Sturm/secats/ewton/consecutel_improve

%Sturm's method

clc; clear; close all;

f = @(x) x.^3 - 5*x + 4;
df = @(x) 3*x.^2 - 5;
x = -4:0.01:4;

F = [1, 0, -5, 4];
DF = [3, 0, -5];

L = length(F);

figure(1)
plot(x,f(x)); hold on; grid on;
plot(roots(F),f(roots(F)),'ro'); hold off;

a=0;
b=4;

c_a = 0;           %counter_a
c_b = 0;           %counter b

ax_1 = f(a);
bx_1 = f(b);

ax_2 = df(a);
bx_2 = df(b);

if ax_1*ax_2 < 0
    c_a = c_a + 1;
end

if bx_1*bx_2 < 0
    c_b = c_b + 1;
end

ax_1 = ax_2;
bx_1 = bx_2;

for n=1:(L-2)

    [q,r] = deconv(F, DF)
    L_x = length(F)-length(DF);
    r2 = -1.*r(end-L_x:end);
    ax_2 = polyval(r2, a);
    bx_2 = polyval(r2, b);

    if ax_1*ax_2 < 0
        c_a = c_a + 1;
    end

    if bx_1*bx_2 < 0
        c_b = c_b + 1;
    end

    ax_1 = ax_2;
    bx_1 = bx_2;

    F = DF;
    DF = r2;

end

How_many_roots = c_a - c_b






















````````````````````````````````
%secant method 1

clc; clear; close all;

dx = 0.01;
x = -10:dx:10;

f = @(x) x.^2-4;
y=f(x);
k = 100;            %number of repetitions
q = 5;              %starting position
a = 1;              %frst limit
b = 5;              %second limit


for n=1:k
    if f(q) == 0
        display("yep, that's the one")
        break
    elseif (f(a)*f(q))<0
        b = q;
        q = a - ((f(a)*(b-a))/(f(b)-f(a)));
    else
        a = q;
        q = b - ((f(b)*(b-a))/(f(b)-f(a)));
    end

    if abs(0-f(q))<0.000000001
        q
        break
    end
end























``````````````````````````````````````````

%secant method 2

clc; clear; close all;

dx = 0.01;
x = -10:dx:10;

f = @(x) x.^2-4;

k = 100;            %number of repetitions
a = 1;             %frst limit
b = 5;              %second limit

gate = 0;

%coefficients=polyfit([0,1],[0,2],1)
%roots(coefficients)
for n=1:k
        cf = polyfit([a,b], [f(a),f(b)], 1);
        x0 = roots(cf);
        if gate == 0
            cf2 = polyfit([x0, b], [f(x0),f(b)], 1);
            a = roots(cf2);
            gate = 1;
            fin = a;
        else
            cf2 = polyfit([a, x0], [f(a),f(x0)], 1);
            b = roots(cf2);
            gate = 0;
            fin = b;
        end
    if abs(0 - f(fin))<0.0000000001
        fin
        break
    end
end










`````````````````````````````````````````````````````````````
%secant method 3

clc; clear; close all;

dx = 0.01;
x = -100000:dx:100000;

f = @(x) x.^3-11*x.^2+14*x+80;

k = 1;            %number of repetitions
a = -6;             %frst limit
b = 10;              %second limit

c = (a+b)/2

for n=1:k

f_ex = (f(c)+(sign(f(b))*sqrt((f(c)^2)-(f(a)*f(b)))))/f(b)
xa = f(a)
xb = f(b)*(f_ex^2)
xc = f(c)*f_ex
x1 = xc + (xc - xa)*(f(xc)*sign(f(xa)-f(xb)))/nthroot((f(xc)^2)-(f(xa)*f(xb)),2)
b = x1;
c = (a+b)/2;
if abs(0-f(b))<0.0000001
    b
    break
end
b
end





















``````````````````````````````````````````
%Newton method

clc; clear; close all;

dx = 0.01;
x = -10:dx:10;
x2 = 1:dx:10;

f = @(x) x.^2 - 4;
df1 = @(x) 2*x;
df2 = @(x) 2+0*x;

k = 50;            %number of repetitions
a = 1;              %frst limit
b = 10;              %second limit

for n=1:k
    if f(a)*df2(a) > 0
        x1 = a - (f(a)/df1(a));
        a = x1;
    else
        x1 = b - (f(b)/df1(b));
        b = x1;
    end

    if abs(0-f(x1))<0.000000000001
        x1;
        break
    end
end

figure(1)
plot(x2,f(x2)); hold on; grid on;
plot(x1,f(x1),'ro'); hold off;



























````````````````````````````````
%consecutively improved approximation

clc; clear; close all;

%f = exp(x) - (1/x)
f2 = @(x) x - (1/4)*(exp(x)-(1/x))
k = 1/4

x=3/4;

A=zeros(1,20);
n=length(A);

for k=1:n
    A(k) = f2(x);
    x=f2(x)
end;



figure(1);
plot(A)