Exercise 2.1. Let f(x) be a twice-continuously differentiable univariate function definedon a domain (a, b). Assume that f has a zero r* € (a, b) and that f"(x) > 0 and f'(x) < 0for all æ € (a, b). Consider the application of Newton's method to find a zero of f, with xochosen so that r, € (a, b) with f(x1) > 0.(a) Show that xị < x*.(b) Show that if ¤k E (a, b) with f(xk) > 0, then xk < *k+1 < x*.(c) Show that if xk € (a, b) and f(xk) > 0, then f(xk+1) > 0.(d) Show that æk < æk+1 < x* for all k > 1, and hence verify that, under the conditionsstated, the Newton sequence {xk} converges to x*.(e) Hence show that Newton's method for the reciprocal r = 1/d (see Exercise 1.4) willconverge for all 0 < xo < 2/d.

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Exercise 2.1. Let f(x) be a twice-continuously differentiable univariate function defined on a domain (a, b). Assume that f has a zero a* € (a, b) and that f"(x) > 0 and f'(x) < 0 for all æ € (a, b). Consider the application of Newton's method to find a zero of f, with ro chosen so that r, € (a, b) with f(x1) > 0. (a) Show that xị < x*. (b) Show that if æk € (a, b) with f(®k) > 0, then æk < *k+1 < x*. (c) Show that if ær E (a, b) and f(x4) > 0, then f(xk+1) > 0.

Exercise 2.1. Let f(x) be a twice-continuously differentiable univariate function defined on a domain (a, b). Assume that f has a zero a* € (a, b) and that f"(x) > 0 and f'(x) < 0 for all æ € (a, b). Consider the application of Newton's method to find a zero of f, with ro chosen so that r, € (a, b) with f(x1) > 0. (a) Show that xị < x. (b) Show that if æk € (a, b) with f(®k) > 0, then æk < k+1 < x*. (c) Show that if ær E (a, b) and f(x4) > 0, then f(xk+1) > 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Exercise 2.1. Let f(x) be a twice-continuously differentiable univariate function defined
on a domain (a, b). Assume that f has a zero r* € (a, b) and that f"(x) > 0 and f'(x) < 0
for all æ € (a, b). Consider the application of Newton's method to find a zero of f, with xo
chosen so that r, € (a, b) with f(x1) > 0.
(a) Show that xị < x*.
(b) Show that if ¤k E (a, b) with f(xk) > 0, then xk < *k+1 < x*.
(c) Show that if xk € (a, b) and f(xk) > 0, then f(xk+1) > 0.
(d) Show that æk < æk+1 < x* for all k > 1, and hence verify that, under the conditions
stated, the Newton sequence {xk} converges to x*.
(e) Hence show that Newton's method for the reciprocal r = 1/d (see Exercise 1.4) will
converge for all 0 < xo < 2/d.

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