Simpson usuli

Parabolalar usuli. Funksiyani kvadratik parabolalar bilan approksimatsiya qilib, aniq integralni yuqori aniqlikda hisoblang.

n mod 2 = 0Bo'linishlar soni 'n' juft butun son bo'lishi shart

Method mapSimpson 1/3 algoritmi

Qadamni hisoblashOraliq n ta teng bo'lakka bo'linadi: h = (b - a) / n.

Tugunlar va yig'indilarChetdagi, toq va juft indeksli funksiya qiymatlari ajratiladi.

Kvadratik kvadraturaI ≈ (h/3) · [Y_chet + 4·ΣY_toq + 2·ΣY_juft] formulasi qo'llanadi.

Input / Parametrlar

Masalani sozlang

f(x) — Integrallanuvchi ifoda (masalan: dx / sqrt(1 + 2x^2), sin(x), x^2).

f(x) integral osti funksiyasi

f(x) =

\int_{0.6}^{1.6} \frac{ dx}{\sqrt{1+2 { x}^{2}}}

Quyi chegara aYuqori chegara b

Bo'laklar soni n (juft son)

Natija / Integral

0.555223

h = 0.10000

Benchmark aniq qiymat (n=10000):0.555224

Xatolik bahosi Δ:1.0466e-6

Analysis / Formulalar

h = \frac{b - a}{n} = \frac{1.6000 - 0.6000}{10} = 0.100000

\begin{aligned} Y_{chet} &= y_0 + y_n = 1.166719 \\ \sum Y_{toq} &= y_1 + y_3 + \dots = 2.773429 \\ \sum Y_{juft} &= y_2 + y_4 + \dots = 2.198124 \end{aligned}

\begin{aligned} I &\approx \frac{h}{3} \Big[ Y_{chet} + 4\sum Y_{toq} + 2\sum Y_{juft} \Big] \\ I &\approx \frac{0.10000}{3} \Big[ 1.1667 + 4(2.7734) + 2(2.1981) \Big] \\ I &\approx \mathbf{0.555223} \end{aligned}

\begin{aligned} I_{aniq} &\approx \int_{0.60}^{1.60} \frac{ dx}{\sqrt{1+2 { x}^{2}}} \approx 0.555224 \\ \Delta &= |I - I_{aniq}| = \mathbf{1.0466 \times 10^{-6}} \end{aligned}

Visual / Grafik

Jadval / Nuqtalar

i	x_i	y_i = f(x_i)	Koeffitsiyent c_i	Turi
0	0.60000	0.762493	1	Chetki (c=1)
1	0.70000	0.710669	4	Toq (c=4)
2	0.80000	0.662266	2	Juft (c=2)
3	0.90000	0.617802	4	Toq (c=4)
4	1.00000	0.577350	2	Juft (c=2)
5	1.10000	0.540738	4	Toq (c=4)
6	1.20000	0.507673	2	Juft (c=2)
7	1.30000	0.477818	4	Toq (c=4)
8	1.40000	0.450835	2	Juft (c=2)
9	1.50000	0.426401	4	Toq (c=4)
10	1.60000	0.404226	1	Chetki (c=1)