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(MATH3311)[2012](s)midterm~=cqp27^_29311.pdf
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Midterm Exam of MATH3311, Spring, 2012

Problems (The numbers in brackets are credits.):

1. (10) Use three-digit rounding arithmetic (in every step) to compute the roots of
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x + x +1=0.
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Use reliable schemes and present the relative errors.

2. Let g(x) be de.ned over [0, 1].
(a)
(3) State the conditions for g(x) to have at least one .xed point over [0, 1].

(b)
(3) State the conditions for g(x) to have a unique .xed point over [0, 1].

(c)
(4) Suppose g(x) satis.es both (a) and (b). For an absolute error of . = 10.8 , how many iterations will it take the .xed-point iteration method to stop?


3.
(10) For the root .nding problem of x2 . 2 = 0. Propose and write down the fastest iteration formula you have learned in lectures.

4.
Point out the convergence rate of various root-.nding methods below, with L stand-ing for linear convergent, SL for super-linearly convergence, and Q for quadratic convergence.

(a)
(2.5) Binomial method.

(b)
(2.5) Fixed-point iteration method.

(c)
(2.5) Newtons iteration method.

(d)
(2.5) Secant method.



5.
(10) Given {xi,f(xi)}n , write down a pseudo code for valuating the Lagrange in-


i=0
terpolation Pn(x) for any x. If the function is f(x)= ex , provide the error term |f(x) . Pn(x)|.
===================== Good Luck! =====================
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