Numerical Methods Exam 2014 PDF

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2014 examination for Numerical Methods. Good luck...

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Examination in School of Mathematical Sciences Semester 2, 2014

104838 MATHS 2104 Numerical Methods II

Official Reading Time: Writing Time: Total Duration:

10 mins 120 mins 130 mins

NUMBER OF QUESTIONS: 5

TOTAL MARKS: 58

Instructions • Attempt all questions. • Begin each answer on a new page. • Examination materials must not be removed from the examination room. Materials • 1 Blue book is provided. • A calculator is required. Only calculators without remote communications or CAS capability are allowed. The memory must be cleared before entering the examination room. • A formula sheet is attached.

DO NOT COMMENCE WRITING UNTIL INSTRUCTED TO DO SO.

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Numerical Methods II

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1. Consider the equispaced data (0, f0 ), (h, f1 ), (2h, f2 ), where fj = f (jh) for j = 0, 1, 2. (a) Write down the Lagrange form of the quadratic interpolating polynomial that passes through these points. [2 marks] (b) Use the interpolant found in part (a) to obtain an estimate of f0′ = f ′ (0).

[2 marks]

(c) Use Taylor’s theorem (not the polynomial interpolation error theorem) to find the order of the error for the estimate in part (b). Hint: You will need to include terms up to O(h3 ) in Taylor’s theorem. [4 marks] (d) The formula is tested by differentiating exp x and the error is plotted in Figure 1. Are the test results consistent with the error derived in (c)? Explain your answer. [2 marks] [TOTAL 10 marks] −2 −3 −4

log

10

|error|

−5 −6 −7 −8 −9 −10 −11 −12 −8

−7

−6

−5

log

10

−4 h

−3

−2

−1

Figure 1: Numerical differentiation error obtained when differentiating exp x using the finitedifference formula obtained in (b).

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Numerical Methods II

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2. (a) Use the trapezoidal method to estimate the integral Z

4

f (x) dx 0

given the discrete data listed in Table 1 below. Show your working. Use all the data points. [2 marks] 0 1 3 4 1 3 5 −1

xj fj

Table 1: Discrete data, fj = f (xj ). (b) Write a fully-vectorised function that uses the trapezoidal method to estimate the integral Z xn f (x) dx (1) x1

using discretely sampled data fj = f (xj ), j = 1, 2, . . . , n. The function must have the interface function s = itrap(f, x) The input arguments x and f are vectors containing the data xj and fj , respectively. Note that the xj need not be evenly spaced. You may assume that the xj are arranged in ascending order. Do not document your function. [4 marks] (c) Use the polynomial interpolation error theorem to show that the error ej in the trapezoidal method estimate of Z xj+1 f (x) dx xj

is bounded by

1 Mh3j , j = 1, . . . , n − 1, 12 where f ′′ (x) ≤ M for x1 ≤ x ≤ xn and hj = xj+1 − xj . |ej | ≤

[6 marks]

(d) Hence show that the error e in the trapezoidal estimate of (1) is bounded by |e| ≤

1 M(xn − x1 )h2 12

where h = max hj .

[4 marks]

j

[TOTAL 16 marks]

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Numerical Methods II

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3. (a) Consider the linear system  2 A = 6 0

Ax = b, where    0 1 x1 1 5 , x = x2 2 7 x3



 5 and b =  15  . −3

The matrix A has the lu factorisation    1 0 0 2 0 1 3 1 0 0 1 2 0 2 1 0 0 3 Use the lu factorisation of A given above to solve Ax = b for x. Show your working. [5 marks] (b) Consider the linear system     x 4 2 −1 5 −1 3     y = −4 . 1 z 1 −2 5 7 

Perform one Jacobi iteration of the resulting system, starting from the initial guess [1, 2, 3]T . Show your working. [2 marks] (c) Consider the boundary-value problem d2 u − u = x2 , dx2

u(0) = 0,

u(1) = 1.

(2)

For constant grid spacing h, the second derivative can be approximated by u′′j = u′′ (xj ) =

uj +1 − 2uj + uj −1 + O(h2 ), h2

(3)

where uj = u(xj ), xj = jh and j is an integer. (i) Let u = [u0 , u1 , u2 , u3 ]T . Use (3) to write down a linear system of the form Du = f that approximates the differential equation and boundary conditions given by (2). Make sure you clearly define the elements of the matrix D and the vector f . [3 marks] (ii) Will Jacobi iteration converge to the solution of this system of equations? Explain your answer. [2 marks] [TOTAL 12 marks]

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Numerical Methods II

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4. (a) Consider the nonlinear equation x = e−x .

(4)

(i) Write down the formula for the Newton iteration to solve (4) in the form xk+1 = xk + · · · [3 marks] (ii) Using an initial guess x0 = 0.5, find a solution of (4) to two decimal places. Show your working. [2 marks] (b) Consider the nonlinear system x2 + y 2 = 1 x + xy + y = 1

(5)

Write a short Matlab script that uses multidimensional Newton iteration to find any of the solutions of (5). Do not document your script. [5 marks] [TOTAL 10 marks]

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Numerical Methods II

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5. Consider the initial value problem, dy = f (t, y), dt

y(t0 ) = y0 .

(6)

The modified Euler method is yk+1/2 = yk + 21 hf(tk , yk ), yk+1 = yk + hf(tk + 21 h, yk+1/2 ),

(7)

where yk = y(tk ) and h is the step size. (a) Suppose that f (t, y) = cos t − y 2 and y(1) = 0. Find an approximate value of y(1.5) using a single step of the modified Euler method (7). Show your working. [4 marks] (b) Use Taylor’s theorem to show that the error in a single step of the modified Euler method is O(h3 ) as h → 0. [6 marks] [TOTAL 10 marks]

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Numerical Methods II

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Useful Formulae The Lagrange form of the interpolation polynomial is pn (x) =

n X

fk Lk (x),

n Y x − xi , Lk (x) = xk − xi i=0 i6=k

k=0

where n is the degree of the polynomial. If f (x) has n + 1 continuous derivatives on the smallest interval I that contains {x, x0 , . . . , xn }, then the error of the polynomial interpolant is n f n+1 (t) Y ǫn (x) = f (x) − pn (x) = (x − xk ) (n + 1)! k=0

where the unknown t ∈ I depends on x.

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