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Practise Exam
1. There will be one page of short answer questions.
2. Let,
 

1
2
5  ,b =  1  .
1
20

10 3
A =  3 40
2
5
To solve the system A~x = ~b using Jacobi iteration, we write the k th iteration as,
~x(k) = T ~x(k−1) + ~c .
Solve for the matrix T and vector ~c.
3. Consider the following nonlinear system of equations,
f1 (x1 , x2 ) = x1 x2 + ex1 x2 ,
f2 (x1 , x2 ) = 3x2 + cos(x1 x2 ) ,
(a) We wish to use Newton’s method to find a root of this system. Give the iteration formula (you don’t need
to invert any matrices for this part, just express the formula in terms of an inverse)
(b) Given ~x(0) = (x1 , x2 ) = (0, 1) find ~x(1) .
Rπ
4. Use composite Simpson’s rule to estimate 0 sin2 (x) dx to within 0.01. Make sure to prove that your error is
within these bounds.
5. Numerical diffentiation
(a) Find the quadratic polynomial interpolating y(x0 ), y(x0 + h3 ) and y(x0 + h). Include the error term.
(b) Use the above polynomial to find an approximation to y ′ (x0 ) using the values at x = x0 , x0 + h3 , x + 0 + h.
Also find the error term
6. Use Richardson Extrapolation to increase the order of the above approximation.
7. Find a, b, c such that y = a2 + b cos 2x + c sin 3x is the least square approximation to y = x in [−π, π] with
respect to the weight function w(x) = 1.
8. Derive the nonlinear system for a, b such that the exponential function y = beax fit the following data points,
(1.0, 1.0), (1.2, 1.4), (1.5, 2.0), best in the least square sense.
Note: You need to derive the nonlinear system for a and b but you do not have to solve it.
9. Derive a second order Taylor’s method to solve the following initial value problem.
y ′ = tey ,
y(0) = 1 .
10. Consider the boundary value problem,
y ′′ = sin(t)y ,
y(0) = 1 ,
y(1) = 2 .
(a) To solve this problem using linear shooting, the equation must be restated as two initial value problems.
List the restated equations.
(b) What is the solution to the boundary value problem in terms of the solutions to the restated initial value
problems.
11. (a) Show that the truncation error for the midpoint method
y0 = y(0) ,
yi+1 = yi + hf (ti +
is O(h3 ).
1
h
h
, yi + f (ti , yi )
2
2
(b) At time ti , the above midpoint method and a third order Runge Kutta method are computed using the
same yi . The numerical value from the above method gives wi+1 = 0.55 and the numerical value from the
third order method gives ỹi+1 = 0.56. What step size should be used to keep the error below 0.001.
12. (a) For the initial value problem y ′ = 0, y(0) = 0 what is the exact solutions?
(b) Find y2 and y3 for the following two step method.
yi+1 = 4y1 − 3yi−1 ,
y0 = 1, y1 = ǫ .
(c) Is the above method stable?
13. Construct a 3-step method to solve the ODE,
y ′ = f (t, y) ,
y(0) = α , y(h) = α1 , y(2h) = α2 ,
of the form,
wi+1 = wi−2 + b0 fi−2 + b1 fi−1 + b2 fi .
Here fi = f (ti , wi ). Construct the method by considering a polynomial interpolating f (t, y(t)) at the points
ti−2 , ti−1 , ti and using this polynomial to find an approximation of the following form,
Z
3h
f (t, y(t)) dy = b0 fi−2 + b1 fi−1 + b2 fi + · · ·
0
14. The 3-step Adams Moulton method is given by,
w0 = α ,
wi+1
w1 = β , w2 = γ ,
h
= wi + (5f (ti+1 , wi+1 ) + 8f (ti , wi ) − f (ti−1 , wi−1 )) .
2
(a) For the initial value problem,
y ′ = y sin(t2 ) ,
y(0) = 1 .
use the 3-step Adams Moulton method with h = .1 to find an approximation to y(.2). You may assume
that y(.1) = 1.0001 has already been calculated from a one-step method.
(b) What would you need to do in order to approximate the solution to,
y ′ = t sin(y 2 ) ,
y(0) = 1 ,
using the 3-step Adams Moulton method(briefly explain any difficulties you would encounter and how you
would solve them).
15. (a) The trapezoid method is given by
yi+1 = yi +
h
(f (ti , yi ) + f (ti+1 , yi+1 )) .
2
Show how to determine the region of absolute stability. You don’t need to actually find it.
(b) The Backwards Euler method is given by
yi+1 = yi + hf (ti+1 , yi+1 )
and its region of absolute stability is outside the unit circle |hλ − 1| > 1. What problems could you
encounter when trying to approximate the solution to y ′ = y using this method? Is there a restriction on
h to ensure convergence to the true solution? If so what is the restriction?
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