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{
$Id: ode.pas,v 1.3 2005/02/14 17:13:21 peter Exp $
This file is part of the Numlib package.
Copyright (c) 1986-2000 by
Kees van Ginneken, Wil Kortsmit and Loek van Reij of the
Computational centre of the Eindhoven University of Technology
FPC port Code by Marco van de Voort (marco@freepascal.org)
documentation by Michael van Canneyt (Michael@freepascal.org)
Solve first order starting value differential eqs, and
sets of first order starting value differential eqs,
Both versions are not suited for stiff differential equations
See the file COPYING.FPC, included in this distribution,
for details about the copyright.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
**********************************************************************}
Unit ode;
{$I DIRECT.INC}
interface
uses typ;
{Solve first order, starting value, differential eqs,
Calc y(b) for dy/dx=f(x,y) and y(a)=ae}
Procedure odeiv1(f: rfunc2r; a, ya: ArbFloat; Var b, yb: ArbFloat;
ae: ArbFloat; Var term: ArbInt);
{ The same as above, for a set of equations. ya and yb are vectors}
Procedure odeiv2(f: oderk1n; a: ArbFloat; Var ya, b, yb: ArbFloat;
n: ArbInt; ae: ArbFloat; Var term: ArbInt);
implementation
Procedure odeiv1(f: rfunc2r; a, ya: ArbFloat; Var b, yb: ArbFloat;
ae: ArbFloat; Var term: ArbInt);
Var last, first, reject, goon : boolean;
x, y, d, h, xl, yl, int, hmin,
absh,k0, k1, k2, k3, k4, k5,
discr, tol, mu, mu1, fh, hl : ArbFloat;
Begin
x := a;
y := ya;
d := b-a;
yb := y;
term := 1;
If ae <= 0 Then
Begin
term := 3;
exit
End;
If d <> 0 Then
Begin
xl := x;
yl := y;
h := d/4;
absh := abs(h);
int := abs(d);
hmin := int*1e-6;
ae := ae/int;
first := true;
goon := true;
while goon Do
Begin
absh := abs(h);
If absh < hmin Then
Begin
If h>0 Then h := hmin
Else h := -hmin;
absh := hmin
End;
If (h >= b-xl) = (h >= 0) Then
Begin
last := true;
h := b-xl;
absh := abs(h)
End
Else last := false;
x := xl;
y := yl;
k0 := f(x,y)*h;
x := xl+h*2/9;
y := yl+k0*2/9;
k1 := f(x,y)*h;
x := xl+h/3;
y := yl+(k0+k1*3)/12;
k2 := f(x,y)*h;
x := xl+h/2;
y := yl+(k0+k2*3)/8;
k3 := f(x,y)*h;
x := xl+h*0.8;
y := yl+(k0*53-k1*135+k2*126+k3*56)/125;
k4 := f(x,y)*h;
If last Then x := b
Else x := xl+h;
y := yl+(k0*133-k1*378+k2*276+k3*112+k4*25)/168;
k5 := f(x,y)*h;
discr := abs(21*(k0-k3)-162*(k2-k3)-125*(k4-k3)+42*(k5-k3))/14;
tol := absh*ae;
mu := 1/(1+discr/tol)+0.45;
reject := discr > tol;
If reject Then
Begin
If absh <= hmin Then
Begin
b := xl;
yb := yl;
term := 2;
exit
End;
h := mu*h
End
Else
Begin
If first Then
Begin
first := false;
hl := h;
h := mu*h
End
Else
Begin
fh := mu*h/hl+mu-mu1;
hl := h;
h := fh*h
End;
mu1 := mu;
y := yl+(-k0*63+k1*189-k2*36-k3*112+k4*50)/28;
k5 := f(x,y)*hl;
y := yl+(k0*35+k2*162+k4*125+k5*14)/336;
If b <> x Then
Begin
xl := x;
yl := y
End
Else
Begin
yb := y;
goon := false
End
End {not reject}
End; {while}
End {d<>0}
End; {odeiv1}
Procedure odeiv2(f: oderk1n; a: ArbFloat; Var ya, b, yb: ArbFloat;
n: ArbInt; ae: ArbFloat; Var term: ArbInt);
Var pya, pyb, yl, k0, k1, k2, k3, k4, k5, y : ^arfloat1;
i, jj, ns : ArbInt;
last, first, reject, goon : boolean;
x, xl, hmin, int, hl, absh, fhm,
discr, tol, mu, mu1, fh, d, h : ArbFloat;
Begin
If (ae <= 0) Or (n < 1) Then
Begin
term := 3;
exit
End;
ns := n*sizeof(ArbFloat);
pya := @ya;
pyb := @yb;
move(pya^[1], pyb^[1], ns);
term := 1;
getmem(yl, ns);
getmem(k0, ns);
getmem(k1, ns);
getmem(k2, ns);
getmem(k3, ns);
getmem(k4, ns);
getmem(k5, ns);
getmem(y, ns);
x := a;
d := b-a;
move(pya^[1], y^[1], ns);
If d <> 0 Then
Begin
xl := x;
move(y^[1], yl^[1], ns);
h := d/4;
absh := abs(h);
int := abs(d);
hmin := int*1e-6;
hl := ae;
ae := ae/int;
first := true;
goon := true;
while goon Do
Begin
absh := abs(h);
If absh < hmin Then
Begin
If h > 0 Then h := hmin
Else h := -hmin;
absh := hmin
End;
If (h >= b-xl) = (h >= 0) Then
Begin
last := true;
h := b-xl;
absh := abs(h)
End
Else last := false;
x := xl;
move(yl^[1], y^[1], ns);
f(x, y^[1], k0^[1]);
For i:=1 To n Do
k0^[i] := k0^[i]*h;
x := xl+h*2/9;
For jj:=1 To n Do
y^[jj] := yl^[jj]+k0^[jj]*2/9;
f(x, y^[1], k1^[1]);
For i:=1 To n Do
k1^[i] := k1^[i]*h;
x := xl+h/3;
For jj:=1 To n Do
y^[jj] := yl^[jj]+(k0^[jj]+k1^[jj]*3)/12;
f(x, y^[1], k2^[1]);
For i:=1 To n Do
k2^[i] := k2^[i]*h;
x := xl+h/2;
For jj:=1 To n Do
y^[jj] := yl^[jj]+(k0^[jj]+k2^[jj]*3)/8;
f(x, y^[1], k3^[1]);
For i:=1 To n Do
k3^[i] := k3^[i]*h;
x := xl+h*0.8;
For jj:=1 To n Do
y^[jj] := yl^[jj]+
(k0^[jj]*53-k1^[jj]*135+k2^[jj]*126+k3^[jj]*56)/125;
f(x, y^[1], k4^[1]);
For i:=1 To n Do
k4^[i] := k4^[i]*h;
If last Then x := b
Else x := xl+h;
For jj:=1 To n Do
y^[jj] := yl^[jj]+(k0^[jj]*133-k1^[jj]*378+k2^[jj]*276+
k3^[jj]*112+k4^[jj]*25)/168;
f(x, y^[1], k5^[1]);
For i:=1 To n Do
k5^[i] := k5^[i]*h;
reject := false;
fhm := 0;
tol := absh*ae;
For jj:=1 To n Do
Begin
discr := abs((k0^[jj]-k3^[jj])*21-(k2^[jj]-k3^[jj])*162-
(k4^[jj]-k3^[jj])*125+(k5^[jj]-k3^[jj])*42)/14;
reject := (discr > tol) Or reject;
fh := discr/tol;
If fh > fhm Then fhm := fh
End; {jj}
mu := 1/(1+fhm)+0.45;
If reject Then
Begin
If absh <= hmin Then
Begin
b := xl;
move(yl^[1], pyb^[1], ns);
term := 2;
freemem(yl, ns);
freemem(k0, ns);
freemem(k1, ns);
freemem(k2, ns);
freemem(k3, ns);
freemem(k4, ns);
freemem(k5, ns);
freemem(y, ns);
exit
End;
h := mu*h
End
Else
Begin
If first Then
Begin
first := false;
hl := h;
h := mu*h
End
Else
Begin
fh := mu*h/hl+mu-mu1;
hl := h;
h := fh*h
End;
mu1 := mu;
For jj:=1 To n Do
y^[jj] := yl^[jj]+(-k0^[jj]*63+k1^[jj]*189
-k2^[jj]*36-k3^[jj]*112+k4^[jj]*50)/28;
f(x, y^[1], k5^[1]);
For i:=1 To n Do
k5^[i] := k5^[i]*hl;
For jj:=1 To n Do
y^[jj] := yl^[jj]+(k0^[jj]*35+k2^[jj]*162+k4^[jj]*125
+k5^[jj]*14)/336;
If b <> x Then
Begin
xl := x;
move(y^[1], yl^[1], ns)
End
Else
Begin
move(y^[1], pyb^[1], ns);
goon := false
End
End {not reject}
End {while}
End; {d<>0}
freemem(yl, ns);
freemem(k0, ns);
freemem(k1, ns);
freemem(k2, ns);
freemem(k3, ns);
freemem(k4, ns);
freemem(k5, ns);
freemem(y, ns)
End; {odeiv2}
End.
{
$Log: ode.pas,v $
Revision 1.3 2005/02/14 17:13:21 peter
* truncate log
}
|
