# Ermittlung der Wichtungsfunktionen nach der Formel (7.254):
> with(orthopoly,P):
> H[n][i]:=2*(1-x[i]^2)/n^2/(P(n-1,x[i]))^2;

                                            2
                                 2 (1 - x[i] )
                    H[n][i] := ------------------
                                2               2
                               n  P(n - 1, x[i])

> n:=1;

                                n := 1

> Nullstellen[n]:=solve(P(n,x)=0);

                         Nullstellen[1] := 0

> H[0]:=2/(P(0,0))^2;

                              H[0] := 2

> n:=2;

                                n := 2

> Nullstellen[n]:=solve(P(n,x)=0);

                                       1/2     1/2
                                      3       3
                    Nullstellen[2] := ----, - ----
                                       3       3

> for i in[%[1]] do
> H[i]:=2*(1-i^2)/n^2/(P(n-1,i))^2 od;

                                1/2
                               3
                             H[----] := 1
                                3

> n:=3;

                                n := 3

> Nullstellen[n]:=solve(P(n,x)=0);

                                        1/2      1/2
                                      15       15
                 Nullstellen[3] := 0, -----, - -----
                                        5        5

> for i in[%[1],%[2]] do
> H[i]:=2*(1-i^2)/n^2/(P(n-1,i))^2 od;

                             H[0] := 8/9


                               1/2
                             15
                           H[-----] := 5/9
                               5

> n:=4;

                                n := 4

> Nullstellen[n]:=solve(P(n,x)=0):
> Nullstellen[n]:=vector([sqrt(525-70*sqrt(30))/35,
> sqrt(525+70*sqrt(30))/35]);

                       [            1/2 1/2              1/2 1/2]
                       [(525 - 70 30   )     (525 + 70 30   )   ]
     Nullstellen[4] := [-------------------, -------------------]
                       [        35                   35         ]

> for i in[%[1],%[2]] do
> H[i]:=factor(2*(1-i^2)/n^2/(P(n-1,i))^2) od;

                              1/2 1/2             1/2
                  (525 - 70 30   )              30
                H[-------------------] := 1/2 + -----
                          35                     36


                              1/2 1/2             1/2
                  (525 + 70 30   )              30
                H[-------------------] := 1/2 - -----
                          35                     36

> n:=5;

                                n := 5

> Nullstellen[n]:=solve(P(n,x)=0):
> Nullstellen[n]:=vector([0,sqrt(245-14*sqrt(70))/21,
> sqrt(245+14*sqrt(70))/21]);

                     [               1/2 1/2              1/2 1/2]
                     [   (245 - 14 70   )     (245 + 14 70   )   ]
   Nullstellen[5] := [0, -------------------, -------------------]
                     [           21                   21         ]

> for i in[%[1],%[2],%[3]] do
> H[i]:=factor(2*(1-i^2)/n^2/(P(n-1,i))^2) od;

                                     128
                             H[0] := ---
                                     225


                             1/2 1/2                1/2
                 (245 - 14 70   )        161   13 70
               H[-------------------] := --- + --------
                         21              450     900


                             1/2 1/2                1/2
                 (245 + 14 70   )        161   13 70
               H[-------------------] := --- - --------
                         21              450     900

> n:=6;

                                n := 6

> Digits:=15:
> Nullstellen[n]:=fsolve(P(n,x)=0,x=0..1);

  Nullstellen[6] :=

        0.238619186083197, 0.661209386466265, 0.932469514203152

> for i in[%[1],%[2],%[3]] do
> H[i]:=2*(1-i^2)/n^2/(P(n-1,i))^2 od;

              H[0.238619186083197] := 0.467913934572690


              H[0.661209386466265] := 0.360761573048132


              H[0.932469514203152] := 0.171324492379175

> n:=7;

                                n := 7

> Nullstellen[n]:=fsolve(P(n,x)=0,x=0..1);

  Nullstellen[7] :=

        0., 0.405845151377397, 0.741531185599394, 0.949107912342759

> for i in[%[1],%[2],%[3],%[4]] do
> H[i]:=2*(1-i^2)/n^2/(P(n-1,i))^2 od;

                      H[0.] := 0.417959183673468


              H[0.405845151377397] := 0.381830050505118


              H[0.741531185599394] := 0.279705391489262


              H[0.949107912342759] := 0.129484966168952

> n:=8;

                                n := 8

> Nullstellen[n]:=fsolve(P(n,x)=0,x=0..1);

  Nullstellen[8] := 0.183434642495650, 0.525532409916329,

        0.796666477413627, 0.960289856497536

> for i in[%[1],%[2],%[3],%[4]] do
> H[i]:=2*(1-i^2)/n^2/(P(n-1,i))^2 od;
> 

              H[0.183434642495650] := 0.362683783378364


              H[0.525532409916329] := 0.313706645877902


              H[0.796666477413627] := 0.222381034453288


              H[0.960289856497536] := 0.101228536290390

> n:=9;

                                n := 9

> Nullstellen[n]:=fsolve(P(n,x)=0,x=0..1);

  Nullstellen[9] := 0., 0.324253423403809, 0.613371432700590,

        0.836031107326636, 0.968160239507626

> for i in[%[1],%[2],%[3],%[4],%[5]] do
> H[i]:=2*(1-i^2)/n^2/(P(n-1,i))^2 od;

                      H[0.] := 0.330239355001260


              H[0.324253423403809] := 0.312347077040016


              H[0.613371432700590] := 0.260610696402912


              H[0.836031107326636] := 0.180648160694868


              H[0.968160239507626] := 0.0812743883616172

> n:=10;

                               n := 10

> Nullstellen[n]:=fsolve(P(n,x)=0,x=0..1);

  Nullstellen[10] := 0.148874338981631, 0.433395394129247,

        0.679409568299024, 0.865063366688985, 0.973906528517172

> for i in[%[1],%[2],%[3],%[4],%[5]] do
> H[i]:=2*(1-i^2)/n^2/(P(n-1,i))^2 od;

              H[0.148874338981631] := 0.295524224714754


              H[0.433395394129247] := 0.269266719309992


              H[0.679409568299024] := 0.219086362516218


              H[0.865063366688985] := 0.149451349150630


              H[0.973906528517172] := 0.0666713443093662

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