void	f07twf_ (const char uplo, const char diag, const Integer n, Complex a[], const Integer lda, Integer *info, const Charlen length_uplo, const Charlen length_diag)

The routine may be called by the names f07twf, nagf_lapacklin_ztrtri or its LAPACK name ztrtri.

3 Description

f07twf forms the inverse of a complex triangular matrix

A

. 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

5 Arguments

1: $uplo$ – 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'

2: $diag$ – 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'

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $a (lda, *)$ – Complex (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: the

n \times n

triangular matrix

A

If $uplo ='U'$ , $A$ is upper triangular and the elements of the array below the diagonal are not referenced.
If $uplo ='L'$ , $A$ is lower triangular and the elements of the array above the diagonal are not referenced.
If $diag ='U'$ , the diagonal elements of $A$ are assumed to be $1$ , and are not referenced.

On exit:

A

is overwritten by

A^{- 1}

, using the same storage format as described above.

5: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f07twf is called.

Constraint:

lda \geq \max (1, n)

6: $info$ – Integer Output

On exit:

info = 0

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

6 Error Indicators and Warnings

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

$info > 0$: Element $⟨ value ⟩$ of the diagonal is exactly zero. $A$ is singular 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 Parallelism and Performance

f07twf 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

The total number of real floating-point operations is approximately

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

The real analogue of this routine is f07tjf.

10 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}) .

f07tw: FL CL CPP AD

NAG FL Interfacef07twf (ztrtri)

▸▿ 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
f07twf (ztrtri)