAUP(*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
TAUP
must be at least
max1,minM,N.
On exit: further details of the unitary matrix P.
- 9: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
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 F08KSF (ZGEBRD) 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
QBPH=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 real floating point operations is approximately 16n23m-n/3 if m≥n or 16m23n-m/3 if m<n.
If
m≫n, it can be more efficient to first call
F08ASF (ZGEQRF) to perform a
QR factorization of
A, and then to call F08KSF (ZGEBRD) to reduce the factor
R to bidiagonal form. This requires approximately
8n2m+n floating point operations.
If
m≪n, it can be more efficient to first call
F08AVF (ZGELQF) to perform an
LQ factorization of
A, and then to call F08KSF (ZGEBRD) to reduce the factor
L to bidiagonal form. This requires approximately
8m2m+n operations.
To form the unitary matrices
PH and/or
Q F08KSF (ZGEBRD) may be followed by calls to
F08KTF (ZUNGBR):
to form the
m by
m unitary matrix
Q
CALL ZUNGBR('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 F08KSF (ZGEBRD);
to form the
n by
n unitary matrix
PH
CALL ZUNGBR('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 F08KSF (ZGEBRD).
To apply
Q or
P to a complex rectangular matrix
C, F08KSF (ZGEBRD) may be followed by a call to
F08KUF (ZUNMBR).
The real analogue of this routine is
F08KEF (DGEBRD).
9 Example
This example reduces the matrix
A to bidiagonal form, where
9.1 Program Text
Program Text (f08ksfe.f90)
9.2 Program Data
Program Data (f08ksfe.d)
9.3 Program Results
Program Results (f08ksfe.r)