Download Math 151A HW #5: Interpolation, Mean Value Theorem, Error, and Root-Finding and more Assignments Mathematics in PDF only on Docsity!
Math 151A Homework #
- Consider the function f (x) = x
5 .
a. Use the points x = {− 1 , 0 , 1 } and Newton’s divided differences to construct the inter-
polation polynomial through those points.
Newton’s divided differences gives us
So we get P (x) = −1 + (x + 1) = x.
b. Now add the point x = 2.
From the above matrix, we get P (x) = 5x
3 − 4 x.
c. Now use the points x = { 0 , 1 } and the derivatives at those points to construct a Hermite
interpolating polynomial through those two points.
For this we use z 0 = 0, z 1 = 0, z 2 = 1 and z 3 = 1 to get
Which gives us P (x) = (x − 0)
2
2 (x − 1) = 3x
2 − 2 x.
d. Plot all of these on the same plot.
0
5
10
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
x** x 5x3-4x 3x3-2x**
- Mean value theorem
Suppose f ∈ C
1 [a, b]. Then what do we know about the following quantity?
f (b) − f (a)
b − a
There exists a ξ ∈ (a, b) such that
f (b) − f (a)
b − a
= f
′ (ξ)
- Round-off error
We can think of chopping as rounding down, and “rounding” as rounding off. Then if we
round up, what do we expect the error to be?
x 0
x
1
x 3
x 2
x 4
L
4,
- Bisection method
Which theorem ensures that the bisection method always brackets a root?
The intermediate value theorems tells us that if f (a) < 0 and f (b) > 0, then there exists a
c ∈ (a, b) with f (c) = 0.
- Binary arithmetic
Suppose my computer uses 4 bits for the mantissa, and 2 bits for the characteristic, rep-
resented by (M M M M, CC). What is the result of adding the numbers represented by
(0100, 00) and (1101, 00)?
In binary form, we get
0100 × 2
0
+ 1101 × 2
0
10001 × 2
0 = 1000 × 2
1
So we get (0100, 00) + (1101, 00) = (1000, 01) or 4 + 13 = 16.
- Order of convergence
Find the order of convergence of the sequence pn = (
1 2
3 n ?
|pn+1 − p|
|pn − p|
α
1 2
3 n+
1 2
)^3
n )α
α· 3 n / 2
3 · 3 n
which requires that α = 3.
- Method of false position
What advantage does the method of false position offer over the secant method?
It ensures that the root is always bracketed.
