symfun

Create symbolic functions

Syntax

f(inputs) = formula

f = symfun(formula,inputs)

f = symfun(fM)

Description

f(inputs) = formula creates the symbolic function f. For example, f(x,y) = x + y. The symbolic variables in inputs are the input arguments. The symbolic expression formula is the body of the function f.

f = symfun(formula,inputs) is the formal way to create a symbolic function.

f = symfun(fM) converts a symbolic matrix function fM of type symfunmatrix to a symbolic function f of type symfun. (since R2024b)

Examples

Create and Define Symbolic Functions

Define the symbolic function f(x,y) = x + y. First, create the function by using syms. Then define the function.

syms f(x,y)
f(x,y) = x + y

f(x, y) = $$x+y$$

Find the value of f at x = 1 and y = 2.

f(1,2)

ans = $$3$$

Define the function again by using the formal syntax.

syms x y
f = symfun(x+y,[x y])

f(x, y) = $$x+y$$

Return Body and Arguments of Symbolic Function

Return the body of a symbolic function by using formula. You can use the body for operations such as indexing into the function. Return the arguments of a symbolic function by using argnames.

Index into the symbolic function [x^2, y^4]. Since a symbolic function is a scalar, you cannot directly index into the function. Instead, index into the body of the function.

syms f(x,y)
f(x,y) = [x^2, y^4];
fbody = formula(f);
fbody(1)

ans = $$x^2$$

fbody(2)

ans = $$y^4$$

Return the arguments of the function.

fvars = argnames(f)

fvars = $$\left(\begin{array}{cc}x& y\end{array}\right)$$

Combine Two Symbolic Functions

Create two symbolic functions.

syms f(x) g(x)
f(x) = 2*x^2 - x;
g(x) = 3*x^2 + 2*x;

Combine the two symbolic functions into another symbolic function $$h(x)$$ with the data type symfun.

h(x) = [f(x); g(x)]

h(x) = 
$$\left(\begin{array}{c}2\hspace{0.17em}{x}^2-x\\ 3\hspace{0.17em}{x}^2+2\hspace{0.17em}x\end{array}\right)$$

Evaluate the function $$h(x)$$ at $$x=1$$ and $$x=2$$.

h(1)

ans = 
$$\left(\begin{array}{c}1\\ 5\end{array}\right)$$

h(2)

ans = 
$$\left(\begin{array}{c}6\\ 16\end{array}\right)$$

You can also combine the two functions into an array of symbolic expressions with the data type sym.

h_expr = [f(x); g(x)]

h_expr = 
$$\left(\begin{array}{c}2\hspace{0.17em}{x}^2-x\\ 3\hspace{0.17em}{x}^2+2\hspace{0.17em}x\end{array}\right)$$

Index into h_expr to access the first and the second symbolic expressions.

h_expr(1)

ans = $$2\hspace{0.17em}{x}^2-x$$

h_expr(2)

ans = $$3\hspace{0.17em}{x}^2+2\hspace{0.17em}x$$

Convert Symbolic Matrix Function to Symbolic Function

Since R2024b

Create 2-by-1 and 2-by-2 symbolic matrix variables to represent the matrices $\mathit{X}$ and $\mathit{A}$.

syms X [2 1] matrix
syms A [2 2] matrix

Create two symbolic matrix functions to represent the functions $\mathit{F}\left(\mathit{X},\mathit{A}\right)$ and $\partial \mathit{F}\left(\mathit{X},\mathit{A}\right)/\partial {\mathit{X}}^{\mathit{T}}$. When creating the symbolic matrix functions, keep existing definitions of the symbolic matrix variables $\mathit{X}$ and $\mathit{A}$ in the workspace. The symbolic matrix functions require matrices of the same sizes as $\mathit{X}$ and $\mathit{A}$ as their input arguments.

syms F(X,A) [1 1] matrix keepargs
syms dF(X,A) [2 1] matrix keepargs

Define the function $\mathit{F}\left(\mathit{X},\mathit{A}\right)={\mathit{X}}^{\mathit{T}}\mathit{A}\mathit{X}$ and find its derivative $\partial \mathit{F}\left(\mathit{X},\mathit{A}\right)/\partial {\mathit{X}}^{\mathit{T}}$. The resulting symbolic matrix functions are in matrix notation in terms of $\mathit{X}$ and $\mathit{A}$.

F(X,A) = X.'*A*X

F(X, A) = $$X^{\mathrm{T}}\hspace{0.17em}A\hspace{0.17em}X$$

dF(X,A) = diff(F,X.')

dF(X, A) = $$A\hspace{0.17em}X+A^{\mathrm{T}}\hspace{0.17em}X$$

Convert the symbolic matrix functions from data type symfunmatrix to symfun. The resulting symbolic functions are in scalar notation in terms of the matrix elements of $\mathit{X}$ and $\mathit{A}$. These functions accept scalars as their input arguments.

Fsymfun = symfun(F)

Fsymfun(X1, X2, A1_1, A1_2, A2_1, A2_2) = $$X_1\hspace{0.17em}\left(A_{1,1}\hspace{0.17em}{X}_1+A_{1,2}\hspace{0.17em}{X}_2\right)+X_2\hspace{0.17em}\left(A_{2,1}\hspace{0.17em}{X}_1+A_{2,2}\hspace{0.17em}{X}_2\right)$$

dFsymfun = symfun(dF)

dFsymfun(X1, X2, A1_1, A1_2, A2_1, A2_2) = 
$$\left(\begin{array}{c}2\hspace{0.17em}{A}_{1,1}\hspace{0.17em}{X}_1+A_{1,2}\hspace{0.17em}{X}_2+A_{2,1}\hspace{0.17em}{X}_2\\ A_{1,2}\hspace{0.17em}{X}_1+A_{2,1}\hspace{0.17em}{X}_1+2\hspace{0.17em}{A}_{2,2}\hspace{0.17em}{X}_2\end{array}\right)$$

Input Arguments

formula — Function body

symbolic expression | vector of symbolic expressions | matrix of symbolic expressions

Function body, specified as a symbolic expression, vector of symbolic expressions, or matrix of symbolic expressions that can be converted to the sym data type.

Example: x + y

inputs — Input argument or arguments of function

symbolic variable | array of symbolic variables

Input argument or arguments of a function, specified as a symbolic variable or array of symbolic variables, respectively.

Example: [x,y]

Data Types: sym

fM — Symbolic matrix function to convert

symbolic matrix function

Since R2024b

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

Alternatively, you can use symfunmatrix2symfun to convert a symbolic matrix function to a symbolic function.

Data Types: symfunmatrix

Output Arguments

f — Symbolic function

symfun object

Symbolic function, returned as a symfun object.

While the data type of the function f is symfun, the data type of the evaluated function, such as f(1,2), is sym.

Version History

Introduced in R2012a

R2024b: Convert symbolic matrix function

You can convert a symbolic matrix function of type symfunmatrix to a symbolic function of type symfun by using symfun. For an example, see Convert Symbolic Matrix Function to Symbolic Function.