MATLAB
In class, we learned how to perform Gaussian Elimination with Backwards Substitution and wrote a function to accomplish this task in MATLAB. This was Gaussian Elimination (hereafter called GE) in its simplest form. However, there is an easy way to make this algorithm much more accurate and robust. We can introduce a process called pivoting (this is an expansion of the pivot row that we learned about in class). This strategy involves placing the element with the largest magnitude in the pivot position, such that the pivot element is the largest of all elements in the pivot column. This way, when we calculate the multiplier used in successive steps to transform elements in each row, the magnitude of the multiplier will always be less than or equal to 1. This will reduce a type of error called round-off error. For more reading, see King and Mody's "Numerical and Statistical Methods in Bioengineering" (2011).
Here is a general explanation of the algorithm:
1. First, we compare all of the elements in the same column. We use the max command in MATLAB to find the largest number in the column.
2. Then, we swap the row containing the largest element with the pivot row. Now, our pivot row contains the largest element.
Recall that the syntax for the max function is [Y,I] = max(X) where X is the vector within which the maximum element is searched, Y is the maximum element of X, and I is the index of the maximum element.
Additionally, we can use array indexing to enable the use of an array as an index for another array.
For this problem, you should solve the following system of equations using GE with pivoting:
0.001x + 6.4y = 4.801
22.3x - 8y = 16.3
To get you started, here are some suggested pseudocode changes that you can make to your algorithm to solve a system of n equations:
