NAG Library Routine Document

F07MAF (DSYSV)

+− 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

F07MAF (DSYSV) computes the solution to a real system of linear equations

A X = B,

where

A

is an

n

n

symmetric matrix and

X

and

B

are

n

r

matrices.

2 Specification

SUBROUTINE F07MAF (

UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO)

INTEGER	N, NRHS, LDA, IPIV(*), LDB, LWORK, INFO
REAL (KIND=nag_wp)	A(LDA,), B(LDB,), WORK(max(1,LWORK))
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name dsysv.

3 Description

F07MAF (DSYSV) uses the diagonal pivoting method to factor

A

A = U D U^{T}

UPLO ='U'

A = L D L^{T}

UPLO ='L'

, where

U

(or

L

) is a product of permutation and unit upper (lower) triangular matrices, and

D

is symmetric and block diagonal with

1

1

and

2

2

diagonal blocks. The factored form of

A

is then used to solve the system of equations

A X = B

Note that, in general, different permutations (pivot sequences) and diagonal block structures are obtained for

UPLO ='U'

'L'

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: A(LDA, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: the

n

n

symmetric matrix

A

If $UPLO ='U'$ , the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If $UPLO ='L'$ , the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: if

INFO = 0

, the block diagonal matrix

D

and the multipliers used to obtain the factor

U

L

from the factorization

A = U D U^{T}

A = L D L^{T}

as computed by F07MDF (DSYTRF).

5: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F07MAF (DSYSV) is called.

Constraint:

LDA \geq \max (1, N)

6: IPIV( $*$ ) – INTEGER arrayOutput

Note: the dimension of the array IPIV must be at least

\max (1, N)

On exit: details of the interchanges and the block structure of

D

. More precisely,

if $IPIV (i) = k > 0$ , $d_{i i}$ is a $1$ by $1$ pivot block and the $i$ th row and column of $A$ were interchanged with the $k$ th row and column;
if $UPLO ='U'$ and $IPIV (i - 1) = IPIV (i) = - l < 0$ , $(\begin{matrix} d_{i - 1, i - 1} & {\bar{d}}_{i, i - 1} \\ {\bar{d}}_{i, i - 1} & d_{i i} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i - 1)$ th row and column of $A$ were interchanged with the $l$ th row and column;
if $UPLO ='L'$ and $IPIV (i) = IPIV (i + 1) = - m < 0$ , $(\begin{matrix} d_{i i} & d_{i + 1, i} \\ d_{i + 1, i} & d_{i + 1, i + 1} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i + 1)$ th row and column of $A$ were interchanged with the $m$ th row and column.

7: 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

8: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07MAF (DSYSV) is called.

Constraint:

LDB \geq \max (1, N)

9: WORK( $\max (1, LWORK)$ ) – REAL (KIND=nag_wp) arrayWorkspace

On exit: if

INFO = 0

WORK (1)

returns the optimal LWORK.

10: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F07MAF (DSYSV) is called.

LWORK \geq 1

, and for best performance

LWORK \geq \max (1, N \times nb)

, where

nb

is the optimal block size for F07MDF (DSYTRF).

LWORK = - 1

, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.

11: 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 $INFO = i$ , $d_{i i}$ is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, so the solution could not be 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.

F07MBF (DSYSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, F04BHF solves

A x = b

and returns a forward error bound and condition estimate. F04BHF calls F07MAF (DSYSV) 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 analogues of F07MAF (DSYSV) are F07MNF (ZHESV) for Hermitian matrices, and F07NNF (ZSYSV) for symmetric matrices.

9 Example

This example solves the equations

A x = b,

where

A

is the symmetric matrix

A = (\begin{matrix} - 1.81 & 2.06 & 0.63 & - 1.15 \\ 2.06 & 1.15 & 1.87 & 4.20 \\ 0.63 & 1.87 & - 0.21 & 3.87 \\ - 1.15 & 4.20 & 3.87 & 2.07 \end{matrix}) and b = (\begin{matrix} 0.96 \\ 6.07 \\ 8.38 \\ 9.50 \end{matrix}) .

Details of the factorization of

A

are also output.

NAG Library Routine DocumentF07MAF (DSYSV)

+− 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

F07MAF (DSYSV)