NAG CL Interface
f07wwc (zpftri)

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1 Purpose

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.
If uplo=Nag_Upper, A=UHU and A-1 is computed by first inverting U and then forming (U-1)U-H.
If uplo=Nag_Lower, A=LLH 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 or 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 or Nag_RFP_ConjTrans.
3: uplo Nag_UploType Input
On entry: specifies how A has been factorized.
uplo=Nag_Upper
A=UHU, where U is upper triangular.
uplo=Nag_Lower
A=LLH, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
4: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
5: ar[n×(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×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: n0.
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
XA-I2c(n)εκ2(A)   and   AX-I2c(n)εκ2(A) ,  
where c(n) is a modest function of n, ε is the machine precision and κ2(A) is the condition number of A defined by
κ2(A)=A2A-12 .  

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
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 83n3.
The real analogue of this function is f07wjc.

10 Example

This example computes the inverse of the matrix A, where
A= ( 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i ) .  
Here A is Hermitian positive definite, stored in RFP format, and must first be factorized by f07wrc.

10.1 Program Text

Program Text (f07wwce.c)

10.2 Program Data

Program Data (f07wwce.d)

10.3 Program Results

Program Results (f07wwce.r)