sym

Create the symbolic variables x and y.

x = sym("x")

x = $$x$$

y = sym("y")

y = $$y$$

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

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 = $${\mathrm{e}}^{\pi }$$

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

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
 

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

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

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