Question

this is numerical analysis. Please do all the questions

3. (a) Consider the quadrature rule path ( * s(a)dx = Af (a – 1) + Bf(a) + Cf(a+h). Find A, B, C which maximize the degree of precision. Hint: First derive the rule for a = 0 and then use a change of variable. (b) State this degree of precision and verify it is not any higher. (c) Suppase g is a function whose 3rd divided differences are all the same. You are given that 9(-2) = 1,9(0) = 3, 9(1) = 4,9(2) = 5. Show, without finding an explicit formula for y, that 4. Hint: First express it as the difference of two integrals. Then use a well-known quadra ture rule for each. This question is unconnected to parts (a) and (b) above, and the rule in part (a) isn't applicable.

Answer 1

3. (a). We want this ath s femo dve = A f(a+h) +B(f@))+c fcoth), first we derive for azo, then S" firms daz A4Ch) +8 $60)+cfch formula to hold for 1 , x , .... s fenn das A(1) +861 +261 h = A +B+C (1) For f(x)=x, we have Shedx= AC-ty +B60)+cchs -Ahtch ACC-A) (2) for fewe! 0 h² 2 C-A= b 2 NG -(3) for f(x) = x² , we have shx?dx= A(M3+ B (0) +ch%) Ah²+ch²= 6²A+C) -3 .. 3 Now A+b+c=h C-A = h 2 we get A+C =h 3 We have 3 equations f 3 unknowns. golving 3 equations A=-h B = 2h C-5h 3 Sfino dx = = f(-6) + 24 (0) + shf(n) Scanned with CamScanner By change of variables, we put 0 CS Hot-a.

ath a f(t-a) ok = -f(+3+ 254 (0)+ 55 f (h) Degree of precision is a fince it holds for polynomials 1, x, x OY Definition of degree of precision - The degree of accuracy precision of a quadra tie formula is the largest positive integer n such that the formula is exact for uk for each k=0, 12,..n. CS Scanned with CamScanner

c) Given that g(x) is given by 2 0 2. ५ 3 4 Note that, y f[xo aj flap , xx f[r, dj, te 2. 1 3 — 4 2 S 1 o By Newton's interpolation ormula g(x) N lt 1. (2+2) g(x)=x+3 g(a) dx = 1 x + 3 da za C s galdr = 4 V2