On Mar 4, 2006, at 4:33 PM, Paul Rubin wrote: > "sam" <[EMAIL PROTECTED]> writes: >> Hello all, I am taking a class in scientific programming at the local >> college. My problem is that the following difference produces >> round off >> errors as the value of x increases. For x >= 19 the diference goes to >> zero.I understand the problem, but am curious as to whether their >> exists a solution. [f(x) = cosh^2(x) - sinh^2(x) = 1] > > The usual way is to notice that the difference goes to zero because > the lowest order terms of the Taylor series for cosh^2 and sinh^2 are > equal and cancel each other out. The solution is to write down the > series for (cosh^2(x) - sinh^2(x)) and add up a few non-cancelled > terms. > All these series should converge very fast.
Here's my analysis: First, keep in mind that the range of values for a Taylor expansion of sinh() and cosh() is limited. Then... Write down the series for (cosh^2(x) - sinh^2(x)), as Paul suggested. c2 = taylor(cosh(x)^2,x,8) s2 = taylor(sinh(x)^2,x,8) The first method gives the expected answer: eval(c2-s2) ==> 1 Write down the series for cosh(x), square that; write the series for sinh(x), square that and subtract. Compare the two (I just did this in Eigenmath). Higher-order terms remain for the second method.. I don't know how many terms my cosh() and sinh() functions use, but if I evaluate (again, in Eigenmath) The second method, which is more akin to what Python is doing (unless there is an explicit 'sinh**2(x)' function) is different. c2 = (taylor(cosh(x),x,8))^2, which gives terms up to and including (x**8)**2 = x**16 s2 = (taylor(sinh(x),x,9))^2, which gives terms up to and including x**18, then subtract the two, the result is eval(c2 - s2) ==> 1 - 1/1814400 * x**10 - ... higher terms. This begs the question, "What is the algorithm used by my Python?" And, the ever-important question, "Does it matter in my final result?" This is important, because I do not, in general, trust the result when subtracting two large numbers. :--David T. -- http://mail.python.org/mailman/listinfo/python-list
