Oltin qism usuli

x= x i -1/2 =x-( x i -x i -1)/2= x-x i/2 -x i -1 /2= x- x i -1 + x i -1 /2-x i/ 2 \u003d t-h men/2
Keyin menkiritilgan intervallarni hisobga olgan holda integralning qiymati yozilishi mumkin:
Qiymatni ifodasida almashtiring a, b koeffitsientlari va v
Shunday qilib,
S men - bu integralning qiymati menth segment. A dan b gacha bo'lgan segmentda integralni olish uchun barcha S ni qo'shish kerak men
Agar h menhar qanday kishi uchun \u003d h men\u003d 1, ..., N, keyin Simpson formulasini soddalashtirish mumkin
(4)
Formulani (4) soddalashtirish mumkin, buning uchun biz yig'indilar belgisi ostida ifodadagi qavslarni ochamiz

Birinchi yig'indidan funktsiyaning nuqtadagi qiymatini tanlaymiz x=a
,
va oxirgi yig'indidan - funktsiyaning nuqtadagi qiymati x=b

Natijada, biz bir xil tarmoq uchun ishlaydigan Simpson formulasini olamiz.


Shuni hisobga olaylik,  , biz Simpson formulasining yakuniy ifodasini olamiz
(5) formulaning birinchi yig'indisida segmentning barcha ichki tugunlaridagi funktsiya qiymatlari yig'indisi, ikkinchi yig'indisi funktsiyalarning o'rta nuqtalaridagi yig'indisi hisoblanadi men- segmentlar.

Agar segmentlarning o'rta nuqtalari tugunlar bilan birga katakchaga kiritilgan bo'lsa, unda yangi qadam h 0 \u003d h / 2 \u003d (b-a) / (2 * n) bo'ladi va (5) formulani quyidagicha yozish mumkin:

Ko'rib chiqing  ... Ushbu integralning qiymatini analitik usulda topish oson va u -0,75 ga teng. 3 va undan past darajadagi polinom ko'rinishidagi integral uchun Simpson usuli aniq qiymat beradi.
Ushbu integralni Simpson usuli bilan hisoblash algoritmi (formula (5)).
1-dan n-1gacha i bo'yicha pastadir
tsikl tugashi
i-da tsikl 1 dan n gacha
tsikl tugashi
s \u003d h * (f0 + 2 * s1 + 4 * s2 + fn) / 6
funktsiya f1
parametrlar x
x ^ 3 + 3 * x ^ 2 + x * 4 - 4 ga qaytish
Tilda Simpson usuli bilan integralni hisoblash dasturining misoli VFP((6) formulaga muvofiq):
O'nlikni 10 ga o'rnating
? "I \u003d" simpson (0,2,20)
TARTIBI simpson
PARAMETRALAR a, b, n
S_even \u003d 0
S_ toq \u003d 0
x \u003d a + h TO b-h QADAM 2 * h uchun
S_ g'alati \u003d S_ g'alati + 4 * f (x)
x \u003d a + 2 * h TO b-h QADAM 2 * h uchun
S_even \u003d S_even + 2 * f (x)
S \u003d f (a) * h / 3 + (S_even + S_even) * h / 3 + f (b) * h / 3
Tilda misol echimi VBA:
"integralning qiymatini antidivivativ bilan hisoblashning to'g'riligini tekshirish tartibi
s_ juft \u003d 0
s_ toq \u003d 0
X \u003d a + h uchun b - h qadam 2 * h
s_ g'alati \u003d s_ g'alati + 4 * f (x)
Debug.Print "s_ odd \u003d" & s_ g'alati
X \u003d a + 2 * h uchun b - h qadam 2 * h
s_ juft \u003d s_ juft + 2 * f (x)
Debug.Print "s_even \u003d" & s_even
s \u003d h / 3 * (f (a) + (s_ juft + s_ toq) + f (b))
Debug.Print "Simpson Method: s \u003d" & s
Debug.Print "Antiderivative value: s_test \u003d" & s_test (b-a)
VBA dasturini ishga tushirish natijasi:
s_ g'alati \u003d 79.9111111111111
s_ juft \u003d 36.0888888888889
Simpson usuli: s \u003d 2.66666666666667
Antivivativ qiymat: s_test \u003d 2.66666666666667