1) Find the root or the fixed point. (40 points)

a) Write out the Newton method for solving nonlinear problem f (x) = x − cos(x), given an initial guess 0 x .

State the order of convergence of the Newton method?(10 points)

b) Roughly reasoning the convergence of iteration sequence (proof of Fixed Point Theorem): (10 points)

Let g(x) be a continuous function defined on[a,b]. Suppose we have (1) g(x) ∈[a,b]for any x ∈[a,b]; and

(2) there exists a small number 0 < k <1, such that | g′(x)|≤ k for all x ∈[a,b]. Then for any initial guess

[ , ] p0 ∈ a b , the sequence defined by ( ), 1, pn = g pn−1 n ≥ converges to the unique fixed point [ , ] * p ∈ a b .

c) Consider the sequence ∞ { k }k=0 x generated by the following iteration method with initial guess x0 = 1. Use

Fixed Point Theorem to show the sequence is convergent to the fixed point [1, 2] * x ∈ . (10 points)

( ) 2

1

2

3 ( ) 2

1 2

k 1 k k k x = − x + x + = g x +

d) Given nonlinear function ( ) 2 3 2 f x = x − x − , show/verify that it has a solution/root in interval

[a,b] = [0,4]. Use Bisection method to generate the first two approximations 1 2 p and p of ∞ {pn }n=1 . (10

points)

2) Let’s consider to approximate f (x) = sin(x) with Lagrange interpolation polynomial. Given the function values

of ) 1, 2 ,sin( 2

2 ) 4 sin(0) = 0,sin(π = π = consider the quadratic Lagrange interpolation polynomial

( ) 2 p x to approximate sin(x). (40 points)

a) Write out the quadratic ( ) 2 p x polynomial (no need to simply the function expression). (15 points)

b) Use the polynomial to obtain an approximation ofsin(π / 6) . (10 points)

c) For any x∈[0,π / 2], write out the error term between sin(x) and its approximation ( ) 2 p x ? Use this

analytical result to estimate the error of the approximation at point x = π / 6.(15 points)

3) Suppose we have a uniform partition of the domain with mesh size h and the grid points are denoted as

x j = j * h . (20 points)

a) Use polynomial approximation idea to derive the 2-points forward Euler scheme listed below. (10points)

′� � ≈

� +1�− � �

ℎ

b) Use this scheme to approximate f ′(1.0) for f (x) = sin(x) with h = 0.1 , compute the actual error. (10points)