Download FMN081: Computational Methods in Mechanics 1st Repetition Exam

Claus Führer, 2005-04-11
FMN081: Computational Methods in Mechanics
Numerical Analysis, Matematikcentrum
1st Repetition Exam
The exam tests your understanding of the course - mainly of its practical
part. It is constructed following the lines of all assignments.
Some questions can just be answered with a few words - other require more
explanations from your side.
You are allowed to use all course related material, i.e. course book, webhand-outs, assignments, your solutions and notes.
The exam consists of 7 questions. You can achieve max 25 points. If you got
12 points you passed the exam.
Question 1 - (4p) Write down one step of Newton’s method applied to
the problem

 2
x1 − 2x1 x2 x3
F (x) =  x1 − x22 − x3  = 0.
2x1 − 2x2 − 2x33
T
Start with x(0) = 1 1 1 .
(You need not to compute the solution after one step, just write the
step.) Give also the Jacobian in the second step of the simplified
Newton method.
Question 2 - (4p) (This task is similar to Assignm. 3.2) Let f be a function which you want to interpolate on n+1 equidistant points x0 , . . . xn
by a polynomial p.
What are the interpolation conditions for p? What is the dimension
of the corresponding Vandermonde matrix? Let f (x) = sin(x), n = 2
and x0 = 0, x1 = π/4, x2 = π/2. Give an estimate for the maximal
interpolation error.
1
Question 3 - (2p) In the following picture the cubic B-splines of a given
grid are plotted.
1
Grid: 0,0,0,0,1,2,3,4,5, 5,5,5
0.8
0.6
0.4
0.2
0
0
2
4
5
What is the support (stöd) of the fourth B-spline curve, N44 , (counted
from the left)? What is meant by local support and why is it important
to have a basis for splines with local support?
Question 4 - (3p) Let P n be the space of all polynomials of max degree
n and assume that xi , i = 0 : n are given points.
Consider the inner product
(p, q) =
n
X
p(xi )q(xi ).
i=0
Show that the Lagrange polynomials are orthogonal with respect to
this inner product.
Question 5 - (4p) Let f (x) = sin(x). Describe a method (by text and
formulas - not by a MATLAB code) to find a quadratic polynomial
which describes this function best in the interval [0, 1] with respect to
the 2-norm. How is the 2-norm of a function in this interval defined?
Question 6 - (5p) In Assignment 7 you considered the baseball problem,
which is a boundary value problem. You solved this problem by a
shooting method. You had to use Newton’s method and an ODE solver
for this problem. Which was the nonlinear system you had to solve?
How did you compute the Jacobian? How many differential equations
had to be solved per Newton iteration step? (Give an explanation.
Don’t give just a number.)
2
Question 7 - (3p) Here is a plot of the stability region of a numerical
method to solve ordinary differential equations:
2
hλ
1
0
-1
-2
-2
-1.5
-1
-0.5
0
0.5
What does this plot say about the stability of the method? What
means λ in the plot? If λ = −20, which is the maximal step size for
which the method is stable? (Give an approximate value.)
Good luck!
3