NAG Library Routine Document

F07WDF (DPFTRF)

+− 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

F07WDF (DPFTRF) computes the Cholesky factorization of a real symmetric positive definite matrix 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 F07WDF (

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 dpftrf.

3 Description

F07WDF (DPFTRF) forms the Cholesky factorization of a real symmetric positive definite matrix

A

either as

A = U^{T} U

UPLO ='U'

A = L L^{T}

UPLO ='L'

, where

U

is an upper triangular matrix and

L

is a lower triangular, stored in RFP format.

4 References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville

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 whether the upper or lower triangular part of

A

is stored.

$UPLO ='U'$: The upper triangular part of $A$ is stored, and $A$ is factorized as $U^{T} U$ , where $U$ is upper triangular.
$UPLO ='L'$: The lower triangular part of $A$ is stored, and $A$ is factorized as $L L^{T}$ , 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

n

n

symmetric matrix

A

, stored in RFP format, as described in Section 3.3.3 in the F07 Chapter Introduction.

On exit: if

INFO = 0

, the factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

, in the same storage format as

A

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 leading minor of order $i$ 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 $A$ .

7 Accuracy

UPLO ='U'

, the computed factor

U

is the exact factor of a perturbed matrix

A + E

, where

|E| \leq c (n) ε |U^{T}| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

UPLO ='L'

, a similar statement holds for the computed factor

L

. It follows that

|e_{i j}| \leq c (n) ε \sqrt{a_{i i} a_{j j}}

8 Further Comments

The total number of floating point operations is approximately

\frac{1}{3} n^{2}

A call to F07WDF (DPFTRF) may be followed by calls to the routines:

F07WEF (DPFTRS) to solve $A X = B$ ;
F07WJF (DPFTRI) to compute the inverse of $A$ .

The complex analogue of this routine is F07WRF (ZPFTRF).

9 Example

This example computes the Cholesky factorization 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}),

and is stored using RFP format.

NAG Library Routine DocumentF07WDF (DPFTRF)

+− 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

F07WDF (DPFTRF)