, stored using RFP format. The RFP storage format is described in Section 3.4.3 in the F07 Chapter Introduction. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

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_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

Nag_ColMajor

2: $transr$ – Nag_RFP_Store Input

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_UploType Input

On entry: specifies whether

A

is upper or lower triangular.

$uplo = Nag_Upper$: $A$ is upper triangular.
$uplo = Nag_Lower$: $A$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

4: $diag$ – Nag_DiagType Input

On entry: indicates whether

A

is a nonunit or unit triangular matrix.

$diag = Nag_NonUnitDiag$: $A$ is a nonunit triangular matrix.
$diag = Nag_UnitDiag$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

diag = Nag_NonUnitDiag

Nag_UnitDiag

5: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

6: $ar [n \times (n + 1) / 2]$ – double Input/Output

On entry: the upper or lower triangular part (as specified by uplo) of the

n

n

symmetric matrix

A

, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Section 3.4.3 in the F07 Chapter Introduction.

On exit:

A

is overwritten by

A^{- 1}

, in the same storage format as

A

7: $fail$ – 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: $n \geq 0$ .
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 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 $A$ is exactly zero. $A$ is singular its inverse cannot be computed.

7 Accuracy

The computed inverse

X

satisfies

|X A - I| \leq c (n) ε |X| |A|,

where

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

Note that a similar bound for

|A X - I|

cannot be guaranteed, although it is almost always satisfied.

The computed inverse satisfies the forward error bound

|X - A^{- 1}| \leq c (n) ε |A^{- 1}| |A| |X| .

See Du Croz and Higham (1992).

8 Parallelism and Performance

f07wkc 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{1}{3} n^{3}

The complex analogue of this function is f07wxc.

10 Example

This example computes the inverse of the matrix

A

, where

A = (\begin{matrix} 4.30 & 0.00 & 0.00 & 0.00 \\ - 3.96 & - 4.87 & 0.00 & 0.00 \\ 0.40 & 0.31 & - 8.02 & 0.00 \\ - 0.27 & 0.07 & - 5.95 & 0.12 \end{matrix})

and is stored using RFP format.

Interfaces: FL CL AD

NAG CL Interface Introduction

F07 (Lapacklin) Chapter Contents

F07 (Lapacklin) Chapter Introduction

f07wk: FL CL AD

NAG CL Interfacef07wkc (dtftri)

▸▿ 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 CL Interface
f07wkc (dtftri)