NAG Library Routine Document

F04CFF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

1 Purpose

F04CFF computes the solution to a complex system of linear equations

A X = B

, where

A

is an

n

n

Hermitian positive definite band matrix of band width

2 k + 1

, 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 F04CFF (

UPLO, N, KD, NRHS, AB, LDAB, B, LDB, RCOND, ERRBND, IFAIL)

INTEGER	N, KD, NRHS, LDAB, LDB, IFAIL
REAL (KIND=nag_wp)	RCOND, ERRBND
COMPLEX (KIND=nag_wp)	AB(LDAB,), 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 band matrix with

k

superdiagonals, and

L

is a lower triangular band matrix with

k

subdiagonals. 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: $KD$ – INTEGERInput

On entry: the number of superdiagonals

k

(and the number of subdiagonals) of the band matrix

A

Constraint:

KD \geq 0

4: $NRHS$ – INTEGERInput

On entry: the number of right-hand sides

r

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

B

Constraint:

NRHS \geq 0

5: $AB (LDAB *)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: the

n

n

Hermitian band matrix

A

. The upper or lower triangular part of the Hermitian matrix is stored in the first

KD + 1

rows of the array. The

j

th column of

A

is stored in the

j

th column of the array AB as follows:

The matrix is stored in rows

1

k + 1

, more precisely,

if $UPLO ='U'$ , the elements of the upper triangle of $A$ within the band must be stored with element $A_{i j}$ in $AB (k + 1 + i - j, j) for \max (1, j - k) \leq i \leq j$ ;
if $UPLO ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $AB (1 + i - j, j) for j \leq i \leq \min (n, j + k) .$

See Section 9 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

6: $LDAB$ – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F04CFF is called.

Constraint:

LDAB \geq KD + 1

7: $B (LDB *)$ – 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

8: $LDB$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F04CFF is called.

Constraint:

LDB \geq \max (1, N)

9: $RCOND$ – 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})

10: $ERRBND$ – 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.

11: $IFAIL$ – 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.

$IFAIL > 0 and IFAIL \leq N$: 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.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

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. F04CFF uses the approximation

{‖E‖}_{1} = ε {‖A‖}_{1}

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

8 Parallelism and Performance

F04CFF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F04CFF 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The band storage scheme for the array AB is illustrated by the following example, when

n = 6

k = 2

, and

UPLO ='U'

On entry:

\begin{matrix} * & * & a_{13} & a_{24} & a_{35} & a_{46} \\ * & a_{12} & a_{23} & a_{34} & a_{45} & a_{56} \\ a_{11} & a_{22} & a_{33} & a_{44} & a_{55} & a_{66} \end{matrix}

On exit:

\begin{matrix} * & * & u_{13} & u_{24} & u_{35} & u_{46} \\ * & u_{12} & u_{23} & u_{34} & u_{45} & u_{56} \\ u_{11} & u_{22} & u_{33} & u_{44} & u_{55} & u_{66} \end{matrix}

Similarly, if

UPLO ='L'

the format of AB is as follows:

On entry:

\begin{matrix} a_{11} & a_{22} & a_{33} & a_{44} & a_{55} & a_{66} \\ a_{21} & a_{32} & a_{43} & a_{54} & a_{65} & * \\ a_{31} & a_{42} & a_{53} & a_{64} & * & * \end{matrix}

On exit:

\begin{matrix} l_{11} & l_{22} & l_{33} & l_{44} & l_{55} & l_{66} \\ l_{21} & l_{32} & l_{43} & l_{54} & l_{65} & * \\ l_{31} & l_{42} & l_{53} & l_{64} & * & * \end{matrix}

Array elements marked

*

need not be set and are not referenced by the routine.

Assuming that

n ≫ k

, the total number of floating-point operations required to solve the equations

A X = B

is approximately

{n (k + 1)}^{2}

for the factorization and

4 n k r

for the solution following the factorization. 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 F04CFF is F04BFF.

10 Example

This example solves the equations

A X = B,

where

A

is the Hermitian positive definite band matrix

A = (\begin{array}{r} 9.39 & 1.08 - 1.73 i & 0 & 0 \\ 1.08 + 1.73 i & 1.69 & - 0.04 + 0.29 i & 0 \\ 0 & - 0.04 - 0.29 i & 2.65 & - 0.33 + 2.24 i \\ 0 & 0 & - 0.33 - 2.24 i & 2.17 \end{array})

and

B = (\begin{array}{r} - 12.42 + 68.42 i & 54.30 - 56.56 i \\ - 9.93 + 0.88 i & 18.32 + 4.76 i \\ - 27.30 - 0.01 i & - 4.40 + 9.97 i \\ 5.31 + 23.63 i & 9.43 + 1.41 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 DocumentF04CFF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine Document

F04CFF