NAG Library Routine Document

F07WXF (ZTFTRI)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07WXF (ZTFTRI) computes the inverse of a complex triangular matrix stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.

2 Specification

SUBROUTINE F07WXF (

TRANSR, UPLO, DIAG, N, A, INFO)

INTEGER	N, INFO
COMPLEX (KIND=nag_wp)	A(N*(N+1)/2)
CHARACTER(1)	TRANSR, UPLO, DIAG

The routine may be called by its LAPACK name ztftri.

3 Description

F07WXF (ZTFTRI) forms the inverse of a complex triangular matrix

A

, stored using RFP format. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

4 References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

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: 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'

3: DIAG – CHARACTER(1)Input

On entry: indicates whether

A

is a nonunit or unit triangular matrix.

$DIAG ='N'$: $A$ is a nonunit triangular matrix.
$DIAG ='U'$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

DIAG ='N'

'U'

4: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

5: A( $N \times (N + 1) / 2$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output

On entry: the

n

n

triangular matrix

A

, stored in RFP format.

On exit:

A

is overwritten by

A^{- 1}

, in the same storage format as

A

6: INFO – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.

$INFO > 0$: If $INFO = i$ , $a (i, i)$ is exactly zero; $A$ is singular and its inverse cannot be computed.

7 Accuracy

The computed inverse

X

satisfies

|X A - I| \leq c (n) ε |X| |A|,

where

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

Note that a similar bound for

|A X - I|

cannot be guaranteed, although it is almost always satisfied.

The computed inverse satisfies the forward error bound

|X - A^{- 1}| \leq c (n) ε |A^{- 1}| |A| |X| .

See Du Croz and Higham (1992).

8 Further Comments

The total number of real floating point operations is approximately

\frac{4}{3} n^{3}

The real analogue of this routine is F07WKF (DTFTRI).

9 Example

This example computes the inverse of the matrix

A

, where

A = (\begin{array}{r} 4.78 + 4.56 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 2.00 - 0.30 i & - 4.11 + 1.25 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 2.89 - 1.34 i & 2.36 - 4.25 i & 4.15 + 0.80 i & 0.00 + 0.00 i \\ - 1.89 + 1.15 i & 0.04 - 3.69 i & - 0.02 + 0.46 i & 0.33 - 0.26 i \end{array})

and is stored using RFP format.

NAG Library Routine DocumentF07WXF (ZTFTRI)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07WXF (ZTFTRI)