<Text-field layout="Heading 1" style="Heading 1">Uebung 01</Text-field><Text-field layout="Heading 2" style="Heading 2">Aufgabe 1</Text-field>restart;f := (x) -> piecewise(x <= 0, 1, x > 0, -1);#f := (x) -> piecewise(x <= (-2*Pi)/3, 3*x/Pi+3, x <= 0, -(3*x)/Pi-1, x <= 2*Pi/3, 3*x/Pi-1, x <= Pi, -(3*x)/Pi + 3);plot(f(x), x=-Pi..Pi);fourier := proc(f,stellen,a,b)
local phi0, phiCos, phiSin, cf0, cfCos, cfSin, lt1, lt2, deg, i, res;
lt1 := x -> (x*(b-a)/Pi+a+b)/2;
lt2 := x -> (2*x-a-b)/(b-a)*Pi;
phi0 := (x) -> 1/sqrt(2*Pi);
phiCos := (x, k) -> 1/sqrt(Pi)*cos(k*x);
phiSin := (x, k) -> 1/sqrt(Pi)*sin(k*x);
cf0 := int(f(x)*phi0(lt2(x)),x=a..b);
cfCos := (k) -> int(f(x)*phiCos(lt2(x),k),x=a..b);
cfSin := (k) -> int(f(x)*phiSin(lt2(x),k),x=a..b);
deg := stellen*2-1;
res := lt1(cf0*phi0);
for i from 1 to deg do
if (i mod 2) = 0 then
res := res + lt1(cfCos(i) * phiCos(x,i));
else
res := res + lt1(cfSin(i) * phiSin(x,i));
end if;
end do;
res := unapply(res,x);
print(plot([ f(x), res(x), abs(f(x)-res(x)) ], x=a..b));
return unapply(res(x),x);
end proc:fourier := proc(f,stellen,a,b)
local phi0, phiCos, phiSin, cf0, cfCos, cfSin, lt1, lt2, deg, i, res;
lt1 := x -> (x*(b-a)/Pi+a+b)/2;
lt2 := x -> (2*x-a-b)/(b-a)*Pi;
#phi0 := (x) -> 1/sqrt(2*Pi);
phiCos := (x, k) -> 1/sqrt(Pi)*cos(k*x);
phiSin := (x, k) -> 1/sqrt(Pi)*sin(k*x);
cf0 := int(f(x)*1/sqrt(2*Pi),x=a..b);
cfCos := (k) -> int(f(x)*phiCos(lt2(x),k),x=a..b);
cfSin := (k) -> int(f(x)*phiSin(lt2(x),k),x=a..b);
deg := stellen*2-1;
res := lt1(cf0*1/sqrt(2*Pi));
for i from 1 to deg do
if (i mod 2) = 0 then
res := res + lt1(cfCos(i) * phiCos(x,i));
else
res := res + lt1(cfSin(i) * phiSin(x,i));
end if;
end do;
res := unapply(res,x);
print(plot([ f(x), res(x), abs(f(x)-res(x)) ], x=a..b));
return unapply(res(x),x);
end proc:fapr := fourier(f,5,-Pi,Pi);evalf(abs(fapr(0)-f(0)));fapr := fourier(f,14,-Pi,Pi);evalf(abs(fapr(0)-f(0)));<Text-field layout="Heading 2" style="Heading 2">Aufgabe 2</Text-field>restart;f := (x) -> exp(x):trigon := proc(f,a,b,m,n)
local xp, lt1, lt2, cfa, cfb, cf0, cfn, s0, sn, lak, lbk, k, s0n, res, fapr;
lt1 := x -> (x*(b-a)/Pi+a+b)/2;
lt2 := x -> (2*x-a-b)/(b-a)*Pi;
xp := [ seq(-Pi+j/m*Pi, j=0..2*m-1) ];
xp := map(lt1, xp);
cfa := k -> (1/m)*sum( f(op(j+1,xp))*cos(k*lt2(op(j+1,xp))), j=0..2*m-1);
cfb := k -> (1/m)*sum( f(op(j+1,xp))*sin(k*lt2(op(j+1,xp))), j=0..2*m-1);
cf0 := (1/m)*sum( f(Member(xp, j+1)), j=0..2*m-1);
cfn := cfa(n);
s0 := cf0/2;
sn := cfn*cos(n*lt2(x));
lak := [ seq(cfa(k), k=1..n-1) ]:
lbk := [ seq(cfb(k), k=1..n-1) ]:
k := 'k':
s0n := sum( op(k,lak)*cos(k*lt2(x)) + op(k,lbk)*sin(k*lt2(x)), k=1..n-1 ):
res :=s0+sn+s0n;
res := evalf(res):
fapr := unapply(res,x);
print(plot([f(x), fapr(x), abs(f(x)-fapr(x))], x=0..5));
return unapply(fapr(x),x);
end proc:fapr := trigon(f,0,5,10,8);<Text-field layout="Heading 2" style="Heading 2">Aufgabe 3</Text-field>restart;with(stats);with(statplots);with(describe);st := [12,2,6,2,19,5,5,34,4,1,4,8,7,1,21,6,11,8,28,6,4,5,1,18,9,5,1,22,1,1,5,3,14,5,3,4,5,1,3,16,2];sst := sort(st);[ median(sst), evalf[2](mean(sst)), evalf(variance(sst)), evalf(standarddeviation(sst))];[ quantile[1/20](sst), evalf[5](quantile[19/20](sst)) ];histogram(sst, area=nops(sst), numbars=7);#In Prozent
histogram(sst, area=100, numbars=7);<Text-field layout="Heading 2" style="Heading 2">Aufgabe 4</Text-field>restart;with(plots):with(stats):with(statplots):with(describe):with(StringTools):n := 50001;
r := evalf[n](Pi):s := Drop( convert(r, string), 2):
l := Explode(s);
nops(l);Count := proc(liste, value)
local res, i;
res := 0;
for i from 1 to nops(liste) do
if op(i,liste) = value then
res := res +1;
end if;
end do;
return res;
end proc:Total := proc(liste)
return sum(op(i,liste),i=1..nops(liste));end proc:frq := [ seq(Count(l, i), i="0".."9" ) ];
Total(frq);
evalf( [standarddeviation(frq), standarddeviation(frq)/mean(frq)] );
evalf[3](frq/Total(frq));display([ listplot(frq), plot(mean(frq), x=0..10) ]);