Algebraic simplification - MATLAB simplify (2024)

Table of Contents

Syntax Description Examples Simplify Expressions Simplify Matrix Elements Get Simpler Results for Logarithms and Powers Get Simpler Results Using More Simplification Steps Get Equivalent Results for Symbolic Expression Separate Real and Imaginary Parts Avoid Imaginary Terms in Exponents Simplify Units Get Simpler Result by Expanding Expression Input Arguments expr — Input expression symbolic expression | symbolic function | symbolic vector | symbolic matrix Name-Value Arguments All — Option to return equivalent results false (default) | true Criterion — Simplification criterion 'default' (default) | 'preferReal' IgnoreAnalyticConstraints — Simplification rules false (default) | true Seconds — Time limit for the simplification process Inf (default) | positive number Steps — Number of simplification steps 1 (default) | positive number Tips Algorithms References Version History See Also Functions Live Editor Tasks Topics MATLAB Command Americas Europe Asia Pacific FAQs References

Algebraic simplification

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Syntax

S = simplify(expr)

S = simplify(expr,Name,Value)

Description

example

S = simplify(expr) performs algebraic simplification of expr. If expr is a symbolic vector or matrix, this function simplifies each element of expr.

example

S = simplify(expr,Name,Value) performs algebraic simplification of expr using additional options specified by one or more Name,Value pair arguments.

Examples

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Simplify Expressions

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Simplify these symbolic expressions:

syms x a b cS = simplify(sin(x)^2 + cos(x)^2)

S = $1$

S = simplify(exp(c*log(sqrt(a+b))))

S = ${(a + b)}^{c / 2}$

Simplify Matrix Elements

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Call simplify for this symbolic matrix. When the input argument is a vector or matrix, simplify tries to find a simpler form of each element of the vector or matrix.

syms xM = [(x^2 + 5*x + 6)/(x + 2), sin(x)*sin(2*x) + cos(x)*cos(2*x);(exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2, sqrt(16)];S = simplify(M)

S = $(\begin{array}{c} x + 3 & \cos (x) \\ \sin (x) & 4 \end{array})$

Get Simpler Results for Logarithms and Powers

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Simplify a symbolic expression that contains logarithms and powers. By default, simplify does not combine powers and logarithms because combining them is not valid for generic complex values.

syms xexpr = (log(x^2 + 2*x + 1) - log(x + 1))*sqrt(x^2);S = simplify(expr)

S = $- (\log (x + 1) - \log ({(x + 1)}^{2})) \sqrt{x^{2}}$

To apply the simplification rules that allow the simplify function to combine powers and logarithms, set 'IgnoreAnalyticConstraints' to true:

S = simplify(expr,'IgnoreAnalyticConstraints',true)

S = $x \log (x + 1)$

Get Simpler Results Using More Simplification Steps

Get Equivalent Results for Symbolic Expression

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Get equivalent results for a symbolic expression by setting the value of 'All' to true.

syms xexpr = cos(x)^2 - sin(x)^2;S = simplify(expr,'All',true)

S = $(\begin{array}{c} \cos (2 x) \\ {\cos (x)}^{2} - {\sin (x)}^{2} \end{array})$

Increase the number of simplification steps to 10. Find the other equivalent results for the same expression.

S = simplify(expr,'Steps',10,'All',true)

S = $(\begin{array}{c} \cos (2 x) \\ 1 - 2 {\sin (x)}^{2} \\ 2 {\cos (x)}^{2} - 1 \\ {\cos (x)}^{2} - {\sin (x)}^{2} \\ \cot (2 x) \sin (2 x) \\ \frac{e^{- 2 x i}}{2} + \frac{e^{2 x i}}{2} \end{array})$

Separate Real and Imaginary Parts

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Attempt to separate real and imaginary parts of an expression by setting the value of 'Criterion' to 'preferReal'.

syms xf = (exp(x + exp(-x*1i)/2 - exp(x*1i)/2)*1i)/2 -... (exp(-x - exp(-x*1i)/2 + exp(x*1i)/2)*1i)/2;S = simplify(f,'Criterion','preferReal','Steps',100)

S = $\sin (\sin (x)) \cosh (x) + \cos (\sin (x)) \sinh (x) i$

If 'Criterion' is not set to 'preferReal', then simplify returns a shorter result but the real and imaginary parts are not separated.

S = simplify(f,'Steps',100)

S = $\sin (\sin (x) + x i)$

When you set 'Criterion' to 'preferReal', the simplifier disfavors expression forms where complex values appear inside subexpressions. In nested subexpressions, the deeper the complex value appears inside an expression, the least preference this form of an expression gets.

Avoid Imaginary Terms in Exponents

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Attempt to avoid imaginary terms in exponents by setting 'Criterion' to 'preferReal'.

Show this behavior by simplifying a complex symbolic expression with and without setting 'Criterion' to 'preferReal'. When 'Criterion' is set to 'preferReal', then simplify places the imaginary term outside the exponent.

expr = sym(1i)^(1i+1);withoutPreferReal = simplify(expr,'Steps',100)

withoutPreferReal = ${(- 1)}^{\frac{1}{2} + \frac{1}{2} i}$

withPreferReal = simplify(expr,'Criterion','preferReal','Steps',100)

withPreferReal = $e^{- \frac{π}{2}} i$

Simplify Units

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Simplify expressions containing symbolic units of the same dimension by using simplify.

u = symunit;expr = 300*u.cm + 40*u.inch + 2*u.m;S = simplify(expr)

S = $\frac{752}{125} m "meter - a physical unit of length."$

simplify automatically chooses the unit to rewrite into. To choose a specific unit, use rewrite.

Get Simpler Result by Expanding Expression

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In most cases, to simplify a symbolic expression using Symbolic Math Toolbox™, you only need to use the simplify function. But for some large and complex expressions, you can obtain a faster and simpler result by using the expand function before applying simplify.

For instance, this workflow gives better results when finding the determinant of a matrix that represents the Kerr metric [1]. Declare the parameters of the Kerr metric.

syms theta real;syms r rs a real positive;

Define the matrix that represents the Kerr metric.

rho = sqrt(r^2 + a^2*cos(theta)^2);delta = r^2 + a^2 - r*rs;g(1,1) = - (1 - r*rs/rho^2);g(1,4) = - (rs*a*r*sin(theta)^2)/rho^2;g(4,1) = - (rs*a*r*sin(theta)^2)/rho^2;g(2,2) = rho^2/delta;g(3,3) = rho^2;g(4,4) = (r^2 + a^2 + rs*a^2*r*sin(theta)^2/rho^2)*sin(theta)^2;

Evaluate the determinant of the Kerr metric.

det_g = det(g)

det_g = $- \frac{{\sin (θ)}^{2} (a^{6} {\cos (θ)}^{4} + a^{4} r^{2} {\cos (θ)}^{4} + 2 a^{4} r^{2} {\cos (θ)}^{2} + rs a^{4} r {\cos (θ)}^{2} {\sin (θ)}^{2} - rs a^{4} r {\cos (θ)}^{2} + 2 a^{2} r^{4} {\cos (θ)}^{2} + a^{2} r^{4} - rs a^{2} r^{3} {\cos (θ)}^{2} + rs a^{2} r^{3} {\sin (θ)}^{2} - rs a^{2} r^{3} + r^{6} - rs r^{5})}{a^{2} + r^{2} - rs r}$

Simplify the determinant using the simplify function.

D = simplify(det_g)

D = $- {\sin (θ)}^{2} (a^{2} {\cos (θ)}^{2} + r^{2}) (- a^{2} {\sin (θ)}^{2} + a^{2} + r^{2})$

Instead, flatten the expression using the expand function, and then apply the simplify function. The result is simpler with this extra step.

D = simplify(expand(det_g))

D = $- {\sin (θ)}^{2} {(- a^{2} {\sin (θ)}^{2} + a^{2} + r^{2})}^{2}$

Input Arguments

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`expr` — Input expression
symbolic expression | symbolic function | symbolic vector | symbolic matrix

Input expression, specified as a symbolic expression, function, vector, or matrix.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 'Seconds',60 limits the simplificationprocess to 60 seconds.

`All` — Option to return equivalent results
`false` (default) | `true`

Option to return equivalent results, specified as the comma-separated pair consisting of 'All' and either of the two logical values. When you use this option, the input argument expr must be a scalar.

`false`	Use the default option to return only the final simplification result.
`true`	Return a column vector of equivalent results for the input expression. You can use this option along with the `'Steps'` option to obtain alternative expressions in the simplification process.

`Criterion` — Simplification criterion
`'default'` (default) | `'preferReal'`

Simplification criterion, specified as the comma-separated pairconsisting of 'Criterion' and one of these charactervectors.

`'default'`	Use the default (internal) simplification criteria.
`'preferReal'`	Favor the forms of `S` containing real valuesover the forms containing complex values. If any form of `S` containscomplex values, the simplifier disfavors the forms where complex valuesappear inside subexpressions. In case of nested subexpressions, thedeeper the complex value appears inside an expression, the least preferencethis form of an expression gets.

`IgnoreAnalyticConstraints` — Simplification rules
`false` (default) | `true`

Simplification rules, specified as the comma-separated pairconsisting of 'IgnoreAnalyticConstraints' and oneof these values.

false Use strict simplification rules. simplify always returns results that are analytically equivalent to the initial expression.

true Apply purely algebraic simplifications to expressions. Setting IgnoreAnalyticConstraints to true can give you simpler solutions, which could lead to results not generally valid. In other words, this option applies mathematical identities that are convenient, but the results might not hold for all possible values of the variables. In some cases, the results might not be equivalent to the initial expression. For details, see Algorithms.

`Seconds` — Time limit for the simplification process
`Inf` (default) | `positive number`

Time limit for the simplification process, specified as thecomma-separated pair consisting of 'Seconds' anda positive value that denotes the maximal time in seconds.

`Steps` — Number of simplification steps
`1` (default) | `positive number`

Number of simplification steps, specified as the comma-separatedpair consisting of 'Steps' and a positive valuethat denotes the maximal number of internal simplification steps.Note that increasing the number of simplification steps can slow downyour computations.

simplify(expr,'Steps',n) is equivalent to simplify(expr,n), where n is the number of simplification steps.

Tips

Simplification of mathematical expression is not aclearly defined subject. There is no universal idea as to which formof an expression is simplest. The form of a mathematical expressionthat is simplest for one problem might be complicated or even unsuitablefor another problem.

Algorithms

When you use IgnoreAnalyticConstraints, then simplify follows some of these rules:

log(a) + log(b)=log(a·b) for all values of a and b. In particular, the following equality is valid for all values of a, b, and c:
(a·b)^c=a^c·b^c.
log(a^b)=b·log(a) for all values of a and b. In particular, the following equality is valid for all values of a, b, and c:
(a^b)^c=a^b·c.
If f and g are standard mathematical functions and f(g(x))=x for all small positive numbers, f(g(x))=x is assumed to be valid for all complex values of x. In particular:
- log(e^x)=x
- asin(sin(x))=x, acos(cos(x))=x, atan(tan(x))=x
- asinh(sinh(x))=x, acosh(cosh(x))=x, atanh(tanh(x))=x
- W_k(x·e^x)=x for all branch indices k of the Lambert W function.

References

[1] Zee, A. Einstein Gravity in a Nutshell. Princeton: Princeton University Press, 2013.

Version History

Introduced before R2006a

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Algebraic simplification - MATLAB simplify (2024)

FAQs

How to get simplified answers in Matlab? ›

S = simplify( expr ) performs algebraic simplification of expr . If expr is a symbolic vector or matrix, this function simplifies each element of expr . S = simplify( expr , Name,Value ) performs algebraic simplification of expr using additional options specified by one or more Name,Value pair arguments.

Action	Keys
Open a previously saved file.	Ctrl+O
Open a new blank file.	Ctrl+N
Undo a modification, returning it to the previous state.	Ctrl+Z
Redo an undone modification, returning it to the changed state.	Ctrl+Y or, in the design area only, Ctrl+Shift+Z

What does GCA mean in MATLAB? ›

gca and gcf

The axis has properties that describe the characteristics of your axes. To list all properties of the figure, you can type get(gcf) (which stands for get current figure). To list all properties of the axis, you can type get(gca) (which stands for get current axis).

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What does Flipdim mean in MATLAB? ›

B = flipdim(A,dim) returns A with dimension dim flipped. When the value of dim is 1 , the array is flipped row-wise down. When dim is 2 , the array is flipped columnwise left to right. flipdim(A,1) is the same as flipud(A) , and flipdim(A,2) is the same as fliplr(A) .

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How do you solve a simple equation in MATLAB? ›

Solve an Equation

If eqn is an equation, solve(eqn, x) solves eqn for the symbolic variable x . Use the == operator to specify the familiar quadratic equation and solve it using solve . solx is a symbolic vector containing the two solutions of the quadratic equation.

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How do you quickly get help about a given function in MATLAB? ›

Select a function name in the Editor, Command Window, or Help browser; right-click; and then select Help on Selection. After you type an open parenthesis for function inputs, pause or press Ctrl + F1. Use the help command. Click the function icon to the left of the command prompt.

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What is simplify fraction in MATLAB? ›

simplifyFraction( expr ) simplifies the rational expression expr such that the numerator and denominator have no divisors in common. simplifyFraction( expr ,'Expand',true) expands the numerator and denominator of the resulting simplified fraction as polynomials without factorization.

How do you output an answer in MATLAB? ›

There are three common ways:

Type the name of a variable without a trailing semi-colon.
Use the “disp” function.
Use the “fprintf” function, which accepts a C printf-style formatting string.