F406 Banded Linear Equations

Subroutine subprograms RBEQN and DBEQN solve a system of N simultaneous linear equations with K right-hand sides, the coefficient matrix being a band matrix with bandwidth 2M+1:

$\sum$_j=1^Na_ijx_jk= b_ik,(i=1,2,...,N, k=1,2,...,K); (a_ij= 0 for |i-j| > M).

Only those coefficients $a$_ij for which $|i-j|\le M$

need be supplied on entry (see Usage).

Structure:

SUBROUTINE subprograms
User Entry Names: RBEQN, DBEQN
Files Referenced: Printer
External References: KERMTR (N001), ABEND (Z035)

Usage:

For $t=R$ (type REAL), $t=D$ (type DOUBLE PRECISION),

    CALL tBEQN(N,M,ABAND,IDIM,IFAIL,K,B)

N

( INTEGER) Number of equations.

M

( INTEGER) Band parameter M.

ABAND

(type according to t) Two-dimensional array whose first dimension has the value IDIM.

IDIM

( INTEGER) First dimension of array ABAND (and of array B if $K>1$ ).

IFAIL

( INTEGER) On exit, IFAIL will be set to -1 if the coefficient matrix is singular, and to 0 otherwise.

K

( INTEGER) Number of right-hand sides in array B.

B

(type according to t) In general, a two-dimensional array whose first dimension has the value IDIM. B may be one-dimensional if $K=1$ .

On entry, ABAND must contain the packed form of the coefficient matrix as described below, and array B must contain the matrix of right-hand sides $b$_ik . Then, provided the coefficient matrix is non-singular, IFAIL will be set to 0 and the solution $x$_ik will replace $b$_ik in B. The contents of ABAND are destroyed. If the coefficient matrix is singular, IFAIL will be set to -1. In this case the contents of ABAND and B are unpredictable.

The storage convention for ABAND is that it must contain, on entry, those coefficients $a$_ij for which $|i\; -\; j|\le M$ , stored "left-justified" as an array of N rows and at most $2M+1$ columns. For example, if $N=4$ and $M=1$ , the coefficient matrix

$($

$a$11	$a$12	$0$	$0$
$a$21	$a$22	$a$23	$0$
$0$	$a$32	$a$33	$a$34
$0$	$0$	$a$43	$a$44

$)is\; stored\; as($

$a$11	$a$12	$X$
$a$21	$a$22	$a$23
$a$32	$a$33	$a$34
$a$43	$a$44	$X$

$)$

where X denotes elements whose value need not to be set.

If ALPHA(I,J) is a function subprogram or statement function which computes $a$_ij , the following Fortran statements will set ABAND correctly:

      DO 2 I =1,N
      L = 1
      DO 1 J = MAX(I-M,1),MIN(I+M,N)
      ABAND(I,L) = ALPHA(I,J)
      L = L+1
    1 CONTINUE
    2 CONTINUE

Method:

Gaussian elimination with row interchanges. The storage organization is as described in the reference.

Error handling:

If the integer arguments do not satisfy the conditions $1\; \le M+1\; \le N\; \le IDIM,\hspace{0.17em}K\; \le 0$ , a message is printed and program execution is terminated by calling ABEND (Z035).

References:

1. J.H. Wilkinson and C. Reinsch (eds.), Handbook for automatic computation, Vol.2: Linear algebra (Springer-Verlag, New York 1971) 54.

F500