NAG CL Interface
f07gwc (zpptri)

1 Purpose

f07gwc computes the inverse of a complex Hermitian positive definite matrix A, where A has been factorized by f07grc, using packed storage.

2 Specification

#include <nag.h>
void  f07gwc (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex ap[], NagError *fail)
The function may be called by the names: f07gwc, nag_lapacklin_zpptri or nag_zpptri.

3 Description

f07gwc is used to compute the inverse of a complex Hermitian positive definite matrix A, the function must be preceded by a call to f07grc, which computes the Cholesky factorization of A, using packed storage.
If uplo=Nag_Upper, A=UHU and A-1 is computed by first inverting U and then forming U-1U-H.
If uplo=Nag_Lower, A=LLH and A-1 is computed by first inverting L and then forming L-HL-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

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: 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.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: ap[dim] Complex Input/Output
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the Cholesky factor of A stored in packed form, as returned by f07grc.
On exit: the factorization is overwritten by the n by n matrix A-1.
The storage of elements Aij depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Aij is stored in ap[j-1×j/2+i-1], for ij;
if order=Nag_ColMajor and uplo=Nag_Lower,
Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Upper,
Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Lower,
Aij is stored in ap[i-1×i/2+j-1], for ij.
5: 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_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.
NE_SINGULAR
Diagonal element value of the Cholesky factor is zero; the Cholesky factor is singular and the inverse of A cannot be computed.

7 Accuracy

The computed inverse X satisfies
XA-I2cnεκ2A   and   AX-I2cnεκ2A ,  
where cn is a modest function of n, ε is the machine precision and κ2A is the condition number of A defined by
κ2A=A2A-12 .  

8 Parallelism and Performance

f07gwc 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 f07gjc.

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 packed form, and must first be factorized by f07grc.

10.1 Program Text

Program Text (f07gwce.c)

10.2 Program Data

Program Data (f07gwce.d)

10.3 Program Results

Program Results (f07gwce.r)