The routine may be called by the names f08bcf, nagf_lapackeig_dtpmqrt or its LAPACK name dtpmqrt.
3Description
f08bcf is intended to be used after a call to f08bbf which performs a factorization of a triangular-pentagonal matrix containing an upper triangular matrix over a pentagonal matrix . The orthogonal matrix is represented as a product of elementary reflectors.
This routine may be used to form the matrix products
where the real rectangular matrix is split into component matrices and .
If is being applied from the left ( or ) then
where is , is , is fixed and is the number of rows of the matrix containing the elementary reflectors (i.e., m as passed to f08bbf); the number of columns of is (i.e., n as passed to f08bbf).
If is being applied from the right ( or ) then
where is , and is and is fixed.
The matrices and are overwriten by the result of the matrix product.
A common application of this routine is in updating the solution of a linear least squares problem as illustrated in Section 10 in f08bbf.
4References
Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore
5Arguments
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: – IntegerInput
On entry: the number of rows of the matrix , that is,
if
then , the number of rows of the matrix ;
if
then , the number of rows of the matrix .
Constraint:
.
4: – IntegerInput
On entry: the number of columns of the matrix , that is,
if
then , the number of columns of the matrix ;
if
then , the number of columns of the matrix .
Constraint:
.
5: – IntegerInput
On entry: , the number of elementary reflectors whose product defines the matrix .
Constraint:
.
6: – IntegerInput
On entry: , the number of rows of the upper trapezoidal part of the pentagonal composite matrix , passed (as b) in a previous call to f08bbf. This must be the same value used in the previous call to f08bbf (see l in f08bbf).
Constraint:
.
7: – IntegerInput
On entry: , the blocking factor used in a previous call to f08bbf to compute the factorization of a triangular-pentagonal matrix containing composite matrices and .
Constraints:
;
if , .
8: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array v
must be at least
.
On entry: the matrix ; this should remain unchanged from the array b returned by a previous call to f08bbf.
9: – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which f08bcf is called.
Constraints:
if , ;
if , .
10: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array t
must be at least
.
On entry: this must remain unchanged from a previous call to f08bbf (see t in f08bbf).
11: – IntegerInput
On entry: the first dimension of the array t as declared in the (sub)program from which f08bcf is called.
Constraint:
.
12: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array c1
must be at least
if and at least if .
On entry: , the first part of the composite matrix :
On exit: c2 is overwritten by the corresponding block of or or or .
15: – IntegerInput
On entry: the first dimension of the array c2 as declared in the (sub)program from which f08bcf is called.
Constraint:
.
16: – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work
must be at least
if and at least if .
17: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7Accuracy
The computed result differs from the exact result by a matrix such that
where is the machine precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08bcf 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.
9Further Comments
The total number of floating-point operations is approximately if and if .