how to find the root of transcedental equation while loop

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shiv gaur
shiv gaur 2022-2-17
评论： shiv gaur 2022-2-17
k0=(2*pi/0.6328)*1e6;
t2=1.5e-6;
n1=1.512;n2=1.521;n3=4.1-1i*0.211;
n4=1;
m=0;
t3=1e-9;
k1=k0*sqrt(n1^2-x^2);
k2=k0*sqrt(n2^2-x^2);
k3=k0*sqrt(n3^2-x^2);
k4=k0*sqrt(n4^2-x^2);
tol = 1e-12;
n = 1;
y=-(k2)*t2+atan(k1/1i*k2)+atan((k3/k2)*tan(atan(k4/1i*k2)-k3*t3))+m*pi;
how to calculate the root of equation i.e the vaue of x numerically by while loop

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回答（1 个）

Walter Roberson
Walter Roberson 2022-2-17
Ran in:
format long g
k0=(2*pi/0.6328)*1e6;
t2=1.5e-6;
n1=1.512;n2=1.521;n3=4.1-1i*0.211;
n4=1;
m=0;
t3=1e-9;
tol = 1e-12;
n = 1;
x = rand
x =
0.599324733046148
y = inf;
while n < 5 & abs(y) > tol
k1=k0*sqrt(n1^2-x^2);
k2=k0*sqrt(n2^2-x^2);
k3=k0*sqrt(n3^2-x^2);
k4=k0*sqrt(n4^2-x^2);
y=-(k2)*t2+atan(k1/1i*k2)+atan((k3/k2)*tan(atan(k4/1i*k2)-k3*t3))+m*pi;
[n, x, y]
n = n + 1;
end
ans =
1 + 0i 0.599324733046148 + 0i -17.6929697847717 + 7.90103155130006e-07i
ans =
2 + 0i 0.599324733046148 + 0i -17.6929697847717 + 7.90103155130006e-07i
ans =
3 + 0i 0.599324733046148 + 0i -17.6929697847717 + 7.90103155130006e-07i
ans =
4 + 0i 0.599324733046148 + 0i -17.6929697847717 + 7.90103155130006e-07i
All of the results are the same because you have not defined any way for x to change.
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shiv gaur
shiv gaur 2022-2-17
from muller method ans is different why is so
here is program
function kps3
T3 = 1e-9:1e-9:1e-6;
for j=1:numel(T3)
t3 = T3(j);
p0 = 0.5;
p1 = 1;
p2 = 1.5;
TOL = 10^-8;
N0 = 100; format long
h1 = p1 - p0;
h2 = p2 - p1;
DELTA1 = (f(p1,t3) - f(p0,t3))/h1;
DELTA2 = (f(p2,t3) - f(p1,t3))/h2;
d = (DELTA2 - DELTA1)/(h2 + h1);
i=3;
while i <= N0
b = DELTA2 + h2*d;
D = (b^2 - 4*f(p2,t3)*d)^(1/2);
if abs(b-D) < abs(b+D)
E = b + D;
else
E = b - D;
end
h = -2*f(p2,t3)/E;
p = p2 + h;
if abs(h) < TOL
%disp(p)
break
end
p0 = p1;
p1 = p2;
p2 = p;
h1 = p1 - p0;
h2 = p2 - p1;
DELTA1 = (f(p1,t3) - f(p0,t3))/h1;
DELTA2 = (f(p2,t3) - f(p1,t3))/h2;
d = (DELTA2 - DELTA1)/(h2 + h1);
i=i+1;
end
if i > N0
formatSpec = string('The method failed after N0 iterations,N0= %d \n');
fprintf(formatSpec,N0);
end
P(j)=real(p);
end
plot(T3,P)
end
function y=f(x, t3)
k0=(2*pi/0.6328)*1e6;
t2=1.5e-6;
n1=1.512;n2=1.521;n3=4.1-%i*0.211;
n4=1;
m=0;
m=0;
k1=k0*sqrt(n1^2-x^2);
k2=k0*sqrt(n2^2-x^2);
k3=k0*sqrt(n3^2-x^2);
k4=k0*sqrt(n4^2-x^2);
y=(k2)*t2-atan(k1/1i*k2)-atan((k3/k2)*tan(atan(k4/1i*k2)-k3*t3))+m*pi;
end
why results are so different using while loop pl sort out the problem using this while loop
shiv gaur
shiv gaur 2022-2-17
I think root will be same for both program use while loop or muller

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