SLATEC Routines --- HWSCRT ---


*DECK HWSCRT
      SUBROUTINE HWSCRT (A, B, M, MBDCND, BDA, BDB, C, D, N, NBDCND,
     +   BDC, BDD, ELMBDA, F, IDIMF, PERTRB, IERROR, W)
C***BEGIN PROLOGUE  HWSCRT
C***PURPOSE  Solves the standard five-point finite difference
C            approximation to the Helmholtz equation in Cartesian
C            coordinates.
C***LIBRARY   SLATEC (FISHPACK)
C***CATEGORY  I2B1A1A
C***TYPE      SINGLE PRECISION (HWSCRT-S)
C***KEYWORDS  CARTESIAN, ELLIPTIC, FISHPACK, HELMHOLTZ, PDE
C***AUTHOR  Adams, J., (NCAR)
C           Swarztrauber, P. N., (NCAR)
C           Sweet, R., (NCAR)
C***DESCRIPTION
C
C     Subroutine HWSCRT solves the standard five-point finite
C     difference approximation to the Helmholtz equation in Cartesian
C     coordinates:
C
C          (d/dX)(dU/dX) + (d/dY)(dU/dY) + LAMBDA*U = F(X,Y).
C
C
C
C     * * * * * * * *    Parameter Description     * * * * * * * * * *
C
C             * * * * * *   On Input    * * * * * *
C
C     A,B
C       The range of X, i.e., A .LE. X .LE. B.  A must be less than B.
C
C     M
C       The number of panels into which the interval (A,B) is
C       subdivided.  Hence, there will be M+1 grid points in the
C       X-direction given by X(I) = A+(I-1)DX for I = 1,2,...,M+1,
C       where DX = (B-A)/M is the panel width. M must be greater than 3.
C
C     MBDCND
C       Indicates the type of boundary conditions at X = A and X = B.
C
C       = 0  If the solution is periodic in X, i.e., U(I,J) = U(M+I,J).
C       = 1  If the solution is specified at X = A and X = B.
C       = 2  If the solution is specified at X = A and the derivative of
C            the solution with respect to X is specified at X = B.
C       = 3  If the derivative of the solution with respect to X is
C            specified at X = A and X = B.
C       = 4  If the derivative of the solution with respect to X is
C            specified at X = A and the solution is specified at X = B.
C
C     BDA
C       A one-dimensional array of length N+1 that specifies the values
C       of the derivative of the solution with respect to X at X = A.
C       When MBDCND = 3 or 4,
C
C            BDA(J) = (d/dX)U(A,Y(J)), J = 1,2,...,N+1  .
C
C       When MBDCND has any other value, BDA is a dummy variable.
C
C     BDB
C       A one-dimensional array of length N+1 that specifies the values
C       of the derivative of the solution with respect to X at X = B.
C       When MBDCND = 2 or 3,
C
C            BDB(J) = (d/dX)U(B,Y(J)), J = 1,2,...,N+1  .
C
C       When MBDCND has any other value BDB is a dummy variable.
C
C     C,D
C       The range of Y, i.e., C .LE. Y .LE. D.  C must be less than D.
C
C     N
C       The number of panels into which the interval (C,D) is
C       subdivided.  Hence, there will be N+1 grid points in the
C       Y-direction given by Y(J) = C+(J-1)DY for J = 1,2,...,N+1, where
C       DY = (D-C)/N is the panel width.  N must be greater than 3.
C
C     NBDCND
C       Indicates the type of boundary conditions at Y = C and Y = D.
C
C       = 0  If the solution is periodic in Y, i.e., U(I,J) = U(I,N+J).
C       = 1  If the solution is specified at Y = C and Y = D.
C       = 2  If the solution is specified at Y = C and the derivative of
C            the solution with respect to Y is specified at Y = D.
C       = 3  If the derivative of the solution with respect to Y is
C            specified at Y = C and Y = D.
C       = 4  If the derivative of the solution with respect to Y is
C            specified at Y = C and the solution is specified at Y = D.
C
C     BDC
C       A one-dimensional array of length M+1 that specifies the values
C       of the derivative of the solution with respect to Y at Y = C.
C       When NBDCND = 3 or 4,
C
C            BDC(I) = (d/dY)U(X(I),C), I = 1,2,...,M+1  .
C
C       When NBDCND has any other value, BDC is a dummy variable.
C
C     BDD
C       A one-dimensional array of length M+1 that specifies the values
C       of the derivative of the solution with respect to Y at Y = D.
C       When NBDCND = 2 or 3,
C
C            BDD(I) = (d/dY)U(X(I),D), I = 1,2,...,M+1  .
C
C       When NBDCND has any other value, BDD is a dummy variable.
C
C     ELMBDA
C       The constant LAMBDA in the Helmholtz equation.  If
C       LAMBDA .GT. 0, a solution may not exist.  However, HWSCRT will
C       attempt to find a solution.
C
C     F
C       A two-dimensional array which specifies the values of the right
C       side of the Helmholtz equation and boundary values (if any).
C       For I = 2,3,...,M and J = 2,3,...,N
C
C            F(I,J) = F(X(I),Y(J)).
C
C       On the boundaries F is defined by
C
C            MBDCND     F(1,J)        F(M+1,J)
C            ------     ---------     --------
C
C              0        F(A,Y(J))     F(A,Y(J))
C              1        U(A,Y(J))     U(B,Y(J))
C              2        U(A,Y(J))     F(B,Y(J))     J = 1,2,...,N+1
C              3        F(A,Y(J))     F(B,Y(J))
C              4        F(A,Y(J))     U(B,Y(J))
C
C
C            NBDCND     F(I,1)        F(I,N+1)
C            ------     ---------     --------
C
C              0        F(X(I),C)     F(X(I),C)
C              1        U(X(I),C)     U(X(I),D)
C              2        U(X(I),C)     F(X(I),D)     I = 1,2,...,M+1
C              3        F(X(I),C)     F(X(I),D)
C              4        F(X(I),C)     U(X(I),D)
C
C       F must be dimensioned at least (M+1)*(N+1).
C
C       NOTE:
C
C       If the table calls for both the solution U and the right side F
C       at a corner then the solution must be specified.
C
C     IDIMF
C       The row (or first) dimension of the array F as it appears in the
C       program calling HWSCRT.  This parameter is used to specify the
C       variable dimension of F.  IDIMF must be at least M+1  .
C
C     W
C       A one-dimensional array that must be provided by the user for
C       work space.  W may require up to 4*(N+1) +
C       (13 + INT(log2(N+1)))*(M+1) locations.  The actual number of
C       locations used is computed by HWSCRT and is returned in location
C       W(1).
C
C
C             * * * * * *   On Output     * * * * * *
C
C     F
C       Contains the solution U(I,J) of the finite difference
C       approximation for the grid point (X(I),Y(J)), I = 1,2,...,M+1,
C       J = 1,2,...,N+1  .
C
C     PERTRB
C       If a combination of periodic or derivative boundary conditions
C       is specified for a Poisson equation (LAMBDA = 0), a solution may
C       not exist.  PERTRB is a constant, calculated and subtracted from
C       F, which ensures that a solution exists.  HWSCRT then computes
C       this solution, which is a least squares solution to the original
C       approximation.  This solution plus any constant is also a
C       solution.  Hence, the solution is not unique.  The value of
C       PERTRB should be small compared to the right side F.  Otherwise,
C       a solution is obtained to an essentially different problem.
C       This comparison should always be made to insure that a
C       meaningful solution has been obtained.
C
C     IERROR
C       An error flag that indicates invalid input parameters.  Except
C       for numbers 0 and 6, a solution is not attempted.
C
C       = 0  No error.
C       = 1  A .GE. B.
C       = 2  MBDCND .LT. 0 or MBDCND .GT. 4  .
C       = 3  C .GE. D.
C       = 4  N .LE. 3
C       = 5  NBDCND .LT. 0 or NBDCND .GT. 4  .
C       = 6  LAMBDA .GT. 0  .
C       = 7  IDIMF .LT. M+1  .
C       = 8  M .LE. 3
C
C       Since this is the only means of indicating a possibly incorrect
C       call to HWSCRT, the user should test IERROR after the call.
C
C     W
C       W(1) contains the required length of W.
C
C *Long Description:
C
C     * * * * * * *   Program Specifications    * * * * * * * * * * * *
C
C
C     Dimension of   BDA(N+1),BDB(N+1),BDC(M+1),BDD(M+1),F(IDIMF,N+1),
C     Arguments      W(see argument list)
C
C     Latest         June 1, 1976
C     Revision
C
C     Subprograms    HWSCRT,GENBUN,POISD2,POISN2,POISP2,COSGEN,MERGE,
C     Required       TRIX,TRI3,PIMACH
C
C     Special        NONE
C     Conditions
C
C     Common         NONE
C     Blocks
C
C     I/O            NONE
C
C     Precision      Single
C
C     Specialist     Roland Sweet
C
C     Language       FORTRAN
C
C     History        Standardized September 1, 1973
C                    Revised April 1, 1976
C
C     Algorithm      The routine defines the finite difference
C                    equations, incorporates boundary data, and adjusts
C                    the right side of singular systems and then calls
C                    GENBUN to solve the system.
C
C     Space          13110(octal) = 5704(decimal) locations on the NCAR
C     Required       Control Data 7600
C
C     Timing and        The execution time T on the NCAR Control Data
C     Accuracy       7600 for subroutine HWSCRT is roughly proportional
C                    to M*N*log2(N), but also depends on the input
C                    parameters NBDCND and MBDCND.  Some typical values
C                    are listed in the table below.
C                       The solution process employed results in a loss
C                    of no more than three significant digits for N and
C                    M as large as 64.  More detailed information about
C                    accuracy can be found in the documentation for
C                    subroutine GENBUN which is the routine that
C                    solves the finite difference equations.
C
C
C                       M(=N)    MBDCND    NBDCND    T(MSECS)
C                       -----    ------    ------    --------
C
C                        32        0         0          31
C                        32        1         1          23
C                        32        3         3          36
C                        64        0         0         128
C                        64        1         1          96
C                        64        3         3         142
C
C     Portability    American National Standards Institute FORTRAN.
C                    The machine dependent constant PI is defined in
C                    function PIMACH.
C
C     Reference      Swarztrauber, P. and R. Sweet, 'Efficient FORTRAN
C                    Subprograms for The Solution Of Elliptic Equations'
C                    NCAR TN/IA-109, July, 1975, 138 pp.
C
C     * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C***REFERENCES  P. N. Swarztrauber and R. Sweet, Efficient Fortran
C                 subprograms for the solution of elliptic equations,
C                 NCAR TN/IA-109, July 1975, 138 pp.
C***ROUTINES CALLED  GENBUN
C***REVISION HISTORY  (YYMMDD)
C   801001  DATE WRITTEN
C   890531  Changed all specific intrinsics to generic.  (WRB)
C   890531  REVISION DATE from Version 3.2
C   891214  Prologue converted to Version 4.0 format.  (BAB)
C   920501  Reformatted the REFERENCES section.  (WRB)
C***END PROLOGUE  HWSCRT