Introduction to Numerical Methods and MATLAB Programming for Engineers

In MATLAB, the basic objects are matrices, i.e. arrays of numbers. Vectors can be thought of as special matrices. A row vector is recorded as a \(1 \times n\) matrix and a column vector is recorded as a \(m \times 1\) matrix. To enter a row vector in MATLAB, type the following in the command window: and press enter. MATLAB will print out the row vector. To enter a column vector type You can access an entry in a vector with and change the value of that entry with You can extract a slice out of a vector with You can change a row vector into a column vector, and vice versa, easily in MATLAB using (This is called transposing the vector and we call ' the transpose operator.) There are also useful shortcuts to make vectors such as

To make MATLAB put fewer blank lines in its output, enter To make MATLAB display more digits, enter Note that this does not change the number of digits MATLAB is using in its calculations; it only changes what is displayed.

Consider the data in Table Table 1.1.1 (Adapted from Ayyup &McCuen 1996, p.174.)

We can enter this data into Matlab with the following commands entered in the command window: Entering the name of the variable retrieves its current values. For instance We can plot data in the form of vectors using the plot command: This will produce a graph with the data points connected by lines. If you would prefer that the data points be represented by symbols you can do so. For instance

A major theme in this course is that often we are interested in a certain function \(y = f(x)\text{,}\) but the only information we have about this function is a discrete set of data \(\{(x_{i},y_{i})\}\text{.}\) Plotting the data, as we did above, can be thought of envisioning the function using just the data. We will find later that we can also do other things with the function, like differentiating and integrating, just using the available data. Numerical methods, the topic of this course, means doing mathematics by computer. Since a computer can only store a finite amount of information, we will almost always be working with a finite, discrete set of values of the function (data), rather than a formula for the function.

If we wish to deal with formulas for functions, Matlab contains a number of built-in functions, including all the usual functions, such as sin( ), exp( ), etc.. The meaning of most of these is clear. The dependent variable (input) always goes in parentheses in MATLAB. For instance should return the value of \(\sin \pi\text{,}\) which is of course \(0\) and will return \(e^{0}\text{,}\) which is \(1\text{.}\) More importantly, the built-in functions can operate not only on single numbers but on vectors. For example will return a plot of \(\sin x\) on the interval \([0,2 \pi]\text{.}\)

Some of the built-in functions in MATLAB include: cos( ), tan( ), sinh( ), cosh( ), log( ) (natural logarithm), log10( ) (log base 10), asin( ) (inverse sine), acos( ), atan( ). To find out more about a function, use the help command; try

If we wish to deal with a function that is a combination of the built-in functions, Matlab has a couple of ways for the user to define functions. One that we will use a lot is the anonymous function, which is a way to define a function in the command window. The following is a typical anonymous function: This produces the function \(f(x) = 2 x^{2} -3x+1\text{.}\) To obtain a single value of this function enter Just as for built-in functions, the function \(f\) as we defined it can operate not only on single numbers but on vectors. Try the following: This is an example of vectorization, i.e. putting several numbers into a vector and treating the vector all at once, rather than one component at a time, and is one of the strengths of MATLAB. The reason \(f(x)\) works when \(x\) is a vector is because we represented \(x^{2}\) by x.^2. The . turns the exponent operator ^ into entry-wise exponentiation, so that [-2 -1.8 -1.6].^2 means \([(-2)^{2}, (-1.8)^{2}, (-1.6)^{2}]\) and yields [4 3.24 2.56]. In contrast, [-2 -1.8 -1.6]^2 means the matrix product \([-2, -1.8,-1.6][-2, -1.8,-1.6]\) and yields only an error. The . is needed in .^, .*, and ./. It is not needed when you * or / by a scalar or for +.

The results can be plotted using the plot command, just as for data: Notice that before plotting the function, we in effect converted it into data. Plotting on any machine always requires this step.