I am trying to solve the following system. Firstly I have a table of numerical values I will call numbers
numbers={{8.03077, 7.8435, 0.155633}, {8.02983, 7.90858, 0.155633}, {8.02489,
7.94988, 0.155633}, {8.0169, 7.97541, 0.155633}, {8.00632, 7.99063,
0.155633}, {7.99326, 7.99916, 0.155633}, {7.9777, 8.00329,
0.155633}, {7.95953, 8.00449, 0.155633}, {7.93859, 8.00365,
0.155633}}
Now I have to solve a system $$\frac{\cos x}{x}=a$$ and $$\frac{\cos y}{y}=b$$ and $$R(y-x)=c$$ where $a,b,c$ are the numerical values in the table above. A is first, b is second and c is third. Basically this is my formula:
Table[FindRoot[{Cos[x]/x == numbers[[j, 1]],
Cos[y]/y == numbers[[j, 2]],
R*(-x + y) ==
numbers[[j, 3]]}, {{x, .01}, {y, .05}, {R, .1}}][[1, 2]], {j, 1,
Length[numbers], 1}];
Now if you run this code, you will find an error saying:
FindRoot::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations. >>
I assume I have to change the starting points of x,y or R but I tried many different combinations yet nothing seems to work. So my question here is: What can I do? Is there a nice mathematical way to determine the optimum starting points? Is there a way to see which parameter is good and which one is absolutely horrible?