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Numerical Methods I – Project 2 Report

Problem

Report Content:

• Programming Fourier Series Form

• Programming Simpson’s Approach

• Programming Trapezoidal Rule’s Approach

• Testing

• Conclusion

• Programming Fourier Series Form:

A helper function is programmed, which evaluates the following expression:

fx=b0+k=1naksinkx+bkcoskx

Given the sequences ak, bk for k=0 to n (a0 is arbitrary), n, and x, the function evaluates the expression, see below the used function (used in both Simpson’s and Trap scripts).

• Programming Simpson’s Approach:

The script is composed of the following steps:

1. Initializing the variable h which holds the partitions width.
2. Initializing the variable fx, holds our function’s value on the points of the partitions a+ih.
3. Looping to implement Simpson’s rule, the image shown below.

The code can be found below:

• Programming Trapezoidal Rule’s Approach:

The script is composed of the following steps:

1. Initializing the variable h which holds the partitions width.
2. Initializing the variable fx, holds our function’s value on the points of the partitions a+ih.
3. Looping to implement Trapezoidal rule, the image shown below.

The code can be found below:

• Testing:

A test for the function f(x)=x is performed, integrating from 0 to 1, with number of partitions N=26, and Fourier Series of order n=30.

As can be concluded by the answers of the scripts, comparing them to the expected answer of 0.5, both approaches gave accurate reliable results.

The following is the test script:

And the following is the result of running the script:

• Conclusion

I would like to say that both the Simpson’s and Trapezoidal approaches were efficient and gave approximately the same result for the same number of partitions.