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Unformatted text preview: Math 151A Notes: • Exceptionally, there is no oﬃce hour with the instructor on Monday, January 29. • A sample matlab code for ﬁxed-point iteration has been posted on the class web-page. HW #4, due on Friday, February 2 - Reading: section 2.4. - The problems below from Section 2.4. #2(a): Use Newton’s method to ﬁnd solutions accurate to within 10−5 for the problem: 1 − 4x cos x + 2x2 + cos 2x = 0. Repeat using the modiﬁed Newton’s method described in eq. (2.11). (for the output, give the ﬁnal answer and the number of steps required in practice). #6(a): Show that the sequence pn =
1 n2 converges linearly to p = 0.
n #8(a): Show that the sequence pn = 10−2 converges quadratically to p = 0. #10: Suppose p is a zero of multiplicity m of f , where f (m) is continuous on an open interval containing p. Show that the following ﬁxed-point method has g ′(p) = 0: mf (x) g (x) = x − ′ . f (x) What is the order of convergence ? #12: Suppose that f has m continuous derivatives. Modify the proof of Thm. 2.10 to show that: f has a zero of multiplicity m at p if and only if 0 = f (p) = f ′ (p) = ... = f (m−1) (p) = 0, but f (m) = 0. 1 ...
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This note was uploaded on 10/28/2010 for the course MATH 151a taught by Professor Staff during the Spring '08 term at UCLA.
- Spring '08