Reduce function is not showing all the roots of a transcendental equation

Question

I am using Mathematica 9. I have done one mathematica code to find the roots of a transcendental equation. The code is given below:

A = BesselJ[1, x]/BesselJ[0, x];
A1 = A*x;
nclm = 1.4271;
ncom = 1.4446;
lamda = 1.55;
a1 = 52.5;
V1 = 2*Pi*a1/lamda;
V = V1*Sqrt[ncom^2 - nclm^2];
w = Sqrt[V^2 - x^2];
B = BesselK[1, w]/BesselK[0, w];
B1 = B*w;
Plot[{A1, B1}, {x, 0, 50}]
Reduce[A1 - B1 == 0 && 0 < x < 50]

After simulate this code, I am getting the all outputs, but the Reduce function is not showing all the roots, Reduce function shows 13 number of roots, but the equation has 15 number of roots as seen from the intersection points of the plot.

As the equation has two more roots after x = 39.0978, but these two roots are not showing.

Is there any way to show that roots?

Answer 1

Use exact numbers (see Rationalize)

A = BesselJ[1, x]/BesselJ[0, x];
A1 = A*x;
nclm = 1.4271 // Rationalize[#, 0] &;
ncom = 1.4446 // Rationalize[#, 0] &;
lamda = 1.55 // Rationalize[#, 0] &;
a1 = 52.5 // Rationalize[#, 0] &;
V1 = 2*Pi*a1/lamda;
V = V1*Sqrt[ncom^2 - nclm^2];
w = Sqrt[V^2 - x^2];
B = BesselK[1, w]/BesselK[0, w];
B1 = B*w;

roots = {Reduce[A1 - B1 == 0 && 0 < x < 50] // N // ToRules}

(*  {{x -> 2.35543}, {x -> 5.40651}, {x -> 8.47513}, {x -> 11.5471}, 
     {x -> 14.6195}, {x -> 17.6912}, {x -> 20.7614}, {x -> 23.8295}, 
     {x -> 26.8946}, {x -> 29.9557}, {x -> 33.0116}, {x -> 36.0601}, 
     {x -> 39.0978}, {x -> 42.1179}, {x -> 45.1024}}  *)

Length[roots]

(*  15  *)

Plot[{A1, B1}, {x, 0, 50},
 Epilog -> {Green, AbsolutePointSize[6],
   Point[{x, A1} /. roots]}]

Answer 2

It is enough to increase WorkingPrecision:

Reduce[A1 - B1 == 0 && 0 < x < 50, WorkingPrecision -> 30]
x == 2.35543 || x == 5.40651 || x == 8.47513 || x == 11.5471 ||  x == 14.6195 || x == 17.6912 || x == 20.7614 || x == 23.8295 ||  x == 26.8946 || x == 29.9557 || x == 33.0116 || x == 36.0601 ||  x == 39.0978 || x == 42.1179 || x == 45.1024