# List of ways to highly accurately solving a mixed equation

1. Apr 2, 2015

### Philosophaie

I would like a list of ways highly accurate to solve the equation:

E - e * sin(E) = M

I can solve this with the Newton Method:

M = 2*pi/3
e = 0.002
E = pi
d = 0.01
Do While SQRT(d^2) > 0.000001
d= (E - e * sin(E) - M) / (1 - e * cos(E))
E = E + d
Loop

This is not very accurate.

Is there some sort of trigonometric expansion or infinite series that would be more accurate?

Last edited: Apr 2, 2015
2. Apr 2, 2015

### Staff: Mentor

This is very puzzling. Are you saying that Newton's method is not very accurate, or that it doesn't converge fast enough for you? Either way, your equation for implementing it doesn't look correct to me. Shouldn't it be E = E - d, not E = E + d? Also, your initial guess for these parameter values would be better chosen to be E = M.

Chet

3. Apr 2, 2015

### Staff: Mentor

There are methods that converge faster than Newton - you can take the second derivative into account, for example. But they are not more accurate - the limit is always exactly the solution to the equation (or the limits of your floating point precision).

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