Polynomial Multiplication and Fast Fourier Transform
(Com S 477/577 Notes)
Yan-Bin Jia
Oct 21, 2012
Suppose we are given two polynomials:
p(x) = a0 + a1 x + + aN 1 xN 1 ,
q (x) = b0 + b1 x + + bN 1 xN 1 .
Their product is dened by
p(x) q (x) = c0 + c1 x +
The Midterm Exam will take place in class on Thursday, Oct 11. It will be a closed-book and closed-notes exam. Scratch papers will be distributed and calculators and drawing tools will be allowed.
The emphasis is on understanding and application of the co
Com S 477/577 Fall 2012
Assignment 6 (40 pts)
Due before class (hard copy only)
Thursday, October 11
All problems fall within the scope of the Midterm Exam on the due day of this assignment.
1. (9 pts) A cycloid is the plane curve traced out by a point on
Parametric Curves
(Com S 477/577 Notes)
Yan-Bin Jia
Oct 2, 2012
1
Introduction
A curve in R2 (or R3 ) is a dierentiable function : [a, b] R2 (or R3 ). The initial point is [a]
and the nal point is [b]. The domain of the curve is the interval [a, b]. A por
Curvature
(Com S 477/577 Notes)
Yan-Bin Jia
Oct 2, 2012
We want to nd a measure of how curved a curve is. Since this curvature should depend only
on the shape of the curve, it should not be changed when the curve is reparametrized. Further,
the measure of
Singular Value Decomposition
(Com S 477/577 Notes)
Yan-Bin Jia
Sep 6, 2012
1
Introduction
Now comes a highlight of linear algebra. Any m n matrix can be factored as
A = U V T
where U is an m m orthogonal matrix1 whose columns are the eigenvectors of AAT ,
Solution of Linear Equations
(Com S 477/577 Notes)
Yan-Bin Jia
Sep 4, 2012
We discussed general methods for solving arbitrary equations. We also looked at the special
class of polynomial equations. Another special class is made up of systems of linear equ
Solution of Nonlinear Equations
(Com S 477/577 Notes)
Yan-Bin Jia
Aug 30, 2010
One of the most frequently occurring problems in scientic work is to nd the roots of equations
of the form
f (x) = 0.
(1)
The function f (x) may be given explicitly, as, for ex
Roots of Polynomials
(Com S 477/577 Notes)
Yan-Bin Jia
Aug 28, 2012
We will look at how to nd roots, or zeros, of polynomials in one variable. The solution of
multi-variate polynomials can often be transformed into a problem that requires the solution of
Polynomial Interpolation
(Com S 477/577 Notes)
Yan-Bin Jia
Aug 23, 2012
1
Interpolation Problem
Consider a family of functions of a single variable x:
(x; a0 , a1 , . . . , an ),
where a0 , . . . , an are the parameters. The problem of interpolation for c
Polynomial Evaluation
(Com S 477/577 Notes)
Yan-Bin Jia
Aug 21, 2012
Polynomials are perhaps the best understood and most applied functions. The foundation comes
from algebra and calculus. Taylors expansion says that a function can be locally expanded aro
Closed Curves and Space Curves
(Com S 477/577 Notes)
Yan-Bin Jia
Oct 4, 2012
So far we have discussed only local properties of (plane) curves. These properties depend only
on the behavior of a curve near a given point, and not on the global shape of the c