NAG Library Routine Document

F07GAF (DPPSV)

+− 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

F07GAF (DPPSV) computes the solution to a real system of linear equations

A X = B,

where

A

is an

n

n

symmetric positive definite matrix stored in packed format and

X

and

B

are

n

r

matrices.

2 Specification

SUBROUTINE F07GAF (

UPLO, N, NRHS, AP, B, LDB, INFO)

INTEGER	N, NRHS, LDB, INFO
REAL (KIND=nag_wp)	AP(), B(LDB,)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name dppsv.

3 Description

F07GAF (DPPSV) uses the Cholesky decomposition to factor

A

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 matrix. The factored form of

A

is then used to solve the system of equations

A X = B

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Parameters

1: UPLO – CHARACTER(1)Input

On entry: if

UPLO ='U'

, the upper triangle of

A

is stored.

UPLO ='L'

, the lower triangle of

A

is stored.

Constraint:

UPLO ='U'

'L'

2: N – INTEGERInput

On entry:

n

, the number of linear equations, i.e., the order of the matrix

A

Constraint:

N \geq 0

3: NRHS – INTEGERInput

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrix

B

Constraint:

NRHS \geq 0

4: 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

n

n

symmetric matrix

A

, packed by columns.

More precisely,

if $UPLO ='U'$ , the upper triangle of $A$ 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 $A$ must be stored with element $A_{i j}$ in $AP (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

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: B(LDB, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array B must be at least

\max (1, NRHS)

On entry: the

n

r

right-hand side matrix

B

On exit: if

INFO = 0

, the

n

r

solution matrix

X

6: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07GAF (DPPSV) is called.

Constraint:

LDB \geq \max (1, N)

7: 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 argument 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 $i$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

7 Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖E‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖A^{- 1}‖}_{1} {‖A‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.

F07GBF (DPPSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, F04BEF solves

A x = b

and returns a forward error bound and condition estimate. F04BEF calls F07GAF (DPPSV) to solve the equations.

8 Further Comments

The total number of floating point operations is approximately

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

, where

r

is the number of right-hand sides.

The complex analogue of this routine is F07GNF (ZPPSV).

9 Example

This example solves the equations

A x = b,

where

A

is the symmetric positive definite matrix

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 b = (\begin{matrix} 8.70 \\ - 13.35 \\ 1.89 \\ - 4.14 \end{matrix}) .

Details of the Cholesky factorization of

A

are also output.

NAG Library Routine DocumentF07GAF (DPPSV)

+− 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

F07GAF (DPPSV)