Uebung 114)b)with(stats): with(plots):

Warning, the name changecoords has been redefined

x_i:=[-2,-1,1,3,3.5,5,7,10]: f_xi:=[8.1,5.2,1.1,-7.3,-8.1,-13.2,-18.7,-27.5]:fb:=fit[leastsquare[[x,y]]]([x_i, f_xi]); f:=unapply( rhs(fb), x);

NiM+SSNmYkc2Ii9JInlHRiUsJiQiK09RKFxXIyEiKiIiIkkieEdGJSQhKyNmXXQsJEYrNiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYkIitPUShcVyMhIioiIiI5JCQhKyNmXXQsJEYvRiVGJUYl

listplot([x_i,f_xi]);

Error, (in plots/pointplot) incorrect first argument

c)x_i2:=[-Pi/4,-Pi/8,0,Pi/4,Pi/8]; xp:=x_i2: fp:=map(cos, xp); xp:=evalf(xp): fp:=evalf(fp): #f_xi2:=[cos(op(1,x_i2)),cos(op(2,x_i2)),cos(op(3,x_i2)),cos(op(4,x_i2)),cos(op(5,x_i2))];

NiM+SSV4X2kyRzYiNycsJEkjUGlHSSpwcm90ZWN0ZWRHRikjISIiIiIlLCRGKCNGKyIiKSIiISwkRigjIiIiRiwsJEYoI0YzRi8=NiM+SSNmcEc2IjcnLCQqJCIiIyMiIiJGKUYqLUkkY29zRzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiU2IywkSSNQaUdGLyNGKyIiKUYrRidGLA==

fb:=fit[leastsquare[[x,y]]]([x_i2, f_xi2]); f:=unapply( rhs(fb), x);

NiM+SSNmYkc2Ii9JInlHRiUsKCokIiIjIyIiIkYqI0YsIiImLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjJJKF9zeXNsaWJHRiU2IywkSSNQaUdGMiNGLCIiKSNGKkYuRi1GLA==NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJCIiIyMiIiJGLiNGMCIiJi1JJGNvc0c2JEkqcHJvdGVjdGVkR0Y2SShfc3lzbGliR0YlNiMsJEkjUGlHRjYjRjAiIikjRi5GMkYxRjBGJUYlRiU=

fc:=(x,y)->1/5*2^(1/2)+2/5*cos(1/8*Pi)+1/5;

NiM+SSNmY0c2ImYqNiRJInhHRiVJInlHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwoKiQiIiMjIiIiRi8jRjEiIiYtSSRjb3NHNiRJKnByb3RlY3RlZEdGN0koX3N5c2xpYkdGJTYjLCRJI1BpR0Y3I0YxIiIpI0YvRjNGMkYxRiVGJUYl

plot([cos(x)](fc);

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

2)newton := proc(p0,TOL,N,f) local converging,fprime,i,pold,p; Digits := 50; fprime := (x) -> D(f)(x); converging := true; i := 0; pold := p0; printf("Startwert: %12.20f\n", evalf(p0)); while converging do i := i+1; p := pold-f(pold)/fprime(pold); printf("i = %d p = %12.20f f(p) := %12.20f\n",i,p,f(p)); pold := p; converging := evalf(abs(f(p)))>TOL and i<N; end do; print(pold); print(evalf(pold)); end proc:x1:=5: p1:=6: q1:=5: r:=p1/q1: f1:=(x,y)->x1^r;

NiM+SSNmMUc2ImYqNiRJInhHRiVJInlHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSlJI3gxR0YlSSJyR0YlRiVGJUYl

newton(1,10^(-5),5,x^(3/5));

Startwert: 1.00000000000000000000 i = 1 p = Error, (in fprintf) number expected for floating point format

2) (Niggli)pow:=proc(a, ex, n) local p,q,f,df,res,i; p:=Member([op(ex)],1): q:=Member([op(ex)],2): f:=x->x^q - a^p; df:=x->q*x^(q-a); res:=[a]: i:=0; while i<n do res:= Append(res, Last(res) - f(Last(res)/df(Last(res)): return (res) end proc:

Warning, premature end of input, use <Shift> + <Enter> to avoid this message.

1)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 thn return 0 elif =0 then return 1 else return a(x,i)...3)Digits=210;

NiMvIiM1IiQ1Iw==

# k:=13 -> 10^-20; # k:=37 -> 10^-50; # k:=78 -> 10^-100; # k:=119 -> 10^-150; # k:=161 -> 10^-200; k:=161;

NiM+SSJrRzYiIiRoIg==

Pi_2:=sum('1/16^k*( 4/(8*k+1) - 2/(8*k+4) - 1/(8*k+5) - 1/(8*k+6) )','k'=0..k); evalf[210](%);

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

Fehler:=evalf[210](Pi-Pi_2);

NiM+SSdGZWhsZXJHNiIkIipxSEA2KSEkNCM=

# k:=20; -> 10^-20 # k:=40; -> 10^-50 # k:=69; -> 10^-100 # k:=95; -> 10^-150 # k:=120; -> 10^-200 k:=120;

NiM+SSJrRzYiIiQ/Ig==

e_2:=sum('1/k!','k'=0..k); evalf[201](%); Fehler_e:=evalf[201](exp(1)-e_2);

NiM+SSRlXzJHNiIjImB3XEphKjRhZ282KilmaSVldz9cRkR4YnhGKTQ8ZWIieXZzJTRGKmYoXEsqPUsiZlAjM1tOKyJmOV1uK2BdJTR3biJcTEl5U3opSC5xTUErNVQqKSozQUJbTGQ0JEh5QiE9W2YwdDZFKlInZSFlek0iIl93KysrKysrKysrKysrKysvWkEsYzEjcCQpSDFZZGt0WD04Y3cjSE9KSFU9JCpbWVhDIypvOSFSJnopKXkoKXBeNUQmcDNyOUVaL1drblxgRTAhZlZdKipmOGMjeVpzdVxuSS0kKkgpKlxMZGcpZVw=NiMkImR3LD41RCY+YDIpKUhRQmoyXFZ6aUdCOFEyaiZmZ19ITS4hSGRWIUhtZjh1Ij0jKmZJK0s+Um11VUZrO0QmeUBRclhmWk5OSW0yQ3hpbnAnXGRmKipwJDRac3YoXGlFTnJ1R2dgQlghZiVHPUc9RiEkKyM=NiM+SSlGZWhsZXJfZUc2IiQiIiFGJw==