Algebraic simplification - MATLAB simplify- MathWorks Deutschland (2024)

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

Open Live Script

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

Open Live Script

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

Open Live Script

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

Open Live Script

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

Open Live Script

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

Open Live Script

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

Open Live Script

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
- MathWorks Deutschland (2024)

FAQs

How to simplify an algebraic expression 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.

Discover More ›

What is the pretty command in Matlab? ›

pretty( X ) prints X in a plain-text format that resembles typeset mathematics. For true typeset rendering, use Live Scripts instead. See What Is a Live Script or Function?

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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.

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What is Syms in Matlab? ›

syms lists the names of all symbolic scalar variables, functions, matrix variables, matrix functions, and arrays in the MATLAB workspace. example. S = syms returns a cell array of the names of all symbolic scalar variables, functions, matrix variables, matrix functions, and arrays.

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How do you simplify algebraic expressions quickly? ›

How to simplify expressions. To simplify expressions first expand any brackets, next multiply or divide any terms and use the laws of indices if necessary, then collect like terms by adding or subtracting and finally rewrite the expression.

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What app can I use to simplify algebraic expressions? ›

Wolfram|Alpha can be used as a simplification calculator to simplify polynomials, Booleans, numbers, rational functions and many other math objects.

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What is the %% used for in MATLAB? ›

Two percent signs, %% , serve as a cell delimiter as described in Create and Run Sections in Code. Add a comment to a block of code: % The purpose of this loop is to compute % the value of ...

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How to write algebraic expressions in MATLAB? ›

Basic Algebraic Operations

syms t G = [cos(t) sin(t); -sin(t) cos(t)] create this transformation matrix.
G = [ cos(t), sin(t)] [ -sin(t), cos(t)] ...
A = G*G. ...
A = G^2. ...
A = [ cos(t)^2 - sin(t)^2, 2*cos(t)*sin(t)] [ -2*cos(t)*sin(t), cos(t)^2 - sin(t)^2] ...
A = simplify(A) ...
A = [ cos(2*t), sin(2*t)] [ -sin(2*t), cos(2*t)] ...
I = G.'

More items...

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What does eye () do in MATLAB? ›

I = eye( n ) returns an n -by- n identity matrix with ones on the main diagonal and zeros elsewhere.

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What is simplify mesh in MATLAB? ›

simplify( mesh ) simplifies the surface mesh mesh by using quadric decimation. simplify( mesh ,Name=Value) specifies options using one or more name-value arguments. For example, SimplificationMethod="vertex-clustering" simplifies the surface mesh by using the vertex-clustering method.

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What is an example of simplify in math? ›

For example, 1/2 (x + 4) can be simplified as x/2 + 2. Let us take one more example to understand it. Example: Simplify the expression: 3/4x + y/2 (4x + 7). By using the distributive property, the given expression can be written as 3/4x + y/2 (4x) + y/2 (7).

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What is == in MATLAB? ›

Description. example. A == B returns a logical array or a table of logical values with elements set to logical 1 ( true ) where inputs A and B are equal; otherwise, the element is logical 0 ( false ).

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What does Dirac mean in MATLAB? ›

dirac returns floating-point results for numeric arguments that are not symbolic objects. dirac acts element-wise on nonscalar inputs. The input arguments x and n must be vectors or matrices of the same size, or else one of them must be a scalar.

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What is ABS in MATLAB? ›

abs( z ) returns the absolute value (or complex modulus) of z . Because symbolic variables are assumed to be complex by default, abs returns the complex modulus (magnitude) by default. If z is an array, abs acts element-wise on each element of z .

Read The Full Story ›

How do you simplify an algebraic formula? ›

Algebraic expressions can be simplified by multiplying out the brackets and collecting like terms, or by factorising with a common factor. Straight line gradients can be calculated using a formula.

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How do you simplify algebraic notation? ›

An expression can be simplified by collecting like terms. Like terms are those which contain the same letter symbol and equal powers. As 2 + 2 + 2 can be written as 3 x 2, a + a + a can be written as 3 x a. However, with algebraic notation, multiply and division symbols are not included.

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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.

Algebraic simplification - MATLAB simplify - MathWorks Deutschland (2024)

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-Befehl

Americas

Europe

Asia Pacific

FAQs

How to simplify an algebraic expression in Matlab? ›

What is an example of simplify in math? ›

References

Algebraic simplification - MATLAB simplify - MathWorks Deutschland (2024)

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-Befehl

Americas

Europe

Asia Pacific

FAQs

How to simplify an algebraic expression in Matlab? ›

What is an example of simplify in math? ›

References

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

`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`