6CCE3EAL Engineering Algorithms 2023
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6CCE3EAL
Engineering Algorithms (Mock)
January 2023 (Period 1)
1. a. A one-dimensional ordinary differential equation (ODE) y\ = f(x,y) with initial condition y(0) = y0 is approximated by the trapezoidal method
yn+1 = yn + ∆x [ f(xn+1,yn+1) + f(xn ,yn )] ,
where xn = n∆x and ∆x is the step size.
i. What is meant by the order of a numerical ODE scheme? [6 marks]
ii. Consider a test problem where f(x,y) = λy . Show that the method
is stable only if
' ' < 1. [14 marks]
b. Given a function f(x) and an equispaced grid of points xn = n∆x, derive a two-point central difference scheme for the derivative of f . Determine the accuracy of the scheme. [10 marks]
c. Consider the matrix equation Ax = b given by
l1 1 1 」 lx1」 l 1 」
3 1 −3 x2 = 5
「1 −2 −5 「x3 「10
i. Find a LU decomposition of A using the Doolittle method. [12 marks]
ii. Using this LU decomposition, solve the system using forward- and back-substitution. [8 marks]
2. a. Consider the function f(x,y) = x2y − xy2 + 3x2y2 .
i. Compute the gradient and Hessian of f(x,y). [8 marks]
ii. Find and classify all critical points of f(x,y). [12 marks]
.
b. Using the method of feasible regions, solve the linear program
min z = 2x1 + x2
s.t. x2 ≤ 10
2x1 + 5x2 ≤ 60
x1 + x2 ≤ 18
3x1 + x2 ≤ 44
x1 ≥ 0,x2 ≥ 0. [14 marks]
c. Consider the function g(x,y) = 1 − (xy − 3)2 .
i. Compute the gradient of g(x,y). [4 marks]
ii. Starting at (x,y) = (0, 0), apply one step of the steepest ascent method to approximate the solution to the problem
max g(x,y).
(x,y)∈R2 [12 marks]
2023-01-05