Transcription

Up to now, we've been doing numeric calculations which are required in many real-world applications.

Take the simple example of finding the solutions to a quadratic equation. We can quickly find

the solutions numerically when we have the coefficients. However, sometimes we get imaginary

roots. How can we determine when the solutions are real?

If we look at the symbolic solution, we see that the expression inside the square root

must be positive to have real roots. Let's see how we get to this solution using

MATLAB and the Symbolic Math Toolbox. The first step is to define symbolic variables

using the syms command. Symbolic variables differ from the numeric

variables we've used so far because we don't have to assign value to them. We simply declare

their existence and we can use them in calculations. Like here, we enter the generic quadratic

equation, and assign the expression to the variable 'y'. Notice that 'y' automatically

becomes a symbolic variable even though we didn't declare it as one.

Now, to find the symbolic solution, we use the SOLVE function while setting 'y' equal

to 0, and specifying 'x' as the variable to solve for. And bingo! The SOLVE function returns

a variable with the two solutions that match the quadratic formula.

But wait a second. There's something different about this last command. Because a single

equal sign is the assignment operator, we used two equal signs to create a symbolic

equation. Now what if we need to apply our solution

to a specific example? In other words, we have to replace the symbolic variables 'a',

'b', and 'c' with fixed values. No problem. The SUBS function replaces symbolic variables

with numbers or other symbolic expressions. Here's how it works. We need three inputs:

the expression we wish to evaluate, the symbolic variables to replace, and the values to substitute

in for the symbolic variables. The result is kept as an exact symbolic solution, rather

than a numeric approximation. Perhaps you've noticed, our solution is not

fully simplified. Let's fix that using the SIMPLIFY function. There we go, an exact solution

to our quadratic equation. In many cases, we still need a numeric representation

of our symbolic results. Again, no problem. We use the variable precision arithmetic,

or VPA, function to view the exact solution as a number. The 2nd input to the VPA function

specifies the number of digits to include in the output, and the output is still a symbolic

variable. We have now worked through the essentials

of solving an equation symbolically, and you can use these techniques to solve a wide variety

of problems. Before we close, here are a few examples from

other applications that often arise, like finding integrals, doing series expansions,

and plotting equations of one or more variables. Now put away the pen and paper, and try solving

a few problems in MATLAB yourself.