Skip to main content

Numerical Methods with Programming

Leonid Kovalev

Section 2.6 Examples and questions

These are additional examples for reviewing the topic we have covered. When reading each example, try to find your own solution before clicking “Answer”. There are also questions for reviewing the concepts of this section.
🔗

Example 2.6.1. Tridiagonal matrix of arbitrary size.

Given a positive integer \(n\text{,}\) construct a tridiagonal matrix which has entries -1, 2, -1 (as in Section 2.2) and size \(n\times n\text{.}\)
Answer.
diag(2*ones(1, n)) - diag(ones(1, n-1), 1) - diag(ones(1, n-1), -1)
or
🔗
2*diag(ones(1, n)) - diag(ones(1, n-1), 1) - diag(ones(1, n-1), -1)
or
🔗
2*eye(n) - diag(ones(1, n-1), 1) - diag(ones(1, n-1), -1)
It is important to recognize why n-1 appears above. The main diagonal has length n but the diagonals next to it are shorter by 1 element, so their length is n-1.
🔗
🔗
🔗

Example 2.6.2. Solving a tridiagonal system.

Solve the system of linear equations
\begin{align*} 3x_1 - x_2 \amp = 1 \\ - x_1 + 3x_2 - x_3 \amp = 0 \\ - x_2 + 3x_3 - x_4 \amp = 0 \\ \cdots \\ - x_{n-2} + 3x_{n-1} - x_{n} \amp = 0 \\ - x_{n-1} + 3x_{n} \amp = 2 \end{align*}
Use the first two digits of your SUID as the value of \(n\text{.}\)
Answer.
n = 28;
A = 3*eye(n) - diag(ones(1, n-1), 1) - diag(ones(1, n-1), -1);
b = zeros(n, 1);
b(1) = 1;
b(end) = 2;
x = A\b;
disp(x)
Explanation. The coefficient matrix has 3 on the main diagonal, and -1 above and below it. This matrix is created with the help of diag as explained in Section 2.2. The right-hand size column vector begins with all-zeros: note that zeros(n, 1) is a column of zeros, while zeros(1, n) is a row of zeros. The first and last entries of b are changed as described in the system.
🔗
🔗
🔗

Example 2.6.3. Table of squares and cubes of small integers.

Create a 3×9 matrix which has the integers 1 through 9 in the first row, their squares in the second row, and their cubes in the third row.
Answer.
The single line A = (1:9).^((1:3)') does the job. The elementwise operation .^ can combine the arrays of sizes 1×9 and 3×1, as explained in Section 2.4. The result is a 3×9 matrix with the expected property: the base (1 through 9) is the column number, the exponent (1 through 3) is the row number.
🔗
🔗
🔗

Example 2.6.4. Plot of a function.

Plot the function \(y = (x - 1)/(x^2 + 1)\) on the interval \([-1, 5]\)
Answer.
x = linspace(-1, 5, 500);	
y = (x - 1)./(x.^2 + 1);
plot(x, y)
Figure 2.6.5. Plot of \(y = (x - 1)/(x^2 + 1)\) on \([-1, 5]\)
🔗
Note the use of semicolons to prevent Matlab from displaying hundreds of numbers that we do not need to see.
🔗
🔗
🔗

Question 2.6.6. x as a function of y.

How would you plot a function where \(x\) is given as a function of \(y\text{,}\) for example \(x = y + \cos y\) with \(-5\le y\le 5\text{?}\)
🔗
🔗

Question 2.6.7. A “wrong” way to multiply matrices.

Can you imagine a situation where you would want to compute A.*B where both A and B are square matrices of the same size?
🔗
An idea: a matrix can be used to store a greyscale image, so that each entry is the brightness of the corresponding pixel. What would elementwise multiplication A.*B mean in this context?
🔗
🔗
🔗
PrevTopNext