# 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