NAG Library Routine Document

F06WPF (ZTFSM)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

1 Purpose

F06WPF (ZTFSM) performs one of the matrix-matrix operations

\begin{array}{l} B \leftarrow α A^{- 1} B, & B \leftarrow α A^{- H} B, \\ B \leftarrow α B A^{- 1} or & B \leftarrow α B A^{- H}, \end{array}

where

A

is a complex triangular matrix stored in Rectangular Full Packed (RFP) format,

B

is an

m

n

complex matrix, and

α

is a complex scalar.

A^{- H}

denotes

{(A^{H})}^{- 1}

or equivalently

{(A^{- 1})}^{H}

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

SUBROUTINE F06WPF (

TRANSR, SIDE, UPLO, TRANS, DIAG, M, N, ALPHA, AR, B, LDB)

INTEGER	M, N, LDB
COMPLEX (KIND=nag_wp)	ALPHA, AR(), B(LDB,)
CHARACTER(1)	TRANSR, SIDE, UPLO, TRANS, DIAG

The routine may be called by its LAPACK name ztfsm.

3 Description

F06WPF (ZTFSM) solves (for

X

) a triangular linear system of one of the forms

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

where

A

is a complex triangular matrix stored in RFP format,

B

X

are

m

n

complex matrices, and

α

is a complex 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 Parameters

1: $TRANSR$ – CHARACTER(1)Input

On entry: specifies whether the normal RFP representation of

A

or its conjugate transpose is stored.

$TRANSR ='N'$: The matrix $A$ is stored in normal RFP format.
$TRANSR ='C'$: The conjugate transpose of the RFP representation of the matrix $A$ is stored.

Constraint:

TRANSR ='N'

'C'

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^{- H}

, i.e., whether or not

A

is transpose conjugated in the linear system to be solved.

$TRANS ='N'$: The operation involves $A^{- 1}$ , i.e., $A$ is not transpose conjugated.
$TRANS ='C'$: The operation involves $A^{- H}$ , i.e., $A$ is transpose conjugated.

Constraint:

TRANS ='N'

'C'

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 AR are not referenced.

Constraint:

DIAG ='N'

'U'

6: $M$ – INTEGERInput

On entry:

m

, the number of rows of the matrix

B

Constraint:

M \geq 0

7: $N$ – INTEGERInput

On entry:

n

, the number of columns of the matrix

B

Constraint:

N \geq 0

8: $ALPHA$ – COMPLEX (KIND=nag_wp)Input

On entry: the scalar

α

9: $AR (*)$ – COMPLEX (KIND=nag_wp) arrayInput

Note: the dimension of the array AR 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

m

triangular matrix

A

SIDE ='L'

or the

n

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

, AR is not referenced.

10: $B (LDB *)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: the

m

n

matrix

B

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 F06WPF (ZTFSM) is called.

Constraint:

LDB \geq \max (1, M)

6 Error Indicators and Warnings

None.

7 Accuracy

Not applicable.

8 Parallelism and Performance

F06WPF (ZTFSM) is not threaded by NAG in any implementation.

F06WPF (ZTFSM) 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 upper triangular part of a symmetric matrix

A

which it converts to RFP format. It also reads in

α

and a

4

3

matrix

B

and then performs the matrix-matrix operation

B \leftarrow α A^{- 1} B

NAG Library Routine DocumentF06WPF (ZTFSM)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine Document

F06WPF (ZTFSM)