In converges to √2 with rate of convergence 1. In converges to √2 with rate of convergence 2 Fixed point iteration for f(x) = x² - 2 Newton's method for f(x) = x² - 2

Verify that x = 1/a is a fixed point of the function g(x) x(2 ax). Determine the order of convergence and the asymptotic error constant of the iteration sequence Pn = 9(Pn-1) toward x = = 1/a. MOT = -

2. Write the formula for finding a root of f(x) = - x³ –- cos (x), where x is a real positive number, by Newton- Raphson's method. Then, compute the second iteration approximation. Discuss the order of convergence of Newton-Raphson's method.

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The approximation of the root x" of the function f(x) = 2x + 5 - e* in the interval [3,4] accurate to within 10- using secant's method starting with x, = 3.2 and x, = 3.27. Needs 1 iteration Does not converge Needs 3 iterations O Needs 2 iterations

1). To find a zero of a function Ax) = x²-3x-2 by the Newton-Raphson iteration, the sequence {p,,}n=0 converges to a root p = -1 in a ( ) way. 2). How many initial points are required to guess in Newton-Raphson slope algorithm for finding a zero of the function f(x)? ( ) 3). Point out, respectively, one of the advantages and drawbacks of the bracketing method, Advantage: ( ); Drawback: ( ). 4). In order to constitute a fourth-order polynomial P4(x), how many distinct data points (xk, Yk) are at least required through which P4(x) will pass ( ). 5). If the Gauss-Legendre rule is used for an exact integral of a cubic polynomial over interval [-2, 2], write out the node abscissas of the least number ( and the related weights ( ). 6). Suppose the integral interval is [a, b]. The step size for the simple trapezoidal rule is ), while the step size for the simple Simpson's % rule is ( ).

Consider the function g(x) = x^2 + 3/16(a) This function has two fixed points, what are they?(b) Consider the fixed point iteration xk+1 = g(xk) for this g. For which of the points you found in(a) can you be sure that the iterations will converge to that fixed point? Justify your answer.(c) For the point(s) you found in (b), roughly how many iterations will be required to reduce theconvergence error by a factor of 10?

2. Given g(x) = 4r 4. %3D Prove that the given form r = for the use of fixed point iteration method. Thus, use the fixed point iteration formula ri+1 Stop the iteration when |ri+1 - r: < 0.005. g(x) satisfies the convergence criteria g(x.) to find the root starting with ro = 3.2.

Solve for the following (with solutions)

To find the root of the function f(x), write down the mapping function, g(x), for Newton’s method. Using x0 = π/3 as the initial guess, perform the first four iterations using Newton’s method. Also calculate the ratio |xn/xn−1| at every step. What does this ratio denote and what does its limiting value tell you about the convergence rate of Newton’s method?