SLATEC Routines --- HSTART ---
*DECK HSTART
SUBROUTINE HSTART (F, NEQ, A, B, Y, YPRIME, ETOL, MORDER, SMALL,
+ BIG, SPY, PV, YP, SF, RPAR, IPAR, H)
C***BEGIN PROLOGUE HSTART
C***SUBSIDIARY
C***PURPOSE Subsidiary to DEABM, DEBDF and DERKF
C***LIBRARY SLATEC
C***TYPE SINGLE PRECISION (HSTART-S, DHSTRT-D)
C***AUTHOR Watts, H. A., (SNLA)
C***DESCRIPTION
C
C HSTART computes a starting step size to be used in solving initial
C value problems in ordinary differential equations.
C **********************************************************************
C Abstract
C
C Subroutine HSTART computes a starting step size to be used by an
C initial value method in solving ordinary differential equations.
C It is based on an estimate of the local Lipschitz constant for the
C differential equation (lower bound on a norm of the Jacobian),
C a bound on the differential equation (first derivative), and
C a bound on the partial derivative of the equation with respect to
C the independent variable.
C (All approximated near the initial point A.)
C
C Subroutine HSTART uses a function subprogram HVNRM for computing
C a vector norm. The maximum norm is presently utilized though it
C can easily be replaced by any other vector norm. It is presumed
C that any replacement norm routine would be carefully coded to
C prevent unnecessary underflows or overflows from occurring, and
C also, would not alter the vector or number of components.
C
C **********************************************************************
C On Input you must provide the following
C
C F -- This is a subroutine of the form
C F(X,U,UPRIME,RPAR,IPAR)
C which defines the system of first order differential
C equations to be solved. For the given values of X and the
C vector U(*)=(U(1),U(2),...,U(NEQ)) , the subroutine must
C evaluate the NEQ components of the system of differential
C equations dU/DX=F(X,U) and store the derivatives in the
C array UPRIME(*), that is, UPRIME(I) = * dU(I)/DX * for
C equations I=1,...,NEQ.
C
C Subroutine F must not alter X or U(*). You must declare
C the name F in an EXTERNAL statement in your program that
C calls HSTART. You must dimension U and UPRIME in F.
C
C RPAR and IPAR are real and integer parameter arrays which
C you can use for communication between your program and
C subroutine F. They are not used or altered by HSTART. If
C you do not need RPAR or IPAR, ignore these parameters by
C treating them as dummy arguments. If you do choose to use
C them, dimension them in your program and in F as arrays
C of appropriate length.
C
C NEQ -- This is the number of (first order) differential equations
C to be integrated.
C
C A -- This is the initial point of integration.
C
C B -- This is a value of the independent variable used to define
C the direction of integration. A reasonable choice is to
C set B to the first point at which a solution is desired.
C You can also use B, if necessary, to restrict the length
C of the first integration step because the algorithm will
C not compute a starting step length which is bigger than
C ABS(B-A), unless B has been chosen too close to A.
C (It is presumed that HSTART has been called with B
C different from A on the machine being used. Also see
C the discussion about the parameter SMALL.)
C
C Y(*) -- This is the vector of initial values of the NEQ solution
C components at the initial point A.
C
C YPRIME(*) -- This is the vector of derivatives of the NEQ
C solution components at the initial point A.
C (defined by the differential equations in subroutine F)
C
C ETOL -- This is the vector of error tolerances corresponding to
C the NEQ solution components. It is assumed that all
C elements are positive. Following the first integration
C step, the tolerances are expected to be used by the
C integrator in an error test which roughly requires that
C ABS(local error) .LE. ETOL
C for each vector component.
C
C MORDER -- This is the order of the formula which will be used by
C the initial value method for taking the first integration
C step.
C
C SMALL -- This is a small positive machine dependent constant
C which is used for protecting against computations with
C numbers which are too small relative to the precision of
C floating point arithmetic. SMALL should be set to
C (approximately) the smallest positive real number such
C that (1.+SMALL) .GT. 1. on the machine being used. the
C quantity SMALL**(3/8) is used in computing increments of
C variables for approximating derivatives by differences.
C also the algorithm will not compute a starting step length
C which is smaller than 100*SMALL*ABS(A).
C
C BIG -- This is a large positive machine dependent constant which
C is used for preventing machine overflows. A reasonable
C choice is to set big to (approximately) the square root of
C the largest real number which can be held in the machine.
C
C SPY(*),PV(*),YP(*),SF(*) -- These are real work arrays of length
C NEQ which provide the routine with needed storage space.
C
C RPAR,IPAR -- These are parameter arrays, of real and integer
C type, respectively, which can be used for communication
C between your program and the F subroutine. They are not
C used or altered by HSTART.
C
C **********************************************************************
C On Output (after the return from HSTART),
C
C H -- Is an appropriate starting step size to be attempted by the
C differential equation method.
C
C All parameters in the call list remain unchanged except for
C the working arrays SPY(*),PV(*),YP(*) and SF(*).
C
C **********************************************************************
C
C***SEE ALSO DEABM, DEBDF, DERKF
C***ROUTINES CALLED HVNRM
C***REVISION HISTORY (YYMMDD)
C 800501 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 891024 Changed references from VNORM to HVNRM. (WRB)
C 891024 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900328 Added TYPE section. (WRB)
C 910722 Updated AUTHOR section. (ALS)
C***END PROLOGUE HSTART