midterm solutions - 1 Let f(0 = 1 f(0 = 0 and f(1 = 2 Find...

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Unformatted text preview: 1. Let f(0) = 1, f’ (0) = 0, and f (1) = 2. Find the. Hermite Interpolation Polynomial, and use the polynomial you obtained to approximate f(0.5). [XV/'0 GU\ :/ %:l 2 1-0 I’D” ROM : 1+ owe—o) + l-(oc-wvc-o') : 1+ m2; gnaw/r11 RM? =— H 0-51: L25 2. Let f E 02h, b] and P1 its interpoiation linear polynomial at 330 = a and $1 = b. (a) Find the Cauchy remainder f (1:) — P1($). (b) Let f = sings, a = 0, b = 7r/2. Find a bound of the error ”f — Pl”co using the Cauchy remainder. (0.)). {m‘vfimz é‘Fgm) (‘X-Ok) (0H9) TPay come. gmém L) 3. Let P n(m) be the interpolation polynomial of degree at most 11 of the function f (as) at the distinct nodes (.120, 3:1. ”.er Suppose we know P (m) = sen—go — onm — $71) + Q($ J giggle polynomial of degree at most 11— 1 Find the divided difi'erence flmo e1 e _] uric-n W“ (”EL Pitt/:01: Miami, gMJM-Q’DH +10% 41.,- :04] Ma) (“'96- .) Tke My {—cm 1717906; Ts Hm. M, ”M W“ on He ode M ' PM) 5? aw‘lcacjxom 0(a) +6200)- : 3W x — (KMW’X Howe) 't 62W) :31“ 3mm? 21% X“+ 62m 91166 aw, i"; of degree and-mi— #4, W 306n19 -Hl€L-ea0['7lfl+5’fl In 3166/ we- name/1., --: Mac” r 3‘46 ., jomme 96a] 73- M;- ‘P [We/"3 1 4. Let f be a function defined on the interval [0, 1],. and Hf] = f f($)da:. Let h. = UN and o acj = jh, for j = 0,1,2‘,...,N. Then you can use the following Composite. 'Ikapezoidal Rule quadrature Thlfl = h (éflfiol + f($1) + + f(-T.N-ll + éfUENJ), to approximate the definite integral. (a) Let f (m) = 522. Use Th to approximate I [ f] with N = 2, for which youmay need the following quantities: 80 = 1, 8025 = 1.2840, 61 = 2.7183. Does Th converge to [[f] as h goes to 0? What is the rate of convergence? (b) Letflx.) = «'5. Does Tth] guarantee a second order convergence to Hf]? EXplain why. ((1,, Na. hil/13‘9‘5° 0:0, 06:05. «2:1. TV; ‘2'. 3-1-(i‘F'D‘ol‘l' -F((KI) + :2L._PC%)) o ‘ 1 I :i—LJi.€ + €05 +-'-2_€) :2 .5 (.gl + (~ng + Ji-2.1(9%)f—’/§7/{ new?» 1056941 on [ffljlmbebamlefll 105 CL WWMZ m [0,1,] —:> latfll: Mal?" ...
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  • Fall '08
  • Staff
  • Numerical Analysis, Trigraph, Polynomial interpolation, Cauchy, Hermite Interpolation Polynomial

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