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unit myMath;
interface
uses math, sysutils;
type TExtPoint = record
case byte of
0: (x,y: extended;); // coordinates or
1: (val,err: extended;); // value and relative error
end;
function myfloattostr(f: double): string;
function multiply(a,b: TExtPoint): TExtPoint;
function divide(a,b: TExtPoint): TExtPoint;
function valuetostr(a: TExtPoint; separator: String): String;
function valuetostr(a: TExtPoint): String;
function valuetostr(a: TExtPoint; separator: String; mitFehler: boolean): String;
function valuetostr(a: TExtPoint; mitFehler: boolean): String;
procedure GaussFitDownhillSimplex(W: array of TExtPoint; var y0,dt,delta: extended);
procedure GaussFitGradientenverfahren(W: array of TExtPoint; var y0,dt,delta: extended);
var Komma: char = ',';
implementation
function myfloattostr(f: double): string;
var s,t: string;
begin
s:=floattostr(f);
t:='';
while pos('.',s)>0 do
begin
t:=t+copy(s,1,pos('.',s)-1)+komma;
delete(s,1,pos('.',s));
end;
myfloattostr:=t+s;
end;
//*****************************************************************************
function multiply(a,b: TExtPoint): TExtPoint;
var erg: TExtPoint;
begin
erg.val:=a.val*b.val;
erg.err:=a.err+b.err;
multiply:=erg;
end;
function divide(a,b: TExtPoint): TExtPoint;
var erg: TExtPoint;
begin
if b.val=0 then writeln('Division durch Null!');
erg.val:=a.val/b.val;
erg.err:=a.err+b.err;
divide:=erg;
end;
function valuetostr(a: TExtPoint; separator: String): String;
begin
valuetostr:=valuetostr(a,separator,true);
end;
function valuetostr(a: TExtPoint): String;
begin
valuetostr:=valuetostr(a,#9);
end;
function valuetostr(a: TExtPoint; separator: String; mitFehler: boolean): String;
begin
if mitFehler then valuetostr:=myfloattostr(a.val)+separator+myfloattostr(a.val*a.err)+separator+myfloattostr(a.err)
else valuetostr:=myfloattostr(a.val);
end;
function valuetostr(a: TExtPoint; mitFehler: boolean): String;
begin
valuetostr:=valuetostr(a,#9,mitFehler);
end;
//*****************************************************************************
function qerr(W: array of TExtPoint; Param: TExtPoint): extended;
var I: Integer;
erg: extended;
begin
erg:=0;
For I:=0 to length(W)-1 do
erg:=erg +
sqr(W[I].y - Param.y*exp(-sqr(W[I].x/Param.x)));
qerr:=erg;
end;
procedure GaussFitDownhillSimplex(W: array of TExtPoint; var y0,dt,delta: extended);
var Simplex: array[0..2] of TExtPoint;
Errs: array[0..2] of Extended;
mitt,np: TExtPoint;
ma,mi,nerr: extended;
I,J: Integer;
const gutFaktor = 2.5;
schlechtFaktor = 0.5;
mittelFaktor = 1.0;
begin
Simplex[0].x:=dt;
Simplex[0].y:=y0;
Simplex[1].x:=dt+delta*1000;
Simplex[1].y:=y0;
Simplex[2].x:=dt;
Simplex[2].y:=y0+delta*1000;
For I:=0 to 2 do
Errs[I]:=qerr(W,Simplex[I]);
repeat
J:=0;
For I:=1 to 2 do
if Errs[I]>Errs[J] then
J:=I;
mitt.x:=0;
mitt.y:=0;
Ma:=Errs[Byte(J=0)];
Mi:=Errs[Byte(J=0)];
For I:=0 to 2 do
if J<>I then
begin
Ma:=Max(Ma,Errs[I]);
Mi:=Min(Mi,Errs[I]);
mitt.x:=mitt.x+Simplex[I].x;
mitt.y:=mitt.y+Simplex[I].y;
end;
mitt.x:=mitt.x/2;
mitt.y:=mitt.y/2;
np.x:=2*mitt.x-Simplex[J].x;
np.y:=2*mitt.y-Simplex[J].y;
nerr:=qerr(W,np);
np.x:=mitt.x-Simplex[J].x;
np.y:=mitt.y-Simplex[J].y;
if nerr<Mi then // Minimum !
begin
Simplex[J].x:=mitt.x+gutFaktor*np.x;
Simplex[J].y:=mitt.y+gutFaktor*np.y;
end
else
if nerr>Ma then // Maximum !
begin
Simplex[J].x:=mitt.x+schlechtFaktor*np.x;
Simplex[J].y:=mitt.y+schlechtFaktor*np.y;
end
else
begin
Simplex[J].x:=mitt.x+mittelFaktor*np.x;
Simplex[J].y:=mitt.y+mittelFaktor*np.y;
end;
Errs[J]:=qerr(W,Simplex[J]);
until (abs((Simplex[0].x-Simplex[1].x)*(Simplex[0].y-Simplex[2].y) -
(Simplex[0].x-Simplex[2].x)*(Simplex[0].y-Simplex[1].y)) <= delta);
J:=0;
For I:=1 to 2 do
if Errs[I]<Errs[J] then
J:=I;
delta:=Errs[J];
dt:=Simplex[J].x;
y0:=Simplex[J].y;
end;
procedure GaussFitGradientenverfahren(W: array of TExtPoint; var y0,dt,delta: extended);
var ex,tmp,err: extended;
i,vz,vzz: integer;
dy0,
ddt,fac: extended;
weiter: boolean;
begin
dt:=1/dt;
err:=0;
fac:=2;
vz:=0;
vzz:=0;
for i:=0 to length(W)-1 do
err:=err+sqr(W[i].y-y0*exp(-sqr(W[i].x*dt)));
repeat
dy0:=0;
ddt:=0;
for i:=0 to length(W)-1 do
begin
ex:=exp(-sqr(W[i].x*dt));
tmp:=W[i].y*ex-y0*sqr(ex);
dy0:=dy0 - tmp;
ddt:=ddt + sqr(W[i].x)*tmp;
end;
ddt:=ddt*dt*y0*2;
weiter:=sqr(ddt)+sqr(dy0)>1;
if weiter then tmp:=1/sqrt(sqr(ddt)+sqr(dy0))
else tmp:=1;
if (vz<>2*byte(ddt>0)+byte(dy0>0)) or
(ddt=0) or (dy0=0) then
begin
fac:=fac/2;
vzz:=0;
end
else
begin
inc(vzz);
for i:=5 to vzz do
fac:=min(1,fac*1.2);
end;
vz:=2*byte(ddt>0)+byte(dy0>0);
tmp:=tmp*fac;
dt:=dt-ddt*tmp;
y0:=y0-dy0*tmp;
tmp:=err;
err:=0;
for i:=0 to length(W)-1 do
err:=err+sqr(W[i].y-y0*exp(-sqr(W[i].x*dt)));
until (tmp-err<delta*err) and (fac<delta);
dt:=1/dt;
delta:=err;
end;
end.
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