f08kgf multiplies an arbitrary real matrix by one of the real orthogonal matrices or which were determined by f08kef when reducing a real matrix to bidiagonal form.
The routine may be called by the names f08kgf, nagf_lapackeig_dormbr or its LAPACK name dormbr.
3Description
f08kgf is intended to be used after a call to f08kef, which reduces a real rectangular matrix to bidiagonal form by an orthogonal transformation: . f08kef 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).
4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
Note: in the descriptions below, denotes the order of or : if , and if , .
1: – 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: – 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: – 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: – IntegerInput
On entry: , the number of rows of the matrix .
Constraint:
.
5: – IntegerInput
On entry: , the number of columns of the matrix .
Constraint:
.
6: – IntegerInput
On entry: if , the number of columns in the original matrix .
If , the number of rows in the original matrix .
Constraint:
.
7: – 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.
8: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08kgf is called.
Constraints:
if , ;
if , .
9: – 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 in its argument tauq if , or in its argument taup if .
10: – 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: – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08kgf is called.
Constraint:
.
12: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
13: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08kgf 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: – 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.
f08kgf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kgf 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
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 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 to form .