f07gj: FL CL CPP AD PY MB

NAG FL Interface
f07gjf (dpptri)

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NAG Library Manual, Mark 30.1

Interfaces: FL CL CPP AD PY MB

NAG FL Interface Introduction

F07 (Lapacklin) Chapter Contents

F07 (Lapacklin) Chapter Introduction

f07gj: FL CL CPP AD PY MB

▸▿ 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

1 Purpose

f07gjf computes the inverse of a real symmetric positive definite matrix

A

, where

A

has been factorized by f07gdf, using packed storage.

2 Specification

Fortran Interface

Subroutine f07gjf (

uplo, n, ap, info)

Integer, Intent (In)	::	n
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	ap(*)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f07gjf_ (const char uplo, const Integer n, double ap[], Integer *info, const Charlen length_uplo)

The routine may be called by the names f07gjf, nagf_lapacklin_dpptri or its LAPACK name dpptri.

3 Description

f07gjf is used to compute the inverse of a real symmetric positive definite matrix

A

, the routine must be preceded by a call to f07gdf, which computes the Cholesky factorization of

A

, using packed storage.

uplo ='U'

A = U^{T} U

and

A^{- 1}

is computed by first inverting

U

and then forming

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

uplo ='L'

A = L L^{T}

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: $uplo$ – Character(1) Input

On entry: specifies how

A

has been factorized.

$uplo ='U'$: $A = U^{T} U$ , where $U$ is upper triangular.
$uplo ='L'$: $A = L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $ap (*)$ – Real (Kind=nag_wp) array Input/Output

Note: the dimension of the array ap must be at least

\max (1, n \times (n + 1) / 2)

On entry: the Cholesky factor of

A

stored in packed form, as returned by f07gdf.

On exit: the factorization is overwritten by the

n \times n

matrix

A^{- 1}

More precisely,

if $uplo ='U'$ , the upper triangle of $A^{- 1}$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A^{- 1}$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

4: $info$ – Integer Output

On exit:

info = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: 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

{‖ 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

Background information to multithreading can be found in the Multithreading documentation.

f07gjf 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.

9 Further Comments

The total number of floating-point operations is approximately

\frac{2}{3} n^{3}

The complex analogue of this routine is f07gwf.

10 Example

This example computes the inverse of the matrix

A

, where

A = (\begin{matrix} 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 \end{matrix}) .

Here

A

is symmetric positive definite, stored in packed form, and must first be factorized by f07gdf.

f07gj: FL CL CPP AD PY MB

NAG FL Interfacef07gjf (dpptri)

▸▿ 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 FL Interface
f07gjf (dpptri)