Everything You Ever Wanted To Know About Polynomials
(but were afraid to ask)
I. What is a polynomial?
Algebraically speaking, a polynomial is a member of the ring
that a polynomial p can be written in the form:
R[x] or C[x].
That means
n
p(x) = a0 + a1
Calculus Basics for M572
R
Continuous Functions: Let n be an open set, then C k () is the set of all
functions dened on such that the function and its rst k derivatives are continuous
on . If k = 0 then can be a closed (or partially closed) set. If is par
Math 571
Inner Product Spaces
1. Preliminaries
An inner product space is a vector space V along with a function , called an inner product which
associates each pair of vectors u, v with a scalar u, v , and which satises:
(1) u, u 0 with equality if and on
Floating Point Numbers and Round-off Error
Floating Point Numbers
Every real number x can be written in normalized form in base as
x = r n , with
1
r < 1, and n an integer
with the obvious exception for x = 0. r is called the mantissa and n the exponent
Final Exam, Math 572
11 May, 2010
Name:
Solutions
1. (10 points): Consider the Crank-Nicolson scheme
un+1 = un +
n+1
u 1 2un+1 + un+1 + un1 2un + un+1
+1
2
2
for approximating the solution to the heat equation u = u on the intervals
t
x2
0 x 1 and 0 < t
Math 572
Winter 2010
Solutions to Exam 2 Review
(1a.)
det(A I) = det
4
3
6
5
= (4 )(5 ) + 18
= 2 2
= ( 2)( + 1)
Therefore the eigenvalues are 1 = 2 and 2 = 1. We compute the eigenvectors using the null()
function in Matlab using the r option:
1
null(A 2