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SYMBOLIC CALCULATION

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.

Simplify

- 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 =

1

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 =

1/4

'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 =

3*x^2+sin(x)

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 =

3*(y+5)^2

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 =

1/3*x^3+(y+5)^3*x

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 =

12500+10*x^2

 

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 =

5^(1/2)-1
-1-5^(1/2)

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

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

Matlab returns:

ans =

-3.2361
1.2361

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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