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- MAT128A: Project Three: Chebyshev Expansions

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1. Project Description

In this project, you will write C functions to evaluate Chebyshev polynomials, to calculate the

Chebyshev coefficients of a function and to evaluate a function given its Chebyshev coefficients.

The first of these functions will be called “chebpolys,” and it will, given an integer N > 0 and a

point x in the interval [−1, 1], return the values of

T0(x), T1(x), T2(x), . . . , TN (x),

where Tn(x) = cos(n arccos(x)) is the Chebyshev polynomial of degree n. You should compute the

Chebyshev polynomials using the formulas

T0(x) = 1, T1(x) = x, Tn+1(x) = 2xTn(x) − Tn−1(x).

The second function will be called “chebcoefs,” and it will, given the capability of evaluating a

function at any point on an interval [a, b] and an integer N > 1, return approximations of the first

N + 1 Chebyshev coefficients of f. The Chebyshev expansion of f on [a, b] is

f(y) = X∞

n=0

0

anTn

ˆ

2

b − a

y −

b + a

b − a

˙

,

where the coefficients {an} are given by the formula

an =

2

π

Z 1

−1

f

ˆ

b − a

2

x +

b + a

2

˙

Tn(x) dx.

You will use the Chebyshev extrema quadrature rule to approximate the coefficients a0, a1, . . . , aN .

That is, if, for each j = 0, 1, . . . , N, we let

xj = cos ˜

j +

1

2

N + 1

π

¸

,

then you should use the quadrature formula

an ≈

2

N + 1

X

N

j=0

f

ˆ

b − a

2

xj +

b + a

2

˙

Tn pxj q.

to approximate the Chebyhev coefficients.

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The third function will be called “chebeval,” and it will, given an integer N > 1, a point y on

the interval [a, b] and the approximations of the coefficients a0, . . . , aN computed by “chebcoefs”,

compute the value of the expansion

X

N

n=0

0

anTn

ˆ

2

b − a

y −

b + a

b − a

˙

,

at the point y.

The file “chebexp.c” gives the calling syntax for the “chebcoefs”, “chebpolys” and “chebval” routines. Your task is to implement the functions as described there.

2. Testing and grading

A public test code is given in the file chebtest1.c. Another test code, called chebtest2.c, will be

used to test your function as well. Half of the project grade will come from the first test file, and

the second half will come from the second. The commands

gcc -o chebtest1 chebexp.c chebtest1.c -lm

./chebtest1

can be used to compiler and execute your program. The first command compiles your program and

creates an executable file named chebtest1. The -lm option tells the compiler to include the math

library which implements cosine, sine and other basic functions. The second command runs the

fourtest1 executable. There are five tests of your function in chebtest1, and the program will tell

your score out of 5. We will also test your code by compiling against chebtest2.c, which we will not

release until after the projects are due. You will get a 0 on your project if it does not compile and

run. Please start work on your project early and come see either myself or our TA, Karry Wong,

if you are having difficulties getting it to compile.

3. Submitting your project

You will submit your project using canvas. You should submit only your chebexp.c file. You must

submit your file by 11:59 PM on the due data. Late assignments will not be accepted.

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