Quiz 2 2017.pdf - Applied Numerical Analysis – Quiz #2...

Applied Numerical Analysis – Quiz #2
Modules 3 and 4
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• This quiz has 10 questions and 4 pages in total.
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Quiz 2 2017, Versie: A

Question 1 Consider i) interpolation and ii) regression of a two-dimensional function f ( x ) where
f : R
2
! R . Assume that the approximating function p ( x ) in both cases is a linear combination of
N distinct basis functions φ
i
( x ), and that f is sampled at M distinct locations. The interpolation
conditions for i) lead to a system of linear equations, with system matrix A .
Consider the existence and uniqueness of solutions to these two problems. What condition is
required on i) and ii) respectively, to guarantee unique solutions for each problem?
Question 2 Steady laminar flow through a channel with parallel walls has a parabolic velocity
profile (“Poiseuille” flow). In a channel of unit height on the interval y 2 [0 , 1] we have two
approximate measurements of the velocity, at y
0
=
1
3
and y
1
=
2
3
with values v
0
= 7 . 3 and v
1
= 8 . 7.
In addition we know that the velocity at both channel walls is exactly zero (by the no-slip boundary
condition), i.e. v (0) = v (1) = 0. Solve a mixed interpolation/regression problem to approximate
v ( y ), using a quadratic polynomial. What is the maximum velocity in the approximation of v ?
A: Unanswered
B: 7.5
C: 8
D: 8.5
E: 9
F: 9.5
G: 10
Question 3 Consider a generalized spline interpolant s ( x ) on the interval [0 , 10]. The interval
is divided into sub-intervals with N + 1 nodes:
0 = x
0
< x
1
< · · · < x
N - 1
< x
N
= 10
at which sample data are given. Assume that the spline consists of polynomials of degree d on
each interval, and that the spline is required to be d - 1-times continuously di ↵ erentiable on the
entire interval.
We impose 6 constraints at the boundaries: f
0
(0) = 1, f
00
(0) = 0, f
000
(0) = 0, f
0
(10) = 0,
f
00
(10) = 0 and f
000
(10) = 0. What is the degree d of the polynomials needed to construct such a
spline?
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