Analytical Plotting with Symbolic Math Toolbox
The fplot family accepts symbolic expressions and equations as inputs enabling easy analytical plotting without explicitly generating numerical data.
This example features the following functions
fplotfplot3fsurffcontourfmeshfimplicitfimplicit3
Interactively plot functions of one variable
syms x
fplot(sin(exp(x)))
fplot([sin(x),cos(x),tan(x)])
Generate an implicit plot of a symbolic expression
syms x y r = 1:10; fimplicit(x^2+y^2 == r)
Parametrically explore multiple functions with subs
syms x a expression = sin(exp(x/a))
expression =
fplot(subs(expression,a,[1,2,4])) hold off legend show
Mix Symbolic and Numeric techniques to develop mathematical models
Explore the spline approximation to
syms f(x) f(x) = x*exp(-x)*sin(5*x) -2; xs = 0:1/3:3; ys = double(subs(f,xs)); fplot(f,[0,3]) hold on plot(xs,ys,'*k','DisplayName','Data Points') fplot(@(x) spline(xs,ys,x),[0 3],'DisplayName','Spline interpolant') hold off grid on legend show
Explore Gibbs Phenomenon
syms x n = 5; approx = cumsum(sin((1:2:2*n-1)*x)./(1:2:2*n-1)); fplot(approx,'LineWidth',1)
Plotting the results of computations
With symbolic input, we can perform computations and plot the results.
syms f(a,x)
assume(a>0);
f(a,x) = a*x^2+a^2*x+2*sqrt(a)f(a, x) =
x_min = solve(diff(f,x), x)
x_min =
fplot(f(a,x_min),[0 5]) xlabel 'a' title 'Minimum value of f depending on a'
assume(a,'clear')Visualize Series and Summations
syms x t6 = taylor(cos(x),x,'Order',6)
t6 =
t8 = taylor(cos(x),x,'Order',8)t8 =
fplot([cos(x) t6 t8]) xlim([-4 4]) ylim([-1.5, 1.5]) title '6th and 8th Order Taylor Series approximations to cos(x)' legend show
Explore functions along with their integrals and derivatives
Some symbolic expressions cannot be converted to MATLAB functions.
syms x
f = x^xf =
int(f,x)
ans =
diff(f,x)
ans =
fplot([f, int(f,x), diff(f,x)],[0 2])
legend show
Generate parametric curves without explicit numerical data
Curves or can be drawn by fplot or fplot3 (just like with plot or plot3 for numerical data) :
syms t fplot3(sin(t)-t/2,cos(t),t^3,'--','LineWidth',2.5) view([-45 45])
Generate surfaces without meshgrid
syms x y fsurf(sin(x)+sin(y)-(x^2+y^2)/20,'ShowContours','on') set(camlight,'Color',[0.5 0.5 1]); set(camlight('left'),'Color', [1 0.6 0.6]); set(camlight('left'),'Color', [1 0.6 0.6]); set(camlight('right'),'Color', [0.8 0.8 0.6]); material shiny view(-19,56)
Generate numeric streamlines from analytical derivatives using meshgrid
syms x y u = diff(diff(sin(x^2+y^2)))
u =
v = diff(diff(cos(x^2+y^2)))
v =
[X, Y] = meshgrid(-3:.1:3,-2:.1:2);
U = subs(u, [x y], {X,Y});
V = subs(v, [x y], {X,Y});
startx = -3:0.1:3;
starty = zeros(size(startx));
h = streamline(X,Y,U,V,X,Y);
for i=1:length(h)-1
h(i).Color = [rand() rand() rand()];
end
Adaptive visualization
Like fplot, fsurf evaluates your symbolic expression more densely where needed, to more accurately show curved areas and asymptotic regions.
fsurf(log(x) + exp(y), [-2 2])
Create Implicit surfaces
Plot the implicit surface . Specify an output to make fimplicit3 return the plot object.
syms x y z f = 1/x^2 - 1/y^2 + 1/z^2; fimplicit3(f)
Visualize multivariate surfaces
Unlike pure symbolic functions (like int, diff, solve), fsurf does not allow specifying the order of variables. To set the order, use symbolic functions:
syms f(t) x(u,v) y(u,v) z(u,v) f(t) = sin(t)*exp(-t^2/3)+1.5; x(u,v) = u
x(u, v) =
y(u,v) = f(u)*sin(v)
y(u, v) =
z(u,v) = f(u)*cos(v)
z(u, v) =
fsurf(x,y,z,[-5 5.1 0 2*pi])
Use fmesh for 3D mesh plots
Plot the parameterized mesh
where
syms s t r = 8 + sin(7*s + 5*t); x = r*cos(s)*sin(t); y = r*sin(s)*sin(t); z = r*cos(t); fmesh(x, y, z, [0 2*pi 0 pi], 'Linewidth', 2) axis equal
Generate a contour plot of a symbolic expression or equation
syms x y g(x,y) g(x,y) = x^3-4*x-y^2
g(x, y) =
fcontour(g,[-3 3 -4 4],'LevelList',-6:6) title 'Some Elliptic Curves'
