SLATEC Routines --- CMGNBN ---

```*DECK CMGNBN
SUBROUTINE CMGNBN (NPEROD, N, MPEROD, M, A, B, C, IDIMY, Y,
+   IERROR, W)
C***BEGIN PROLOGUE  CMGNBN
C***PURPOSE  Solve a complex block tridiagonal linear system of
C            equations by a cyclic reduction algorithm.
C***LIBRARY   SLATEC (FISHPACK)
C***CATEGORY  I2B4B
C***TYPE      COMPLEX (GENBUN-S, CMGNBN-C)
C***KEYWORDS  CYCLIC REDUCTION, ELLIPTIC PDE, FISHPACK,
C             TRIDIAGONAL LINEAR SYSTEM
C           Swarztrauber, P. N., (NCAR)
C           Sweet, R., (NCAR)
C***DESCRIPTION
C
C     Subroutine CMGNBN solves the complex 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.  N must be greater
C       than 2.
C
C     A,B,C
C       One-dimensional complex 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 CMGNBN.  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 complex array that specifies the values of the
C       right side of the linear system of equations given above.  Y
C       must be dimensioned at least M*N.
C
C     W
C       A one-dimensional complex array that must be provided by the
C       user for work space.  W may require up to 4*N +
C       (10 + INT(log2(N)))*M LOCATIONS.  The actual number of locations
C       used is computed by CMGNBN 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 which 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 1979
C     Revision
C
C     Subprograms    CMGNBN,CMPOSD,CMPOSN,CMPOSP,CMPCSG,CMPMRG,
C     Required       CMPTRX,CMPTR3,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 June, 1977
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 CMGNBN 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.
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 a right side Y was computed.  Using this
C                    array Y subroutine CMGNBN was called to produce an
C                    approximate solution Z.  Then the relative error,
C                    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          77     1.E-12
C                         31        1         1          45     4.E-13
C                         31        1         3          91     2.E-12
C                         32        0         0          59     7.E-14
C                         32        1         1          65     5.E-13
C                         32        1         3          97     2.E-13
C                         33        0         0          80     6.E-13
C                         33        1         1          67     5.E-13
C                         33        1         3          76     3.E-12
C                         63        0         0         350     5.E-12
C                         63        1         1         215     6.E-13
C                         63        1         3         412     1.E-11
C                         64        0         0         264     1.E-13
C                         64        1         1         287     3.E-12
C                         64        1         3         421     3.E-13
C                         65        0         0         338     2.E-12
C                         65        1         1         292     5.E-13
C                         65        1         3         329     1.E-11
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  CMPOSD, CMPOSN, CMPOSP
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  CMGNBN
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