NAG Library Routine Document

F07GJF (DPPTRI)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07GJF (DPPTRI) computes the inverse of a real symmetric positive definite matrix

A

, where

A

has been factorized by F07GDF (DPPTRF), using packed storage.

2 Specification

SUBROUTINE F07GJF (

UPLO, N, AP, INFO)

INTEGER	N, INFO
REAL (KIND=nag_wp)	AP(*)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name dpptri.

3 Description

F07GJF (DPPTRI) is used to compute the inverse of a real symmetric positive definite matrix

A

, the routine must be preceded by a call to F07GDF (DPPTRF), 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 Parameters

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 – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: AP( $*$ ) – REAL (KIND=nag_wp) arrayInput/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 (DPPTRF).

On exit: the factorization is overwritten by the

n

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 – INTEGEROutput

On exit:

INFO = 0

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

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

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

$INFO > 0$: If $INFO = i$ , the $i$ th diagonal element 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 Further Comments

The total number of floating point operations is approximately

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

The complex analogue of this routine is F07GWF (ZPPTRI).

9 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 (DPPTRF).

NAG Library Routine DocumentF07GJF (DPPTRI)