NAG Library Routine Document
F08KUF (ZUNMBR)
1 Purpose
F08KUF (ZUNMBR) multiplies an arbitrary complex
by
matrix
by one of the complex unitary matrices
or
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
to real bidiagonal form
by a unitary transformation:
.
F08KSF (ZGEBRD) represents the matrices
and
as products of elementary reflectors.
This routine may be used to form one of the matrix products
overwriting the result on
(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, denotes the order of or : if , and if , .
- 1: VECT – CHARACTER(1)Input
On entry: indicates whether
or
or
or
is to be applied to
.
- or is applied to .
- or is applied to .
Constraint:
or .
- 2: SIDE – CHARACTER(1)Input
On entry: indicates how
or
or
or
is to be applied to
.
- or or or is applied to from the left.
- or or or is applied to from the right.
Constraint:
or .
- 3: TRANS – CHARACTER(1)Input
On entry: indicates whether
or
or
or
is to be applied to
.
- or is applied to .
- or is applied to .
Constraint:
or .
- 4: M – INTEGERInput
On entry: , the number of rows of the matrix .
Constraint:
.
- 5: N – INTEGERInput
On entry: , the number of columns of the matrix .
Constraint:
.
- 6: K – INTEGERInput
On entry: if
, the number of columns in the original matrix
.
If , the number of rows in the original matrix .
Constraint:
.
- 7: A(LDA,) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
A
must be at least
if
and at least
if
.
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 , ;
- if , .
- 9: TAU() – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
TAU
must be at least
.
On entry: further details of the elementary reflectors, as returned by
F08KSF (ZGEBRD) in its parameter
TAUQ if
, or in its parameter
TAUP if
.
- 10: C(LDC,) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
C
must be at least
.
On entry: the matrix .
On exit:
C is overwritten by
or
or
or
or
or
or
or
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:
.
- 12: WORK() – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
, the real part of
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
, 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, if and at least if , where is the optimal block size.
Constraints:
- if , or ;
- if , or .
- 14: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , argument 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
such that
where
is the
machine precision.
The total number of real floating point operations is approximately
- if and , ;
- if and , ;
- if and , ;
- if and , ,
where
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 may be preceded by a or factorization of .
In the first example,
, and
The routine first performs a
factorization of
as
and then reduces the factor
to bidiagonal form
:
. Finally it forms
and calls F08KUF (ZUNMBR) to form
.
In the second example,
, and
The routine first performs an
factorization of
as
and then reduces the factor
to bidiagonal form
:
. Finally it forms
and calls F08KUF (ZUNMBR) to form
.
9.1 Program Text
Program Text (f08kufe.f90)
9.2 Program Data
Program Data (f08kufe.d)
9.3 Program Results
Program Results (f08kufe.r)