1. Prove for any Xo E R that the iteration In+1 = g(xn) converges to a unique fix point a where g(x) = cos X. Find the value
1. Prove for any Xo E R that the iteration In+1 = g(xn) converges to a...
2. Consider g(x) (2 -x). Show that for all starting point ro E (0,2), the Picard's fixed-point iteration converges to the fixed point 1. Are sufficient conditions for convergence of Picard's iteration satisfied? 2. Consider g(x) (2 -x). Show that for all starting point ro E (0,2), the Picard's fixed-point iteration converges to the fixed point 1. Are sufficient conditions for convergence of Picard's iteration satisfied?
[ARCHIMEDES] Suppose that Xo = 2/3, yo = 3, xn = 2xn-1 Yn-1 xn-1 + Yn-1 and Yn = 1xn Yn-1 for ne N. a) Prove that xnx and Yn 1 y, as n = , for some x, y E R. b) Prove that x = y and 3.14155 < x < 3.14161. (The actual value of x is a.)
gol The fixed-point iteration Pn+1 = g(P) converges to a fixed point p = 0 of g(x) = x for all 0 < po < 1. The order of convergence of the sequence {n} is a > 0 if there exists > O such that lim Pn+1-pl =X. -00 P -plº Use the definition (6) to find the order of convergence of the sequence in (5).
Prove the following Green's identity for function..... 4. (a) Prove the following Green's identity for functions f.g E Co(2) where2C R'" where the notation : ▽ Vf n, where n is the outward pointing unit normal vector. You may use the divergence theorem, as well as the identity (b) Let G(x.xo) denote the Green's function for the Laplacian on Ω with Dirichlet boundary con- ditions, that is, 4,G(x, xo) = δ(x-xo), for x 62 (x,x;)= 0 for x Eon By...
1. (Exercise 4.10, modified) Given a series Σ 1 ak with ak 0 for all k and lim Qk+1 k0oak we will prove that the series converges absolutely. (This is part of the ratio test sce the handout.) (a) Fix a valuc q with r <<1. Use the definition of r to prove that there exists a valuc N such that for any k 2 N. (b) Prove that Σο, laNIqk-1 converges, where N is the value from part (a)....
just answer e through h 8. (11 pts) Let (Xn) be a sequence in Rº such that VnEN, Xn+1 = A· Xn+ where A = (5/8 5/3) and Xo = (-1) (a) (1 pt) Find X1. (b) (2 pts) Find the corresponding equilibrium point. (c) (1 pt) Determine the two eigenvalues 11 and 12 of A. (d) (1 pt) For each eigenvalue, find an eigenvector. (e) (1 pt) Is the equilibrium point a sink? Justify. (f) (1 pt) Deduce the...
2. Find a root ofthe functionf(x)=cos(x) +sin(x)-2x2 to fourdeci mal places for!f(xn +1 )1< 0.001 and Ixn-1-Xnl0.001 for each of the following rootfinding methods and initial guesses: a) Newton's Method, for xo = 0.2. b) Secant Method, for x-,-0.2 and xo = 0.5. c) Considering the following fixed point problern for xo=0.2 cos(xn)sin(n) d) Write a code to approximate the root of f(x) for each a), b) andc 2. Find a root ofthe functionf(x)=cos(x) +sin(x)-2x2 to fourdeci mal places for!f(xn...
Can you help with this? Thank you always. Suppose that the function f : R-+ R is continuous at the point xo and that f(xo) > 0. Prove that there is an interval 1 (x,-1/n, xo + 1 /n), where n is a natural number, such that f (x) >0 for all x in I. (Hint: Argue by contradiction.) Suppose that the function f : R-+ R is continuous at the point xo and that f(xo) > 0. Prove that...
Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-values described in this procedure, as follows: (a) Write the linear approximation 1 (x) to the curve at the point (Xn,f(xn). (b) Find where this linear approximation passes through the x-axis by solving L(x)0 for x. xn + 1-1,-I n). is the recursion formula for Newton's Method. : Xo Xo Problem 1. Show that the recursively-defined sequence x*i-x, - gives the sequence of x-values described...
Suppose that 1/2 where Z is any random variable with E22c, say, with c> 0 and a E R fixed, and X is any other random variable. (a) Let e > 0. Use Chebyshev's inequality to show that (b) For what values of does the argument in part (a) prove that Xn converges in probability to X? (c) For the values of α identified in part (b), what other mode of convergence of Xn to X is assured (without any...