NAG FL Interface
f08bvf (ztzrzf)
1
Purpose
f08bvf reduces the by () complex upper trapezoidal matrix to upper triangular form by means of unitary transformations.
2
Specification
Fortran Interface
Integer, Intent (In) |
:: |
m, n, lda, lwork |
Integer, Intent (Out) |
:: |
info |
Complex (Kind=nag_wp), Intent (Inout) |
:: |
a(lda,*), tau(*) |
Complex (Kind=nag_wp), Intent (Out) |
:: |
work(max(1,lwork)) |
|
C Header Interface
#include <nag.h>
void |
f08bvf_ (const Integer *m, const Integer *n, Complex a[], const Integer *lda, Complex tau[], Complex work[], const Integer *lwork, Integer *info) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08bvf_ (const Integer &m, const Integer &n, Complex a[], const Integer &lda, Complex tau[], Complex work[], const Integer &lwork, Integer &info) |
}
|
The routine may be called by the names f08bvf, nagf_lapackeig_ztzrzf or its LAPACK name ztzrzf.
3
Description
The
by
(
) complex upper trapezoidal matrix
given by
where
is an
by
upper triangular matrix and
is an
by
matrix, is factorized as
where
is also an
by
upper triangular matrix and
is an
by
unitary matrix.
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
https://www.netlib.org/lapack/lug
5
Arguments
-
1:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
2:
– Integer
Input
-
On entry: , the number of columns of the matrix .
Constraint:
.
-
3:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
a
must be at least
.
On entry: the leading
by
upper trapezoidal part of the array
a must contain the matrix to be factorized.
On exit: the leading
by
upper triangular part of
a contains the upper triangular matrix
, and elements
to
n of the first
rows of
a, with the array
tau, represent the unitary matrix
as a product of
elementary reflectors (see
Section 3.3.6 in the
F08 Chapter Introduction).
-
4:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08bvf is called.
Constraint:
.
-
5:
– Complex (Kind=nag_wp) array
Output
-
Note: the dimension of the array
tau
must be at least
.
On exit: the scalar factors of the elementary reflectors.
-
6:
– Complex (Kind=nag_wp) array
Workspace
-
On exit: if
, the real part of
contains the minimum value of
lwork required for optimal performance.
-
7:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08bvf 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, , where is the optimal block size.
Constraint:
or .
-
8:
– Integer
Output
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7
Accuracy
The computed factorization is the exact factorization of a nearby matrix
, where
and
is the
machine precision.
8
Parallelism and Performance
f08bvf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is approximately .
The real analogue of this routine is
f08bhf.
10
Example
This example solves the linear least squares problems
for the minimum norm solutions
and
, where
is the
th column of the matrix
,
and
The solution is obtained by first obtaining a factorization with column pivoting of the matrix , and then the factorization of the leading by part of is computed, where is the estimated rank of . A tolerance of is used to estimate the rank of from the upper triangular factor, .
Note that the block size (NB) of assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
10.1
Program Text
10.2
Program Data
10.3
Program Results