Matlab sym()

Introduction to Matlab sym()

When any symbolic variable is created being associated with an assumption, the symbolic variable and its assumption get stored separately by MATLAB. The sym() in MATLAB is used to create numbered symbolic variables or symbolic variables in different MATLAB functions. Symbolic numbers generated from the sym() function are the exact representations of the input numbers.

Sym() can also be used to refer to an existing symbolic variable and current assumption associated with the variable. If the expected symbolic variable is not used in any MATLAB function previously, then sym() creates the variable without associating any assumptions.

Syntax of Matlab sym()

Syntax of matlab sym() is given: below:

Note:

In the case of multidimensional arrays, sym() function generates automatically generated elements, having the prefix followed by the index for the element being added with ‘ _’ as a delimiter.

Example: a1_3_2.

Adding the command ‘clear x’ actually does not clear the symbolic object with respect to assumptions associated with the variable, such as real, positive, or any assumptions, etc. Hence, to remove assumptions, one of the below options is recommended:

• assume(x,’clear’): all assumptions that affect x, are being removed.
• clear all: clears all MATLAB workspace objects resetting the symbolic engine.
• assume and assumeAlso: Add further flexibility for applying assumptions on the variable.

Expressions such aspi = sym(pi),theta = sym(‘1/10’) etc. create symbolic numbers which enable the program to avoid floating-point approximations. The variable is created as a workspace variable to hold the symbolic value given within the sym() function. It temporarily replaces the name for built-in numeric function having the same name.

Examples of Matlab sym()

Following are the examples of matlab sym() are given below:

Example #1

var1=sym('var1')var2=sym('var2')var3=sym('var3')

Output:

Example #2

var = sym('var',[1 4])

Output:

Example #3

MAT = sym('MAT',[3 4])

Output:

Example #4 – Create Symbolic Multidimensional Arrays

MATLAB enables its program to create multidimensional array containing symbolic elements at one time, using sym() function.

MAT = sym('mat',[2 2 2])

Output:

Example #5 – Converting Expressions to Symbolic Expressions

Application of sym() function on the complete expressions makes the conversion inaccurate as MATLAB converts the expression to a floating-point number, which results in reducing accuracy and recovery of the lost accuracy is not possible by sym() all the time. Hence in order to extract symbolic expressions with respect to expressions, sym() needs to be applied on subexpressions instead of the entire expression to achieve better accuracy.

var_inaccurate1 = sym(1/1234567)
var_accurate1 = 1/sym(1234567)
var_inaccurate2 = sym(sqrt(1234567))

Output:

Example #6 – Generating Large Symbolic Numbers

In case of generating symbolic number having 15 or more digits, quotation marks are used to represent the numbers with optimum accuracy.

var_inaccurateNum = sym(11111111111111111111)
var_accurateNum = sym('11111111111111111111')
sym('12335567 + 5i')

Output:

Note: Sym() treats the imaginary symbol ‘i’ as an identifier in a character vector input. Hence in order to provide imaginary number I as an input, it is suggested to use 1i instead.

Example #7 – Generating Symbolic Expressions out of Function Handles

MATLAB supports creating any symbolic expression or a symbolic matrix from anonymous functions that are associated with MATLAB handles.

mat_expr = @(x)(sin(x) + cos(x));
mat_sym_expr = sym(mat_expr)
mat_matrix = @(x)(x*pascal(3));
mat_sym_matrix = sym(mat_matrix)

Output:

Example 8 – Setting of Assumptions While Generating Symbolic Variables

MATLAB supports creating symbolic variables adding a different set of assumptions about characteristics of the variable such as the number being real or positive or rational etc.

p = sym('p','real');
q = sym('q','positive');
r = sym('r','rational');
s = sym('s',{'positive','integer'});
assume([p q r s],'clear')
assumptions

Output:

Example #9 – Selecting Conversion Technique for Floating-Point Values

The functionality of the sym() function is extended to choose any conversion technique. It is achieved by specifying the second argument which is optional by nature. The value for the arguments can be ‘r’, ‘f’, ‘d’, or ‘e’. However, the default value remains ‘r’.

p = sym(pi)
q = sym(pi,'f')
r = sym(pi,'d')
s = sym(pi,'e')

Output:

Conclusion

When one or more elements of a numeric vector or matrix is/are replaced with a symbolic number, MATLAB treats the number as a double-precision number, but those elements of a numeric vector or matrix do not support replacement with a symbolic variable, function or expression as these elements cannot be converted to double-precision numbers. The functions Sym() and syms() share similar functionality but differs as:

The syms() function creates a symbolic object, automatically assigning to a MATLAB variable having the same name, whereas, a symbolic object referred by sym() function can be assigned to a MATLAB variable with the same name and to a different name as well.

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