SLATEC Routines --- STEPS ---
*DECK STEPS
SUBROUTINE STEPS (F, NEQN, Y, X, H, EPS, WT, START, HOLD, K, KOLD,
+ CRASH, PHI, P, YP, PSI, ALPHA, BETA, SIG, V, W, G, PHASE1, NS,
+ NORND, KSTEPS, TWOU, FOURU, XOLD, KPREV, IVC, IV, KGI, GI,
+ RPAR, IPAR)
C***BEGIN PROLOGUE STEPS
C***PURPOSE Integrate a system of first order ordinary differential
C equations one step.
C***LIBRARY SLATEC (DEPAC)
C***CATEGORY I1A1B
C***TYPE SINGLE PRECISION (STEPS-S, DSTEPS-D)
C***KEYWORDS ADAMS METHOD, DEPAC, INITIAL VALUE PROBLEMS, ODE,
C ORDINARY DIFFERENTIAL EQUATIONS, PREDICTOR-CORRECTOR
C***AUTHOR Shampine, L. F., (SNLA)
C Gordon, M. K., (SNLA)
C MODIFIED BY H.A. WATTS
C***DESCRIPTION
C
C Written by L. F. Shampine and M. K. Gordon
C
C Abstract
C
C Subroutine STEPS is normally used indirectly through subroutine
C DEABM . Because DEABM suffices for most problems and is much
C easier to use, using it should be considered before using STEPS
C alone.
C
C Subroutine STEPS integrates a system of NEQN first order ordinary
C differential equations one step, normally from X to X+H, using a
C modified divided difference form of the Adams Pece formulas. Local
C extrapolation is used to improve absolute stability and accuracy.
C The code adjusts its order and step size to control the local error
C per unit step in a generalized sense. Special devices are included
C to control roundoff error and to detect when the user is requesting
C too much accuracy.
C
C This code is completely explained and documented in the text,
C Computer Solution of Ordinary Differential Equations, The Initial
C Value Problem by L. F. Shampine and M. K. Gordon.
C Further details on use of this code are available in "Solving
C Ordinary Differential Equations with ODE, STEP, and INTRP",
C by L. F. Shampine and M. K. Gordon, SLA-73-1060.
C
C
C The parameters represent --
C F -- subroutine to evaluate derivatives
C NEQN -- number of equations to be integrated
C Y(*) -- solution vector at X
C X -- independent variable
C H -- appropriate step size for next step. Normally determined by
C code
C EPS -- local error tolerance
C WT(*) -- vector of weights for error criterion
C START -- logical variable set .TRUE. for first step, .FALSE.
C otherwise
C HOLD -- step size used for last successful step
C K -- appropriate order for next step (determined by code)
C KOLD -- order used for last successful step
C CRASH -- logical variable set .TRUE. when no step can be taken,
C .FALSE. otherwise.
C YP(*) -- derivative of solution vector at X after successful
C step
C KSTEPS -- counter on attempted steps
C TWOU -- 2.*U where U is machine unit roundoff quantity
C FOURU -- 4.*U where U is machine unit roundoff quantity
C RPAR,IPAR -- parameter arrays which you may choose to use
C for communication between your program and subroutine F.
C They are not altered or used by STEPS.
C The variables X,XOLD,KOLD,KGI and IVC and the arrays Y,PHI,ALPHA,G,
C W,P,IV and GI are required for the interpolation subroutine SINTRP.
C The remaining variables and arrays are included in the call list
C only to eliminate local retention of variables between calls.
C
C Input to STEPS
C
C First call --
C
C The user must provide storage in his calling program for all arrays
C in the call list, namely
C
C DIMENSION Y(NEQN),WT(NEQN),PHI(NEQN,16),P(NEQN),YP(NEQN),PSI(12),
C 1 ALPHA(12),BETA(12),SIG(13),V(12),W(12),G(13),GI(11),IV(10),
C 2 RPAR(*),IPAR(*)
C
C **Note**
C
C The user must also declare START , CRASH , PHASE1 and NORND
C logical variables and F an EXTERNAL subroutine, supply the
C subroutine F(X,Y,YP) to evaluate
C DY(I)/DX = YP(I) = F(X,Y(1),Y(2),...,Y(NEQN))
C and initialize only the following parameters.
C NEQN -- number of equations to be integrated
C Y(*) -- vector of initial values of dependent variables
C X -- initial value of the independent variable
C H -- nominal step size indicating direction of integration
C and maximum size of step. Must be variable
C EPS -- local error tolerance per step. Must be variable
C WT(*) -- vector of non-zero weights for error criterion
C START -- .TRUE.
C YP(*) -- vector of initial derivative values
C KSTEPS -- set KSTEPS to zero
C TWOU -- 2.*U where U is machine unit roundoff quantity
C FOURU -- 4.*U where U is machine unit roundoff quantity
C Define U to be the machine unit roundoff quantity by calling
C the function routine R1MACH, U = R1MACH(4), or by
C computing U so that U is the smallest positive number such
C that 1.0+U .GT. 1.0.
C
C STEPS requires that the L2 norm of the vector with components
C LOCAL ERROR(L)/WT(L) be less than EPS for a successful step. The
C array WT allows the user to specify an error test appropriate
C for his problem. For example,
C WT(L) = 1.0 specifies absolute error,
C = ABS(Y(L)) error relative to the most recent value of the
C L-th component of the solution,
C = ABS(YP(L)) error relative to the most recent value of
C the L-th component of the derivative,
C = MAX(WT(L),ABS(Y(L))) error relative to the largest
C magnitude of L-th component obtained so far,
C = ABS(Y(L))*RELERR/EPS + ABSERR/EPS specifies a mixed
C relative-absolute test where RELERR is relative
C error, ABSERR is absolute error and EPS =
C MAX(RELERR,ABSERR) .
C
C Subsequent calls --
C
C Subroutine STEPS is designed so that all information needed to
C continue the integration, including the step size H and the order
C K , is returned with each step. With the exception of the step
C size, the error tolerance, and the weights, none of the parameters
C should be altered. The array WT must be updated after each step
C to maintain relative error tests like those above. Normally the
C integration is continued just beyond the desired endpoint and the
C solution interpolated there with subroutine SINTRP . If it is
C impossible to integrate beyond the endpoint, the step size may be
C reduced to hit the endpoint since the code will not take a step
C larger than the H input. Changing the direction of integration,
C i.e., the sign of H , requires the user set START = .TRUE. before
C calling STEPS again. This is the only situation in which START
C should be altered.
C
C Output from STEPS
C
C Successful Step --
C
C The subroutine returns after each successful step with START and
C CRASH set .FALSE. . X represents the independent variable
C advanced one step of length HOLD from its value on input and Y
C the solution vector at the new value of X . All other parameters
C represent information corresponding to the new X needed to
C continue the integration.
C
C Unsuccessful Step --
C
C When the error tolerance is too small for the machine precision,
C the subroutine returns without taking a step and CRASH = .TRUE. .
C An appropriate step size and error tolerance for continuing are
C estimated and all other information is restored as upon input
C before returning. To continue with the larger tolerance, the user
C just calls the code again. A restart is neither required nor
C desirable.
C
C***REFERENCES L. F. Shampine and M. K. Gordon, Solving ordinary
C differential equations with ODE, STEP, and INTRP,
C Report SLA-73-1060, Sandia Laboratories, 1973.
C***ROUTINES CALLED HSTART, R1MACH
C***REVISION HISTORY (YYMMDD)
C 740101 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890831 Modified array declarations. (WRB)
C 890831 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 STEPS