Quote:
Originally Posted by Don Blazys
To: Modest,
Well, Newtons method can be unreliable if the zero is too near an asymptote or extremum
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Yes, but it did successfully converge.
Quote:
Originally Posted by Don Blazys
and that derivative looks far too complicated and involved for such a simple function!
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I agree with you 100%. It, nevertheless, worked.
Quote:
Originally Posted by Don Blazys
I am now working on finding a simpler and more reliable derivative,
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Glad to hear it. If I were more practiced in my calculus I'd be doing the same.
Quote:
Originally Posted by Don Blazys
because my calculation of the second root or "Prime generating constant" is:
2.566,543,832,171,7....
and thats using nothing but the "graph" function of a hand held calculator!
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Your hand-held is off by 0.0000000000007255... It falls about half-way between your result and my result.
Quote:
Originally Posted by Don Blazys
so it is clear that more reliable calculations are in order.
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No, not really. You've missed the essential part. I'm not relying on the unwieldy derivative above to check the zero. I'm doing that directly. I only used the derivative to help me find the zero computationally. If you want to check my value, you can plug it directly into your equation and see how close it is to zero. I'm sure you already know this being you're much better at this sort of thing than I am. But, what you may not know is that there are many calculators available besides your hand-held that will do this for you. For example, you can use google.
Do a google search typing in these two strings exactly:
((sin((2.566543832171388844467529)^(1/2)))^(-1)-1)^(-1) / ((pi)^2 + (ln(ln(2*((2* ((2.566543832171388844467529)^(-1) +1))^(-1) +1))))^(-1))-1
((sin((2.566543832172425504475)^(1/2)))^(-1)-1)^(-1) / ((pi)^2 + (ln(ln(2*((2* ((2.566543832172425504475)^(-1) +1))^(-1) +1))))^(-1))-1
Google calculator will solve these, the first is your x and the second is mine. The results are linked here:
Google calculator with my x
Google calculator with your x
Google solves my approximated value out to
and your approximated value to
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Google has greater precision than your hand-held, but not as good as PERL, yet it agrees my value is closer to the root. Others can check the result, but it looks significantly different from the value needed to generate primes with the given method.
~modest
