NAG Library Routine Document

F07WJF (DPFTRI)

+− 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

F07WJF (DPFTRI) computes the inverse of a real symmetric positive definite matrix using the Cholesky factorization computed by F07WDF (DPFTRF) stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.

2 Specification

SUBROUTINE F07WJF (

TRANSR, UPLO, N, A, INFO)

INTEGER	N, INFO
REAL (KIND=nag_wp)	A(N*(N+1)/2)
CHARACTER(1)	TRANSR, UPLO

The routine may be called by its LAPACK name dpftri.

3 Description

F07WJF (DPFTRI) is used to compute the inverse of a real symmetric positive definite matrix

A

, the routine must be preceded by a call to F07WDF (DPFTRF), which computes the Cholesky factorization of

A

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

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

5 Parameters

1: TRANSR – CHARACTER(1)Input

On entry: specifies whether the RFP representation of

A

is normal or transposed.

$TRANSR ='N'$: The matrix $A$ is stored in normal RFP format.
$TRANSR ='T'$: The matrix $A$ is stored in transposed RFP format.

Constraint:

TRANSR ='N'

'T'

2: 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'

3: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

4: A( $N \times (N + 1) / 2$ ) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the Cholesky factorization of

A

stored in RFP format, as returned by F07WDF (DPFTRF).

On exit: the factorization is overwritten by the

n

n

matrix

A^{- 1}

stored in RFP format.

5: 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 F07WWF (ZPFTRI).

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 RFP format, and must first be factorized by F07WDF (DPFTRF).

NAG Library Routine DocumentF07WJF (DPFTRI)

+− 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

NAG Library Routine Document

F07WJF (DPFTRI)