# hwscrt.f

          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