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Based on Maple kernel, symbolic Math Toolbox performs calculation symbolically in Matlab environment. The following examples introduce some basic operations available in Basic Symbolic Math toolbox version 2.1.3.

Example 1: Simplifying an expression.


- In Matlab command window, we will first need to define alpha as a symbolic expression.

>> alpha = sym('alpha')

alternately, you may also enter:

>> syms alpha

The commands "sym" and "syms" are Matlab's reserved words. When "syms" is used by itself at the command prompt, all defined symbolic values will be listed.

- Next, we will enter the expression of z.

>> z = sin(alpha)^2 + cos(alpha)^2;

Note that ";" is used to suppress echo.

>> simplify(z)

Matlab will yield "1", as expected:

ans =


You may also specify the format of the output in symbolic calculation by adding the option as shown in the example below.

>> syms rho
>> rho=0.25
>> sym(rho,'r')

Matlab returns:

ans =


'r' stands for rational form. Similarly, you may use 'e', 'd' format. Please refer to Matlab's help files or click here for more info on format.


Example 2: Derivative.

We wish to take the derivative of function f(x):


Matlab command entries:

>> syms x
>> f=x^3-cos(x);
>> g=diff(f)

Matlab returns:

g =


Note that the command "diff" was used to obtain the derivative of function f.

Since function f has only one independent variable, the diff command performed the calculation based on x. If there are more than one independent variable in a function, you should include the "intended" variable in the following format:

diff(f, x)

where x is the "intended" variable. For example,

We wish to obtain the derivative of the following function

Matlab command entries:

>> syms x y
>> f=x^2+(y+5)^3;
>> diff(f,y)

Matlab returns:

ans =


Note that in this case, the command diff(f,y) is equivalent to


Example 3: Integral

To integrate function f(x,y) as shown in Example 2, we will use the command "int" as shown below.

>> int(f,x)

Matlab returns:

ans =


The syntax of the integral command can be viewed by typing >> help int in Matlab command window.

If we wish to perform the following definite integral:

Matlab command entry:

>> int(f,y,0,10)

Matlab returns:

ans =



Example 4: Finding roots.

Consider the following polynomial:

Suppose we wish to find the roots of this polynomial. In Matlab Command window:

>> syms x
>> f=2*x^2 + 4*x -8;
>> solve(f,x)

Matlab returns:

ans =


Alternately, you may use the following lines in Matlab to perform the same calculation:

>> f=[2 4 -8];
>> roots(f)

Matlab returns:

ans =


Note that the results from both approaches are the same.


Example 5: Matrix Symbolic Calculation

This example demonstrates how Matlab handles matrix calculation symbolically.

First we need to define the symbolic variables:

>> syms a b c d e f g h

Matrix A is then defined as:

>> A=[a b; c d]

Matlab's echo:

A =

[ a, b]
[ c, d]

Next, matrix B is defined as:

>> B=[e f;g h]

Matlab's echo:

B =

[ e, f]
[ g, h]

The addition of these two matrices yields:

>> C=A+B

C =

[ a+e, b+f]
[ c+g, d+h]

and the product of A and B is:

>> D=A*B

D =

[ a*e+b*g, a*f+b*h]
[ c*e+d*g, c*f+d*h]

If we wish to evaluate a specific matrix numerically, we simply assign the numeric values to the appropriate variable then use the command eval as demonstrate below.

>> a=1;b=2;c=3;d=4;e=5;f=6;e=7;f=8;g=9;h=0;

>> eval(A)

ans =

1 2
3 4

>> eval(B)

ans =

7 8
9 0

>> eval(C)

ans =

8 10
12 4

The inverse of A can be expressed symbolically:

>> D=inv(A)

D =

[ d/(a*d-b*c), -b/(a*d-b*c)]
[ -c/(a*d-b*c), a/(a*d-b*c)]

Numerically, D is expressed by

>> Dn=eval(inv(A))

Dn =

-2.0000 1.0000
1.5000 -0.5000

As a verification, we may evaluate D directly:

>> De=eval(D)

De =

-2.0000 1.0000
1.5000 -0.5000

You may also try

>> Df=inv(eval(A))

to verify if you get the same result.



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