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 <nag.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 the names f06wbf, nagf_blas_dtfsm or its LAPACK name dtfsm.

3 Description

f06wbf solves (for

X

) a triangular linear system of one of the forms

\begin{array}{l} A X = α B, & A^{T} X = α B, \\ X A = α B or & X A^{T} = α B, \end{array}

where

A

is a real triangular matrix stored in RFP format,

B

X

are

m \times 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'

'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 $A X = α B$ or $A^{T} X = α B$ .
$side ='R'$: $B$ is post-multiplied from the right. The system to be solved has the form $X A = α B$ or $X A^{T} = α B$ .

Constraint:

side ='L'

'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'

'L'

4: $trans$ – Character(1) Input

On entry: specifies whether the operation involves

A^{- 1}

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'

'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'

'U'

6: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

B

Constraint:

m \geq 0

7: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

B

Constraint:

n \geq 0

8: $alpha$ – Real (Kind=nag_wp) Input

On entry: the scalar

α

9: $a (*)$ – Real (Kind=nag_wp) array Input

Note: the dimension of the array a must be at least

\max (1, m \times (m + 1) / 2)

side ='L'

and at least

\max (1, n \times (n + 1) / 2)

side ='R'

On entry:

A

, the

m \times m

triangular matrix

A

side ='L'

or the

n \times n

triangular matrix

A

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: $b (ldb, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array b must be at least

\max (1, n)

On entry: the

m \times n

matrix

B

alpha = 0

, b need not be set.

On exit: the updated matrix

B

, or similarly the solution matrix

X

11: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f06wbf is called.

Constraint:

ldb \geq \max (1, m)

6 Error Indicators and Warnings

None.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f06wbf 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 \times 4

matrix

B

and then performs the matrix-matrix operation

B \leftarrow α A^{- 1} B

f06wb: FL CL CPP AD PY MB

NAG FL Interfacef06wbf (dtfsm)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f06wbf (dtfsm)