Project Two: Numerical Computation of Fourier Coefficients

Numerical Analysis, Section 2
Project Two: Numerical Computation of Fourier Coefficients

1. Project description
In this project, we will write C functions to calculate the Fourier coefficients of a function and to
evaluate a function given its Fourier coefficients.
The first of these functions will be called “fourcoefs” and it will, given the capability of evaluating
a function f at any point on the interval [−π, π] and an integer N 0, use the (2N + 1)-point
periodic trapezoidal rule to approximate the Fourier coefficients
a−N , a−N+1, . . . , a−1, a0, a1, . . . , aN−1, aN
of f. These are defined by the formula
am =
1
2π
Z π
−π
f(t) exp(−imt) dt.
The (2N + 1)-point periodic trapezoidal rule is
1
2π
Z π
−π
g(t) dt ≈
1
2N + 1
X
2N
j=0
g
ˆ
−π +
2π
2N + 1
j
˙
,
so that the desired approximations
aĄ−N , aČ−N+1, . . . , aĄ−1, ar0, ar1, . . . , aĆN−1, aĂN
of the Fourier coefficients of f are given by
aĂm =
1
2N + 1
X
2N
j=0
f
ˆ
−π +
2π
2N + 1
j
˙
exp ˆ
−im ˆ
−π +
2π
2N + 1
j
˙˙
=
1
2N + 1
X
2N
j=0
f
ˆ
−π +
2π
2N + 1
j
˙
exp pimπq exp ˆ
−im ˆ
2π
2N + 1
j
˙˙
=
(−1)m
2N + 1
X
2N
j=0
f
ˆ
−π +
2π
2N + 1
j
˙
exp ˆ
−
2πi
2N + 1
mj˙
You can use your code from Project 1 to construct the nodes and weights of the trapezoidal rule,
but be aware that the (2N + 1)-point periodic trapezoidal rule on the interval [−π, π] coincides
with the (2N + 2)-point nonperiodic trapezoidal rule on the interval [−π, π].
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The second function you will write will be called “foureval” and it will, given the approximations
(1) of the Fourier coefficients computed by the “fourcoefs” functions, evaluate the Fourier series
X
N
n=−N
ran exp(int)
at a specified point t.
The file “fourexp.c” gives the calling syntax for the “fourcoefs” and “foureval” functions. Your task
is to implement the functions as described there.
2. Testing and grading
A public test code is given in the file “fourtest1.c”. Another test code, called “fourtest2.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 fourtest1 fourexp.c fourtest1.c -lm
./fourtest1
can be used to compiler and execute your program. The first command compiles your program and
creates an executable file named “fourtest1”. 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 fourtest1, and the program will
tell your score out of 5. We will also test your code by compiling against “fourtest2.c”, which we
will not release until after the projects are due.
More explicitly, we will grade your project by running the sequence of command
gcc -o fourtest1 fourexp.c fourtest1.c -lm
./fourtest1
gcc -o fourtest2 fourexp.c fourtest2.c -lm
./fourtest2
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.
Submitting your project:
You will submit your project using canvas. You should submit only your “fourexp.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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