Numpy matrix exponentiation gives negative value

Question

I wanted to use NumPy in a Fibonacci question because of its efficiency in matrix multiplication. You know that there is a method for finding Fibonacci numbers with the matrix [[1, 1], [1, 0]].

I wrote some very simple code but after increasing n, the matrix is starting to give negative numbers.

import numpy
def fib(n):
    return (numpy.matrix("1 1; 1 0")**n).item(1)

print fib(90)
# Gives -1581614984

What could be the reason for this?

Note: linalg.matrix_power also gives negative values.

Note2: I tried numbers from 0 to 100. It starts to give negative values after 47. Is it a large integer issue because NumPy is coded in C ? If so, how could I solve this ?

Edit: Using regular python list matrix with linalg.matrix_power also gave negative results. Also let me add that not all results are negative after 47, it occurs randomly.

Edit2: I tried using the method @AlbertoGarcia-Raboso suggested. It resolved the negative number problem, however another issues occured. It gives the answer as -5.168070885485832e+19 where I need -51680708854858323072L. So I tried using int(), it converted it to L, but now it seems the answer is incorrect because of a loss in precision.

Answer 1

The reason you see negative values appearing is because NumPy has defaulted to using the np.int32 dtype for your matrix.

The maximum positive integer this dtype can represent is 2³¹-1 which is 2147483647. Unfortunately, this is less the 47th Fibonacci number, 2971215073. The resulting overflow is causing the negative number to appear:

>>> np.int32(2971215073)
-1323752223

Using a bigger integer type (like np.int64) would fix this, but only temporarily: you'd still run into problems if you kept on asking for larger and larger Fibonacci numbers.

The only sure fix is to use an unlimited-size integer type, such as Python's int type. To do this, modify your matrix to be of np.object type:

def fib_2(n):
    return (np.matrix("1 1; 1 0", dtype=np.object)**n).item(1)

The np.object type allows a matrix or array to hold any mix of native Python types. Essentially, instead of holding machine types, the matrix is now behaving like a Python list and simply consists of pointers to integer objects in memory. Python integers will be used in the calculation of the Fibonacci numbers now and overflow is not an issue.

>>> fib_2(300)
222232244629420445529739893461909967206666939096499764990979600

This flexibility comes at the cost of decreased performance: NumPy's speed originates from direct storage of integer/float types which can be manipulated by your hardware.