MATLAB Short Course 2: Symbolic manipulation, graphics
Contents
Define the symbolic variables
syms x syms y z
limit, integration, differentiation
limit((1-cos(x))/x^2,x,0)
int(1/sqrt(1+x^2),x)
diff(1/sqrt(1+x^2),x)
diff(1/sqrt(1+x^2),x,2) % last 2 means taking derivatives twice
diff(1/sqrt(1+x^2),x,3)
ans = 1/2 ans = asinh(x) ans = -x/(x^2 + 1)^(3/2) ans = (3*x^2)/(x^2 + 1)^(5/2) - 1/(x^2 + 1)^(3/2) ans = (9*x)/(x^2 + 1)^(5/2) - (15*x^3)/(x^2 + 1)^(7/2)
Taylor expansion
taylor(cos(x),x,0)
taylor(cos(x),x,0,'Order',10)
ans = x^4/24 - x^2/2 + 1 ans = x^8/40320 - x^6/720 + x^4/24 - x^2/2 + 1
solve polynomial equations
syms x p q solve(x^2+p*x+q==0,x) % notice the double equal sign "=="
ans = - p/2 - (p^2 - 4*q)^(1/2)/2 (p^2 - 4*q)^(1/2)/2 - p/2
the result may not always what you expect
roots of the cubic equation
syms p q x solve(x^3+p*x+q==0,x)
ans = root(z^3 + p*z + q, z, 1) root(z^3 + p*z + q, z, 2) root(z^3 + p*z + q, z, 3)
get numbers from symbolic solutions via "double"
solve(x^3+x+1==0,x)
double(ans) % ans is the last (named or un-named) result
ans = root(z^3 + z + 1, z, 1) root(z^3 + z + 1, z, 2) root(z^3 + z + 1, z, 3) ans = -0.6823 + 0.0000i 0.3412 - 1.1615i 0.3412 + 1.1615i
symbolic sum/product
syms k n symsum(k,k,1,n) symsum(1/n^2,n,1,inf) symprod((n^2-1)/n^2,n,2,inf)
ans = (n*(n + 1))/2 ans = pi^2/6 ans = 1/2
MATLAB plot
the key is to generate the data points
x = -10:0.1:10; y = sin(x); plot(x,y) % change the plot style plot(x,y,'*') plot(x,y,'-*') % polar plot t = linspace(0,2*pi,100); % same as: t = 0:0.01:2*pi; rho = sin(t).*cos(2*t); % note the use of "." pointwise product polar(t,rho)
plot surfaces
x = linspace(-1,1,11);
y = x;
[X, Y] = meshgrid(x,y); % have a look at both X and Y
Z = 1-X.^2-Y.^2;
surf(x,y,Z)
surface on a denser mesh
x = linspace(-1,1,51);
y = x;
[X, Y] = meshgrid(x,y);
Z = 1-X.^2-Y.^2;
mesh(x,y,Z) % or surf(x,y,Z)
Contour for the heart shaped curve
x = linspace(-2,2,101); y = x; [X, Y] = meshgrid(x,y); Z = (X.^2+Y.^2-1).^3-X.^2.*Y.^3; contour(Z,[0 0])
