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MATH260001

Numerical Analysis

Semester Two 201819

1.   (a) Show that the function f (x) = exp(x) -   has only one zero, x* .  Write down the general step of Newton’s  method for solving  f (x)  =  0.   Starting with an initial guess of x0  = 0, calculate the next three approximations to the solution of f (x) = exp(x) -  = 0 using Newton’s method.  For each iteration calculate the error from the exact solution.

Write down the map xn+1  = g(xn ) corresponding to the Newton’s method solution of f (x) = exp(x) -  = 0 and show that g\ (x) = 0 at the xed point.

State the xed point theorem giving the conditions that guarantee the iteration scheme xn+1  = g(xn ) is stable and has a unique xed point in the interval [a, b].

Show that g(x) satisfies the conditions of this theorem in a neighbourhood of x* . (b) Write down the Lagrange form of the interpolating polynomial P (x) that satisfies

P (x) = f (x) for x = xi , i = 1, . . . , n and x1  < x2  < . . . < xn .         (1)

Find the form of the polynomial that interpolates f (x) =  北(1)  through the points x1  = 0.5, x2  = 1 and x3  = 3, then simplify it. Use this polynomial to estimate the value of f (2) and find the error from the actual value.

Using the error formula for Lagrange interpolation,

1    n                         dn f

f (x) = P (x) + n! i=1(x - xi ) dxn  (ξ) ,    for some ξ · (x1 , xn ),         (2)

find upper and lower bounds for the error in your estimate above.  How does this compare with the actual error?

2.   (a) Simpson’s rule for integrating a function over the interval [-h, h] can be written in the form,

h

f (x)dx = w1 f (-h) + w2 f (0) + w3 f (h) + E,                   (3)

-h

where w1 , w2  and w3  are constants, and E represents the error term.  Use the method of undetermined coefficients to nd the constants w1 , w2 and w3 . Calculate the Simpson’s rule approximation to the integral

m/2

I =         cos(x)dx,                                          (4)

0

and calculate the error from the actual answer.

(b) Show that Simpson’s rule is exact for all cubic polynomials and hence using a suitable quartic polynomial obtain the error term in the form Kf (i)(ξ) (where f (i)  denotes the fourth derivative of f) for some value ξ · (-h, h) where K is a constant to be determined. Show that the error in the integral I is consistent with this error term.

3.   (a)  Consider the ordinary differential equation,

dy

dt

By integrating equation (5) over the interval [tn , tn+1] and approximating f (t, y) as a constant derive the forward Euler approximation,

yn+1  = yn + hf (tn , yn ),                                       (6)

where h = tk+1 - tk  (for all k).

Verify that y(t) = cos(t) is the solution of the initial value problem

dy

dt

Use the forward Euler method to calculate an approximation to y(π/3) using a step-size of h  =  π/6 and calculate the absolute error from the exact solution y(π/3) = 0.5.

(b)  By finding a suitable quadrature formula for integration of equation (5) over the interval [tn , tn+1] derive the three-step Adams-Bashforth scheme,

yn+1  = yn +  [23f (tn , yn ) - 16f (tn-1 , yn-1 ) + 5f (tn-2 , yn-2 )].       (8)

4.   (a)  Find the LU-factorisation of the matrix,

 -22 ( 1

3

1

2

1(2) 1 ,

showing your method of working. Demonstrate how this factorisation may be used to solve the linear system,

-212        =  -433 .                                 (9)

(b)  Explain what is meant by (row) pivoting in the context of matrix factorisation. Show that the linear system,

   2--1(2)     =  -421 .                               (10)

cannot  be solved  by  LU-factorisation without  pivoting even though the  matrix is non-singular.  Show how pivoting overcomes this problem and hence nd the solution.