NAG Library Routine Document

F04CEF

+− 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

F04CEF computes the solution to a complex system of linear equations

A X = B

, where

A

is an

n

n

Hermitian positive definite matrix, stored in packed format, and

X

and

B

are

n

r

matrices. An estimate of the condition number of

A

and an error bound for the computed solution are also returned.

2 Specification

SUBROUTINE F04CEF (

UPLO, N, NRHS, AP, B, LDB, RCOND, ERRBND, IFAIL)

INTEGER	N, NRHS, LDB, IFAIL
REAL (KIND=nag_wp)	RCOND, ERRBND
COMPLEX (KIND=nag_wp)	AP(), B(LDB,)
CHARACTER(1)	UPLO

3 Description

The Cholesky factorization is used to factor

A

A = U^{H} U

, if

UPLO ='U'

, or

A = L L^{H}

, if

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

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5 Parameters

1: UPLO – CHARACTER(1)Input

On entry: if

UPLO ='U'

, the upper triangle of the matrix

A

is stored.

UPLO ='L'

, the lower triangle of the matrix

A

is stored.

Constraint:

UPLO ='U'

'L'

2: N – INTEGERInput

On entry: the number of linear equations

n

, i.e., the order of the matrix

A

Constraint:

N \geq 0

3: NRHS – INTEGERInput

On entry: the number of right-hand sides

r

, i.e., the number of columns of the matrix

B

Constraint:

NRHS \geq 0

4: AP( $*$ ) – COMPLEX (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

Hermitian matrix

A

. The upper or lower triangular part of the Hermitian matrix is packed column-wise in a linear array. The

j

th column of

A

is stored in the array AP as follows:

if $UPLO ='U'$ , $AP (i + (j - 1) j / 2) = a_{i j}$ for $1 \leq i \leq j$ ;
if $UPLO ='L'$ , $AP (i + (j - 1) (2 n - j) / 2) = a_{i j}$ for $j \leq i \leq n$ .

See Section 8 below for further details.

On exit: if

IFAIL = 0

N + 1

, the factor

U

L

from the Cholesky factorization

A = U^{H} U

A = L L^{H}

, in the same storage format as

A

5: B(LDB, $*$ ) – COMPLEX (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

matrix of right-hand sides

B

On exit: if

IFAIL = 0

N + 1

, 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 F04CEF is called.

Constraint:

LDB \geq \max (1, N)

7: RCOND – REAL (KIND=nag_wp)Output

On exit: if

IFAIL = 0

N + 1

, an estimate of the reciprocal of the condition number of the matrix

A

, computed as

RCOND = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})

8: ERRBND – REAL (KIND=nag_wp)Output

On exit: if

IFAIL = 0

N + 1

, an estimate of the forward error bound for a computed solution

\hat{x}

, such that

{‖\hat{x} - x‖}_{1} / {‖x‖}_{1} \leq ERRBND

, where

\hat{x}

is a column of the computed solution returned in the array B and

x

is the corresponding column of the exact solution

X

. If RCOND is less than machine precision, then ERRBND is returned as unity.

9: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL < 0 and IFAIL \neq - 999$: If $IFAIL = - i$ , the $i$ th argument had an illegal value.

$IFAIL = - 999$: Allocation of memory failed. The real allocatable memory required is N, and the complex allocatable memory required is $2 \times N$ . Allocation failed before the solution could be computed.

$IFAIL > 0 and IFAIL \leq N$: If $IFAIL = i$ , the leading minor of order $i$ of $A$ is not positive definite. The factorization could not be completed, and the solution has not been computed.

$IFAIL = N + 1$: RCOND is less than machine precision, so that the matrix $A$ is numerically singular. A solution to the equations $A X = B$ has nevertheless 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. F04CEF uses the approximation

{‖E‖}_{1} = ε {‖A‖}_{1}

to estimate ERRBND. See Section 4.4 of Anderson et al. (1999) for further details.

8 Further Comments

The packed storage scheme is illustrated by the following example when

n = 4

and

UPLO ='U'

. Two-dimensional storage of the Hermitian matrix

A

\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{22} & a_{23} & a_{24} \\ a_{33} & a_{34} \\ a_{44} \end{matrix} (a_{i j} = {\bar{a}}_{j i})

Packed storage of the upper triangle of

A

AP = [\begin{matrix} a_{11}, & a_{12}, & a_{22}, & a_{13}, & a_{23}, & a_{33}, & a_{14}, & a_{24}, & a_{34}, & a_{44} \end{matrix}]

The total number of floating point operations required to solve the equations

A X = B

is proportional to

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

. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The real analogue of F04CEF is F04BEF.

9 Example

This example solves the equations

A X = B,

where

A

is the Hermitian positive definite matrix

A = (\begin{array}{r} 3.23 & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 \end{array})

and

B = (\begin{array}{r} 3.93 - 6.14 i & 1.48 + 6.58 i \\ 6.17 + 9.42 i & 4.65 - 4.75 i \\ - 7.17 - 21.83 i & - 4.91 + 2.29 i \\ 1.99 - 14.38 i & 7.64 - 10.79 i \end{array}) .

An estimate of the condition number of

A

and an approximate error bound for the computed solutions are also printed.

NAG Library Routine DocumentF04CEF

+− 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

F04CEF