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Orthogonal
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Recurrence Relationship Approach in Matlab
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# Orthogonal
This is the public folder for the "Orthogonal Polynomials" team.
\ No newline at end of file
This is the public folder for the "Orthogonal Polynomials" team.
Reccurance Relationship Approach in Matlab ( Based on previous work done by Aflredo in Maple)
a=0;
b=0;
N = 50;
s= 1i;
digits = 100;
t = s
*
N;
r = [];
R = [];
r(1)= 0;
R(1)=((a+b+1)/2)
*(exp(-2*
t)/exp(-t));
for n = 1:N+1
r(n+1)=((1/2
*t)*
(4
*R(n)^2+2*
R(n)
*(2*
t-2
*(n-1)-1-a-b)-2*
a
*
t)-r(n));
R(n+1)=(r(n+1)
*t*
(a+r(n+1))
*(t+R(n))) /(((-r(n+1)^2)*
t+((-2
*(n-1)-b-a-2)*
R(n)-a
*t))*
r(n+1)+R(n)
*(n)*
(n+b));
end
aa(1) = (1/t)
*(a+b+1-t-2*
R(1));
bb(1)=(1/(t
*(t+R(1))))*
(-(a+b)
*r(1)-(t/R(1)*
((r(1)^2)+a
*
r(1))));
for n = 2:N+1
aa(n) = (1/t)
*(2*
(n-1) + a+b + 1-t-2
*
R(n));
bb(n) = (1/(t
*(t+R(n)))) *
((n-1)
*(n-1+b)-(2*
(n-1)+a+b)
*r(n)-(t/R(n)*
((r(n)^2)+a
*
r(n))));
end
x = 1:length(aa);
plot(x, imag(aa), "go")
grid
J(1,1) = aa(1);
J(1,2) = 1;
for k = 2:N-1
J(k,k-1) = bb(k);
J(k,k) = aa(k);
J(k,k+1) = 1;
end
J(N,N-1) = bb(N);
J(N,N) = aa(N);
v = eig(J);
imv = imag(v);
relv = real(v);
q = 1:length(v);
plot(imv, relv, "bo")
grid
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