Homework #7
Due Thursday, 03/06
1. (a) What system of nonlinear equations would you solve in order to nd w1 , w2 , w3 , x1 , x2 and x3 such that the integration formula I3 (f ) = w1 f (x1 ) + w2 f (x2 ) + w3 f (x3 ) is exactly equal to
1 1
f (x)dx for eve
Homework # 6
Due Monday, 02/25
1. In order to derive Simpsons rule, we proved that, if p(x) is the unique polynomial of degree 2 which interpolates the function f (x) at x0 , x1 and x2 , then
x2
p(x)dx =
x0
h (f (x0 ) + 4f (x1 ) + f (x2 ) 3
In this proble
Homework # 5
Due Thursday, 02/14
1. The two important proofs in Polynomial Interpolation are the proof of existence and uniquness of the interpolating polynomial the error estimate (a) Make sure you understand these two proofs. Read them one more time. Ma
Homework # 4
Due Thursday, 02/07
1. In this problem, we will see that sometimes it is possible to avoid loosing signicant digits (the main idea is to replace substraction by some other operations). (a) Show that the two roots of x2 26x + 1 are 13 + 168 an
Homework #3
Due Thursday, 01/31
1. We have seen in class that, if f (x) has a root of multiplicity m 2, then Newtons method converges only with order 1. Here is a modied version of Newtons method: pn+1 = pn m f (pn ) f (pn )
Show that, if it converges, th
Homework # 2
Due Thursday, 01/24
1. Find the solution of x3 = x2 + x + 1 using: (a) the bisection method (What initial interval did you choose? How did you choose it?) (b) Newtons method (What initial iterate did you choose? How did you choose it?) (c) th
Homework # 1
Due Thursday, 01/17
NOTE: Each time you are ask to write a code which gives the solution with accuracy , it means that you have to use the stopping criterium |pn pn1 | Note that this does not guaranty that |pn p| ! So you are actually not sur