# def type of mpmath matrices in cython

I'm trying to speed up the exact iterative calculation of the 1E9 th Fibonacci number, which has 208987640 digits. Working with Python and mpmath I've reached an acceptable time, but I'm looking to further reduce it using Cython. In this case the overload is due to the huge numbers, stored in Knuth's 2 x 2 matrices, that have been defined using Numpy and mpmath. For Cython to be helpful in this subject, I should define those matrices as static, but I don't know the type I should use to define it. Is there a special type for this case?

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The following works without any Cython in 26s (but I do not know what you mean with the "Knuth matrices")

sage: m = matrix(2, [1,1,1,0])
sage: all((m**i)[0,0] == fibonacci(i+1) for i in range(10))
True
sage: %time A = m **999999999
CPU times: user 25.5 s, sys: 688 ms, total: 26.2 s
Wall time: 26.2 s
sage: A[0,0].ndigits()
208987640


You can also check that the second columns are previous Fibonacci numbers:

sage: A[0,1] == A[1,0]
True
sage: A[1,0] + A[1,1] == A[0,0]
True


In the background the integer matrices in Sage uses flint which performs very well.

EDIT: if you only want the number of digits there is of course much faster

sage: s = RIF(5).sqrt()
sage: a = log(1/s) + 1000000000 * log((1+s)/2)
sage: b = a / RIF(10).log()
sage: b.unique_ceil()
208987640

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