I tried using this command but found it cannot deal with small enough numbers for my purposes. It returns "FAIL". I am trying to write a program that uses Cauchy's gradient method/ method of steepest ascent/descent to find local maxima and minima. Basically I find the gradient of a 3D function, restrict the domain to the line in the direction of the gradient, and travel along that line until I reach the first maximum (or minimum). That's what I need something like NextZero for.

However, I need at least 10 decimal places of accuracy. NextZero doesn't work so well when I'm close to the extremum and the next zero is occuring very close to 0.

For example, this is a function it chokes on: 0.1165096616e-4+1.12290098*10^(-10)*t-0.28086e-5*(-0.9827864558e-1-9.362*10^(-7)*t)^2+0.74930e-4*cos(1.714985063+0.7493e-5*t)*cos(0.9827864558e-1+9.362*10^(-7)*t)-0.93620e-5*sin(1.714985063+0.7493e-5*t)*sin(0.9827864558e-1+9.362*10^(-7)*t)

A graph of the derivative of this function clearly shows that it has a root somewhere between t=.12 and .13. But NextZero cannot find this zero. Perhaps its numerical methods are too coarse.

Also, unfortunately, I am not always dealing with polynomials. Much of the Root Finding package seems to have been written exclusively for polynomials. It would help me greatly if it were extended to other types of functions.

Are there any other ideas?

The problem is that the formula for a Taylor series centered at a is sum(f^n(a)/n!*(x-a)^n,n=0..infinity), plugging in a definite value [i]n[/i] would give me the [i]nth[/i] degree Taylor polynomial, but I am interested in an infinite Taylor series and thus require an abritrary formula for the [i]nth[/i] derivative of the function.