Exam in Numerical analysis Course code C0002M Exam date 2010-03-24
Total number of exercises: 6 Hours 09.00 – 14.00
Teacher: Ove Edlund
Teacher on call: Ove Edlund Phone: 070-2828661
Conditions for grades (LTU): U:0–13, 3:14–19, 4:20–25, 5:26–30
Conditions for grades (ETCS): F:0–13, E:14–15, D:16–18, C:19–22, B:23–26, A:27–30 The result will be reported on the student portal. The marked solutions may
be fetched at “studenttorget” when indicated at “studentwebben”.
Allowed accessories: Pocket calculator, The special notes attached to the exam
Full written solutions should be given to all problems. The reasoning, used notation and calculations may not be so sparsely presented that they are difficult to follow.
Also partially solved problems may give points.
Just the answer gives 0 points.
Department of Mathematics
1 (3)
Problem 1
Given the function
f (x) = 4 ln x − x . Calculate an approximative solution to f (x) = 0 by
(a) doing 4 steps of the Bisection method starting with the bracket [1, 2]. (2 p) (b) doing 3 steps with Newton’s method starting in x0 = 1. (2 p)
Problem 2
Given the matrices
Z =
1 −1 2
−1 1 0
1 −3 1 1 −3 −1
, Q = 1 2
1 1 1
−1 −1 1
1 −1 1 1 −1 −1
, R =
2 −4 1
0 2 1
0 0 2
.
(a) Show that Q and R is a QR-factorization of Z. (2 p) (b) Use the QR-factorization to find the a that gives the best possible
approximation, in least squares sense, to the equality
1 −1 2
−1 1 0
1 −3 1 1 −3 −1
a1 a2 a3
=
8
−2 8 6
. (2 p)
(c) Prove mathematically that the formula you used in (b) gives the least
squares solution. (3 p)
Problem 3
(a) Find a Newton-interpolation polynomial that interpolates the following points
x 0 1 2 4
y 1 0 3 3 . (2 p)
(b) Find the Lagrange-polynomial that interpolates the same points. (2 p) (c) Is there any difference between the two polynomials in (a) and (b)?
Motivate! (1 p)
2 (3)
Problem 4
Given the integral
Z 2 0
x−xdx . Approximate it with
(a) Simpsons composite rule using 4 intervals. It is known that
x→0+lim x−x = 1. (2 p)
(b) Gauss quadrature using three points. (2 p)
Problem 5
(a) Combine f (x), f (x − h) and f (x + 2 h) into a formula to estimate f0(x)
with an as high order of the error as possible. (2 p) (b) Use your formula from (a) to estimate f0(2) when f (x) = e−x
ln x, using
both h = 0.1 and h = 0.3. (1 p)
(c) Derive a suitable extrapolation formula to improve the order of the error in a situation like (b) and apply it to improve the approximation
of the derivative in (b). (2 p)
Problem 6
Describe how you would solve the following IVP using Heuns method y00− (1 − y2) y0 + y = x, y(0) = 1, y0(0) = 0
More specifically: How would you find an approximation to y(3) that is not useless, using Heuns method?
You should only describe the methodology, and not do the actual calcu-
lation. (5 p)
3 (3)
