# SLATEC Routines --- GENBUN ---

```*DECK GENBUN
SUBROUTINE GENBUN (NPEROD, N, MPEROD, M, A, B, C, IDIMY, Y,
+   IERROR, W)
C***BEGIN PROLOGUE  GENBUN
C***PURPOSE  Solve by a cyclic reduction algorithm the linear system
C            of equations that results from a finite difference
C            approximation to certain 2-d elliptic PDE's on a centered
C            grid .
C***LIBRARY   SLATEC (FISHPACK)
C***CATEGORY  I2B4B
C***TYPE      SINGLE PRECISION (GENBUN-S, CMGNBN-C)
C***KEYWORDS  ELLIPTIC, FISHPACK, PDE, TRIDIAGONAL
C           Swarztrauber, P. N., (NCAR)
C           Sweet, R., (NCAR)
C***DESCRIPTION
C
C     Subroutine GENBUN solves the linear system of equations
C
C          A(I)*X(I-1,J) + B(I)*X(I,J) + C(I)*X(I+1,J)
C
C          + X(I,J-1) - 2.*X(I,J) + X(I,J+1) = Y(I,J)
C
C               for I = 1,2,...,M  and  J = 1,2,...,N.
C
C     The indices I+1 and I-1 are evaluated modulo M, i.e.,
C     X(0,J) = X(M,J) and X(M+1,J) = X(1,J), and X(I,0) may be equal to
C     0, X(I,2), or X(I,N) and X(I,N+1) may be equal to 0, X(I,N-1), or
C     X(I,1) depending on an input parameter.
C
C
C     * * * * * * * *    Parameter Description     * * * * * * * * * *
C
C             * * * * * *   On Input    * * * * * *
C
C     NPEROD
C       Indicates the values that X(I,0) and X(I,N+1) are assumed to
C       have.
C
C       = 0  If X(I,0) = X(I,N) and X(I,N+1) = X(I,1).
C       = 1  If X(I,0) = X(I,N+1) = 0  .
C       = 2  If X(I,0) = 0 and X(I,N+1) = X(I,N-1).
C       = 3  If X(I,0) = X(I,2) and X(I,N+1) = X(I,N-1).
C       = 4  If X(I,0) = X(I,2) and X(I,N+1) = 0.
C
C     N
C       The number of unknowns in the J-direction.  N must be greater
C       than 2.
C
C     MPEROD
C       = 0 if A(1) and C(M) are not zero.
C       = 1 if A(1) = C(M) = 0.
C
C     M
C       The number of unknowns in the I-direction.  M must be greater
C       than 2.
C
C     A,B,C
C       One-dimensional arrays of length M that specify the
C       coefficients in the linear equations given above.  If MPEROD = 0
C       the array elements must not depend upon the index I, but must be
C       constant.  Specifically, the subroutine checks the following
C       condition
C
C             A(I) = C(1)
C             C(I) = C(1)
C             B(I) = B(1)
C
C       for I=1,2,...,M.
C
C     IDIMY
C       The row (or first) dimension of the two-dimensional array Y as
C       it appears in the program calling GENBUN.  This parameter is
C       used to specify the variable dimension of Y.  IDIMY must be at
C       least M.
C
C     Y
C       A two-dimensional array that specifies the values of the right
C       side of the linear system of equations given above.  Y must be
C       dimensioned at least M*N.
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 + (10 + INT(log2(N)))*M
C       locations.  The actual number of locations used is computed by
C       GENBUN and is returned in location W(1).
C
C
C             * * * * * *   On Output     * * * * * *
C
C     Y
C       Contains the solution X.
C
C     IERROR
C       An error flag that indicates invalid input parameters.  Except
C       for number zero, a solution is not attempted.
C
C       = 0  No error.
C       = 1  M .LE. 2
C       = 2  N .LE. 2
C       = 3  IDIMY .LT. M
C       = 4  NPEROD .LT. 0 or NPEROD .GT. 4
C       = 5  MPEROD .LT. 0 or MPEROD .GT. 1
C       = 6  A(I) .NE. C(1) or C(I) .NE. C(1) or B(I) .NE. B(1) for
C            some I=1,2,...,M.
C       = 7  A(1) .NE. 0 or C(M) .NE. 0 and MPEROD = 1
C
C     W
C       W(1) contains the required length of W.
C
C *Long Description:
C
C     * * * * * * *   Program Specifications    * * * * * * * * * * * *
C
C     Dimension of   A(M),B(M),C(M),Y(IDIMY,N),W(see parameter list)
C     Arguments
C
C     Latest         June 1, 1976
C     Revision
C
C     Subprograms    GENBUN,POISD2,POISN2,POISP2,COSGEN,MERGE,TRIX,TRI3,
C     Required       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 April 1, 1973
C                    Revised August 20,1973
C                    Revised January 1, 1976
C
C     Algorithm      The linear system is solved by a cyclic reduction
C                    algorithm described in the reference.
C
C     Space          4944(decimal) = 11520(octal) 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 GENBUN is roughly proportional
C                    to M*N*log2(N), but also depends on the input
C                    parameter NPEROD.  Some typical values are listed
C                    in the table below.  More comprehensive timing
C                    charts may be found in the reference.
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(I) = C(I) = -0.5*B(I) = 1,       I=1,2,...,M
C
C                    and, when MPEROD = 1
C
C                       A(1) = C(M) = 0
C                       A(M) = 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 GENBUN was
C                    called to produce an approximate solution Z.  Then
C                    the relative error, defined as
C
C                       E = MAX(ABS(Z(I,J)-X(I,J)))/MAX(ABS(X(I,J)))
C
C                    where the two maxima are taken over all I=1,2,...,M
C                    and J=1,2,...,N, was computed.  The value of E is
C                    given in the table below for some typical values of
C                    M and N.
C
C
C                       M (=N)    MPEROD    NPEROD    T(MSECS)    E
C                       ------    ------    ------    --------  ------
C
C                         31        0         0          36     6.E-14
C                         31        1         1          21     4.E-13
C                         31        1         3          41     3.E-13
C                         32        0         0          29     9.E-14
C                         32        1         1          32     3.E-13
C                         32        1         3          48     1.E-13
C                         33        0         0          36     9.E-14
C                         33        1         1          30     4.E-13
C                         33        1         3          34     1.E-13
C                         63        0         0         150     1.E-13
C                         63        1         1          91     1.E-12
C                         63        1         3         173     2.E-13
C                         64        0         0         122     1.E-13
C                         64        1         1         128     1.E-12
C                         64        1         3         199     6.E-13
C                         65        0         0         143     2.E-13
C                         65        1         1         120     1.E-12
C                         65        1         3         138     4.E-13
C
C     Portability    American National Standards Institute Fortran.
C                    The machine dependent constant PI is defined in
C                    function PIMACH.
C
C     Required       COS
C     Resident
C     Routines
C
C     Reference      Sweet, R., 'A Cyclic Reduction Algorithm For
C                    Solving Block Tridiagonal Systems Of Arbitrary
C                    Dimensions,' SIAM J. on Numer. Anal.,
C                    14(Sept., 1977), PP. 706-720.
C
C     * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C***REFERENCES  R. Sweet, A cyclic reduction algorithm for solving
C                 block tridiagonal systems of arbitrary dimensions,
C                 SIAM Journal on Numerical Analysis 14, (September
C                 1977), pp. 706-720.
C***ROUTINES CALLED  POISD2, POISN2, POISP2
C***REVISION HISTORY  (YYMMDD)
C   801001  DATE WRITTEN
C   861211  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  GENBUN
```