Introduction to @-functions in "standard" Matlab

You should always start with this:

clearvars; % clear all variables in the work space

Standard Matlab (as opposed to "symbolic Matlab") uses machine numbers with about 16 digit accuracy. Therefore it gives approximate results (not exact results).

Using a function g(x)

Example: Define the function

g = @(x) x*sin(x);

Find

y = g(3)
y =          0.423360024179602

Matlab shows by default only 5 of 16 digits. We can see 15 digits by first using format long g

format long g
y
y =          0.423360024179602

If you perform several operations, not all of the displayed 15 digits will be correct (roundoff errors accumulate).

Plotting a function using fplot

Plot the function g for

fplot(g,[0,20])
Warning: Function behaves unexpectedly on array inputs. To improve performance, properly vectorize your function to return an output with the same size and shape as the input arguments.

You can ignore this warning. Everything is working fine. You can get rid of this type of warning by using warning('off','MATLAB:fplot:NotVectorized')

xlabel('x') % label x-axis

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

grid on % show grid

Definite integral

For

use integral(g,a,b,'Arr',true) (with 'Arr',true this works even if we define g with * / ^ instead of .* ./ .^ )

I = integral(g,0,8,'Arr',true)
I =           2.15335851709229

Finding a solution of a nonlinear equation or

We use xs=fzero(g,x0) where x0 is an initial guess.

In our example there are infinitely many solutions.

Find a solution x near 3:

xs = fzero(g,3)
xs =           3.14159265358979

Find x such that

, using initial guess 7:

Here we need to find a zero of the function

xs = fzero( @(x)g(x)-5 , 7 )
xs =           7.06889140339507

Using a function G(x,y)

We can also use functions of more than one variable.

Example: Define the function

G = @(x,y) x^4+y^4-4*(x^2+y^2)+4;

Evaluate

z = G(1,2)
z =      1

Plot the graph of

for

as a surface over the

-plane:

warning('off','MATLAB:fplot:NotVectorized') % get rid of the NotVectorized warning

fsurf(G,[-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

for

fcontour(G,[-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(G,[x1,x2,y1,y2],'LevelStep',dz):

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

axis equal;

title('points with G(x,y)=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(G,[x1,x2,y1,y2],'LevelList',[0]) or fimplicit(G,[x1,x2,y1,y2]):

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

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