NAG Library Routine Document
F08KGF (DORMBR)
1 Purpose
F08KGF (DORMBR) multiplies an arbitrary real
by
matrix
by one of the real orthogonal matrices
or
which were determined by
F08KEF (DGEBRD) when reducing a real matrix to bidiagonal form.
2 Specification
SUBROUTINE F08KGF ( |
VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO) |
INTEGER |
M, N, K, LDA, LDC, LWORK, INFO |
REAL (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 dormbr.
3 Description
F08KGF (DORMBR) is intended to be used after a call to
F08KEF (DGEBRD), which reduces a real rectangular matrix
to bidiagonal form
by an orthogonal transformation:
.
F08KEF (DGEBRD) 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 real 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,) – REAL (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
F08KEF (DGEBRD).
- 8: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08KGF (DORMBR) is called.
Constraints:
- if , ;
- if , .
- 9: TAU() – REAL (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
F08KEF (DGEBRD) in its parameter
TAUQ if
, or in its parameter
TAUP if
.
- 10: C(LDC,) – REAL (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 F08KGF (DORMBR) is called.
Constraint:
.
- 12: WORK() – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
,
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 F08KGF (DORMBR) 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 floating point operations is approximately
- if and , ;
- if and , ;
- if and , ;
- if and , ,
where
is the value of the parameter
K.
The complex analogue of this routine is
F08KUF (ZUNMBR).
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 F08KGF (DORMBR) 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 F08KGF (DORMBR) to form
.
9.1 Program Text
Program Text (f08kgfe.f90)
9.2 Program Data
Program Data (f08kgfe.d)
9.3 Program Results
Program Results (f08kgfe.r)