MATH260001 Numerical Analysis 201819
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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 fixed point.
State the fixed point theorem giving the conditions that guarantee the iteration scheme xn+1 = g(xn ) is stable and has a unique fixed 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 find 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 (i①)(ξ) (where f (i①) 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 find the solution.
2022-08-22