NAG Library Routine Document

f07wwf  (zpftri)


    1  Purpose
    7  Accuracy


f07wwf (zpftri) computes the inverse of a complex Hermitian positive definite matrix using the Cholesky factorization computed by f07wrf (zpftrf) stored in Rectangular Full Packed (RFP) format.


Fortran Interface
Subroutine f07wwf ( transr, uplo, n, ar, info)
Integer, Intent (In):: n
Integer, Intent (Out):: info
Complex (Kind=nag_wp), Intent (Inout):: ar(n*(n+1)/2)
Character (1), Intent (In):: transr, uplo
C Header Interface
#include nagmk26.h
void  f07wwf_ ( const char *transr, const char *uplo, const Integer *n, Complex ar[], Integer *info, const Charlen length_transr, const Charlen length_uplo)
The routine may be called by its LAPACK name zpftri.


f07wwf (zpftri) 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 3.3.3 in the F07 Chapter Introduction. The routine must be preceded by a call to f07wrf (zpftrf), which computes the Cholesky factorization of A.
If uplo='U', A=UHU and A-1 is computed by first inverting U and then forming U-1U-H.
If uplo='L', A=LLH and A-1 is computed by first inverting L and then forming L-HL-1.


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


1:     transr – Character(1)Input
On entry: specifies whether the normal RFP representation of A or its conjugate transpose is stored.
The matrix A is stored in normal RFP format.
The conjugate transpose of the RFP representation of the matrix A is stored.
Constraint: transr='N' or 'C'.
2:     uplo – Character(1)Input
On entry: specifies how A has been factorized.
A=UHU, where U is upper triangular.
A=LLH, where L is lower triangular.
Constraint: uplo='U' or 'L'.
3:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     arn×n+1/2 – Complex (Kind=nag_wp) arrayInput/Output
On entry: the Cholesky factorization of A stored in RFP format, as returned by f07wrf (zpftrf).
On exit: the factorization is overwritten by the n by n matrix A-1 stored in RFP format.
5:     info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
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 routine 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 f07mwf (zhetri).


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 .  

Parallelism and Performance

f07wwf (zpftri) 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.

Further Comments

The total number of real floating-point operations is approximately 83n3.
The real analogue of this routine is f07wjf (dpftri).


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 f07wrf (zpftrf).

Program Text

Program Text (f07wwfe.f90)

Program Data

Program Data (f07wwfe.d)

Program Results

Program Results (f07wwfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017