Simplify Symbolic Expressions - MATLAB & Simulink (2024)

Simplify Using Options

By default, simplify uses strict simplification rules and ensures that simplified expressions are always mathematically equivalent to initial expressions. For example, it does not combine logarithms for complex values in general.

syms xsimplify(log(x^2) + log(x))

ans =log(x^2) + log(x)

You can apply additional simplification rules which are notcorrect for all values of parameters and all cases, but using which simplify canreturn shorter results. For this approach, use IgnoreAnalyticConstraints.For example, simplifying the same expression withIgnoreAnalyticConstraints,you get the result with combined logarithms.

simplify(log(x^2) + log(x),'IgnoreAnalyticConstraints',true)

ans =3*log(x)

IgnoreAnalyticConstraints provides a shortcutallowing you to simplify expressions under commonly used assumptionsabout values of the variables. Alternatively, you can set appropriateassumptions on variables explicitly. For example, combining logarithmsis not valid for complex values in general. If you assume that x isa real value, simplify combines logarithms without IgnoreAnalyticConstraints.

assume(x,'real')simplify(log(x^2) + log(x))

ans =log(x^3)

For further computations, clear the assumption on x by recreating it using syms.

syms x

Another approach that can improve simplification of an expressionor function is the syntax simplify(f,'Steps',n),where n is a positive integer that controls howmany steps simplify takes. Specifying more simplificationsteps can help you simplify the expression better, but it takes moretime. By default, n = 1. For example, create andsimplify this expression. The result is shorter than the originalexpression, but it can be simplified further.

syms xy = (cos(x)^2 - sin(x)^2)*sin(2*x)*(exp(2*x) - 2*exp(x) + 1)/... ((cos(2*x)^2 - sin(2*x)^2)*(exp(2*x) - 1));simplify(y)

ans =(sin(4*x)*(exp(x) - 1))/(2*cos(4*x)*(exp(x) + 1))

Specify the number of simplification steps for the same expression.First, use 25 steps.

simplify(y,'Steps',25)

ans =(tan(4*x)*(exp(x) - 1))/(2*(exp(x) + 1))

Use 50 steps to simplify the expression even further.

simplify(y,'Steps',50)

ans =(tan(4*x)*tanh(x/2))/2

Suppose, you already simplified an expression or function, but you want the other forms of the same expression. To do this, you can set the 'All' option to true. The syntax simplify(f,'Steps',n,'All',true) shows other equivalent results of the same expression in the simplification steps.

syms xy = cos(x) + sin(x)simplify(y,'Steps',10,'All',true)

ans = 2^(1/2)*sin(x + pi/4) 2^(1/2)*cos(x - pi/4) cos(x) + sin(x) 2^(1/2)*((exp(- x*1i - (pi*1i)/4)*1i)/2 - (exp(x*1i + (pi*1i)/4)*1i)/2)

To return even more equivalent results, increase the number of steps to 25.

simplify(y,'Steps',25,'All',true)

ans = 2^(1/2)*sin(x + pi/4) 2^(1/2)*cos(x - pi/4) cos(x) + sin(x) -2^(1/2)*(2*sin(x/2 - pi/8)^2 - 1) 2^(1/2)*(exp(- x*1i + (pi*1i)/4)/2 + exp(x*1i - (pi*1i)/4)/2) 2^(1/2)*((exp(- x*1i - (pi*1i)/4)*1i)/2 - (exp(x*1i + (pi*1i)/4)*1i)/2)

Simplify Using Assumptions

Some expressions cannot be simplified in general, but becomemuch shorter under particular assumptions. For example, simplifyingthis trigonometric expression without additional assumptions returnsthe original expression.

syms nsimplify(sin(2*n*pi))

ans =sin(2*pi*n)

However, if you assume that variable n representsan integer, the same trigonometric expression simplifies to 0.

assume(n,'integer')simplify(sin(2*n*pi))

ans =0

For further computations, clear the assumption.

syms n

Simplify Fractions

You can use the general simplification function, simplify,to simplify fractions. However, Symbolic Math Toolbox offers amore efficient function specifically for this task: simplifyFraction.The statement simplifyFraction(f) represents theexpression f as a fraction, where both the numeratorand denominator are polynomials whose greatest common divisor is 1.For example, simplify these expressions.

syms x ysimplifyFraction((x^3 - 1)/(x - 1))

ans =x^2 + x + 1

simplifyFraction((x^3 - x^2*y - x*y^2 + y^3)/(x^3 + y^3))

ans =(x^2 - 2*x*y + y^2)/(x^2 - x*y + y^2)

By default, simplifyFraction does not expandexpressions in the numerator and denominator of the returned result.To expand the numerator and denominator in the resulting expression,use the Expand option. For comparison, first simplifythis fraction without Expand.

simplifyFraction((1 - exp(x)^4)/(1 + exp(x))^4)

ans =(exp(2*x) - exp(3*x) - exp(x) + 1)/(exp(x) + 1)^3

Now, simplify the same expressions with Expand.

simplifyFraction((1 - exp(x)^4)/(1 + exp(x))^4,'Expand',true)

ans =(exp(2*x) - exp(3*x) - exp(x) + 1)/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1)