NAG Library Routine Document
F08KUF (ZUNMBR)
1 Purpose
F08KUF (ZUNMBR) multiplies an arbitrary complex
m by
n matrix
C by one of the complex unitary matrices
Q or
P which were determined by
F08KSF (ZGEBRD) when reducing a complex matrix to bidiagonal form.
2 Specification
| SUBROUTINE F08KUF ( |
VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO) |
| INTEGER |
M, N, K, LDA, LDC, LWORK, INFO |
| COMPLEX (KIND=nag_wp) |
A(LDA,*), TAU(*), C(LDC,*), WORK(max(1,LWORK)) |
| CHARACTER(1) |
VECT, SIDE, TRANS |
|
The routine may be called by its
LAPACK
name zunmbr.
3 Description
F08KUF (ZUNMBR) is intended to be used after a call to
F08KSF (ZGEBRD), which reduces a complex rectangular matrix
A to real bidiagonal form
B by a unitary transformation:
A=QBPH.
F08KSF (ZGEBRD) represents the matrices
Q and
PH as products of elementary reflectors.
This routine may be used to form one of the matrix products
overwriting the result on
C (which may be any complex rectangular matrix).
4 References
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
Note: in the descriptions below, r denotes the order of Q or PH: if SIDE='L', r=M and if SIDE='R', r=N.
- 1: VECT – CHARACTER(1)Input
-
On entry: indicates whether
Q or
QH or
P or
PH is to be applied to
C.
- VECT='Q'
- Q or
QH is applied to C.
- VECT='P'
- P or
PH is applied to C.
Constraint:
VECT='Q' or 'P'.
- 2: SIDE – CHARACTER(1)Input
-
On entry: indicates how
Q or
QH or
P or
PH is to be applied to
C.
- SIDE='L'
- Q or
QH or P or
PH is applied to C from the left.
- SIDE='R'
- Q or
QH or P or
PH is applied to C from the right.
Constraint:
SIDE='L' or 'R'.
- 3: TRANS – CHARACTER(1)Input
-
On entry: indicates whether
Q or
P or
QH or
PH is to be applied to
C.
- TRANS='N'
- Q or P is applied to C.
- TRANS='C'
- QH or PH is applied to C.
Constraint:
TRANS='N' or 'C'.
- 4: M – INTEGERInput
-
On entry: m, the number of rows of the matrix C.
Constraint:
M≥0.
- 5: N – INTEGERInput
-
On entry: n, the number of columns of the matrix C.
Constraint:
N≥0.
- 6: K – INTEGERInput
-
On entry: if
VECT='Q', the number of columns in the original matrix
A.
If VECT='P', the number of rows in the original matrix A.
Constraint:
K≥0.
- 7: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
A
must be at least
max1,minr,K if
VECT='Q' and at least
max1,r if
VECT='P'.
On entry: details of the vectors which define the elementary reflectors, as returned by
F08KSF (ZGEBRD).
- 8: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which F08KUF (ZUNMBR) is called.
Constraints:
- if VECT='Q', LDA≥max1,r;
- if VECT='P', LDA≥max1,minr,K
.
- 9: TAU(*) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
TAU
must be at least
max1,minr,K.
On entry: further details of the elementary reflectors, as returned by
F08KSF (ZGEBRD) in its parameter
TAUQ if
VECT='Q', or in its parameter
TAUP if
VECT='P'.
- 10: C(LDC,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
C
must be at least
max1,N.
On entry: the matrix C.
On exit:
C is overwritten by
QC or
QHC or
CQ or
CHQ or
PC or
PHC or
CP or
CHP as specified by
VECT,
SIDE and
TRANS.
- 11: LDC – INTEGERInput
-
On entry: the first dimension of the array
C as declared in the (sub)program from which F08KUF (ZUNMBR) is called.
Constraint:
LDC≥max1,M.
- 12: 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.
- 13: LWORK – INTEGERInput
-
On entry:
the dimension of the array
WORK as declared in the (sub)program from which F08KUF (ZUNMBR) 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≥N×nb if SIDE='L' and at least M×nb if SIDE='R', where nb is the optimal block size.
Constraints:
- if SIDE='L', LWORK≥max1,N or LWORK=-1;
- if SIDE='R', LWORK≥max1,M or LWORK=-1.
- 14: 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 result differs from the exact result by a matrix
E such that
where
ε is the
machine precision.
8 Further Comments
The total number of real floating point operations is approximately
- if SIDE='L' and m≥k, 8nk2m-k;
- if SIDE='R' and n≥k, 8mk2n-k;
- if SIDE='L' and m<k, 8m2n;
- if SIDE='R' and n<k, 8mn2,
where
k is the value of the parameter
K.
The real analogue of this routine is
F08KGF (DORMBR).
9 Example
For this routine two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix A may be preceded by a QR or LQ factorization of A.
In the first example,
m>n, and
The routine first performs a
QR factorization of
A as
A=QaR and then reduces the factor
R to bidiagonal form
B:
R=QbBPH. Finally it forms
Qa and calls F08KUF (ZUNMBR) to form
Q=QaQb.
In the second example,
m<n, and
The routine first performs an
LQ factorization of
A as
A=LPaH and then reduces the factor
L to bidiagonal form
B:
L=QBPbH. Finally it forms
PbH and calls F08KUF (ZUNMBR) to form
PH=PbHPaH.
9.1 Program Text
Program Text (f08kufe.f90)
9.2 Program Data
Program Data (f08kufe.d)
9.3 Program Results
Program Results (f08kufe.r)