Perform Symbolic Computations - MATLAB & Simulink (2024)

Expressions with One Variable

To differentiate a symbolic expression, use the diff command.The following example illustrates how to take a first derivative ofa symbolic expression:

syms xf = sin(x)^2;diff(f)

ans =2*cos(x)*sin(x)

Partial Derivatives

For multivariable expressions, you can specify the differentiationvariable. If you do not specify any variable, MATLAB^® choosesa default variable by its proximity to the letter x:

syms x yf = sin(x)^2 + cos(y)^2;diff(f)

ans =2*cos(x)*sin(x)

For the complete set of rules MATLAB applies for choosinga default variable, see Find a Default Symbolic Variable.

To differentiate the symbolic expression f withrespect to a variable y, enter:

syms x yf = sin(x)^2 + cos(y)^2;diff(f, y)

ans =-2*cos(y)*sin(y)

Second Partial and Mixed Derivatives

To take a second derivative of the symbolic expression f withrespect to a variable y, enter:

syms x yf = sin(x)^2 + cos(y)^2;diff(f, y, 2)

ans =2*sin(y)^2 - 2*cos(y)^2

You get the same result by taking derivative twice: diff(diff(f,y)). To take mixed derivatives, use two differentiationcommands. For example:

syms x yf = sin(x)^2 + cos(y)^2;diff(diff(f, y), x)

ans =0

Indefinite Integrals of One-Variable Expressions

Suppose you want to integrate a symbolic expression. The firststep is to create the symbolic expression:

syms xf = sin(x)^2;

To find the indefinite integral, enter

int(f)

ans =x/2 - sin(2*x)/4

Indefinite Integrals of Multivariable Expressions

If the expression depends on multiple symbolic variables, youcan designate a variable of integration. If you do not specify anyvariable, MATLAB chooses a default variable by the proximityto the letter x:

syms x y nf = x^n + y^n;int(f)

ans =x*y^n + (x*x^n)/(n + 1)

For the complete set of rules MATLAB applies for choosinga default variable, see Find a Default Symbolic Variable.

You also can integrate the expression f = x^n + y^n withrespect to y

syms x y nf = x^n + y^n;int(f, y)

ans =x^n*y + (y*y^n)/(n + 1)

If the integration variable is n, enter

syms x y nf = x^n + y^n;int(f, n)

ans =x^n/log(x) + y^n/log(y)

Definite Integrals

To find a definite integral, pass the limits of integrationas the final two arguments of the int function:

syms x y nf = x^n + y^n;int(f, 1, 10)

ans =piecewise(n == -1, log(10) + 9/y, n ~= -1,... (10*10^n - 1)/(n + 1) + 9*y^n)

If MATLAB Cannot Find a Closed Form of an Integral

If the int function cannot compute an integral,it returns an unresolved integral:

syms xint(sin(sinh(x)))

ans =int(sin(sinh(x)), x)

Solve Algebraic Equations with One Symbolic Variable

Use the double equal sign (==) to define an equation. Then youcan solve the equation bycalling the solve function. For example, solve this equation:

syms xsolve(x^3 - 6*x^2 == 6 - 11*x)

ans = 1 2 3

If you do not specify the right side of the equation, solve assumesthat it is zero:

syms xsolve(x^3 - 6*x^2 + 11*x - 6)

ans = 1 2 3

Solve Algebraic Equations with Several Symbolic Variables

If an equation contains several symbolic variables, you canspecify a variable for which this equation should be solved. For example,solve this multivariable equation with respect to y:

syms x ysolve(6*x^2 - 6*x^2*y + x*y^2 - x*y + y^3 - y^2 == 0, y)

ans = 1 2*x -3*x

If you do not specify any variable, you get the solution ofan equation for the alphabetically closest to x variable.For the complete set of rules MATLAB applies for choosing a defaultvariable see Find a Default Symbolic Variable.

Solve Systems of Algebraic Equations

You also can solve systems of equations. For example:

syms x y z[x, y, z] = solve(z == 4*x, x == y, z == x^2 + y^2)

x = 0 2 y = 0 2 z = 0 8

Substitute Symbolic Variables with Numbers

You can substitute a symbolic variable with a numeric value by using the subs function. For example, evaluate the symbolic expression f at the point x=1/3:

syms xf = 2*x^2 - 3*x + 1;subs(f, 1/3)

ans =2/9

The subs function does not change the originalexpression f:

f

f =2*x^2 - 3*x + 1

Substitute in Multivariate Expressions

When your expression contains more than one variable, you canspecify the variable for which you want to make the substitution.For example, to substitute the value x=3 in the symbolic expression

syms x yf = x^2*y + 5*x*sqrt(y);

enter the command

subs(f, x, 3)

ans =9*y + 15*y^(1/2)

Substitute One Symbolic Variable for Another

You also can substitute one symbolic variable for another symbolicvariable. For example to replace the variable y withthe variable x, enter

subs(f, y, x)

ans =x^3 + 5*x^(3/2)

Substitute a Matrix into a Polynomial

You can also substitute a matrix into a symbolic polynomialwith numeric coefficients. There are two ways to substitute a matrixinto a polynomial: element by element and according to matrix multiplicationrules.

Element-by-Element Substitution.To substitute a matrix at each element, use the subs command:

syms xf = x^3 - 15*x^2 - 24*x + 350;A = [1 2 3; 4 5 6];subs(f,A)

ans =[ 312, 250, 170][ 78, -20, -118]

You can do element-by-element substitution for rectangular orsquare matrices.

Substitution in a Matrix Sense.If you want to substitute a matrix into a polynomial using standardmatrix multiplication rules, a matrix must be square. For example,you can substitute the magic square A into a polynomial f:

Substitute the Elements of a Symbolic Matrix

To substitute a set of elements in a symbolic matrix, also usethe subs command. Suppose you want to replace someof the elements of a symbolic circulant matrix A

syms a b cA = [a b c; c a b; b c a]

A =[ a, b, c][ c, a, b][ b, c, a]

To replace the (2, 1) element of A with beta andthe variable b throughout the matrix with variable alpha,enter

alpha = sym('alpha');beta = sym('beta');A(2,1) = beta;A = subs(A,b,alpha)

The result is the matrix:

A =[ a, alpha, c][ beta, a, alpha][ alpha, c, a]

For more information, see Substitute Elements in Symbolic Matrices.

Explicit Function Plot

Create a 2-D line plot by using fplot. Plot the expression $$x^3-6x^2+11x-6$$.

syms xf = x^3 - 6*x^2 + 11*x - 6;fplot(f)

Add labels for the x- and y-axes. Generate the title by using texlabel(f). Show the grid by using grid on. For details, see Add Title and Axis Labels to Chart.

xlabel('x')ylabel('y')title(texlabel(f))grid on

Implicit Function Plot

Plot equations and implicit functions using fimplicit.

Plot the equation $$(x^2+y^2{)}^4=(x^2-y^2{)}^2$$ over $$-1<x<1$$.

syms x yeqn = (x^2 + y^2)^4 == (x^2 - y^2)^2;fimplicit(eqn, [-1 1])

3-D Plot

Plot 3-D parametric lines by using fplot3.

Plot the parametric line

$$\begin{array}{c}x=t^2\sin (10t)\\ y=t^2\cos (10t)\\ z=t.\end{array}$$

syms tfplot3(t^2*sin(10*t), t^2*cos(10*t), t)

Create Surface Plot

Create a 3-D surface by using fsurf.

Plot the paraboloid $$z=x^2+y^2$$.

syms x yfsurf(x^2 + y^2)