ferrari/derivative_aprox/derivative/cardano

%ferrari

clc; clear; close all;

f = [1, 2, 3, 4]; %we supose that "a" is 1
f2 = [1, 1, 2, 3, 4]; %f1 but with "a"

a3=f(1); a2=f(2); a1=f(3); a0=f(4);

B(1) = 1;           %we have to add this to be able to use cardano function
B(2) = -a2/2;
B(3) = (a3*a1 - 4*a0)/4;
B(4) = (4*a2*a0 - (a3^2)*a0 -a1^2)/8;
z = cardano(B)
k=z(3);
p = sqrt(2*k +(a3^2)/4 - a2);
q = (k*a3 - a1)/(2*p);
D1 = (a3/2 + p)^2 - 4*(k + q); %here we have some simple delta. surprising isnt it?
D2 = (a3/2 - p)^2 - 4*(k - q);
x1= ( -(a3/2 + p) - D1^(1/2) )/2;
x2 = ( -(a3/2 + p) + D1^(1/2) )/2;
x3 = ( -(a3/2 - p) - D2^(1/2) )/2;
x4 = ( -(a3/2 - p) + D2^(1/2) )/2;
X = [x1; x2; x3; x4;]

R = roots(f2)
figure(1);
plot(R,'o'); hold on; grid on;
plot(X,'*r'); hold off;

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%derivative approximation

clc; clear; close all;

dx =0.01;
x = -2:dx:2;

f = @(x) x.^2-3;
fx = [1,0,-3];

n = length(fx)-1; %polynomial order
df3 =(n:-1:1).*fx(1:end-1)

V = polyval(df3,x);

y=f(x);
df1 = (y(2:end)-y(1:end-1))/dx;
df2 = (df1(1:end-1)+df1(2:end))/2;

figure(1)
plot(x(1:end-2),df1(1:end-1)); hold on; grid on;
plot(x(1:end-2),df2,'b-');
plot(x(1:end-2),V(1:end-2),'o'); %diferences are here because of the shift




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%derivative

clc; clear; close all;
%f = x.^2+2*x+3
A = [1, 2, 3];
A = A(:).';
n = length(A)-1; %polynomial order
b = (n:-1:1).*A(1:end-1)




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%cardanos method

clc; clear; close all;


%f= @(x) 1*x.^3+ 2*x.^2 + 3*x + 4;

f = [1, 2, 3, 4];

a3=f(1); a2=f(2); a1=f(3); a0=f(4);
p = (3*a1-((a2)^2))/9;
q = ((a2^3)/27) - ((a1*a2)/6) + (a0/2);
D = q^2 + p^3

if D>= 0
   u = nthroot((-q + sqrt(D)),3);
   v = nthroot((-q - sqrt(D)),3);
   Y(1)= u+v;
   Y(2) = -0.5*(u+v) + ((j*sqrt(3))/2)*(u-v);
   Y(3) = -0.5*(u+v) - ((j*sqrt(3))/2)*(u-v);
else
    fi = acos(-q/sqrt(-p^3));
    Y(1) = 2* sqrt(-p) * cos ((fi)/3);
    Y(2) = 2* sqrt(-p) * cos ((fi + 2*pi)/3);
    Y(3) = 2* sqrt(-p) * cos ((fi + 4*pi)/3);
end

X(1) = Y(1) - a2/3;
X(2) = Y(2) - a2/3;
X(3) = Y(3) - a2/3;

R = roots(f)

figure(1);
plot(R,'o'); hold on; grid on;
plot(X,'*r'); hold off;












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function X = cardano(f)

a3=f(1); a2=f(2); a1=f(3); a0=f(4);
p = (3*a1-((a2)^2))/9;
q = ((a2^3)/27) - ((a1*a2)/6) + (a0/2);
D = q^2 + p^3;

if D>= 0
   u = nthroot((-q + sqrt(D)),3);
   v = nthroot((-q - sqrt(D)),3);
   Y(1)= u+v;
   Y(2) = -0.5*(u+v) + ((j*sqrt(3))/2)*(u-v);
   Y(3) = -0.5*(u+v) - ((j*sqrt(3))/2)*(u-v);
else
    fi = acos(-q/sqrt(-p^3));
    Y(1) = 2* sqrt(-p) * cos ((fi)/3);
    Y(2) = 2* sqrt(-p) * cos ((fi + 2*pi)/3);
    Y(3) = 2* sqrt(-p) * cos ((fi + 4*pi)/3);
end

X(1) = Y(1) - a2/3;
X(2) = Y(2) - a2/3;
X(3) = Y(3) - a2/3;
end