sym

Create symbolic variables, expressions, functions, matrices

Syntax

x = sym("x")

A = sym("a",[n1 ... nM])

A = sym("a",n)

sym(___,set)

sym(___,"clear")

sym(num)

sym(num,flag)

sym(strnum)

symexpr = sym(h)

symexpr = sym(M)

Description

x = sym("x") creates symbolic scalar variable x.

A = sym("a",[n1 ... nM]) creates an n1-by-...-by-nM symbolic array filled with automatically generated elements. For example, A = sym("a",[1 3]) creates the row vector A = [a1 a2 a3]. The generated elements a1, a2, and a3 do not appear in the MATLAB^® workspace. For multidimensional arrays, these elements have the prefix a followed by the element’s index using _ as a delimiter, such as a1_3_2.

A = sym("a",n) creates an n-by-n symbolic matrix filled with automatically generated elements.

sym(___,set) creates a symbolic variable or array and sets the assumption that the variable or all array elements belong to set. Here, set can be "real", "positive", "integer", or "rational". You also can combine multiple assumptions by specifying a string array or cell array of character vectors. For example, assume a positive rational value by specifying set as ["positive" "rational"] or {'positive','rational'}.

sym(___,"clear") clears assumptions set on a symbolic variable or array. You can specify "clear" after the input arguments in any of the previous syntaxes, except combining "clear" and set. You cannot set and clear an assumption in the same function call to sym.

sym(num) converts a number or numeric matrix specified by num to a symbolic number or symbolic matrix.

sym(num,flag) uses the technique specified by flag to convert floating-point numbers to symbolic numbers.

sym(strnum) converts the character vector or string specified by strnum to an accurate symbolic number without approximation.

symexpr = sym(h) creates a symbolic expression or matrix symexpr from an anonymous MATLAB function associated with the function handle h.

symexpr = sym(M) converts a symbolic matrix variable M of type symmatrix to an array of symbolic scalar variables symexpr of type sym.

Examples

Create Symbolic Variables

Create the symbolic variables x and y.

x = sym("x")

x = $$x$$

y = sym("y")

y = $$y$$

Create Symbolic Vectors

Create a 1-by-4 symbolic vector a with automatically generated elements $$a_1,...,a_4$$.

a = sym("a",[1 4])

a = $$\left(\begin{array}{cccc}{a}_1& a_2& a_3& a_4\end{array}\right)$$

You can specify the format for the element names by using a format operator within the first argument. sym replaces %d with the index of the element to generate the element names.

b = sym("x_%d",[1 4])

b = $$\left(\begin{array}{cccc}{x}_1& x_2& x_3& x_4\end{array}\right)$$

However, these syntaxes do not create symbolic variables $$a_1$$, ..., $$a_4$$, $$x_1$$, ..., $$x_4$$ in the MATLAB workspace. To access the elements of a and b, use the standard indexing methods.

a(1)

ans = $$a_1$$

b(2:3)

ans = $$\left(\begin{array}{cc}{x}_2& x_3\end{array}\right)$$

Create Symbolic Matrices

Create a 3-by-4 symbolic matrix with automatically generated elements. The sym function generates matrix elements of the form $$A_{i,j}$$. Here, sym generates the elements $$A_{1,1}$$, ..., $$A_{3,4}$$.

A = sym("A",[3 4])

A = 
$$\left(\begin{array}{cccc}{A}_{1,1}& A_{1,2}& A_{1,3}& A_{1,4}\\ A_{2,1}& A_{2,2}& A_{2,3}& A_{2,4}\\ A_{3,1}& A_{3,2}& A_{3,3}& A_{3,4}\end{array}\right)$$

Create a 4-by-4 matrix with the element names $$x_{1,1}$$, ..., $$x_{4,4}$$ by using a format operator within the first argument. sym replaces %d with the index of the element to generate the element names.

B = sym("x_%d_%d",4)

B = 
$$\left(\begin{array}{cccc}{x}_{1,1}& x_{1,2}& x_{1,3}& x_{1,4}\\ x_{2,1}& x_{2,2}& x_{2,3}& x_{2,4}\\ x_{3,1}& x_{3,2}& x_{3,3}& x_{3,4}\\ x_{4,1}& x_{4,2}& x_{4,3}& x_{4,4}\end{array}\right)$$

These syntaxes do not create symbolic variables $$A_{1,1}$$, ..., $$A_{3,4}$$, $$x_{1,1}$$, ..., $$x_{4,4}$$ in the MATLAB workspace. To access an element of a matrix, use parentheses.

A(2,3)

ans = $$A_{2,3}$$

B(4,2)

ans = $$x_{4,2}$$

Create Symbolic Multidimensional Arrays

Create a 2-by-2-by-2 symbolic array with automatically generated elements $$a_{1,1,1},\dots ,a_{2,2,2}$$.

A = sym("a",[2 2 2])

A(:,:,1) = 
$$\left(\begin{array}{cc}{a}_{1,1,1}& a_{1,2,1}\\ a_{2,1,1}& a_{2,2,1}\end{array}\right)$$

A(:,:,2) = 
$$\left(\begin{array}{cc}{a}_{1,1,2}& a_{1,2,2}\\ a_{2,1,2}& a_{2,2,2}\end{array}\right)$$

Create Symbolic Numbers

Convert numeric values to symbolic numbers or expressions. Use sym on subexpressions instead of the entire expression for better accuracy. Using sym on entire expressions is inaccurate because MATLAB® first converts the expression to a floating-point number, which loses accuracy. sym cannot always recover this lost accuracy.

inaccurate1 = sym(1/1234567)

inaccurate1 = 
$$\frac{7650239286923505}{9444732965739290427392}$$

accurate1 = 1/sym(1234567)

accurate1 = 
$$\frac{1}{1234567}$$

inaccurate2 = sym(sqrt(1234567))

inaccurate2 = 
$$\frac{4886716562018589}{4398046511104}$$

accurate2 = sqrt(sym(1234567))

accurate2 = $$\sqrt{1234567}$$

inaccurate3 = sym(exp(pi))

inaccurate3 = 
$$\frac{6513525919879993}{281474976710656}$$

accurate3 = exp(sym(pi))

accurate3 = $${\mathrm{e}}^{\pi }$$

Create Large Symbolic Numbers

When creating symbolic numbers with 15 or more digits, use quotation marks to accurately represent the numbers.

inaccurateNum = sym(11111111111111111111)

inaccurateNum = $$11111111111111110656$$

accurateNum = sym("11111111111111111111")

accurateNum = $$11111111111111111111$$

When you use quotation marks to create symbolic complex numbers, specify the imaginary part of a number as 1i, 2i, and so on.

sym("1234567 + 1i")

ans = $$1234567+\mathrm{i}$$

Convert Function Handles to Symbolic Expressions

Convert anonymous functions associated with MATLAB® handles to a symbolic expression and a symbolic matrix.

h_expr = @(x)(sin(x) + cos(x));
sym_expr = sym(h_expr)

sym_expr = $$\cos \left(x\right)+\sin \left(x\right)$$

h_matrix = @(x)(x*pascal(3));
sym_matrix = sym(h_matrix)

sym_matrix = 
$$\left(\begin{array}{ccc}x& x& x\\ x& 2\hspace{0.17em}x& 3\hspace{0.17em}x\\ x& 3\hspace{0.17em}x& 6\hspace{0.17em}x\end{array}\right)$$

Set Assumptions While Creating Variables

Create the symbolic variables x, y, z, and t while simultaneously setting assumptions that x is real, y is positive, z is rational, and t is a positive integer.

x = sym("x","real");
y = sym("y","positive");
z = sym("z","rational");
t = sym("t",["positive" "integer"]);

Check the assumptions on x, y, z, and t using assumptions.

assumptions

ans = $$\left(\begin{array}{ccccc}t\in \mathbb{Z}& x\in \mathbb{R}& z\in \mathbb{Q}& 1\le t& 0<y\end{array}\right)$$

For further computations, clear the assumptions using assume.

assume([x y z t],"clear")
assumptions

 
ans =
 
Empty sym: 1-by-0
 

Set Assumptions on Matrix Elements

Create a symbolic matrix and set assumptions on each element of that matrix.

A = sym("A%d%d",[2 2],"positive")

A = 
$$\left(\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right)$$

Solve an equation involving the first element of A. MATLAB assumes that this element is positive.

solve(A(1,1)^2-1, A(1,1))

ans = $$1$$

Check the assumptions on the elements of A by using assumptions.

assumptions(A)

ans = $$\left(\begin{array}{cccc}0<A_{11}& 0<A_{12}& 0<A_{21}& 0<A_{22}\end{array}\right)$$

Clear all previously set assumptions on elements of the symbolic matrix by using assume.

assume(A,"clear");
assumptions(A)

 
ans =
 
Empty sym: 1-by-0
 

Solve the same equation again.

solve(A(1,1)^2-1, A(1,1))

ans = 
$$\left(\begin{array}{c}-1\\ 1\end{array}\right)$$

Choose Conversion Technique for Floating-Point Values

Convert pi to a symbolic value.

Choose the conversion technique by specifying the optional second argument, which can be "r", "f", "d", or "e". The default is "r". See the Input Arguments section for details about the conversion techniques.

r = sym(pi)

r = $$\pi$$

f = sym(pi,"f")

f = 
$$\frac{884279719003555}{281474976710656}$$

d = sym(pi,"d")

d = $$3.1415926535897931159979634685442$$

e = sym(pi,"e")

e = 
$$\pi -\frac{198\hspace{0.17em}\mathrm{eps}}{359}$$

Convert Hessian Matrix

Create 3-by-3 and 3-by-1 symbolic matrix variables.

syms A [3 3] matrix
syms X [3 1] matrix

Find the Hessian matrix of ${\mathit{X}}^{\mathit{T}}\mathit{AX}$.

f = X.'*A*X;
M = diff(f,X,X.')

M = $$A^{\mathrm{T}}+A$$

Convert the result from a symbolic matrix variable to a matrix of symbolic scalar variables.

S = sym(M)

S = 
$$\left(\begin{array}{ccc}2\hspace{0.17em}{A}_{1,1}& A_{1,2}+A_{2,1}& A_{1,3}+A_{3,1}\\ A_{1,2}+A_{2,1}& 2\hspace{0.17em}{A}_{2,2}& A_{2,3}+A_{3,2}\\ A_{1,3}+A_{3,1}& A_{2,3}+A_{3,2}& 2\hspace{0.17em}{A}_{3,3}\end{array}\right)$$

Alternatively, you can use symmatrix2sym to convert a symbolic matrix variable to an array of symbolic scalar variables.

S = symmatrix2sym(M)

S = 
$$\left(\begin{array}{ccc}2\hspace{0.17em}{A}_{1,1}& A_{1,2}+A_{2,1}& A_{1,3}+A_{3,1}\\ A_{1,2}+A_{2,1}& 2\hspace{0.17em}{A}_{2,2}& A_{2,3}+A_{3,2}\\ A_{1,3}+A_{3,1}& A_{2,3}+A_{3,2}& 2\hspace{0.17em}{A}_{3,3}\end{array}\right)$$

Input Arguments

"x" — Variable name

string | character vector

Variable name, specified as a string or character vector. Argument x must be a valid variable name. That is, x must begin with a letter and contain only alphanumeric characters and underscores. To verify that the name is a valid variable name, use isvarname.

Example: "x", "y123", 'z_1'

"a" — Prefix for automatically generated matrix elements

character vector | string

Prefix for automatically generated matrix elements, specified as a string or character vector. Argument a must be a valid variable name. That is, a must begin with a letter and contain only alphanumeric characters and underscores. To verify that the name is a valid variable name, use isvarname.

If you specify the argument a and its vector, matrix, or array dimensions in the argument [n1 ... nM], then a can include formatting operators such as "a_%d_%d". For examples, see Create Symbolic Vectors and Create Symbolic Matrices.

Example: "a", "b", 'a_bc'

[n1 ... nM] — Vector, matrix, or array dimensions

vector of integers

Vector, matrix, or array dimensions, specified as a vector of integers. As a shortcut, you can create a square matrix by specifying only one integer. For example, A = sym("A",3) creates a square 3-by-3 matrix.

Example: [2 3]

set — Assumptions on symbolic variable or matrix

string array | character vector | cell array of character vectors

Assumptions on the symbolic variable or matrix, specified as a character vector, string array, or cell array of character vectors. The available assumptions are "integer", "rational", "real", or "positive".

You can combine multiple assumptions by specifying a string array or cell array of character vectors. For example, assume a positive rational value by specifying set as ["positive" "rational"] or {'positive','rational'}.

Example: "integer"

num — Numeric value to be converted to symbolic number or matrix

number | symbolic constant | matrix of numbers

Numeric value to be converted to symbolic number or matrix, specified as a number, symbolic constant, or matrix of numbers.

Example: pi

flag — Conversion technique

"r" (default) | "d" | "e" | "f"

Conversion technique, specified as one of the characters listed in this table.

"r"	When sym uses the rational mode, it converts floating-point numbers obtained by evaluating expressions of the form p/q, p*pi/q, sqrt(p), 2^q, and 10^q (for modest-sized integers p and q) to the corresponding symbolic form. For example, sym(1/10,"r") returns 1/10. This mode effectively compensates for the round-off error involved in the original evaluation but might not represent the floating-point value precisely. If sym cannot find a simple rational approximation, then it uses the same technique as it would use with the flag "f".
"d"	When sym uses the decimal mode, it takes the number of digits from the current setting of digits. Conversions with fewer than 16 digits lose some accuracy, while more than 16 digits might not be warranted. For example, sym(4/3,"d") with 10-digit accuracy returns 1.333333333, while with 20-digit accuracy it returns 1.3333333333333332593. The latter does not end in 3s, but it is an accurate decimal representation of the floating-point number nearest to 4/3.
"e"	When sym uses the estimate error mode, it supplements a result obtained in the rational mode by a term involving the variable eps. This term estimates the difference between the theoretical rational expression and its actual floating-point value. For example, sym(3*pi/4,"e") returns (3*pi)/4 - (103*eps)/249.
"f"	When sym uses the floating-point to rational mode, it returns the symbolic form for all values in the form N*2^e or -N*2^e, where N >= 0 is a nonnegative integer and e is an integer. The returned symbolic number is a precise rational number that is equal to the floating-point value. For example, sym(1/10,"f") returns 3602879701896397/36028797018963968.

strnum — String representing symbolic number

string | character vector

String representing a symbolic number, specified as a string or character vector.

Example: '1/10'

h — Anonymous function

MATLAB function handle

Anonymous function, specified as a MATLAB function handle. For more information, see Anonymous Functions.

Example: h = @(x)sin(x)

M — Symbolic matrix variable to convert

symbolic matrix variable

Symbolic matrix variable to convert, specified as a symbolic matrix variable.

Alternatively, you can use symmatrix2sym to convert a symbolic matrix variable to an array of symbolic scalar variables.

Example: syms A 2 matrix; M = A^2 + eye(2)

Data Types: symmatrix

Output Arguments

x — Variable

symbolic scalar variable

Variable, returned as a symbolic scalar variable.

A — Vector or matrix with automatically generated elements

symbolic vector | symbolic matrix

Vector or matrix with automatically generated elements, returned as a symbolic vector or matrix of symbolic scalar variables. The elements of this vector or matrix do not appear in the MATLAB workspace.

symexpr — Expression or matrix converted from anonymous MATLAB function or symbolic matrix variable

symbolic expression | symbolic matrix

Expression or matrix converted from an anonymous MATLAB function or a symbolic matrix variable, returned as a symbolic expression or matrix of symbolic scalar variables.

Data Types: sym

Tips

• Statements like pi = sym(pi) and delta = sym("1/10") create symbolic numbers that avoid the floating-point approximations inherent in the values of pi and 1/10. The pi created in this way stores the symbolic number in a workspace variable named pi, which temporarily replaces the built-in numeric function with the same name. Use clear pi to restore the floating-point representation of pi.

• sym always treats i in character vector input as an identifier. To input the imaginary number i, use 1i instead.

• clear x does not clear the symbolic object of its assumptions, such as real, positive, or any assumptions set by assume, sym, or syms. To remove assumptions, use one of these options:

  • assume(x,"clear") removes all assumptions affecting x.

  • clear all clears all objects in the MATLAB workspace and resets the symbolic engine.

  • assume and assumeAlso provide more flexibility for setting assumptions on variables.

• When you replace one or more elements of a numeric vector or matrix with a symbolic number, MATLAB converts that number to a double-precision number.

  A = eye(3);
  A(1,1) = sym(pi)

  A =
      3.1416         0         0
           0    1.0000         0
           0         0    1.0000

  You cannot replace elements of a numeric vector or matrix with a symbolic variable, expression, or function because these elements cannot be converted to double-precision numbers. For example, A(1,1) = sym("a") throws an error.

• When you use the syntax A = sym("a",[n1 ... nM]), the sym function assigns only the symbolic array A to the MATLAB workspace. To also assign the automatically generated elements of A, use the syms function instead. For example, syms a [1 3] creates the row vector a = [a1 a2 a3] and the symbolic variables a1, a2, and a3 in the MATLAB workspace.

Alternative Functionality

Alternative Approaches for Creating Symbolic Variables

To create several symbolic variables in one function call, use syms. Using syms also clears assumptions from the named variables.

Version History

Introduced before R2006a

R2022b: Convert symbolic matrix variables

You can convert a symbolic matrix variable M of type symmatrix to an array of symbolic scalar variables symexpr of type sym by using symexpr = sym(M). For an example, see Convert Hessian Matrix.

R2020a: sym("pi") creates symbolic variable

sym("pi") now creates a symbolic variable named pi instead of a symbolic number representing the mathematical constant π. In previous releases, both sym("pi") and sym(pi) create symbolic numbers representing the constant π.

For example, the command a = sym("pi") creates a symbolic variable named pi and assigns it to the workspace variable a.

a = sym("pi")
class(a)
symType(a)
vpa(2*a)

a =
pi

ans = 
    'sym'

ans = 
    "variable"

ans = 
2.0*pi

To create a symbolic number representing the constant π, use a = sym(pi) instead.

a = sym(pi)
class(a)
symType(a)
vpa(2*a)

a =
pi

ans = 
    'sym'

ans = 
    "constant"

ans =
6.283185307179586476925286766559

This behavior also applies to the mathematical constants catalan and eulergamma.

R2018a: Support for character vectors has been removed

Support for character vectors that are not valid variable names and that do not define a number has been removed. To create symbolic expressions, first create symbolic variables, and then use operations on them. For example, use syms x; x + 1 instead of sym('x + 1'), exp(sym(pi)) instead of sym('exp(pi)'), and syms f(var1,...,varN) instead of f(var1,...,varN) = sym('f(var1,...,varN)').