SLATEC Routines --- DBVSUP ---

```*DECK DBVSUP
SUBROUTINE DBVSUP (Y, NROWY, NCOMP, XPTS, NXPTS, A, NROWA, ALPHA,
+   NIC, B, NROWB, BETA, NFC, IGOFX, RE, AE, IFLAG, WORK, NDW,
+   IWORK, NDIW, NEQIVP)
C***BEGIN PROLOGUE  DBVSUP
C***PURPOSE  Solve a linear two-point boundary value problem using
C            superposition coupled with an orthonormalization procedure
C            and a variable-step integration scheme.
C***LIBRARY   SLATEC
C***CATEGORY  I1B1
C***TYPE      DOUBLE PRECISION (BVSUP-S, DBVSUP-D)
C***KEYWORDS  ORTHONORMALIZATION, SHOOTING,
C             TWO-POINT BOUNDARY VALUE PROBLEM
C***AUTHOR  Scott, M. R., (SNLA)
C           Watts, H. A., (SNLA)
C***DESCRIPTION
C
C **********************************************************************
C
C     Subroutine DBVSUP solves a linear two-point boundary-value problem
C     of the form
C                        DY/DX = MATRIX(X,U)*Y(X) + G(X,U)
C                A*Y(XINITIAL) = ALPHA ,  B*Y(XFINAL) = BETA
C
C     coupled with the solution of the initial value problem
C
C                        DU/DX = F(X,U)
C                      U(XINITIAL) = ETA
C
C **********************************************************************
C     ABSTRACT
C        The method of solution uses superposition coupled with an
C     orthonormalization procedure and a variable-step integration
C     scheme.  Each time the superposition solutions start to
C     lose their numerical linear independence, the vectors are
C     reorthonormalized before integration proceeds.  The underlying
C     principle of the algorithm is then to piece together the
C     intermediate (orthogonalized) solutions, defined on the various
C     subintervals, to obtain the desired solutions.
C
C **********************************************************************
C     INPUT to DBVSUP
C **********************************************************************
C
C     NROWY = actual row dimension of Y in calling program.
C             NROWY must be .GE. NCOMP
C
C     NCOMP = number of components per solution vector.
C             NCOMP is equal to number of original differential
C             equations.  NCOMP = NIC + NFC.
C
C     XPTS = desired output points for solution. They must be monotonic.
C            XINITIAL = XPTS(1)
C            XFINAL = XPTS(NXPTS)
C
C     NXPTS = number of output points.
C
C     A(NROWA,NCOMP) = boundary condition matrix at XINITIAL
C                      must be contained in (NIC,NCOMP) sub-matrix.
C
C     NROWA = actual row dimension of A in calling program,
C             NROWA must be .GE. NIC.
C
C     ALPHA(NIC+NEQIVP) = boundary conditions at XINITIAL.
C                         If NEQIVP .GT. 0 (see below), the boundary
C                         conditions at XINITIAL for the initial value
C                         equations must be stored starting in
C                         position (NIC + 1) of ALPHA.
C                         Thus,  ALPHA(NIC+K) = ETA(K).
C
C     NIC = number of boundary conditions at XINITIAL.
C
C     B(NROWB,NCOMP) = boundary condition matrix at XFINAL.
C                      Must be contained in (NFC,NCOMP) sub-matrix.
C
C     NROWB = actual row dimension of B in calling program,
C             NROWB must be .GE. NFC.
C
C     BETA(NFC) = boundary conditions at XFINAL.
C
C     NFC = number of boundary conditions at XFINAL.
C
C     IGOFX =0 -- The inhomogeneous term G(X) is identically zero.
C           =1 -- The inhomogeneous term G(X) is not identically zero.
C                 (if IGOFX=1, then Subroutine DGVEC (or DUVEC) must be
C                  supplied).
C
C     RE = relative error tolerance used by the integrator.
C          (see one of the integrators)
C
C     AE = absolute error tolerance used by the integrator.
C          (see one of the integrators)
C **NOTE-  RE and AE should not both be zero.
C
C     IFLAG = a status parameter used principally for output.
C             However, for efficient solution of problems which
C             are originally defined as COMPLEX*16 valued (but
C             converted to double precision systems to use this code),
C             the user must set IFLAG=13 on input. See the comment
C
C     WORK(NDW) = floating point array used for internal storage.
C
C     NDW = actual dimension of work array allocated by user.
C           An estimate for NDW can be computed from the following
C            NDW = 130 + NCOMP**2 * (6 + NXPTS/2 + expected number of
C                                           orthonormalizations/8)
C           For the disk or tape storage mode,
C            NDW = 6 * NCOMP**2 + 10 * NCOMP + 130
C  However, when the ADAMS integrator is to be used, the estimates are
C            NDW = 130 + NCOMP**2 * (13 + NXPTS/2 + expected number of
C                                           orthonormalizations/8)
C    and     NDW = 13 * NCOMP**2 + 22 * NCOMP + 130   , respectively.
C
C     IWORK(NDIW) = integer array used for internal storage.
C
C     NDIW = actual dimension of IWORK array allocated by user.
C            An estimate for NDIW can be computed from the following
C            NDIW = 68 + NCOMP * (1 + expected number of
C                                            orthonormalizations)
C **NOTE --  the amount of storage required is problem dependent and may
C            be difficult to predict in advance.  Experience has shown
C            that for most problems 20 or fewer orthonormalizations
C            should suffice. If the problem cannot be completed with the
C            allotted storage, then a message will be printed which
C            estimates the amount of storage necessary. In any case, the
C            user can examine the IWORK array for the actual storage
C            requirements, as described in the output information below.
C
C     NEQIVP = number of auxiliary initial value equations being added
C              to the boundary value problem.
C **NOTE -- Occasionally the coefficients  matrix  and/or  G  may be
C           functions which depend on the independent variable  X  and
C           on  U, the solution of an auxiliary initial value problem.
C           In order to avoid the difficulties associated with
C           interpolation, the auxiliary equations may be solved
C           simultaneously with the given boundary value problem.
C           This initial value problem may be linear or nonlinear.
C                 See SAND77-1328 for an example.
C
C
C     The user must supply subroutines DFMAT, DGVEC, DUIVP and DUVEC,
C     when needed (they must be so named), to evaluate the derivatives
C     as follows
C
C        A. DFMAT must be supplied.
C
C              SUBROUTINE DFMAT(X,Y,YP)
C              X = independent variable (input to DFMAT)
C              Y = dependent variable vector (input to DFMAT)
C              YP = DY/DX = derivative vector (output from DFMAT)
C
C            Compute the derivatives for the homogeneous problem
C              YP(I) = DY(I)/DX = MATRIX(X) * Y(I)  , I = 1,...,NCOMP
C
C            When (NEQIVP .GT. 0) and  matrix  is dependent on  U  as
C            well as on  X, the following common statement must be
C            included in DFMAT
C                    COMMON /DMLIVP/ NOFST
C            for convenience, the  U  vector is stored at the bottom
C            of the  Y  array.  Thus, during any call to DFMAT,
C            U(I) is referenced by  Y(NOFST + I).
C
C
C            Subroutine DBVDER calls DFMAT NFC times to evaluate the
C            homogeneous equations and, if necessary, it calls DFMAT
C            once in evaluating the particular solution. since X remains
C            unchanged in this sequence of calls it is possible to
C            realize considerable computational savings for complicated
C            and expensive evaluations of the matrix entries. To do this
C            the user merely passes a variable, say XS, via common where
C            XS is defined in the main program to be any value except
C            the initial X. Then the non-constant elements of matrix(x)
C            appearing in the differential equations need only be
C            computed if X is unequal to XS, whereupon XS is reset to X.
C
C
C        B. If  NEQIVP .GT. 0 ,  DUIVP must also be supplied.
C
C              SUBROUTINE DUIVP(X,U,UP)
C              X = independent variable (input to DUIVP)
C              U = dependent variable vector (input to DUIVP)
C              UP = DU/DX = derivative vector (output from DUIVP)
C
C            Compute the derivatives for the auxiliary initial value eqs
C              UP(I) = DU(I)/DX, I = 1,...,NEQIVP.
C
C            Subroutine DBVDER calls DUIVP once to evaluate the
C            derivatives for the auxiliary initial value equations.
C
C
C        C. If  NEQIVP = 0  and  IGOFX = 1 ,  DGVEC must be supplied.
C
C              SUBROUTINE DGVEC(X,G)
C              X = independent variable (input to DGVEC)
C              G = vector of inhomogeneous terms G(X) (output from
C              DGVEC)
C
C            Compute the inhomogeneous terms G(X)
C                G(I) = G(X) values for I = 1,...,NCOMP.
C
C            Subroutine DBVDER calls DGVEC in evaluating the particular
C            solution provided G(X) is not identically zero. Thus, when
C            IGOFX=0, the user need not write a DGVEC subroutine. Also,
C            the user does not have to bother with the computational
C            savings scheme for DGVEC as this is automatically achieved
C            via the DBVDER subroutine.
C
C
C        D. If  NEQIVP .GT. 0  and  IGOFX = 1 ,  DUVEC must be supplied.
C
C             SUBROUTINE DUVEC(X,U,G)
C             X = independent variable (input to DUVEC)
C             U = dependent variable vector from the auxiliary initial
C                 value problem    (input to DUVEC)
C             G = array of inhomogeneous terms G(X,U)(output from DUVEC)
C
C            Compute the inhomogeneous terms G(X,U)
C                G(I) = G(X,U) values for I = 1,...,NCOMP.
C
C            Subroutine DBVDER calls DUVEC in evaluating the particular
C            solution provided G(X,U) is not identically zero.  Thus,
C            when IGOFX=0, the user need not write a DUVEC subroutine.
C
C
C
C     The following is optional input to DBVSUP to give user more
C     flexibility in use of code.  See SAND75-0198, SAND77-1328,
C
C ****CAUTION -- The user must zero out IWORK(1),...,IWORK(15)
C                prior to calling DBVSUP. These locations define
C                optional input and must be zero unless set to special
C                values by the user as described below.
C
C     IWORK(1) -- number of orthonormalization points.
C                 A value need be set only if IWORK(11) = 1
C
C     IWORK(9) -- integrator and orthonormalization parameter
C                 (default value is 1)
C                 1 = RUNGE-KUTTA-FEHLBERG code using GRAM-SCHMIDT test.
C                 2 = ADAMS code using GRAM-SCHMIDT test.
C
C     IWORK(11) -- orthonormalization points parameter
C                  (default value is 0)
C                  0 - orthonormalization points not pre-assigned.
C                  1 - orthonormalization points pre-assigned in
C                      the first IWORK(1) positions of work.
C
C     IWORK(12) -- storage parameter
C                  (default value is 0)
C                  0 - all storage in core.
C                  LUN - homogeneous and inhomogeneous solutions at
C                      output points and orthonormalization information
C                      are stored on disk.  The logical unit number to
C                      be used for disk I/O (NTAPE) is set to IWORK(12).
C
C     WORK(1),... -- pre-assigned orthonormalization points, stored
C                    monotonically, corresponding to the direction
C                    of integration.
C
C
C
C                 ******************************************************
C                 *** COMPLEX*16 VALUED PROBLEM ***
C                 ******************************************************
C **NOTE***
C       Suppose the original boundary value problem is NC equations
C     of the form
C                   DW/DX = MAT(X,U)*W(X) + H(X,U)
C                 R*W(XINITIAL)=GAMMA , S*W(XFINAL)=DELTA
C     where all variables are COMPLEX*16 valued. The DBVSUP code can be
C     used by converting to a double precision system of size 2*NC. To
C     solve the larger dimensioned problem efficiently, the user must
C     initialize IFLAG=13 on input and order the vector components
C     according to Y(1)=DOUBLE PRECISION(W(1)),...,Y(NC)=DOUBLE
C     PRECISION(W(NC)),Y(NC+1)=IMAG(W(1)),...., Y(2*NC)=IMAG(W(NC)).
C     Then define
C                        ...............................................
C                        . DOUBLE PRECISION(MAT)    -IMAG(MAT) .
C            MATRIX  =   .                         .
C                        . IMAG(MAT)     DOUBLE PRECISION(MAT) .
C                        ...............................................
C
C     The matrices A,B and vectors G,ALPHA,BETA must be defined
C     similarly. Further details can be found in SAND78-1501.
C
C
C **********************************************************************
C     OUTPUT from DBVSUP
C **********************************************************************
C
C     Y(NROWY,NXPTS) = solution at specified output points.
C
C     IFLAG Output Values
C            =-5 algorithm ,for obtaining starting vectors for the
C                special COMPLEX*16 problem structure, was unable to
C                obtain the initial vectors satisfying the necessary
C                independence criteria.
C            =-4 rank of boundary condition matrix A is less than NIC,
C                as determined by DLSSUD.
C            =-2 invalid input parameters.
C            =-1 insufficient number of storage locations allocated for
C                WORK or IWORK.
C
C            =0 indicates successful solution.
C
C            =1 a computed solution is returned but uniqueness of the
C               solution of the boundary-value problem is questionable.
C               For an eigenvalue problem, this should be treated as a
C               successful execution since this is the expected mode
C               of return.
C            =2 a computed solution is returned but the existence of the
C               solution to the boundary-value problem is questionable.
C            =3 a nontrivial solution approximation is returned although
C               the boundary condition matrix B*Y(XFINAL) is found to be
C               nonsingular (to the desired accuracy level) while the
C               right hand side vector is zero. To eliminate this type
C               of return, the accuracy of the eigenvalue parameter
C               must be improved.
C            ***NOTE-We attempt to diagnose the correct problem behavior
C               and report possible difficulties by the appropriate
C               error flag.  However, the user should probably resolve
C               the problem using smaller error tolerances and/or
C               perturbations in the boundary conditions or other
C               parameters. This will often reveal the correct
C               interpretation for the problem posed.
C
C            =13 maximum number of orthonormalizations attained before
C                reaching XFINAL.
C            =20-flag from integrator (DDERKF or DDEABM) values can
C                range from 21 to 25.
C            =30 solution vectors form a dependent set.
C
C     WORK(1),...,WORK(IWORK(1)) = orthonormalization points
C                                  determined by DBVPOR.
C
C     IWORK(1) = number of orthonormalizations performed by DBVPOR.
C
C     IWORK(2) = maximum number of orthonormalizations allowed as
C                calculated from storage allocated by user.
C
C                actual storage requirements for WORK and IWORK
C                arrays.  In particular,
C                       required storage for  work array is
C        IWORK(3) + IWORK(4)*(expected number of orthonormalizations)
C
C                       required storage for IWORK array is
C        IWORK(5) + IWORK(6)*(expected number of orthonormalizations)
C
C     IWORK(8) = final value of exponent parameter used in tolerance
C                test for orthonormalization.
C
C     IWORK(16) = number of independent vectors returned from DMGSBV.
C                It is only of interest when IFLAG=30 is obtained.
C
C     IWORK(17) = numerically estimated rank of the boundary
C                 condition matrix defined from B*Y(XFINAL)
C
C **********************************************************************
C
C     Necessary machine constants are defined in the Function
C     Routine D1MACH. The user must make sure that the values
C     set in D1MACH are relevant to the computer being used.
C
C **********************************************************************
C **********************************************************************
C
C***REFERENCES  M. R. Scott and H. A. Watts, SUPORT - a computer code
C                 for two-point boundary-value problems via
C                 orthonormalization, SIAM Journal of Numerical
C                 Analysis 14, (1977), pp. 40-70.
C               B. L. Darlow, M. R. Scott and H. A. Watts, Modifications
C                 of SUPORT, a linear boundary value problem solver
C                 Part I - pre-assigning orthonormalization points,
C                 auxiliary initial value problem, disk or tape storage,
C                 Report SAND77-1328, Sandia Laboratories, Albuquerque,
C                 New Mexico, 1977.
C               B. L. Darlow, M. R. Scott and H. A. Watts, Modifications
C                 of SUPORT, a linear boundary value problem solver
C                 Part II - inclusion of an Adams integrator, Report
C                 SAND77-1690, Sandia Laboratories, Albuquerque,
C                 New Mexico, 1977.
C               M. E. Lord and H. A. Watts, Modifications of SUPORT,
C                 a linear boundary value problem solver Part III -
C                 orthonormalization improvements, Report SAND78-0522,
C                 Sandia Laboratories, Albuquerque, New Mexico, 1978.
C               H. A. Watts, M. R. Scott and M. E. Lord, Computational
C                 solution of complex*16 valued boundary problems,
C                 Report SAND78-1501, Sandia Laboratories,
C                 Albuquerque, New Mexico, 1978.
C***ROUTINES CALLED  DEXBVP, DMACON, XERMSG
C***COMMON BLOCKS    DML15T, DML17B, DML18J, DML5MC, DML8SZ
C***REVISION HISTORY  (YYMMDD)
C   750601  DATE WRITTEN
C   890531  Changed all specific intrinsics to generic.  (WRB)
C   890831  Modified array declarations.  (WRB)
C   890921  Realigned order of variables in certain COMMON blocks.
C           (WRB)
C   890921  REVISION DATE from Version 3.2
C   891214  Prologue converted to Version 4.0 format.  (BAB)
C   900510  Convert XERRWV calls to XERMSG calls, remove some extraneous