NAG Library Routine Document

f06wbf (dtfsm)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

f06wbf (dtfsm) performs one of the matrix-matrix operations
BαA-1B , BαA-TB , BαBA-1   or BαBA-T ,  
where A is a real triangular matrix stored in Rectangular Full Packed (RFP) format, B is an m by n real matrix, and α is a real scalar. A-T  denotes AT-1  or equivalently A-1T .
No test for singularity or near-singularity of A is included in this routine. Such tests must be performed before calling this routine.

2
Specification

Fortran Interface
Subroutine f06wbf ( transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb)
Integer, Intent (In):: m, n, ldb
Real (Kind=nag_wp), Intent (In):: alpha, a(*)
Real (Kind=nag_wp), Intent (Inout):: b(ldb,*)
Character (1), Intent (In):: transr, side, uplo, trans, diag
C Header Interface
#include nagmk26.h
void  f06wbf_ (const char *transr, const char *side, const char *uplo, const char *trans, const char *diag, const Integer *m, const Integer *n, const double *alpha, const double a[], double b[], const Integer *ldb, const Charlen length_transr, const Charlen length_side, const Charlen length_uplo, const Charlen length_trans, const Charlen length_diag)
The routine may be called by its LAPACK name dtfsm.

3
Description

f06wbf (dtfsm) solves (for X) a triangular linear system of one of the forms
AX=αB , ATX=αB , XA=αB   or XAT=αB ,  
where A is a real triangular matrix stored in RFP format, B, X are m by n real matrices, and α is a real scalar. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.

4
References

Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

5
Arguments

1:     transr – Character(1)Input
On entry: specifies whether the RFP representation of A is normal or transposed.
transr='N'
The matrix A is stored in normal RFP format.
transr='T'
The matrix A is stored in transposed RFP format.
Constraint: transr='N' or 'T'.
2:     side – Character(1)Input
On entry: specifies whether B is operated on from the left or the right, or similarly whether A (or its transpose) appears to the left or right of the solution matrix in the linear system to be solved.
side='L'
B is pre-multiplied from the left. The system to be solved has the form AX=αB or ATX=αB.
side='R'
B is post-multiplied from the right. The system to be solved has the form XA=αB or XAT=αB.
Constraint: side='L' or 'R'.
3:     uplo – Character(1)Input
On entry: specifies whether A is upper or lower triangular.
uplo='U'
A is upper triangular.
uplo='L'
A is lower triangular.
Constraint: uplo='U' or 'L'.
4:     trans – Character(1)Input
On entry: specifies whether the operation involves A-1 or A-T, i.e., whether or not A is transposed in the linear system to be solved.
trans='N'
The operation involves A-1, i.e., A is not transposed.
trans='T'
The operation involves A-T, i.e., A is transposed.
Constraint: trans='N' or 'T'.
5:     diag – Character(1)Input
On entry: specifies whether A has nonunit or unit diagonal elements.
diag='N'
The diagonal elements of A are stored explicitly.
diag='U'
The diagonal elements of A are assumed to be 1, the corresponding elements of a are not referenced.
Constraint: diag='N' or 'U'.
6:     m – IntegerInput
On entry: m, the number of rows of the matrix B.
Constraint: m0.
7:     n – IntegerInput
On entry: n, the number of columns of the matrix B.
Constraint: n0.
8:     alpha – Real (Kind=nag_wp)Input
On entry: the scalar α.
9:     a* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array a must be at least max1,m×m+1/2 if side='L' and at least max1,n×n+1/2 if side='R'.
On entry: A, the m by m triangular matrix A if side='L' or the n by n triangular matrix A if side='R', stored in RFP format (as specified by transr). The storage format is described in detail in Section 3.3.3 in the F07 Chapter Introduction. If alpha=0.0, a is not referenced.
10:   bldb* – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least max1,n.
On entry: the m by n matrix B.
If alpha=0, b need not be set.
On exit: the updated matrix B, or similarly the solution matrix X.
11:   ldb – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f06wbf (dtfsm) is called.
Constraint: ldb max1,m .

6
Error Indicators and Warnings

None.

7
Accuracy

Not applicable.

8
Parallelism and Performance

f06wbf (dtfsm) 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.

9
Further Comments

None.

10
Example

This example reads in the lower triangular part of a symmetric matrix A which it converts to RFP format. It also reads in α and a 6 by 4 matrix B and then performs the matrix-matrix operation BαA-1B.

10.1
Program Text

Program Text (f06wbfe.f90)

10.2
Program Data

Program Data (f06wbfe.d)

10.3
Program Results

Program Results (f06wbfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017