Appendix 1 - GLS: Solution of a linear system

A method for solving a linear system of equations is required by both the interior and exterior boundary value problems. In this work the subroutine for this purpose is GLS and it is called by subroutines LIBEM2, LIBEM3, LIBEMA, LEBEM3 and LEBEMA. The subroutine is located in file GLS.FOR .

The use of the direct boundary element methods and the general nature of the boundary condition generally results in a linear system of equations of the following form

A x = B y + c
where x and y are the unknown vectors with n components and A and B are real n ×n matrices and c is an n-vector. The xi and yi are also related by the following equations

ai xi + bi yi = fi          for   i = 1,,,n.

This is not a standard matrix-vector problem but it can be turned into one by rearranging the rows. This solution procedure is carried out in subroutine GLS and the subroutine LINSL is called for solving the resulting standard problem by an LU factorisation method that is supplied without calling an external routine.

The parameter list of subroutine GLS is as follows. 

      SUBROUTINE GLS(MAXN,N,A,B,C,ALPHA,BETA,F,X,Y,LFAIL,
     * WKSPC1,WKSPC2)

C Input parameters
C ----------------
C The limit on the dimension of the matrices A and B 
      INTEGER    MAXN
C The dimension of the matrices
      INTEGER    N
C The matrix A
      REAL*8     A(MAXN,MAXN)
C The matrix B
      REAL*8     B(MAXN,MAXN)
C The vector c
      REAL*8     C(MAXN)
C The {\alpha}_i
      REAL*8     ALPHA(MAXN)
C The {\beta}_i
      REAL*8     BETA(MAXN)
C The f_i
      REAL*8     F(MAXN)

C Output parameters
C -----------------
C The solution vector x
      REAL*8     X(MAXN)
C The solution vector y
      REAL*8     Y(MAXN)

C Work space
C ----------
      REAL*8     WKSPC1(MAXN)
      LOGICAL    WKSPC2(MAXN)

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