Maple Programme fuer Pruefung 4 <Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">lagrange(liste1,liste2)</Font></Text-field><Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">funktion</Font></Text-field>lagrange:= proc(liste1, liste2)
local funktion, summand, i , j:
funktion := 0:
summand := 0:
for i from 1 to nops(liste1) do
summand := op(i, liste2):
for j from 1 to nops(liste1) do
if (i <> j) then
summand := summand * (x - op(j,liste1))/(op(i,liste1) - op(j,liste1)):
end if:
end do:
funktion := funktion + summand:
end do:
end proc:<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">beispiel</Font></Text-field>lagrange([x[0],x[1],x[2]],[f(x[0]),f(x[1]),f(x[2])]);simplify(evalf(lagrange([-Pi/4,0,Pi/4],[cos(-Pi/4),cos(0), cos(Pi/4)])));<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">neville(index1, index2, liste1, liste2)</Font></Text-field><Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">funktion</Font></Text-field>nevQ := proc(i,j)
if j > 0 then
return( sort( expand(
( (x-op(1+i-j,xp))*nevQ(i,j-1) -
(x-op(1+i,xp))*nevQ(i-1,j-1) ) /
( op(1+i,xp)- op(1+i-j,xp) )
)));
else
return( op(1+i,fp) );
end if;
end proc: <Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">beispiel</Font></Text-field>f := x -> x*ln(x):
xp := [8.0,8.2,8.5,8.6,8.8,9.0]:
fp := map(f, xp):nevQ(5,5);<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">Kettenbruch</Font></Text-field>restart:<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">funktion</Font></Text-field>r := proc(x,i)
if i > 0 then return evalf( 1/(r(x,i-1) - a(x,i-1)) )
else return evalf(x)
end if;
end proc:
a := (x,i) -> trunc(r(x,i)):p := proc(x,i)
if i=-2 then return 0
elif i=-1 then return 1
elif i=0 then return a(x,0)
else return a(x,i)*p(x,i-1)+p(x,i-2)
end if;
end proc:q := proc(x,i)
if i=-2 then return 1
elif i=-1 then return 0
elif i=0 then return 1
else return a(x,i)*q(x,i-1)+q(x,i-2)
end if;
end proc:<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">beispiel</Font></Text-field>Approximation von Pim:=[ seq(p(Pi,k)/q(Pi,k), k=0..11)];Fehler:abs(evalf(op(nops(m)-1,m))-evalf(Pi));<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">irrZByStep(startwert,radikant,exp,wurzelgrad,iterationen) irrZByPrecision(startwert,radikant,exp,wurzelgrad,genauikeit)</Font></Text-field>restart:<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">funktion</Font></Text-field>irrZByStep := proc(x0, a, p, q, i)
local xi, j, f:
Digits:=20:
xi := x0:
f := (x) -> x**q - a**p:
for j from 1 to i do
xi := xi - (f(xi)/D(f)(xi)):
printf("Iteration : %3d = %10.20f\tgenau = %10.20f\n", j, evalf(xi), evalf(a**(p/q))):
end do:
end proc:irrZByPrecision := proc(x0, a, p, q, k)
local xi, j, f:
Digits := abs(log[10](k))*2:
xi := x0:
j := 1:
f := (x) -> x**q - a**p:
while (abs(evalf(xi**p - a**(p/q))) > evalf(k)) do
xi := xi - (f(xi)/D(f)(xi)):
j := j + 1:
printf("Iteration : %3d = %10.20f\t**p = %10.20f\tgenau = %10.20f\n",j, evalf(xi), evalf(xi**p), evalf(a**(p/q))):
end do:
printf("Genauigkeit %e erhalten nach %d Iterationsschriftten :\t10.50f", k, j, evalf(xi**p)):
end proc:<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">Beispiel</Font></Text-field>sqrt(2);irrZByStep(1, 2, 1, 2, 10);irrZByPrecision(1, 2, 1, 2, 10**(-4));<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">hilb(funktion, polynomgrad, von , bis)</Font></Text-field><Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">funktion</Font></Text-field>hilb := proc ( f , gr, min , max )
local m1, i, j, a1, l1, k, m2, i2, j2, a2, l2, k2, h, g, res, loes, grad, a:
with(LinearAlgebra):
grad := gr+1;
m2 := Matrix(grad);
for i2 from 1 to grad do
for j2 from 1 to grad do
m2[i2,j2] := Int((x^(j2+i2-2)),x=min..max);
end do;
end do;
a2 := Vector(grad,symbol=a);
l2 := Matrix(grad,1);
for k2 from 1 to grad do
l2[k2,1] := Int(f(x)*x^(k2-1),x=min..max);
end do;
print(m2.a2=l2);
m1 := Matrix(grad);
for i from 1 to grad do
for j from 1 to grad do
m1[i,j] := int((x^(j+i-2)),x=min..max);
end do;
end do;
a1 := Vector(grad,symbol=a);
l1 := Vector(grad);
for k from 1 to grad do
l1[k] := int(f(x)*x^(k-1),x=min..max);
end do;
print(m1.a1=l1);
for i from 1 to grad do
h := 0;
for j from 1 to grad do
h := h + a[j]*m1[i,j];
end do;
g[i] := h = l1[i];
print(g[i]);
end do;
res:=solve({seq(g[i],i=1..grad)},{seq(a[i],i=1..grad)});
loes := 0;
for i from 1 to grad do
loes := loes + a[i]*x^(i-1);
end do;
printf("\t\t\tDie Loesung lautet\n:");
print(eval(evalf(loes),evalf(res)));
return (eval(loes,res));
end proc:<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">beispiel</Font></Text-field>g := unapply(hilb(x->x*ln(x), 3, 1, 4),x);<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">Tschebyschew (+ Lagrange)</Font></Text-field>restart;<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">funktion</Font></Text-field>tschebyschew := proc ( i , min , max )
return seq(1/2*((max-min)*cos((2*k-1)/(2*i)*Pi)+min+max),k=1..i);
end proc:lagrange := proc( liste1 , liste2)
local res, zaehler, nenner, laenge, i, n;
Digits:=20;
res := 0;
if nops(liste1) = nops(liste2) then
laenge := nops(liste1);
for n from 1 to laenge do
zaehler := 1;
nenner := 1;
for i from 1 to laenge do
if i <> n then
zaehler := zaehler * (x - op(i,liste1));
nenner := nenner * (op(n,liste1)-op(i,liste1));
end if;
end do;
res := res + ((zaehler/nenner)*op(n,liste2));
end do;
else
printf("Die Listen sind nicht gleich lang");
end if;
res := sort(expand(res));
return res;
end proc:<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">beispiel</Font></Text-field>Funktion N\344hern:l1 := [1.0,2.0,3.0,4.0];l2 := map(x->x*ln(x),l1);fa:=lagrange(l1,l2);N\344herung mit Tschebyschlampel1 := [tschebyschew(8,0,3)];l2 := map(x->cos(x));fb:=sort(evalf(lagrange(l1,l2)));plot({abs(x*ln(x)-fa),abs(x*ln(x)-fb)},x=1..4);<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">Mueter</Font></Text-field>muetter:=proc()
local i;
for i from 0 to 100
do
printf("din Muetter\n");
end do:
end:muetter();TTdSMApJM1JUQUJMRV9TQVZFLzU5OTUxMlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiJSIiLSUkSW50RzYkKiYlInhHIiIiLSUjbG5HNiNGCitGLC9GKztGLCIiJS1GKDYkKiZGKyIiI0YtRixGMC1GKDYkKiZGKyIiJEYtRixGMC1GKDYkKiZGK0YyRi1GLEYwNiIKTTdSMApJM1JUQUJMRV9TQVZFLzU2NzQwOFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSwqJiUiYUc2IyIiIiIiJCZGKTYjIiIjIyIjOkYvCiZGKTYjRiwiI0AmRik2IyIiJSMiJGIjRjcsKkYoRjBGLUY0RjJGOEY1IyIlQjUiIiYsKkYoRjRGLUY4RjJGO0Y1IyIlbDhGLywqRihGOEYKLUY7RjJGP0Y1IyImJFE7IiIoNiIKTTdSMApJM1JUQUJMRV9TQVZFLzU2NzY0OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSwmLSUjbG5HNiMiIiMiIzsjISM6IiIlIiIiLCZGCigjIiRHIiIiJCEiKEYwLCZGKEYzIyEkYiNGLEYwLCZGKCMiJVs/IiImIyElQjUiI0RGMDYiCg==