*DECK DHSTRT
SUBROUTINE DHSTRT (DF, NEQ, A, B, Y, YPRIME, ETOL, MORDER, SMALL,
+ BIG, SPY, PV, YP, SF, RPAR, IPAR, H)
C***BEGIN PROLOGUE DHSTRT
C***SUBSIDIARY
C***PURPOSE Subsidiary to DDEABM, DDEBDF and DDERKF
C***LIBRARY SLATEC
C***TYPE DOUBLE PRECISION (HSTART-S, DHSTRT-D)
C***AUTHOR Watts, H. A., (SNLA)
C***DESCRIPTION
C
C DHSTRT computes a starting step size to be used in solving initial
C value problems in ordinary differential equations.
C
C **********************************************************************
C ABSTRACT
C
C Subroutine DHSTRT 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 DHSTRT uses a function subprogram DHVNRM 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 DF -- This is a subroutine of the form
C DF(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=DF(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 DF must not alter X or U(*). You must declare
C the name DF in an external statement in your program that
C calls DHSTRT. You must dimension U and UPRIME in DF.
C
C RPAR and IPAR are DOUBLE PRECISION and INTEGER parameter
C arrays which you can use for communication between your
C program and subroutine DF. They are not used or altered by
C DHSTRT. If you do not need RPAR or IPAR, ignore these
C parameters by treating them as dummy arguments. If you do
C choose to use them, dimension them in your program and in
C DF as arrays 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 DHSTRT has been called with B
C different from A on the machine being used. Also see the
C 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 DF)
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 DOUBLE PRECISION
C number such that (1.+SMALL) .GT. 1. on the machine being
C used. The quantity SMALL**(3/8) is used in computing
C increments of variables for approximating derivatives by
C differences. Also the algorithm will not compute a
C starting step length which is smaller than
C 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 DOUBLE PRECISION number which can be held in
C the machine.
C
C SPY(*),PV(*),YP(*),SF(*) -- These are DOUBLE PRECISION work
C arrays of length NEQ which provide the routine with needed
C storage space.
C
C RPAR,IPAR -- These are parameter arrays, of DOUBLE PRECISION and
C INTEGER type, respectively, which can be used for
C communication between your program and the DF subroutine.
C They are not used or altered by DHSTRT.
C
C **********************************************************************
C On Output (after the return from DHSTRT),
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 DDEABM, DDEBDF, DDERKF
C***ROUTINES CALLED DHVNRM
C***REVISION HISTORY (YYMMDD)
C 820301 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890831 Modified array declarations. (WRB)
C 890911 Removed unnecessary intrinsics. (WRB)
C 891024 Changed references from DVNORM to DHVNRM. (WRB)
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 DHSTRT
C
INTEGER IPAR, J, K, LK, MORDER, NEQ
DOUBLE PRECISION A, ABSDX, B, BIG, DA, DELF, DELY,
1 DFDUB, DFDXB, DHVNRM,
2 DX, DY, ETOL, FBND, H, PV, RELPER, RPAR, SF, SMALL, SPY,
3 SRYDPB, TOLEXP, TOLMIN, TOLP, TOLSUM, Y, YDPB, YP, YPRIME
DIMENSION Y(*),YPRIME(*),ETOL(*),SPY(*),PV(*),YP(*),
1 SF(*),RPAR(*),IPAR(*)
EXTERNAL DF
C
C ..................................................................
C
C BEGIN BLOCK PERMITTING ...EXITS TO 160
C***FIRST EXECUTABLE STATEMENT DHSTRT
DX = B - A
ABSDX = ABS(DX)
RELPER = SMALL**0.375D0
C
C ...............................................................
C
C COMPUTE AN APPROXIMATE BOUND (DFDXB) ON THE PARTIAL
C DERIVATIVE OF THE EQUATION WITH RESPECT TO THE
C INDEPENDENT VARIABLE. PROTECT AGAINST AN OVERFLOW.
C ALSO COMPUTE A BOUND (FBND) ON THE FIRST DERIVATIVE
C LOCALLY.
C
DA = SIGN(MAX(MIN(RELPER*ABS(A),ABSDX),
1 100.0D0*SMALL*ABS(A)),DX)
IF (DA .EQ. 0.0D0) DA = RELPER*DX
CALL DF(A+DA,Y,SF,RPAR,IPAR)
DO 10 J = 1, NEQ
YP(J) = SF(J) - YPRIME(J)
10 CONTINUE
DELF = DHVNRM(YP,NEQ)
DFDXB = BIG
IF (DELF .LT. BIG*ABS(DA)) DFDXB = DELF/ABS(DA)
FBND = DHVNRM(SF,NEQ)
C
C ...............................................................
C
C COMPUTE AN ESTIMATE (DFDUB) OF THE LOCAL LIPSCHITZ
C CONSTANT FOR THE SYSTEM OF DIFFERENTIAL EQUATIONS. THIS
C ALSO REPRESENTS AN ESTIMATE OF THE NORM OF THE JACOBIAN
C LOCALLY. THREE ITERATIONS (TWO WHEN NEQ=1) ARE USED TO
C ESTIMATE THE LIPSCHITZ CONSTANT BY NUMERICAL DIFFERENCES.
C THE FIRST PERTURBATION VECTOR IS BASED ON THE INITIAL
C DERIVATIVES AND DIRECTION OF INTEGRATION. THE SECOND
C PERTURBATION VECTOR IS FORMED USING ANOTHER EVALUATION OF
C THE DIFFERENTIAL EQUATION. THE THIRD PERTURBATION VECTOR
C IS FORMED USING PERTURBATIONS BASED ONLY ON THE INITIAL
C VALUES. COMPONENTS THAT ARE ZERO ARE ALWAYS CHANGED TO
C NON-ZERO VALUES (EXCEPT ON THE FIRST ITERATION). WHEN
C INFORMATION IS AVAILABLE, CARE IS TAKEN TO ENSURE THAT
C COMPONENTS OF THE PERTURBATION VECTOR HAVE SIGNS WHICH ARE
C CONSISTENT WITH THE SLOPES OF LOCAL SOLUTION CURVES.
C ALSO CHOOSE THE LARGEST BOUND (FBND) FOR THE FIRST
C DERIVATIVE.
C
C PERTURBATION VECTOR SIZE IS HELD
C CONSTANT FOR ALL ITERATIONS. COMPUTE
C THIS CHANGE FROM THE
C SIZE OF THE VECTOR OF INITIAL
C VALUES.
DELY = RELPER*DHVNRM(Y,NEQ)
IF (DELY .EQ. 0.0D0) DELY = RELPER
DELY = SIGN(DELY,DX)
DELF = DHVNRM(YPRIME,NEQ)
FBND = MAX(FBND,DELF)
IF (DELF .EQ. 0.0D0) GO TO 30
C USE INITIAL DERIVATIVES FOR FIRST PERTURBATION
DO 20 J = 1, NEQ
SPY(J) = YPRIME(J)
YP(J) = YPRIME(J)
20 CONTINUE
GO TO 50
30 CONTINUE
C CANNOT HAVE A NULL PERTURBATION VECTOR
DO 40 J = 1, NEQ
SPY(J) = 0.0D0
YP(J) = 1.0D0
40 CONTINUE
DELF = DHVNRM(YP,NEQ)
50 CONTINUE
C
DFDUB = 0.0D0
LK = MIN(NEQ+1,3)
DO 140 K = 1, LK
C DEFINE PERTURBED VECTOR OF INITIAL VALUES
DO 60 J = 1, NEQ
PV(J) = Y(J) + DELY*(YP(J)/DELF)
60 CONTINUE
IF (K .EQ. 2) GO TO 80
C EVALUATE DERIVATIVES ASSOCIATED WITH PERTURBED
C VECTOR AND COMPUTE CORRESPONDING DIFFERENCES
CALL DF(A,PV,YP,RPAR,IPAR)
DO 70 J = 1, NEQ
PV(J) = YP(J) - YPRIME(J)
70 CONTINUE
GO TO 100
80 CONTINUE
C USE A SHIFTED VALUE OF THE INDEPENDENT VARIABLE
C IN COMPUTING ONE ESTIMATE
CALL DF(A+DA,PV,YP,RPAR,IPAR)
DO 90 J = 1, NEQ
PV(J) = YP(J) - SF(J)
90 CONTINUE
100 CONTINUE
C CHOOSE LARGEST BOUNDS ON THE FIRST DERIVATIVE
C AND A LOCAL LIPSCHITZ CONSTANT
FBND = MAX(FBND,DHVNRM(YP,NEQ))
DELF = DHVNRM(PV,NEQ)
C ...EXIT
IF (DELF .GE. BIG*ABS(DELY)) GO TO 150
DFDUB = MAX(DFDUB,DELF/ABS(DELY))
C ......EXIT
IF (K .EQ. LK) GO TO 160
C CHOOSE NEXT PERTURBATION VECTOR
IF (DELF .EQ. 0.0D0) DELF = 1.0D0
DO 130 J = 1, NEQ
IF (K .EQ. 2) GO TO 110
DY = ABS(PV(J))
IF (DY .EQ. 0.0D0) DY = DELF
GO TO 120
110 CONTINUE
DY = Y(J)
IF (DY .EQ. 0.0D0) DY = DELY/RELPER
120 CONTINUE
IF (SPY(J) .EQ. 0.0D0) SPY(J) = YP(J)
IF (SPY(J) .NE. 0.0D0) DY = SIGN(DY,SPY(J))
YP(J) = DY
130 CONTINUE
DELF = DHVNRM(YP,NEQ)
140 CONTINUE
150 CONTINUE
C
C PROTECT AGAINST AN OVERFLOW
DFDUB = BIG
160 CONTINUE
C
C ..................................................................
C
C COMPUTE A BOUND (YDPB) ON THE NORM OF THE SECOND DERIVATIVE
C
YDPB = DFDXB + DFDUB*FBND
C
C ..................................................................
C
C DEFINE THE TOLERANCE PARAMETER UPON WHICH THE STARTING STEP
C SIZE IS TO BE BASED. A VALUE IN THE MIDDLE OF THE ERROR
C TOLERANCE RANGE IS SELECTED.
C
TOLMIN = BIG
TOLSUM = 0.0D0
DO 170 K = 1, NEQ
TOLEXP = LOG10(ETOL(K))
TOLMIN = MIN(TOLMIN,TOLEXP)
TOLSUM = TOLSUM + TOLEXP
170 CONTINUE
TOLP = 10.0D0**(0.5D0*(TOLSUM/NEQ + TOLMIN)/(MORDER+1))
C
C ..................................................................
C
C COMPUTE A STARTING STEP SIZE BASED ON THE ABOVE FIRST AND
C SECOND DERIVATIVE INFORMATION
C
C RESTRICT THE STEP LENGTH TO BE NOT BIGGER
C THAN ABS(B-A). (UNLESS B IS TOO CLOSE
C TO A)
H = ABSDX
C
IF (YDPB .NE. 0.0D0 .OR. FBND .NE. 0.0D0) GO TO 180
C
C BOTH FIRST DERIVATIVE TERM (FBND) AND SECOND
C DERIVATIVE TERM (YDPB) ARE ZERO
IF (TOLP .LT. 1.0D0) H = ABSDX*TOLP
GO TO 200
180 CONTINUE
C
IF (YDPB .NE. 0.0D0) GO TO 190
C
C ONLY SECOND DERIVATIVE TERM (YDPB) IS ZERO
IF (TOLP .LT. FBND*ABSDX) H = TOLP/FBND
GO TO 200
190 CONTINUE
C
C SECOND DERIVATIVE TERM (YDPB) IS NON-ZERO
SRYDPB = SQRT(0.5D0*YDPB)
IF (TOLP .LT. SRYDPB*ABSDX) H = TOLP/SRYDPB
200 CONTINUE
C
C FURTHER RESTRICT THE STEP LENGTH TO BE NOT
C BIGGER THAN 1/DFDUB
IF (H*DFDUB .GT. 1.0D0) H = 1.0D0/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,100.0D0*SMALL*ABS(A))
IF (H .EQ. 0.0D0) H = SMALL*ABS(B)
C
C NOW SET DIRECTION OF INTEGRATION
H = SIGN(H,DX)
C
RETURN
END