NAG CL Interface
f07fjc (dpotri)

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1 Purpose

f07fjc computes the inverse of a real symmetric positive definite matrix A, where A has been factorized by f07fdc.

2 Specification

#include <nag.h>
void  f07fjc (Nag_OrderType order, Nag_UploType uplo, Integer n, double a[], Integer pda, NagError *fail)
The function may be called by the names: f07fjc, nag_lapacklin_dpotri or nag_dpotri.

3 Description

f07fjc is used to compute the inverse of a real symmetric positive definite matrix A, the function must be preceded by a call to f07fdc, which computes the Cholesky factorization of A.
If uplo=Nag_Upper, A=UTU and A-1 is computed by first inverting U and then forming (U-1)U-T.
If uplo=Nag_Lower, A=LLT and A-1 is computed by first inverting L and then forming L-T(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

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=UTU, where U is upper triangular.
uplo=Nag_Lower
A=LLT, 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: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
On entry: the upper triangular matrix U if uplo=Nag_Upper or the lower triangular matrix L if uplo=Nag_Lower, as returned by f07fdc.
On exit: U is overwritten by the upper triangle of A-1 if uplo=Nag_Upper; L is overwritten by the lower triangle of A-1 if uplo=Nag_Lower.
5: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array a.
Constraint: pdamax(1,n).
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.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
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-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.
f07fjc 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 floating-point operations is approximately 23n3.
The complex analogue of this function is f07fwc.

10 Example

This example computes the inverse of the matrix A, where
A= ( 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 ) .  
Here A is symmetric positive definite and must first be factorized by f07fdc.

10.1 Program Text

Program Text (f07fjce.c)

10.2 Program Data

Program Data (f07fjce.d)

10.3 Program Results

Program Results (f07fjce.r)