Finding the intersection points between two curves

I need to find the intersection points of this equation:

this is my code. I dont know what is the best way . Also when I plot in terms of I have some periods but when I plot in terms of β I have only one period. How I can plot it in terms of beta but with nulti perids?

many thanks in advance

h=20;
K=13.6;
l=1148e-6;
g1=fplot(@(beta)tan(beta*l));
hold on 
g2=fplot(@(beta)2*h*beta*K/(K^2*beta^2-h^2));

Warning: Function behaves unexpectedly on array inputs. To improve performance, properly vectorize your function to return an output with the same size and shape as the input arguments.

Answer 1

[Star Strider]

Several options —

h=20;
K=13.6;
l=1148e-6;
f1 = @(beta)tan(beta*l);
f2 = @(beta)2*h*beta*K./(K^2*beta.^2-h^2);

figure
g1 = fplot(f1);
hold on 
g2 = fplot(f2);
g3 = fplot(@(beta) f2(beta)-f1(beta), '--');                            % Use 'fsolve' On 'g3' To Find The Intersections

Another option:

format long
Xint = interp1(g3.YData, g3.XData, 0)

Xint = 
     0

.

[raha ahmadi]

Dear Star Strider As always very thanks. Is there any way to get more answers in the other periods? I need multiple answers With best wishes

[Star Strider]

There do not appear to be any other intersections —

h=20;
K=13.6;
l=1148e-6;
f1 = @(beta)tan(beta*l);
f2 = @(beta)2*h*beta*K./(K^2*beta.^2-h^2);

figure
g1 = fplot(f1);
hold on 
g2 = fplot(f2);
g3 = fplot(@(beta) f2(beta)-f1(beta), [-50 50], '--');                            % Use 'fsolve' On 'g3' To Find The Intersections
grid

format long
Xint = interp1(g3.YData, g3.XData, 0)

Xint = 
     0

If there sere other intersections, the easiest way to find them would be:

zxi = find(diff(sign(g3.YData)))

zxi = 1×4

99 170 171 242

And to my surprise, it seems that there are more roots!

So to get more precise results:

for k = 1:numel(zxi)
    idxrng = (-1:1)+zxi(k);                                         % Index Range For Interpolation
    rootval(k) = interp1(g3.YData(idxrng), g3.XData(idxrng), 0);
end
rootval 

rootval = 1×4

-1.472819271901502 0 0 1.464295806398758

The slope of ‘g1’ prevents the roots from being symmetrical about the origin.

.

[Star Strider]

My pleasure!

If you want to calculate those roots —

h=20;
K=13.6;
l=1148e-6;
g1=fplot(@(betaL)tan(betaL));
hold on 
g2=fplot(@(betaL)2*h.*betaL/l*K./(h^2-K^2.*betaL/l.^2));
hold on
f3 = @(betaL) (tan(betaL)) - (2*h.*betaL/l*K./(h^2-K^2.*betaL/l.^2));
g3 = fplot(f3, [-1 1]*20, ':');
zxi = find(diff(sign(g3.YData)));
for k = 1:numel(zxi)
    idxrng = (-1:1)+zxi(k);                                         % Index Range For Interpolation
    xv(k,:) = interp1(g3.YData(idxrng), g3.XData(idxrng), 0);
    yv(k,:) = interp1(g3.XData, g3.YData, xv(k));
end
plot(xv(abs(yv)<1), zeros(size(xv(abs(yv)<1))), 'xr')
hold off

Intersections = table(xv(abs(yv)<1),yv(abs(yv)<1), 'VariableNames',{'x','y'})

Intersections = 14×2 table
       x            y     
    _______    ___________

    -18.853      3.747e-16
    -15.711    -1.4554e-15
    -12.569     9.7145e-17
     -9.428      2.498e-16
    -6.2861     6.5919e-17
    -3.1452     7.4593e-17
          0              0
          0              0
     3.1386    -1.8735e-16
     6.2802     3.1225e-16
     9.4214    -9.1593e-16
     12.563    -9.5063e-16
     15.704    -5.1348e-16
     18.846    -8.3267e-16

.

[raha ahmadi]

Dear Star Strider

I really thank you for your help. I learned alot from your code. It is very helpful for me

the period of the tangent function in this case is relatively large so I m confused. Because in each period of tan function I have a intersection point.

Wish you all the best

[Star Strider]

As always, my pleasure!

Thank you for your compliment!

I calculated the intersections. I do not understand.

.

[raha ahmadi]

Hi Star Strider sorry about the delay in response, I did not see your comment. I think I know the reason. In the first picture which you and I plotted the blue graph is the tangent function (in spite of what we expect). In this picture the period is very long and if you extend the axes limits you can see the difference between scales. But in the last picture you sent, by changing the scale of argument of tangent made the period of tangent very short.(remember that in the first picture scaling factor is about 1e-6 and this make period very long) I attached the two pictures for convenience

[Star Strider]

O.K.

I do not understand the problem or how this relates to the tangent function.

figure
hfp = fplot(@(x) 548*x./(187.7*x.^2-400), [-1 1]*30);
hold on
rootx = interp1(hfp.YData, hfp.XData, 0)

rootx = 0

plot(rootx, 0, 'xr', 'MarkerSize',7.5)
hold off
grid
legend('$f(x)=\frac{548\ x}{187.7\ x^2-400}$', 'Intercept', 'Interpreter','latex', 'Location','best')

The one root is easy enough to calculate.

.

[raha ahmadi]

Hi. very sorry for delay in response. Maybe it relates to roots function. root can find the roots of the polynomials. if you use it for sin for example you only get one root too. How can we fix it?

clc
clear 
close all
x=[-1000,1000];
y=sin (x)
roots(y)

[Star Strider]

There is nothing to fix.

Look at the results:

x=[-1000,1000]

x = 1×2

-1000 1000

y=sin (x)

y = 1×2

-0.8269 0.8269

roots(y)

ans = 1

so, plotting ‘y’ using polyval:

xv = linspace(-10,10);
pv = polyval(y, xv);

figure
plot(xv, pv)
grid

The reason is readily apparent! The ‘y’ vector corresponds to:

This is a linear relationship with one zero-crossing (root) at .

However,

x = -1000:1000;
y = sin (x);
roots(y)

ans = 
   1.1126 + 0.0000i
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  -0.1633 - 0.9866i
  -0.9851 + 0.1718i
  -0.9851 - 0.1718i
   0.1649 + 0.9863i
   0.1649 - 0.9863i
   0.1618 + 0.9868i
   0.1618 - 0.9868i
  -0.9857 + 0.1688i
  -0.9857 - 0.1688i
  -0.9862 + 0.1657i
  -0.9862 - 0.1657i
   0.9997 + 0.0254i
   0.9997 - 0.0254i
   0.9990 + 0.0443i
   0.9990 - 0.0443i
   0.9992 + 0.0412i
   0.9992 - 0.0412i
   0.9996 + 0.0285i
   0.9996 - 0.0285i
  -0.1602 + 0.9871i
  -0.1602 - 0.9871i
   0.1587 + 0.9873i
   0.1587 - 0.9873i
  -0.9867 + 0.1626i
  -0.9867 - 0.1626i
  -0.9872 + 0.1595i
  -0.9872 - 0.1595i
  -0.9877 + 0.1564i
  -0.9877 - 0.1564i
   0.1556 + 0.9878i
   0.1556 - 0.9878i
  -0.1571 + 0.9876i
  -0.1571 - 0.9876i
   0.1525 + 0.9883i
   0.1525 - 0.9883i
  -0.9882 + 0.1533i
  -0.9882 - 0.1533i
  -0.1540 + 0.9881i
  -0.1540 - 0.9881i
   0.1494 + 0.9888i
   0.1494 - 0.9888i
  -0.1509 + 0.9886i
  -0.1509 - 0.9886i
  -0.1478 + 0.9890i
  -0.1478 - 0.9890i
   0.1463 + 0.9892i
   0.1463 - 0.9892i
   0.1432 + 0.9897i
   0.1432 - 0.9897i
  -0.1447 + 0.9895i
  -0.1447 - 0.9895i
  -0.9887 + 0.1502i
  -0.9887 - 0.1502i
   0.9995 + 0.0317i
   0.9995 - 0.0317i
   0.9993 + 0.0380i
   0.9993 - 0.0380i
   0.1401 + 0.9901i
   0.1401 - 0.9901i
  -0.1416 + 0.9899i
  -0.1416 - 0.9899i
   0.1370 + 0.9906i
   0.1370 - 0.9906i
  -0.1385 + 0.9904i
  -0.1385 - 0.9904i
  -0.9891 + 0.1470i
  -0.9891 - 0.1470i
   0.9976 + 0.0696i
   0.9976 - 0.0696i
   0.1339 + 0.9910i
   0.1339 - 0.9910i
   0.1308 + 0.9914i
   0.1308 - 0.9914i
  -0.1353 + 0.9908i
  -0.1353 - 0.9908i
  -0.9896 + 0.1439i
  -0.9896 - 0.1439i
  -0.9900 + 0.1408i
  -0.9900 - 0.1408i
  -0.1322 + 0.9912i
  -0.1322 - 0.9912i
   0.1277 + 0.9918i
   0.1277 - 0.9918i
  -0.9905 + 0.1377i
  -0.9905 - 0.1377i
  -0.1291 + 0.9916i
  -0.1291 - 0.9916i
  -0.9909 + 0.1346i
  -0.9909 - 0.1346i
   0.1246 + 0.9922i
   0.1246 - 0.9922i
  -0.1260 + 0.9920i
  -0.1260 - 0.9920i
  -0.1229 + 0.9924i
  -0.1229 - 0.9924i
  -0.9913 + 0.1315i
  -0.9913 - 0.1315i
  -0.9917 + 0.1284i
  -0.9917 - 0.1284i
   0.1214 + 0.9926i
   0.1214 - 0.9926i
   0.1183 + 0.9930i
   0.1183 - 0.9930i
   0.1152 + 0.9933i
   0.1152 - 0.9933i
  -0.1198 + 0.9928i
  -0.1198 - 0.9928i
  -0.1167 + 0.9932i
  -0.1167 - 0.9932i
  -0.1135 + 0.9935i
  -0.1135 - 0.9935i
   0.9994 + 0.0348i
   0.9994 - 0.0348i
   0.1121 + 0.9937i
   0.1121 - 0.9937i
   0.1090 + 0.9940i
   0.1090 - 0.9940i
   0.1058 + 0.9944i
   0.1058 - 0.9944i
  -0.1104 + 0.9939i
  -0.1104 - 0.9939i
   0.1027 + 0.9947i
   0.1027 - 0.9947i
  -0.1073 + 0.9942i
  -0.1073 - 0.9942i
  -0.1042 + 0.9946i
  -0.1042 - 0.9946i
  -0.1010 + 0.9949i
  -0.1010 - 0.9949i
  -0.9921 + 0.1253i
  -0.9921 - 0.1253i
  -0.9925 + 0.1222i
  -0.9925 - 0.1222i
  -0.9929 + 0.1190i
  -0.9929 - 0.1190i
  -0.9933 + 0.1159i
  -0.9933 - 0.1159i
  -0.0979 + 0.9952i
  -0.0979 - 0.9952i
  -0.9936 + 0.1128i
  -0.9936 - 0.1128i
  -0.9940 + 0.1097i
  -0.9940 - 0.1097i
  -0.9943 + 0.1066i
  -0.9943 - 0.1066i
  -0.9946 + 0.1034i
  -0.9946 - 0.1034i
   0.0996 + 0.9950i
   0.0996 - 0.9950i
   0.0965 + 0.9953i
   0.0965 - 0.9953i
   0.0933 + 0.9956i
   0.0933 - 0.9956i
  -0.0917 + 0.9958i
  -0.0917 - 0.9958i
  -0.0948 + 0.9955i
  -0.0948 - 0.9955i
  -0.9950 + 0.1003i
  -0.9950 - 0.1003i
  -0.9953 + 0.0972i
  -0.9953 - 0.0972i
   0.0902 + 0.9959i
   0.0902 - 0.9959i
  -0.0885 + 0.9961i
  -0.0885 - 0.9961i
   0.0871 + 0.9962i
   0.0871 - 0.9962i
   0.0777 + 0.9970i
   0.0777 - 0.9970i
   0.0808 + 0.9967i
   0.0808 - 0.9967i
   0.0840 + 0.9965i
   0.0840 - 0.9965i
  -0.9956 + 0.0941i
  -0.9956 - 0.0941i
  -0.9959 + 0.0909i
  -0.9959 - 0.0909i
  -0.9961 + 0.0878i
  -0.9961 - 0.0878i
  -1.0000 + 0.0000i
  -0.9964 + 0.0847i
  -0.9964 - 0.0847i
  -0.9967 + 0.0816i
  -0.9967 - 0.0816i
  -0.9999 + 0.0126i
  -0.9999 - 0.0126i
  -1.0000 + 0.0094i
  -1.0000 - 0.0094i
  -1.0000 + 0.0031i
  -1.0000 - 0.0031i
  -0.9982 + 0.0596i
  -0.9982 - 0.0596i
  -0.9978 + 0.0659i
  -0.9978 - 0.0659i
  -0.9974 + 0.0722i
  -0.9974 - 0.0722i
  -0.9980 + 0.0628i
  -0.9980 - 0.0628i
  -1.0000 + 0.0063i
  -1.0000 - 0.0063i
  -0.9976 + 0.0690i
  -0.9976 - 0.0690i
  -0.9969 + 0.0784i
  -0.9969 - 0.0784i
  -0.9972 + 0.0753i
  -0.9972 - 0.0753i
  -0.9984 + 0.0565i
  -0.9984 - 0.0565i
  -0.9999 + 0.0157i
  -0.9999 - 0.0157i
  -0.9992 + 0.0408i
  -0.9992 - 0.0408i
  -0.9990 + 0.0439i
  -0.9990 - 0.0439i
  -0.9993 + 0.0377i
  -0.9993 - 0.0377i
  -0.9986 + 0.0534i
  -0.9986 - 0.0534i
  -0.9994 + 0.0345i
  -0.9994 - 0.0345i
  -0.9989 + 0.0471i
  -0.9989 - 0.0471i
  -0.9995 + 0.0314i
  -0.9995 - 0.0314i
  -0.9987 + 0.0502i
  -0.9987 - 0.0502i
  -0.9996 + 0.0283i
  -0.9996 - 0.0283i
  -0.9997 + 0.0251i
  -0.9997 - 0.0251i
  -0.9998 + 0.0188i
  -0.9998 - 0.0188i
  -0.9998 + 0.0220i
  -0.9998 - 0.0220i
   0.0746 + 0.9972i
   0.0746 - 0.9972i
   0.0714 + 0.9974i
   0.0714 - 0.9974i
   0.0683 + 0.9977i
   0.0683 - 0.9977i
   0.0652 + 0.9979i
   0.0652 - 0.9979i
  -0.0854 + 0.9963i
  -0.0854 - 0.9963i
  -0.0823 + 0.9966i
  -0.0823 - 0.9966i
  -0.0792 + 0.9969i
  -0.0792 - 0.9969i
  -0.0760 + 0.9971i
  -0.0760 - 0.9971i
  -0.0698 + 0.9976i
  -0.0698 - 0.9976i
  -0.0729 + 0.9973i
  -0.0729 - 0.9973i
   0.0620 + 0.9981i
   0.0620 - 0.9981i
   0.0589 + 0.9983i
   0.0589 - 0.9983i
  -0.0666 + 0.9978i
  -0.0666 - 0.9978i
  -0.0635 + 0.9980i
  -0.0635 - 0.9980i
   0.0495 + 0.9988i
   0.0495 - 0.9988i
   0.0526 + 0.9986i
   0.0526 - 0.9986i
   0.0464 + 0.9989i
   0.0464 - 0.9989i
   0.0432 + 0.9991i
   0.0432 - 0.9991i
   0.0558 + 0.9984i
   0.0558 - 0.9984i
   0.0401 + 0.9992i
   0.0401 - 0.9992i
  -0.0604 + 0.9982i
  -0.0604 - 0.9982i
  -0.0541 + 0.9985i
  -0.0541 - 0.9985i
  -0.0572 + 0.9984i
  -0.0572 - 0.9984i
  -0.0510 + 0.9987i
  -0.0510 - 0.9987i
  -0.0478 + 0.9989i
  -0.0478 - 0.9989i
  -0.0415 + 0.9991i
  -0.0415 - 0.9991i
  -0.0447 + 0.9990i
  -0.0447 - 0.9990i
   0.0369 + 0.9993i
   0.0369 - 0.9993i
  -0.0384 + 0.9993i
  -0.0384 - 0.9993i
   0.0338 + 0.9994i
   0.0338 - 0.9994i
  -0.0353 + 0.9994i
  -0.0353 - 0.9994i
  -0.0259 + 0.9997i
  -0.0259 - 0.9997i
  -0.0290 + 0.9996i
  -0.0290 - 0.9996i
  -0.0227 + 0.9997i
  -0.0227 - 0.9997i
  -0.0321 + 0.9995i
  -0.0321 - 0.9995i
   0.0307 + 0.9995i
   0.0307 - 0.9995i
   0.0212 + 0.9998i
   0.0212 - 0.9998i
   0.0181 + 0.9998i
   0.0181 - 0.9998i
   0.0244 + 0.9997i
   0.0244 - 0.9997i
   0.0150 + 0.9999i
   0.0150 - 0.9999i
   0.0024 + 1.0000i
   0.0024 - 1.0000i
  -0.0007 + 1.0000i
  -0.0007 - 1.0000i
  -0.0196 + 0.9998i
  -0.0196 - 0.9998i
   0.0275 + 0.9996i
   0.0275 - 0.9996i
   0.0087 + 1.0000i
   0.0087 - 1.0000i
   0.0055 + 1.0000i
   0.0055 - 1.0000i
  -0.0102 + 0.9999i
  -0.0102 - 0.9999i
  -0.0039 + 1.0000i
  -0.0039 - 1.0000i
  -0.0070 + 1.0000i
  -0.0070 - 1.0000i
  -0.0133 + 0.9999i
  -0.0133 - 0.9999i
  -0.0164 + 0.9999i
  -0.0164 - 0.9999i
   0.0118 + 0.9999i
   0.0118 - 0.9999i
   0.8988 + 0.0000i

Except for the first element (1.1126), the absolute values of all the others are uniformly 1.

This time, ‘y’ corresponds to:

syms x 
yp = vpa(poly2sym(y,x),5)

yp = 

figure
fplot(yp, [-1000, 1000])
grid
axis([-pi  pi    -5  5])

The vertical dashed lines indicate singularities .

The sin function has an infiinty of roots, those being radians, where n is an integer. It is necessary to use a zero-finding algorithm (such as fzero or fsolve) or interpolation (interp1) to locate them, not the roots function.

.

[raha ahmadi]

your answer is perfect. I m really appresiate.

hope you all the best

[Star Strider]

Thank you!

As always, my pleasure!

.

Answer 2

[darova]

I'd try fsolve for solving

Read help carefully:

fplot(f,[0 10])

[raha ahmadi]

Dear darova

Thank you for your help, I read fsolve but I think it solves a system of nonlinear equations. I used fzero instead but I only got one answer I need solve it in some periods and get more roots

Best regards