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NAG Library Function Document

nag_dpftri (f07wjc)

▸▿ 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

nag_dpftri (f07wjc) computes the inverse of a real symmetric positive definite matrix using the Cholesky factorization computed by nag_dpftrf (f07wdc) stored in Rectangular Full Packed (RFP) format.

2

Specification

#include <nag.h>

#include <nagf07.h>

void	nag_dpftri (Nag_OrderType order, Nag_RFP_Store transr, Nag_UploType uplo, Integer n, double ar[], NagError *fail)

3

Description

nag_dpftri (f07wjc) is used to compute the inverse of a real symmetric 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 function must be preceded by a call to nag_dpftrf (f07wdc), which computes the Cholesky factorization of

A

uplo = Nag_Upper

A = U^{T} U

and

A^{- 1}

is computed by first inverting

U

and then forming

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

uplo = Nag_Lower

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

Arguments

1: $order$ – Nag_OrderTypeInput

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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $transr$ – Nag_RFP_StoreInput

On entry: specifies whether the RFP representation of

A

is normal or transposed.

$transr = Nag_RFP_Normal$: The matrix $A$ is stored in normal RFP format.
$transr = Nag_RFP_Trans$: The matrix $A$ is stored in transposed RFP format.

Constraint:

transr = Nag_RFP_Normal

Nag_RFP_Trans

3: $uplo$ – Nag_UploTypeInput

On entry: specifies how

A

has been factorized.

$uplo = Nag_Upper$: $A = U^{T} U$ , where $U$ is upper triangular.
$uplo = Nag_Lower$: $A = L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

4: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

5: $ar [n \times (n + 1) / 2]$ – doubleInput/Output

On entry: the Cholesky factorization of

A

stored in RFP format, as returned by nag_dpftrf (f07wdc).

On exit: the factorization is overwritten by the

n

n

matrix

A^{- 1}

stored in RFP format.

6: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_MAT_NOT_POS_DEF: 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 function specifically designed to invert a symmetric matrix stored in RFP format which is not positive definite; the matrix must be treated as a full symmetric matrix, by calling nag_dsytri (f07mjc).
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

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

nag_dpftri (f07wjc) 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

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

The complex analogue of this function is nag_zpftri (f07wwc).

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 RFP format, and must first be factorized by nag_dpftrf (f07wdc).

NAG Library Function Document

nag_dpftri (f07wjc)

▸▿ 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

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