NAG Library Function Document

nag_ztftri (f07wxc)

, stored using RFP format. The RFP storage format is described in Section 3.3.3 in the f07 Chapter Introduction. 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 Arguments

1: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $transr$ – Nag_RFP_StoreInput

On entry: specifies whether the normal RFP representation of

A

or its conjugate transpose is stored.

$transr = Nag_RFP_Normal$: The matrix $A$ is stored in normal RFP format.
$transr = Nag_RFP_ConjTrans$: The conjugate transpose of the RFP representation of the matrix $A$ is stored.

Constraint:

transr = Nag_RFP_Normal

Nag_RFP_ConjTrans

3: $uplo$ – Nag_UploTypeInput

On entry: specifies whether

A

is upper or lower triangular.

$uplo = Nag_Upper$: $A$ is upper triangular.
$uplo = Nag_Lower$: $A$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

4: $diag$ – Nag_DiagTypeInput

On entry: indicates whether

A

is a nonunit or unit triangular matrix.

$diag = Nag_NonUnitDiag$: $A$ is a nonunit triangular matrix.
$diag = Nag_UnitDiag$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

diag = Nag_NonUnitDiag

Nag_UnitDiag

5: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

6: $ar [n \times (n + 1) / 2]$ – ComplexInput/Output

On entry: the upper or lower triangular part (as specified by uplo) of the

n

n

Hermitian matrix

A

, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Section 3.3.3 in the f07 Chapter Introduction.

On exit:

A

is overwritten by

A^{- 1}

, in the same storage format as

A

7: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_SINGULAR: Diagonal element $〈value〉$ of $A$ 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

nag_ztftri (f07wxc) is not threaded by NAG in any implementation.

nag_ztftri (f07wxc) 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 function. 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 function is nag_dtftri (f07wkc).

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})

and is stored using RFP format.

NAG Library Function Documentnag_ztftri (f07wxc)

▸▿ 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 Library Function Document

nag_ztftri (f07wxc)