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Matlab
Matlab is a powerful mathematical tool and this tutorial is intended to be an
introduction to some of the functions that you might find useful. For more detail
about other parts of the program, refer to the User's Manual and the help files.
Basics
The following is a picture of what will appear when you start up Matlab.
As you can see, simple computations are very easy. Just type in
exactly what you want computed and press enter. If no variables are
used, the answer will automatically be assigned to the variable
ans. This variable can be used later, but don't forget that it will
automatically be overwritten when a new computation is done.
Variables are also easy to use. Simply type a variable name, the equal
sign, and then some value to be assigned.
At any time, you can type "whos" at the command line to see what
variables have been assigned.
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Calculus
Matlab provides an easy way to compute
both derivatives and integrals. To perform
symbolic differentiation or integration, you
must first declare a symbolic variable. This
is done by typing "syms x" where x is the
variable name. In the following example, x
is declared as symbolic and then used to
find the indefinite integral and derivative of
a function.
The definite integral can also be computed
by adding the lower and upper bounds,
separated by commas, after the function to
integrate. (Note that a function of a valid
symbol is still required)
For more information on the symbolic
calculus functions, see the
toolbox/symbolic section of the help
window.
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Complex Numbers
The variable i is already defined as the square root of -1. We can verify this
by calculating i * i.
The functions abs and angle allow us to convert the complex number from
rectangular to polar form.
Angle returns the phase angle in radians, but converting to degrees we see
that the answer is what we expected
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Matrices
Matrices are the basis of Matlab, so manipulating them is very
simple. First, you input matrices by placing the values in brackets, with
semicolons separating the rows.
• Matrices of the same dimensions can be added and
subtracted, and conformable matrices can be multiplied.
• Finding the determinant or the inverse of a matrix is also
simple.
• For example, if we have the equations
7v1 - 4v2 - 2v3 = 3
-4v1 + 9v2 - 2v3 = 0
-2v1 - 2v2 + 5v3 = -12
• It would be very easy to solve for the unknowns. First, create a 3x3
matrix with the coefficients of v1, v2, and v3, then create a 1x3
matrix with the right hand side of the equations. Finally, multiply the
inverse of the first matrix with the second matrix, and the resulting
matrix contains the answers.
• This method also will work with symbolic variables. For example,
you can solve for equations when using a Laplace transform by
declaring s as a symbol ("syms s") and then entering the values into
the matrix.
Polynomials and Rational Functions
• Matlab also provides tools for manipulating polynomials and rational
functions. To use these tools, the polynomial should be represented
as a vector with the leftmost number being the highest power and
the rightmost number being the constant. For example, x² + 2x + 1
would be represented as [1 2 1].
• The roots function gives the roots of the polynomial and polyval
evaluates the polynomial at the given value. Multiplying and dividing
polynomials can be done with conv and deconv
• Note that deconv will return two vectors, the first contains the
coefficients for the quotient polynomial, and the second contains the
coefficients for the remainder polynomial. The following example
divides x3 + 3x² + 3x + 2 by x + 1
• If the left hand side of the equation didn't contain two variables, the
answer would only have the quotient and the remainder would be
discarded.
• Matlab also has a function that will give the partial fraction
decomposition of a rational function. This is very useful when
working with Laplace transforms. The function residue takes two
polynomials and returns the residues, the poles, and the direct term
(quotient). The partial fraction expansion of (2s + 5) / (s3 + 5s² + 8s
+ 4) is found in the figure.
• There is a pole at –1 and a repeated pole at –2. There is no direct
term since the order of the numerator was less than the order of the
denominator.
(2s + 5) / (s3 + 5s² + 8s + 4) = -3 / (s + 2) -1 / (s + 2)² + 3 / (s + 1)