f07wwc computes the inverse of a complex Hermitian positive definite matrix using the Cholesky factorization computed by f07wrc stored in Rectangular Full Packed (RFP) format.

2 Specification

#include <nag.h>

void	f07wwc (Nag_OrderType order, Nag_RFP_Store transr, Nag_UploType uplo, Integer n, Complex ar[], NagError *fail)

The function may be called by the names: f07wwc, nag_lapacklin_zpftri or nag_zpftri.

3 Description

f07wwc is used to compute the inverse of a complex Hermitian positive definite matrix

A

, stored in RFP format. The RFP storage format is described in Section 3.4.3 in the F07 Chapter Introduction. The function must be preceded by a call to f07wrc, which computes the Cholesky factorization of

A

uplo = Nag_Upper

A = U^{H} U

and

A^{- 1}

is computed by first inverting

U

and then forming

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

uplo = Nag_Lower

A = L L^{H}

and

A^{- 1}

is computed by first inverting

L

and then forming

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

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_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $transr$ – Nag_RFP_Store Input

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_UploType Input

On entry: specifies how

A

has been factorized.

$uplo = Nag_Upper$: $A = U^{H} U$ , where $U$ is upper triangular.
$uplo = Nag_Lower$: $A = L L^{H}$ , where $L$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

5: $ar [n \times (n + 1) / 2]$ – Complex Input/Output

On entry: the Cholesky factorization of

A

stored in RFP format, as returned by f07wrc.

On exit: the factorization is overwritten by the

n \times n

matrix

A^{- 1}

stored in RFP format.

6: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MAT_NOT_POS_DEF: The leading minor of order $⟨ value ⟩$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$ . There is no function specifically designed to invert a Hermitian matrix stored in RFP format which is not positive definite; the matrix must be treated as a full Hermitian matrix, by calling f07mwc.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The computed inverse

X

satisfies

{‖ X A - I ‖}_{2} \leq c (n) ε κ_{2} (A) and {‖ A X - I ‖}_{2} \leq c (n) ε κ_{2} (A),

where

c (n)

is a modest function of

n

ε

is the machine precision and

κ_{2} (A)

is the condition number of

A

defined by

κ_{2} (A) = {‖ A ‖}_{2} {‖ A^{- 1} ‖}_{2} .

8 Parallelism and Performance

f07wwc 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{8}{3} n^{3}

The real analogue of this function is f07wjc.

10 Example

This example computes the inverse of the matrix

A

, where

A = (\begin{array}{r} 3.23 + 0.00 i & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 + 0.00 i & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 + 0.00 i & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 + 0.00 i \end{array}) .

Here

A

is Hermitian positive definite, stored in RFP format, and must first be factorized by f07wrc.

f07ww: FL CL CPP AD

NAG CL Interfacef07wwc (zpftri)

▸▿ 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 CL Interface
f07wwc (zpftri)