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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 _]
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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.
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- Fall '08
- Staff
- Numerical Analysis, Trigraph, Polynomial interpolation, Cauchy, Hermite Interpolation Polynomial