Introduction to symbolic expressions in Matlab (Symbolic Math Toolbox)

You should always start with this:

clearvars; % clear all variables in the work space

The Symbolic Math Toolbox allows to define symbolic expressions in terms of symbolic variables such as t,x,y. All results are exact symbolic results.

If you type

format long g                      % show 15 digits of numerical results
[ sin(pi), sin(pi/3), sin(pi/7) ]
ans = 1×3
      1.22464679914735e-16         0.866025403784439         0.433883739117558

Matlab uses a numerical approximation with 16 digits accuracy for π, and then computes numerical results.

We can define

Pi = sym(pi)       % gives symbolic version of pi
Pi = π

now

[ sin(Pi), sin(Pi/3), sin(Pi/7) ]

ans =

gives symbolic results. We can then evaluate this using vpa (uses by default 32 digits):

vpa(ans)

ans =

Note that we have to use sym(...) for integers to get symbolic results:

[ 1/3, exp(1) ]

ans = 1×2

         0.333333333333333          2.71828182845905

[ 1/sym(3) , exp(sym(1)) ]

ans =

Defining a symbolic expression g depending on x

Example: Define

syms x              % define x as symbolic variable
v = x*sin(x);

Note that v is not a function, but just denotes the expression

Find the derivative

diff(v,x)

ans =

Find the fifth derivative

diff(v,x,5)

ans =

Find an antiderivative:

int(v,x)

ans =

Use int(v,x,a,b) to find the definite integral

int(v,x,0,Pi/3) % we defined Pi = sym(pi)

ans =

Find

, i.e., in v plug in

y = subs(v,x,3)

y =

Find numerical value using vpa

vpa(y) finds a numerical value. By default this shows 32 digits, you can change this with digits(n)

vpa(y)
ans = 0.42336002417960166630223440842433

Plot the symbolic expression v using fplot

Plot the function g for

fplot(v,[0,20])

xlabel('x') % label x-axis

title('function g(x) = x\cdot sin x') % title on top of graph

grid on % show grid

Find a solution x of the nonlinear equation or

In our example there are infinitely many solutions.

solx = solve(v,x)
solx = 0

This returns 0 which is definitely one solution.

We can try to find more solutions:

[solx,param,conditions] = solve(v,x,'ReturnConditions',true)

solx =

param =

conditions =

This shows that 0 and

are solutions, where

can be arbitrary. (Showing 0 separately is redundant as it is included in

with

)

This only works in simple equations.

Let us try to solve

sol = solve(v==5,x)

Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve.

sol =

This is indeed an approximation of a solution. But we may be interested in the solution near

. With vpasolve we can specify an initial guess:

vpasolve(v==5,x,7)
ans = 7.0688914033950668006875995619168

Using an expression v of two variables

We can also use functions of more than one variable.

Example: Define the function

syms x y                   % declare symbolic variables x,y
v = x^4+y^4-4*(x^2+y^2)+4;

We can find partial derivatives:

Find

diff(v,x)

ans =

Find

diff(v,x,x,y)
ans = 0

Evaluate

, i.e. in v plug in

subs(v,[x,y],[1,2])
ans = 1

Plot the graph of v for

as a surface over the

-plane:

fsurf(v,[-2 2 -2 2])

xlabel('x'); ylabel('y'); title('function x^4+y^4-4(x^2+y^2)+4')

Make a contour plot of v for

fcontour(v,[-2 2 -2 2])

colorbar % show colorbar on the right which explains colors of contours

axis equal % use the same unit length for x and y

xlabel('x'); ylabel('y'); title('contour plot of x^4+y^4-4(x^2+y^2)+4')

Here Matlab chooses the contour levels automatically. If you want more or fewer contours you can specify the spacing of the contours by using fcontour(v,[x1,x2,y1,y2],'LevelStep',dz):

fcontour(v,[-2 2 -2 2],'LevelStep',0.5)

colorbar; axis equal

If you only want contours for a few specific z-values you can use

fcontour(v,[-2.1 2.1 -2.1 2.1],'LevelList',[0 1])

axis equal;

title('points with v=0 (blue) and G(x,y)=1 (yellow)')

The blue points satisfy

, the yellow points satisfy

If you only want to see the points with

you can use fcontour(v,[x1,x2,y1,y2],'LevelList',[0]) or fimplicit(v,[x1,x2,y1,y2]):

fimplicit(v,[-2 2 -2 2]);

axis equal; xlabel('x'); ylabel('y'); title('points satisfying x^4+y^4-4(x^2+y^2)+4=0')

NOTE: The plotting commands fplot, fsurf, fcontour, fimplicit work for both Matlab @-functions and symbolic expressions.