NAG Library Routine Document

F04CBF

▸▿ 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

F04CBF computes the solution to a complex system of linear equations

A X = B

, where

A

is an

n

n

band matrix, with

k_{l}

subdiagonals and

k_{u}

superdiagonals, 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 F04CBF (

N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, RCOND, ERRBND, IFAIL)

INTEGER	N, KL, KU, NRHS, LDAB, IPIV(N), LDB, IFAIL
REAL (KIND=nag_wp)	RCOND, ERRBND
COMPLEX (KIND=nag_wp)	AB(LDAB,), B(LDB,)

3 Description

The

L U

decomposition with partial pivoting and row interchanges is used to factor

A

A = P L U

, where

P

is a permutation matrix,

L

is the product of permutation matrices and unit lower triangular matrices with

k_{l}

subdiagonals, and

U

is upper triangular with

(k_{l} + k_{u})

superdiagonals. 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: $N$ – INTEGERInput

On entry: the number of linear equations

n

, i.e., the order of the matrix

A

Constraint:

N \geq 0

2: $KL$ – INTEGERInput

On entry: the number of subdiagonals

k_{l}

, within the band of

A

Constraint:

KL \geq 0

3: $KU$ – INTEGERInput

On entry: the number of superdiagonals

k_{u}

, within the band of

A

Constraint:

KU \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

matrix

A

The matrix is stored in rows

k_{l} + 1

2 k_{l} + k_{u} + 1

; the first

k_{l}

rows need not be set, more precisely, the element

A_{i j}

must be stored in

AB (k_{l} + k_{u} + 1 + i - j, j) = A_{i j} for ​ \max (1, j - k_{u}) \leq i \leq \min (n, j + k_{l}) .

See Section 9 for further details.

On exit: if

IFAIL \geq 0

, AB is overwritten by details of the factorization.

The upper triangular band matrix

U

, with

k_{l} + k_{u}

superdiagonals, is stored in rows

1

k_{l} + k_{u} + 1

of the array, and the multipliers used to form the matrix

L

are stored in rows

k_{l} + k_{u} + 2

2 k_{l} + k_{u} + 1

6: $LDAB$ – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F04CBF is called.

Constraint:

LDAB \geq 2 \times KL + KU + 1

7: $IPIV (N)$ – INTEGER arrayOutput

On exit: if

IFAIL \geq 0

, the pivot indices that define the permutation matrix

P

; at the

i

th step row

i

of the matrix was interchanged with row

IPIV (i)

IPIV (i) = i

indicates a row interchange was not required.

8: $B (LDB *)$ – 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

9: $LDB$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F04CBF is called.

Constraint:

LDB \geq \max (1, N)

10: $RCOND$ – REAL (KIND=nag_wp)Output

On exit: if

IFAIL \geq 0

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

A

, computed as

RCOND = ({‖A‖}_{1} {‖A^{- 1}‖}_{1})

11: $ERRBND$ – 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.

12: $IFAIL$ – 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 $IFAIL = i$ , $u_{i i}$ is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution could not be 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. F04CBF 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

F04CBF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F04CBF 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 real allocatable memory required is N, and the complex allocatable memory required is

2 \times N

. In this case the factorization and the solution

X

have been computed, but RCOND and ERRBND have not been computed.

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

n = 6

k_{l} = 1

, and

k_{u} = 2

. Storage of the band matrix

A

in the array AB:

\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} \\ a_{21} & a_{32} & a_{43} & a_{54} & a_{65} & * \end{matrix}

Array elements marked

*

need not be set and are not referenced by the routine. Array elements marked + need not be set, but are defined on exit from the routine and contain the elements

u_{14}

u_{25}

and

u_{36}

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

A X = B

depends upon the pivoting required, but if

n ≫ k_{l} + k_{u}

then it is approximately bounded by

O (n k_{l} (k_{l} + k_{u}))

for the factorization and

O (n (2 k_{l} + k_{u}), 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 F04CBF is F04BBF.

10 Example

This example solves the equations

A X = B,

where

A

is the band matrix

A = (\begin{array}{r} - 1.65 + 2.26 i & - 2.05 - 0.85 i & 0.97 - 2.84 i & 0 \\ 0.00 + 6.30 i & - 1.48 - 1.75 i & - 3.99 + 4.01 i & 0.59 - 0.48 i \\ 0 & - 0.77 + 2.83 i & - 1.06 + 1.94 i & 3.33 - 1.04 i \\ 0 & 0 & 4.48 - 1.09 i & - 0.46 - 1.72 i \end{array})

and

B = (\begin{array}{r} - 1.06 + 21.50 i & 12.85 + 2.84 i \\ - 22.72 - 53.90 i & - 70.22 + 21.57 i \\ 28.24 - 38.60 i & - 20.73 - 1.23 i \\ - 34.56 + 16.73 i & 26.01 + 31.97 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 DocumentF04CBF

▸▿ 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

F04CBF