f08ckf multiplies a general real
matrix
by the real orthogonal matrix
from an
factorization computed by
f08chf.
f08ckf is intended to be used following a call to
f08chf, which performs an
factorization of a real matrix
and represents the orthogonal matrix
as a product 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.
A common application of this routine is in solving underdetermined linear least squares problems, as described in the
F08 Chapter Introduction, and illustrated in
Section 10 in
f08chf.
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
-
1:
– Character(1)
Input
-
On entry: indicates how
or
is to be applied to
.
- or is applied to from the left.
- or is applied to from the right.
Constraint:
or .
-
2:
– Character(1)
Input
-
On entry: indicates whether
or
is to be applied to
.
- is applied to .
- is applied to .
Constraint:
or .
-
3:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
4:
– Integer
Input
-
On entry: , the number of columns of the matrix .
Constraint:
.
-
5:
– Integer
Input
-
On entry: , the number of elementary reflectors whose product defines the matrix .
Constraints:
- if , ;
- if , .
-
6:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
a
must be at least
if
and at least
if
.
On entry: the
th row of
a must contain the vector which defines the elementary reflector
, for
, as returned by
f08chf.
On exit: is modified by f08ckf but restored on exit.
-
7:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08ckf is called.
Constraint:
.
-
8:
– Real (Kind=nag_wp) array
Input
-
Note: the dimension of the array
tau
must be at least
.
On entry:
must contain the scalar factor of the elementary reflector
, as returned by
f08chf.
-
9:
– Real (Kind=nag_wp) array
Input/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
as specified by
side and
trans.
-
10:
– Integer
Input
-
On entry: the first dimension of the array
c as declared in the (sub)program from which
f08ckf is called.
Constraint:
.
-
11:
– Real (Kind=nag_wp) array
Workspace
-
On exit: if
,
contains the minimum value of
lwork required for optimal performance.
-
12:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08ckf 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 .
-
13:
– Integer
Output
-
On exit:
unless the routine detects an error (see
Section 6).
The computed result differs from the exact result by a matrix
such that
where
is the
machine precision.
Background information to multithreading can be found in the
Multithreading documentation.
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 complex analogue of this routine is
f08cxf.