# pois3d.f

          SUBROUTINE POIS3D (LPEROD, L, C1, MPEROD, M, C2, NPEROD, N, A, B,
         +   C, LDIMF, MDIMF, F, IERROR, W)
    C***BEGIN PROLOGUE  POIS3D
    C***PURPOSE  Solve a three-dimensional block tridiagonal linear system
    C            which arises from a finite difference approximation to a
    C            three-dimensional Poisson equation using the Fourier
    C            transform package FFTPAK written by Paul Swarztrauber.
    C***LIBRARY   SLATEC (FISHPACK)
    C***CATEGORY  I2B4B
    C***TYPE      SINGLE PRECISION (POIS3D-S)
    C***KEYWORDS  ELLIPTIC PDE, FISHPACK, HELMHOLTZ, POISSON
    C***AUTHOR  Adams, J., (NCAR)
    C           Swarztrauber, P. N., (NCAR)
    C           Sweet, R., (NCAR)
    C***DESCRIPTION
    C
    C     Subroutine POIS3D solves the linear system of equations
    C
    C       C1*(X(I-1,J,K)-2.*X(I,J,K)+X(I+1,J,K))
    C     + C2*(X(I,J-1,K)-2.*X(I,J,K)+X(I,J+1,K))
    C     + A(K)*X(I,J,K-1)+B(K)*X(I,J,K)+C(K)*X(I,J,K+1) = F(I,J,K)
    C
    C     for  I=1,2,...,L , J=1,2,...,M , and K=1,2,...,N .
    C
    C     The indices K-1 and K+1 are evaluated modulo N, i.e.
    C     X(I,J,0) = X(I,J,N) and X(I,J,N+1) = X(I,J,1). The unknowns
    C     X(0,J,K), X(L+1,J,K), X(I,0,K), and X(I,M+1,K) are assumed to take
    C     on certain prescribed values described below.
    C
    C    * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
    C
    C
    C    * * * * * * * *    Parameter Description     * * * * * * * * * *
    C
    C
    C            * * * * * *   On Input    * * * * * *
    C
    C     LPEROD   Indicates the values that X(0,J,K) and X(L+1,J,K) are
    C              assumed to have.
    C
    C              = 0  If X(0,J,K) = X(L,J,K) and X(L+1,J,K) = X(1,J,K).
    C              = 1  If X(0,J,K) = X(L+1,J,K) = 0.
    C              = 2  If X(0,J,K) = 0  and X(L+1,J,K) = X(L-1,J,K).
    C              = 3  If X(0,J,K) = X(2,J,K) and X(L+1,J,K) = X(L-1,J,K).
    C              = 4  If X(0,J,K) = X(2,J,K) and X(L+1,J,K) = 0.
    C
    C     L        The number of unknowns in the I-direction. L must be at
    C              least 3.
    C
    C     C1       The real constant that appears in the above equation.
    C
    C     MPEROD   Indicates the values that X(I,0,K) and X(I,M+1,K) are
    C              assumed to have.
    C
    C              = 0  If X(I,0,K) = X(I,M,K) and X(I,M+1,K) = X(I,1,K).
    C              = 1  If X(I,0,K) = X(I,M+1,K) = 0.
    C              = 2  If X(I,0,K) = 0 and X(I,M+1,K) = X(I,M-1,K).
    C              = 3  If X(I,0,K) = X(I,2,K) and X(I,M+1,K) = X(I,M-1,K).
    C              = 4  If X(I,0,K) = X(I,2,K) and X(I,M+1,K) = 0.
    C
    C     M        The number of unknowns in the J-direction. M must be at
    C              least 3.
    C
    C     C2       The real constant which appears in the above equation.
    C
    C     NPEROD   = 0  If A(1) and C(N) are not zero.
    C              = 1  If A(1) = C(N) = 0.
    C
    C     N        The number of unknowns in the K-direction. N must be at
    C              least 3.
    C
    C
    C     A,B,C    One-dimensional arrays of length N that specify the
    C              coefficients in the linear equations given above.
    C
    C              If NPEROD = 0 the array elements must not depend upon the
    C              index K, but must be constant.  Specifically, the
    C              subroutine checks the following condition
    C
    C                          A(K) = C(1)
    C                          C(K) = C(1)
    C                          B(K) = B(1)
    C
    C                  for K=1,2,...,N.
    C
    C     LDIMF    The row (or first) dimension of the three-dimensional
    C              array F as it appears in the program calling POIS3D.
    C              This parameter is used to specify the variable dimension
    C              of F.  LDIMF must be at least L.
    C
    C     MDIMF    The column (or second) dimension of the three-dimensional
    C              array F as it appears in the program calling POIS3D.
    C              This parameter is used to specify the variable dimension
    C              of F.  MDIMF must be at least M.
    C
    C     F        A three-dimensional array that specifies the values of
    C              the right side of the linear system of equations given
    C              above.  F must be dimensioned at least L x M x N.
    C
    C     W        A one-dimensional array that must be provided by the
    C              user for work space.  The length of W must be at least
    C              30 + L + M + 2*N + MAX(L,M,N) +
    C              7*(INT((L+1)/2) + INT((M+1)/2)).
    C
    C
    C            * * * * * *   On Output   * * * * * *
    C
    C     F        Contains the solution X.
    C
    C     IERROR   An error flag that indicates invalid input parameters.
    C              Except for number zero, a solution is not attempted.
    C              = 0  No error
    C              = 1  If LPEROD .LT. 0 or .GT. 4
    C              = 2  If L .LT. 3
    C              = 3  If MPEROD .LT. 0 or .GT. 4
    C              = 4  If M .LT. 3
    C              = 5  If NPEROD .LT. 0 or .GT. 1
    C              = 6  If N .LT. 3
    C              = 7  If LDIMF .LT. L
    C              = 8  If MDIMF .LT. M
    C              = 9  If A(K) .NE. C(1) or C(K) .NE. C(1) or B(I) .NE.B(1)
    C                      for some K=1,2,...,N.
    C              = 10 If NPEROD = 1 and A(1) .NE. 0 or C(N) .NE. 0
    C
    C              Since this is the only means of indicating a possibly
    C              incorrect call to POIS3D, the user should test IERROR
    C              after the call.
    C
    C *Long Description:
    C
    C    * * * * * * *   Program Specifications    * * * * * * * * * * * *
    C
    C     Dimension of   A(N),B(N),C(N),F(LDIMF,MDIMF,N),
    C     Arguments      W(see argument list)
    C
    C     Latest         December 1, 1978
    C     Revision
    C
    C     Subprograms    POIS3D,POS3D1,TRIDQ,RFFTI,RFFTF,RFFTF1,RFFTB,
    C     Required       RFFTB1,COSTI,COST,SINTI,SINT,COSQI,COSQF,COSQF1
    C                    COSQB,COSQB1,SINQI,SINQF,SINQB,CFFTI,CFFTI1,
    C                    CFFTB,CFFTB1,PASSB2,PASSB3,PASSB4,PASSB,CFFTF,
    C                    CFFTF1,PASSF1,PASSF2,PASSF3,PASSF4,PASSF,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        Written by Roland Sweet at NCAR in July 1977
    C
    C     Algorithm      This subroutine solves three-dimensional block
    C                    tridiagonal linear systems arising from finite
    C                    difference approximations to three-dimensional
    C                    Poisson equations using the Fourier transform
    C                    package FFTPAK written by Paul Swarztrauber.
    C
    C     Space          6561(decimal) = 14641(octal) locations on the
    C     Required       NCAR Control Data 7600
    C
    C     Timing and        The execution time T on the NCAR Control Data
    C     Accuracy       7600 for subroutine POIS3D is roughly proportional
    C                    to L*M*N*(log2(L)+log2(M)+5), but also depends on
    C                    input parameters LPEROD and MPEROD.  Some typical
    C                    values are listed in the table below when NPEROD=0.
    C                       To measure the accuracy of the algorithm a
    C                    uniform random number generator was used to create
    C                    a solution array X for the system given in the
    C                    'PURPOSE' with
    C
    C                       A(K) = C(K) = -0.5*B(K) = 1,       K=1,2,...,N
    C
    C                    and, when NPEROD = 1
    C
    C                       A(1) = C(N) = 0
    C                       A(N) = C(1) = 2.
    C
    C                    The solution X was substituted into the given sys-
    C                    tem and, using double precision, a right side Y was
    C                    computed.  Using this array Y subroutine POIS3D was
    C                    called to produce an approximate solution Z.  Then
    C                    the relative error, defined as
    C
    C                    E = MAX(ABS(Z(I,J,K)-X(I,J,K)))/MAX(ABS(X(I,J,K)))
    C
    C                    where the two maxima are taken over I=1,2,...,L,
    C                    J=1,2,...,M and K=1,2,...,N, was computed.  The
    C                    value of E is given in the table below for some
    C                    typical values of L,M and N.
    C
    C
    C                       L(=M=N)   LPEROD    MPEROD    T(MSECS)    E
    C                       ------    ------    ------    --------  ------
    C
    C                         16        0         0         272     1.E-13
    C                         15        1         1         287     4.E-13
    C                         17        3         3         338     2.E-13
    C                         32        0         0        1755     2.E-13
    C                         31        1         1        1894     2.E-12
    C                         33        3         3        2042     7.E-13
    C
    C
    C     Portability    American National Standards Institute FORTRAN.
    C                    The machine dependent constant PI is defined in
    C                    function PIMACH.
    C
    C     Required       COS,SIN,ATAN
    C     Resident
    C     Routines
    C
    C     Reference      NONE
    C
    C    * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
    C
    C***REFERENCES  (NONE)
    C***ROUTINES CALLED  POS3D1
    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***END PROLOGUE  POIS3D