*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
C
DIMENSION Y(*),YPRIME(*),ETOL(*),SPY(*),PV(*),YP(*),SF(*),
1 RPAR(*),IPAR(*)
EXTERNAL F
C
C.......................................................................
C
C***FIRST EXECUTABLE STATEMENT HSTART
DX = B - A
ABSDX = ABS(DX)
RELPER = SMALL**0.375
YNORM = HVNRM(Y,NEQ)
C
C.......................................................................
C
C COMPUTE A WEIGHTED APPROXIMATE BOUND (DFDXB) ON THE PARTIAL
C DERIVATIVE OF THE EQUATION WITH RESPECT TO THE
C INDEPENDENT VARIABLE. PROTECT AGAINST AN OVERFLOW. ALSO
C COMPUTE A WEIGHTED BOUND (FBND) ON THE FIRST DERIVATIVE LOCALLY.
C
DA = SIGN(MAX(MIN(RELPER*ABS(A),ABSDX),100.*SMALL*ABS(A)),DX)
IF (DA .EQ. 0.) DA = RELPER*DX
CALL F(A+DA,Y,SF,RPAR,IPAR)
C
IF (MORDER .EQ. 1) GO TO 20
POWER = 2./(MORDER+1)
DO 10 J=1,NEQ
WTJ = ETOL(J)**POWER
SPY(J) = SF(J)/WTJ
YP(J) = YPRIME(J)/WTJ
10 PV(J) = SPY(J) - YP(J)
GO TO 40
C
20 DO 30 J=1,NEQ
SPY(J) = SF(J)/ETOL(J)
YP(J) = YPRIME(J)/ETOL(J)
30 PV(J) = SPY(J) - YP(J)
C
40 DELF = HVNRM(PV,NEQ)
DFDXB = BIG
IF (DELF .LT. BIG*ABS(DA)) DFDXB = DELF/ABS(DA)
YPNORM = HVNRM(YP,NEQ)
FBND = MAX(HVNRM(SPY,NEQ),YPNORM)
C
C.......................................................................
C
C COMPUTE AN ESTIMATE (DFDUB) OF THE LOCAL LIPSCHITZ CONSTANT FOR
C THE SYSTEM OF DIFFERENTIAL EQUATIONS. THIS ALSO REPRESENTS AN
C ESTIMATE OF THE NORM OF THE JACOBIAN LOCALLY.
C THREE ITERATIONS (TWO WHEN NEQ=1) ARE USED TO ESTIMATE THE
C LIPSCHITZ CONSTANT BY NUMERICAL DIFFERENCES. THE FIRST
C PERTURBATION VECTOR IS BASED ON THE INITIAL DERIVATIVES AND
C DIRECTION OF INTEGRATION. THE SECOND PERTURBATION VECTOR IS
C FORMED USING ANOTHER EVALUATION OF THE DIFFERENTIAL EQUATION.
C THE THIRD PERTURBATION VECTOR IS FORMED USING PERTURBATIONS BASED
C ONLY ON THE INITIAL VALUES. COMPONENTS THAT ARE ZERO ARE ALWAYS
C CHANGED TO NON-ZERO VALUES (EXCEPT ON THE FIRST ITERATION). WHEN
C INFORMATION IS AVAILABLE, CARE IS TAKEN TO ENSURE THAT COMPONENTS
C OF THE PERTURBATION VECTOR HAVE SIGNS WHICH ARE CONSISTENT WITH
C THE SLOPES OF LOCAL SOLUTION CURVES.
C ALSO CHOOSE THE LARGEST BOUND (FBND) FOR THE FIRST DERIVATIVE.
C NO ATTEMPT IS MADE TO KEEP THE PERTURBATION VECTOR SIZE CONSTANT.
C
IF (YPNORM .EQ. 0.) GO TO 60
C USE INITIAL DERIVATIVES FOR FIRST PERTURBATION
ICASE = 1
DO 50 J=1,NEQ
SPY(J) = YPRIME(J)
50 YP(J) = YPRIME(J)
GO TO 80
C CANNOT HAVE A NULL PERTURBATION VECTOR
60 ICASE = 2
DO 70 J=1,NEQ
SPY(J) = YPRIME(J)
70 YP(J) = ETOL(J)
C
80 DFDUB = 0.
LK = MIN(NEQ+1,3)
DO 260 K=1,LK
C SET YPNORM AND DELX
YPNORM = HVNRM(YP,NEQ)
IF (ICASE .EQ. 1 .OR. ICASE .EQ. 3) GO TO 90
DELX = SIGN(1.0,DX)
GO TO 120
C TRY TO ENFORCE MEANINGFUL PERTURBATION VALUES
90 DELX = DX
IF (ABS(DELX)*YPNORM .GE. RELPER*YNORM) GO TO 100
DELXB = BIG
IF (RELPER*YNORM .LT. BIG*YPNORM) DELXB = RELPER*YNORM/YPNORM
DELX = SIGN(DELXB,DX)
100 DO 110 J=1,NEQ
IF (ABS(DELX*YP(J)) .GT. ETOL(J)) DELX=SIGN(ETOL(J)/YP(J),DX)
110 CONTINUE
C DEFINE PERTURBED VECTOR OF INITIAL VALUES
120 DO 130 J=1,NEQ
130 PV(J) = Y(J) + DELX*YP(J)
IF (K .EQ. 2) GO TO 150
C EVALUATE DERIVATIVES ASSOCIATED WITH PERTURBED
C VECTOR AND COMPUTE CORRESPONDING DIFFERENCES
CALL F(A,PV,YP,RPAR,IPAR)
DO 140 J=1,NEQ
140 PV(J) = YP(J) - YPRIME(J)
GO TO 170
C USE A SHIFTED VALUE OF THE INDEPENDENT VARIABLE
C IN COMPUTING ONE ESTIMATE
150 CALL F(A+DA,PV,YP,RPAR,IPAR)
DO 160 J=1,NEQ
160 PV(J) = YP(J) - SF(J)
C CHOOSE LARGEST BOUND ON THE WEIGHTED FIRST
C DERIVATIVE
170 IF (MORDER .EQ. 1) GO TO 190
DO 180 J=1,NEQ
180 YP(J) = YP(J)/ETOL(J)**POWER
GO TO 210
190 DO 200 J=1,NEQ
200 YP(J) = YP(J)/ETOL(J)
210 FBND = MAX(FBND,HVNRM(YP,NEQ))
C COMPUTE BOUND ON A LOCAL LIPSCHITZ CONSTANT
DELF = HVNRM(PV,NEQ)
IF (DELF .EQ. 0.) GO TO 220
DELY = ABS(DELX)*YPNORM
IF (DELF .GE. BIG*DELY) GO TO 270
DFDUB = MAX(DFDUB,DELF/DELY)
C
220 IF (K .EQ. LK) GO TO 280
C CHOOSE NEXT PERTURBATION VECTOR
DO 250 J=1,NEQ
IF (K .EQ. LK-1) GO TO 230
ICASE = 3
DY = ABS(PV(J))
IF (DY .EQ. 0.) DY = MAX(DELF,ETOL(J))
GO TO 240
230 ICASE = 4
DY = MAX(RELPER*ABS(Y(J)),ETOL(J))
240 IF (SPY(J) .EQ. 0.) SPY(J) = YP(J)
IF (SPY(J) .NE. 0.) DY = SIGN(DY,SPY(J))
250 YP(J) = DY
260 CONTINUE
C
C PROTECT AGAINST AN OVERFLOW
270 DFDUB = BIG
C
C.......................................................................
C
C COMPUTE A BOUND (YDPB) ON THE NORM OF THE SECOND DERIVATIVE
C
280 YDPB = DFDXB + DFDUB*FBND
C
C.......................................................................
C
C COMPUTE A STARTING STEP SIZE BASED ON THE ABOVE FIRST AND SECOND
C DERIVATIVE INFORMATION
C
C RESTRICT THE STEP LENGTH TO BE NOT BIGGER THAN
C ABS(B-A). (UNLESS B IS TOO CLOSE TO A)
H = ABSDX
C
IF (YDPB .NE. 0. .OR. FBND .NE. 0.) GO TO 290
C
C BOTH FIRST DERIVATIVE TERM (FBND) AND SECOND
C DERIVATIVE TERM (YDPB) ARE ZERO
GO TO 310
C
290 IF (YDPB .NE. 0.) GO TO 300
C
C ONLY SECOND DERIVATIVE TERM (YDPB) IS ZERO
IF (1.0 .LT. FBND*ABSDX) H = 1./FBND
GO TO 310
C
C SECOND DERIVATIVE TERM (YDPB) IS NON-ZERO
300 SRYDPB = SQRT(0.5*YDPB)
IF (1.0 .LT. SRYDPB*ABSDX) H = 1./SRYDPB
C
C FURTHER RESTRICT THE STEP LENGTH TO BE NOT
C BIGGER THAN 1/DFDUB
310 IF (H*DFDUB .GT. 1.) H = 1./DFDUB
C
C FINALLY, RESTRICT THE STEP LENGTH TO BE NOT
C SMALLER THAN 100*SMALL*ABS(A). HOWEVER, IF
C A=0. AND THE COMPUTED H UNDERFLOWED TO ZERO,
C THE ALGORITHM RETURNS SMALL*ABS(B) FOR THE
C STEP LENGTH.
H = MAX(H,100.*SMALL*ABS(A))
IF (H .EQ. 0.) H = SMALL*ABS(B)
C
C NOW SET DIRECTION OF INTEGRATION
H = SIGN(H,DX)
C
RETURN
END