NAG Library Routine Document

F04CCF

+− 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

F04CCF computes the solution to a complex system of linear equations

A X = B

, where

A

is an

n

n

tridiagonal matrix 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 F04CCF (

N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, RCOND, ERRBND, IFAIL)

INTEGER	N, NRHS, IPIV(N), LDB, IFAIL
REAL (KIND=nag_wp)	RCOND, ERRBND
COMPLEX (KIND=nag_wp)	DL(), D(), DU(), DU2(N-2), 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 unit lower triangular with at most one nonzero subdiagonal element, and

U

is an upper triangular band matrix with two superdiagonals. The factored form of

A

is then used to solve the system of equations

A X = B

Note that the equations

A^{T} X = B

may be solved by interchanging the order of the arguments DU and DL.

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: NRHS – INTEGERInput: On entry: the number of right-hand sides $r$ , i.e., the number of columns of the matrix $B$ .
Constraint: $NRHS \geq 0$ .
3: DL( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output: Note: the dimension of the array DL must be at least $\max (1, N - 1)$ .
On entry: must contain the $(n - 1)$ subdiagonal elements of the matrix $A$ .

On exit: if $IFAIL \geq 0$ , DL is overwritten by the $(n - 1)$ multipliers that define the matrix $L$ from the $L U$ factorization of $A$ .
4: D( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output: Note: the dimension of the array D must be at least $\max (1, N)$ .
On entry: must contain the $n$ diagonal elements of the matrix $A$ .

On exit: if $IFAIL \geq 0$ , D is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $L U$ factorization of $A$ .
5: DU( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output: Note: the dimension of the array DU must be at least $\max (1, N - 1)$ .
On entry: must contain the $(n - 1)$ superdiagonal elements of the matrix $A$

On exit: if $IFAIL \geq 0$ , DU is overwritten by the $(n - 1)$ elements of the first superdiagonal of $U$ .
6: DU2( $N - 2$ ) – COMPLEX (KIND=nag_wp) arrayOutput: On exit: if $IFAIL \geq 0$ , DU2 returns the $(n - 2)$ elements of the second superdiagonal of $U$ .
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 $IPIV (i)$ . $IPIV (i)$ will always be either $i$ or $(i + 1)$ ; $IPIV (i) = i$ 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 $n$ by $r$ matrix of right-hand sides $B$ .

On exit: if $IFAIL = 0$ or $N + 1$ , the $n$ by $r$ solution matrix $X$ .
9: LDB – INTEGERInput: On entry: the first dimension of the array B as declared in the (sub)program from which F04CCF is called.
Constraint: $LDB \geq \max (1, N)$ .
10: RCOND – REAL (KIND=nag_wp)Output: On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$ , computed as $RCOND = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})$ .
11: ERRBND – REAL (KIND=nag_wp)Output: On exit: if $IFAIL = 0$ or $N + 1$ , 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.

$IFAIL = - 999$: Allocation of memory failed. 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.

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

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. F04CCF 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 total number of floating point operations required to solve the equations

A X = B

is proportional to

n 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 F04CCF is F04BCF.

9 Example

This example solves the equations

A X = B,

where

A

is the tridiagonal matrix

A = (\begin{array}{r} - 1.3 + 1.3 i & 2.0 - 1.0 i & 0 & 0 & 0 \\ 1.0 - 2.0 i & - 1.3 + 1.3 i & 2.0 + 1.0 i & 0 & 0 \\ 0 & 1.0 + 1.0 i & - 1.3 + 3.3 i & - 1.0 + 1.0 i & 0 \\ 0 & 0 & 2.0 - 3.0 i & - 0.3 + 4.3 i & 1.0 - 1.0 i \\ 0 & 0 & 0 & 1.0 + 1.0 i & - 3.3 + 1.3 i \end{array})

and

B = (\begin{array}{r} 2.4 - 5.0 i & 2.7 + 6.9 i \\ 3.4 + 18.2 i & - 6.9 - 5.3 i \\ - 14.7 + 9.7 i & - 6.0 - 0.6 i \\ 31.9 - 7.7 i & - 3.9 + 9.3 i \\ - 1.0 + 1.6 i & - 3.0 + 12.2 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 DocumentF04CCF

+− 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

F04CCF