SLATEC Routines --- HSTCSP ---
*DECK HSTCSP
SUBROUTINE HSTCSP (INTL, A, B, M, MBDCND, BDA, BDB, C, D, N,
+ NBDCND, BDC, BDD, ELMBDA, F, IDIMF, PERTRB, IERROR, W)
C***BEGIN PROLOGUE HSTCSP
C***PURPOSE Solve the standard five-point finite difference
C approximation on a staggered grid to the modified Helmholtz
C equation in spherical coordinates assuming axisymmetry
C (no dependence on longitude).
C***LIBRARY SLATEC (FISHPACK)
C***CATEGORY I2B1A1A
C***TYPE SINGLE PRECISION (HSTCSP-S)
C***KEYWORDS ELLIPTIC, FISHPACK, HELMHOLTZ, PDE, SPHERICAL
C***AUTHOR Adams, J., (NCAR)
C Swarztrauber, P. N., (NCAR)
C Sweet, R., (NCAR)
C***DESCRIPTION
C
C HSTCSP solves the standard five-point finite difference
C approximation on a staggered grid to the modified Helmholtz
C equation spherical coordinates assuming axisymmetry (no dependence
C on longitude).
C
C (1/R**2)(d/dR)(R**2(dU/dR)) +
C
C 1/(R**2*SIN(THETA))(d/dTHETA)(SIN(THETA)(dU/dTHETA)) +
C
C (LAMBDA/(R*SIN(THETA))**2)U = F(THETA,R)
C
C where THETA is colatitude and R is the radial coordinate.
C This two-dimensional modified Helmholtz equation results from
C the Fourier transform of the three-dimensional Poisson equation.
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C
C * * * * * * * * Parameter Description * * * * * * * * * *
C
C
C * * * * * * On Input * * * * * *
C
C INTL
C = 0 On initial entry to HSTCSP or if any of the arguments
C C, D, N, or NBDCND are changed from a previous call.
C
C = 1 If C, D, N, and NBDCND are all unchanged from previous
C call to HSTCSP.
C
C NOTE: A call with INTL = 0 takes approximately 1.5 times as much
C time as a call with INTL = 1. Once a call with INTL = 0
C has been made then subsequent solutions corresponding to
C different F, BDA, BDB, BDC, and BDD can be obtained
C faster with INTL = 1 since initialization is not repeated.
C
C A,B
C The range of THETA (colatitude), i.e. A .LE. THETA .LE. B. A
C must be less than B and A must be non-negative. A and B are in
C radians. A = 0 corresponds to the north pole and B = PI
C corresponds to the south pole.
C
C * * * IMPORTANT * * *
C
C If B is equal to PI, then B must be computed using the statement
C
C B = PIMACH(DUM)
C
C This insures that B in the user's program is equal to PI in this
C program which permits several tests of the input parameters that
C otherwise would not be possible.
C
C * * * * * * * * * * * *
C
C M
C The number of grid points in the interval (A,B). The grid points
C in the THETA-direction are given by THETA(I) = A + (I-0.5)DTHETA
C for I=1,2,...,M where DTHETA =(B-A)/M. M must be greater than 4.
C
C MBDCND
C Indicates the type of boundary conditions at THETA = A and
C THETA = B.
C
C = 1 If the solution is specified at THETA = A and THETA = B.
C (See notes 1, 2 below)
C
C = 2 If the solution is specified at THETA = A and the derivative
C of the solution with respect to THETA is specified at
C THETA = B (See notes 1, 2 below).
C
C = 3 If the derivative of the solution with respect to THETA is
C specified at THETA = A (See notes 1, 2 below) and THETA = B.
C
C = 4 If the derivative of the solution with respect to THETA is
C specified at THETA = A (See notes 1, 2 below) and the
C solution is specified at THETA = B.
C
C = 5 If the solution is unspecified at THETA = A = 0 and the
C solution is specified at THETA = B. (See note 2 below)
C
C = 6 If the solution is unspecified at THETA = A = 0 and the
C derivative of the solution with respect to THETA is
C specified at THETA = B (See note 2 below).
C
C = 7 If the solution is specified at THETA = A and the
C solution is unspecified at THETA = B = PI.
C
C = 8 If the derivative of the solution with respect to
C THETA is specified at THETA = A (See note 1 below)
C and the solution is unspecified at THETA = B = PI.
C
C = 9 If the solution is unspecified at THETA = A = 0 and
C THETA = B = PI.
C
C NOTES: 1. If A = 0, do not use MBDCND = 1,2,3,4,7 or 8,
C but instead use MBDCND = 5, 6, or 9.
C
C 2. if B = PI, do not use MBDCND = 1,2,3,4,5 or 6,
C but instead use MBDCND = 7, 8, or 9.
C
C When A = 0 and/or B = PI the only meaningful boundary
C condition is dU/dTHETA = 0. (See D. Greenspan, 'Numerical
C Analysis of Elliptic Boundary Value Problems,' Harper and
C Row, 1965, Chapter 5.)
C
C BDA
C A one-dimensional array of length N that specifies the boundary
C values (if any) of the solution at THETA = A. When
C MBDCND = 1, 2, or 7,
C
C BDA(J) = U(A,R(J)) , J=1,2,...,N.
C
C When MBDCND = 3, 4, or 8,
C
C BDA(J) = (d/dTHETA)U(A,R(J)) , J=1,2,...,N.
C
C When MBDCND has any other value, BDA is a dummy variable.
C
C BDB
C A one-dimensional array of length N that specifies the boundary
C values of the solution at THETA = B. When MBDCND = 1, 4, or 5,
C
C BDB(J) = U(B,R(J)) , J=1,2,...,N.
C
C When MBDCND = 2,3, or 6,
C
C BDB(J) = (d/dTHETA)U(B,R(J)) , J=1,2,...,N.
C
C When MBDCND has any other value, BDB is a dummy variable.
C
C C,D
C The range of R , i.e. C .LE. R .LE. D.
C C must be less than D. C must be non-negative.
C
C N
C The number of unknowns in the interval (C,D). The unknowns in
C the R-direction are given by R(J) = C + (J-0.5)DR,
C J=1,2,...,N, where DR = (D-C)/N. N must be greater than 4.
C
C NBDCND
C Indicates the type of boundary conditions at R = C
C and R = D.
C
C = 1 If the solution is specified at R = C and R = D.
C
C = 2 If the solution is specified at R = C and the derivative
C of the solution with respect to R is specified at
C R = D. (See note 1 below)
C
C = 3 If the derivative of the solution with respect to R is
C specified at R = C and R = D.
C
C = 4 If the derivative of the solution with respect to R is
C specified at R = C and the solution is specified at
C R = D.
C
C = 5 If the solution is unspecified at R = C = 0 (See note 2
C below) and the solution is specified at R = D.
C
C = 6 If the solution is unspecified at R = C = 0 (See note 2
C below) and the derivative of the solution with respect to R
C is specified at R = D.
C
C NOTE 1: If C = 0 and MBDCND = 3,6,8 or 9, the system of equations
C to be solved is singular. The unique solution is
C determined by extrapolation to the specification of
C U(THETA(1),C). But in these cases the right side of the
C system will be perturbed by the constant PERTRB.
C
C NOTE 2: NBDCND = 5 or 6 cannot be used with MBDCND = 1, 2, 4, 5,
C or 7 (the former indicates that the solution is
C unspecified at R = 0; the latter indicates that the
C solution is specified). Use instead NBDCND = 1 or 2.
C
C BDC
C A one dimensional array of length M that specifies the boundary
C values of the solution at R = C. When NBDCND = 1 or 2,
C
C BDC(I) = U(THETA(I),C) , I=1,2,...,M.
C
C When NBDCND = 3 or 4,
C
C BDC(I) = (d/dR)U(THETA(I),C), I=1,2,...,M.
C
C When NBDCND has any other value, BDC is a dummy variable.
C
C BDD
C A one-dimensional array of length M that specifies the boundary
C values of the solution at R = D. When NBDCND = 1 or 4,
C
C BDD(I) = U(THETA(I),D) , I=1,2,...,M.
C
C When NBDCND = 2 or 3,
C
C BDD(I) = (d/dR)U(THETA(I),D) , I=1,2,...,M.
C
C When NBDCND has any other value, BDD is a dummy variable.
C
C ELMBDA
C The constant LAMBDA in the modified Helmholtz equation. If
C LAMBDA is greater than 0, a solution may not exist. However,
C HSTCSP will attempt to find a solution.
C
C F
C A two-dimensional array that specifies the values of the right
C side of the modified Helmholtz equation. For I=1,2,...,M and
C J=1,2,...,N
C
C F(I,J) = F(THETA(I),R(J)) .
C
C F must be dimensioned at least M X N.
C
C IDIMF
C The row (or first) dimension of the array F as it appears in the
C program calling HSTCSP. This parameter is used to specify the
C variable dimension of F. IDIMF must be at least M.
C
C W
C A one-dimensional array that must be provided by the user for
C work space. With K = INT(log2(N))+1 and L = 2**(K+1), W may
C require up to (K-2)*L+K+MAX(2N,6M)+4(N+M)+5 locations. The
C actual number of locations used is computed by HSTCSP and is
C returned in the location 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 (THETA(I),R(J)) for
C I=1,2,...,M, J=1,2,...,N.
C
C PERTRB
C If a combination of periodic, derivative, or unspecified
C boundary conditions is specified for a Poisson equation
C (LAMBDA = 0), a solution may not exist. PERTRB is a con-
C stant, calculated and subtracted from F, which ensures
C that a solution exists. HSTCSP then computes this
C solution, which is a least squares solution to the
C original approximation. This solution plus any constant is also
C a solution; hence, the solution is not unique. The value of
C PERTRB should be small compared to the right side F.
C Otherwise, a solution is obtained to an essentially different
C problem. This comparison should always be made to insure that
C a meaningful solution has been obtained.
C
C IERROR
C An error flag that indicates invalid input parameters.
C Except for numbers 0 and 10, a solution is not attempted.
C
C = 0 No error
C
C = 1 A .LT. 0 or B .GT. PI
C
C = 2 A .GE. B
C
C = 3 MBDCND .LT. 1 or MBDCND .GT. 9
C
C = 4 C .LT. 0
C
C = 5 C .GE. D
C
C = 6 NBDCND .LT. 1 or NBDCND .GT. 6
C
C = 7 N .LT. 5
C
C = 8 NBDCND = 5 or 6 and MBDCND = 1, 2, 4, 5, or 7
C
C = 9 C .GT. 0 and NBDCND .GE. 5
C
C = 10 ELMBDA .GT. 0
C
C = 11 IDIMF .LT. M
C
C = 12 M .LT. 5
C
C = 13 A = 0 and MBDCND =1,2,3,4,7 or 8
C
C = 14 B = PI and MBDCND .LE. 6
C
C = 15 A .GT. 0 and MBDCND = 5, 6, or 9
C
C = 16 B .LT. PI and MBDCND .GE. 7
C
C = 17 LAMBDA .NE. 0 and NBDCND .GE. 5
C
C Since this is the only means of indicating a possibly
C incorrect call to HSTCSP, the user should test IERROR after
C the call.
C
C W
C W(1) contains the required length of W. Also W contains
C intermediate values that must not be destroyed if HSTCSP
C will be called again with INTL = 1.
C
C *Long Description:
C
C * * * * * * * Program Specifications * * * * * * * * * * * *
C
C Dimension of BDA(N),BDB(N),BDC(M),BDD(M),F(IDIMF,N),
C Arguments W(See argument list)
C
C Latest June 1979
C Revision
C
C Subprograms HSTCSP,HSTCS1,BLKTRI,BLKTR1,INDXA,INDXB,INDXC,
C Required PROD,PRODP,CPROD,CPRODP,PPADD,PSGF,BSRH,PPSGF,
C PPSPF,COMPB,TEVLS,R1MACH
C
C Special NONE
C Conditions
C
C Common CBLKT
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 May, 1977
C
C Algorithm This subroutine defines the finite-difference
C equations, incorporates boundary data, adjusts the
C right side when the system is singular and calls
C BLKTRI which solves the linear system of equations.
C
C Space 5269(decimal) = 12225(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 HSTCSP is roughly proportional
C to M*N*log2(N), but depends on the input parameter
C INTL. Some values are listed in the table below.
C The solution process employed results in a loss
C of no more than FOUR significant digits for N and M
C as large as 64. More detailed information about
C accuracy can be found in the documentation for
C subroutine BLKTRI which is the routine that
C actually solves the finite difference equations.
C
C
C M(=N) INTL MBDCND(=NBDCND) T(MSECS)
C ----- ---- --------------- --------
C
C 32 0 1-6 132
C 32 1 1-6 88
C 64 0 1-6 546
C 64 1 1-6 380
C
C Portability American National Standards Institute Fortran.
C The machine accuracy is set using function R1MACH.
C
C Required COS,SIN,ABS,SQRT
C Resident
C Routines
C
C Reference Swarztrauber, P.N., 'A Direct Method For The
C Discrete Solution Of Separable Elliptic Equations,'
C SIAM J. Numer. Anal. 11(1974), pp. 1136-1150.
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 P. N. Swarztrauber, A direct method for the discrete
C solution of separable elliptic equations, SIAM Journal
C on Numerical Analysis 11, (1974), pp. 1136-1150.
C***ROUTINES CALLED HSTCS1, PIMACH
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 HSTCSP