Question

Q1: The error function erf(z) is defined as:

erf (0) = dt

Using integration step of 0.05, evaluate erf(0.5) and compare with the tabulated value erf(0.5)=0.5205 using:

1. Three-point Gaussian quadrature formula
2. Trapezoidal Rule (h=0.05)
3. Simpson’s 1/3 Rule  
4. Simpson’s 3/8 Rule

Compare the results. Note: change the step to suit the assigned methods that are restricted by the number of panels. That is, increase the number of panels to a level that enables you to use the method(s) that cannot be used at the integration step assigned. The limit of integration stays the same.

Answer 1

7 2 Giyen 2015 (2) 1 e dt Thoree- point Jaussian Quadsaluse formula (b-a)X + (bra) Here 11 ろ 670 at 06, a = 0, b=0.5 P % (0.5)x+ (0.5) ta t 5 1a4+1 구 4 ta Yu (241) when t=0 26,2.-1 t-0.5,20= 1 data dal :: Pit) = f(x) 4 of(x+1) Y6 (x+1)? -42 e ୧ 0.5 -t² t's =1 e T(x+1) e dz e da ) - ) By Gaussian Three point formula Ś Plan des 94 546)* [(=J&)* 4 () CS Scanned with CamScanner

-PO) 44 e 16 (0.4.2) e-Yc e P(0) 0.9394 16 (+131, +3) P(+15%) : e $(+F75) 0.82133 $(-1315) = 0.99.68 0.8813 f(x) dx = 347.60.9394) + (0.996846 1 s f(x) da 1.8450 * S f(a) dx = 1.8450 4 11 0.46126 se na buceta na moyon to 9 2 = 0.52043 erf (0.5) - 0.52048 CS Scanned with CamScanner

Totapezoidal avle 스 S f(x) dx Cr(io) 1 2F(x) +: + 2f(x) + it f(ual 2 a where boa 스의 b 005 aeo, X-005 ( 0 5 03 - 10 005 ㅂ - 0-05 , 0.1 01 0 15 3 6·25 0.3 035 0 00150.5 장 자 기. 것이 게 씨 2 이 CO 5 dt - 17. 0.05 2 6:05 | (io) +2(K) +2f(is) f af(33) tz f(x) 다 2 flas) + 2f(x) + 2. R (7) S(xs) + 2 f(24) = (xo) [1: + 201175)+ 2 (0.99) + 2(0.977) + 2 ( 0.97) + 2 (0-9294) + 2(0.9130 + 2 (0.8847) + 2 (0.852)) + 2 (o.813 17 t 0.7488) | 1+ 1-995 + 1.98 + 1-954 + 1.42 4 40 t 1.8988 1.82 48 + 1 4649 + 1-1여 + 1-5332 + o.1386) 18:44 26 ] 40 05 깃 e dt 461065 . CS Scanned with CamScanner

0.5 42 e X0=461065 Nr 0.5 12۔ 0.52025 ad PA et? dt WT dt s f(x) dx = (9.47661 aca, Simpson's yes no slule b tor tYnt 5 [c4ot9n) + 4(4+93+ 4+ + 2(42+Yu+YA+ +Y. t b=0.5 b=ooos | Yo a plo) i 4, = P(2) = 0.9975 Yu- f(0) = 0.99 43= P(3) = 0.9777 94 = $(4). 0.9607 95 = f(3) = 0.9394 46 = $(6)= 0.9139 97 = $(7) - 0.8847 0.8542 yp = f(8) 99 = f(9) Yids Plio) = 0.7788 0.8166 CS Scanned with CamScanner

t6.9607 +09394 +0.8847 0.5 11+ 0.7788) +4C0.9975+ 0.9144 0,8166) + 0.9139+ 85427 2 +0.93944 0.8847 + (6.99+0.960770 [ 24.68] 1 60 0:5 el dit -43 61 00.46133 Set o.s 42 dt = 2 2 Slo X0.46133 No OL os -42 al é de 0-52055 Simpson's rule 3/8 b s f(x) dx = 36 [(yotyn) +3(41+92 +94 +45 + - +40-1) + 2( 43+ 46+994 Ya-3)] a h=0.05 aco, b=0.5 0.5 0.5 De f(t) dit dt "Y? 3x0.05 (1+014 788) +36 0.7788)+3 (0.999540.99 8 +0.6542) +210.9474+0-41394 0.5 M 0.5 3 X 24.0747 160 0.4514 s f(t) dt 5 CS Scanned with CamScanner

s e tatt 읊 4514 시 0.5093 OS 2 92 는 O.S03 에부 SEIÀ ef (0·5 (t) 3-point Gaussian | Trape toidal to value Simpson's Simpso rule * rule 318 05 인 e alt O S2025 0.52055 0.50ml 0.5 2018 이념 ) CS Scanned with CamScanner
we get accurate value by using Simpsons 1/3 rule