1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
|
(*----------
* 1 задача
*----------*)
ApproxIntegral[M_, a_, b_, f_] := With[
{
h=(b-a)/(2M),
xk=Function[k,a+k*(b-a)/(2M)]
},
(h/3) (f[a] + f[b]) +
(2h/3) Sum[f[xk[2k]], {k, 1, M-1}] +
(4h/3) Sum[f[xk[2k-1]], {k, 1, M}]
]
Integrand[x_] := 1/((1 + 2x^2)^5)
m15 := N[ApproxIntegral[15, -1, 1, Integrand], {Infinity,20}]
m50 := N[ApproxIntegral[50, -1, 1, Integrand], {Infinity,20}]
correct := N[Integrate[Integrand[x], {x, -1, 1}], {Infinity,20}]
Print["1 задача)"]
Print["При M=15: ", m15]
Print["Разлика с точен: ", N[correct-m15,{Infinity,20}]]
Print["При M=50: ", m50]
Print["Разлика с точен: ", N[correct-m50,{Infinity,20}]]
Print["Точна стойност: ", correct]
(*----------
* 2 задача
*----------*)
(* Намиране Pn *)
Pncoeff [xs_, f_] := With[
{
k = Length[xs],
x0 = First[xs],
xk = Last[xs]
},
Piecewise[{
{
f[xk],
k == 1
},
{
(Pncoeff[Drop[xs, 1], f] - Pncoeff[Drop[xs, -1], f])/(xk - x0),
x0 != xk
},
{
(D[f[y], {y, k-1}] /. y->x0)/((k-1)!),
x0 == xk
}}]
]
Pn[f_,xsWithDup_] := With[
{ xs = xsWithDup },
Function[x,Sum[
Pncoeff[Take[xs, i], f] *
Product[x - xs[[j]], {j, 1, i-1}],
{i, 1, Length[xs]}
]]
]
f[x_]:= Piecewise[{
{Cos[3x], x <= Pi},
{x - Pi - 1, x > Pi}
}]
values := {9Pi/10, 9Pi/10, 9Pi/10, 10Pi/11, 10Pi/11, 11Pi/10}
newtonpolyn := Pn[f, values]
Print["2 задача)"]
Print["Полином: ", Simplify[newtonpolyn[x]]]
(* InterpolatingPolynomial[{{9Pi/10, f[9Pi/10]}, {10Pi/11, f[10Pi/11]}, {11Pi/10, f[11Pi/10]}}, x] *)
(* Графика разлика *)
Print["Графика на разлика:"]
Plot[Abs[f[z] - newtonpolyn[z]], {z, -6.5, 12}]
|
