ЧА-6

f1[t_] := 1/(1 + t);
f2[t_] := 1/(2 + t);
f3[t_] := 1/(3 + t);
f[t_] := Exp[-t];
s1 = Solve[{a*f1[1] + b*f2[1] + c*f3[1] == f[1],
   a*f1[2] + b*f2[2] + c*f3[2] == f[2],
   a*f1[3] + b*f2[3] + c*f3[3] == f[3]}, {a, b, c}]
L[t_] := Simplify[a*f1[t] + b*f2[t] + c*f3[t] /. s1];
L[t]
Plot[f[t] - L[t], {t, 1, 3}]



n = 5;
f[t_] := Sqrt[t];
Do[x[k] = k/n, {k, 0, n}];
Do[c[k] = ((f[x[k + 1]] - f[x[k]])/(x[k + 1] - x[k]) - (f[x[k]] -
         f[x[k - 1]])/(x[k] - x[k - 1]))/2, {k, 1, n - 1}];
c[0] = ((f[x[0]] + f[x[n]])/(x[n] - x[0]) + (f[x[1]] -
        f[x[0]])/(x[1] - x[0]))/2;
c[n] = ((f[x[0]] + f[x[n]])/(x[n] - x[0]) - (f[x[n]] -
        f[x[n - 1]])/(x[n] - x[n - 1]))/2;
I1[t_] := Sum[c[k]*Abs[t - x[k]], {k, 0, n}];
Plot[f[t] - I1[t], {t, 0, 1}, PlotRange -> All]
Plot[{f[t], I1[t]}, {t, 0, 1}]

n = 50;
f[t_] := Sqrt[t];
Do[x[k] = k/n, {k, 0, n}];
Do[c[k] = ((f[x[k + 1]] - f[x[k]])/(x[k + 1] - x[k]) - (f[x[k]] -
         f[x[k - 1]])/(x[k] - x[k - 1]))/2, {k, 1, n - 1}];
c[0] = ((f[x[0]] + f[x[n]])/(x[n] - x[0]) + (f[x[1]] -
        f[x[0]])/(x[1] - x[0]))/2;
c[n] = ((f[x[0]] + f[x[n]])/(x[n] - x[0]) - (f[x[n]] -
        f[x[n - 1]])/(x[n] - x[n - 1]))/2;
I1[t_] := Sum[c[k]*Abs[t - x[k]], {k, 0, n}];
Plot[f[t] - I1[t], {t, 0, 1}, PlotRange -> All]
Plot[{f[t], I1[t]}, {t, 0, 1}]


n = 50;
f[t_] := Sqrt[t];
Do[x[k] = (k/n)^4, {k, 0, n}];
Do[c[k] = ((f[x[k + 1]] - f[x[k]])/(x[k + 1] - x[k]) - (f[x[k]] -
         f[x[k - 1]])/(x[k] - x[k - 1]))/2, {k, 1, n - 1}];
c[0] = ((f[x[0]] + f[x[n]])/(x[n] - x[0]) + (f[x[1]] -
        f[x[0]])/(x[1] - x[0]))/2;
c[n] = ((f[x[0]] + f[x[n]])/(x[n] - x[0]) - (f[x[n]] -
        f[x[n - 1]])/(x[n] - x[n - 1]))/2;
I1[t_] := Sum[c[k]*Abs[t - x[k]], {k, 0, n}];
Plot[f[t] - I1[t], {t, 0, 1}, PlotRange -> All]
Plot[{f[t], I1[t]}, {t, 0, 1}]