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"""
Calculates the Fibonacci sequence using iteration, recursion, memoization,
and a simplified form of Binet's formula
NOTE 1: the iterative, recursive, memoization functions are more accurate than
the Binet's formula function because the Binet formula function uses floats
NOTE 2: the Binet's formula function is much more limited in the size of inputs
that it can handle due to the size limitations of Python floats
NOTE 3: the matrix function is the fastest and most memory efficient for large n
See benchmark numbers in __main__ for performance comparisons/
https://en.wikipedia.org/wiki/Fibonacci_number for more information
"""
import functools
from collections.abc import Iterator
from math import sqrt
from time import time
import numpy as np
from numpy import ndarray
def time_func(func, *args, **kwargs):
"""
Times the execution of a function with parameters
"""
start = time()
output = func(*args, **kwargs)
end = time()
if int(end - start) > 0:
print(f"{func.__name__} runtime: {(end - start):0.4f} s")
else:
print(f"{func.__name__} runtime: {(end - start) * 1000:0.4f} ms")
return output
def fib_iterative_yield(n: int) -> Iterator[int]:
"""
Calculates the first n (1-indexed) Fibonacci numbers using iteration with yield
>>> list(fib_iterative_yield(0))
[0]
>>> tuple(fib_iterative_yield(1))
(0, 1)
>>> tuple(fib_iterative_yield(5))
(0, 1, 1, 2, 3, 5)
>>> tuple(fib_iterative_yield(10))
(0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55)
>>> tuple(fib_iterative_yield(-1))
Traceback (most recent call last):
...
ValueError: n is negative
"""
if n < 0:
raise ValueError("n is negative")
a, b = 0, 1
yield a
for _ in range(n):
yield b
a, b = b, a + b
def fib_iterative(n: int) -> list[int]:
"""
Calculates the first n (0-indexed) Fibonacci numbers using iteration
>>> fib_iterative(0)
[0]
>>> fib_iterative(1)
[0, 1]
>>> fib_iterative(5)
[0, 1, 1, 2, 3, 5]
>>> fib_iterative(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
>>> fib_iterative(-1)
Traceback (most recent call last):
...
ValueError: n is negative
"""
if n < 0:
raise ValueError("n is negative")
if n == 0:
return [0]
fib = [0, 1]
for _ in range(n - 1):
fib.append(fib[-1] + fib[-2])
return fib
def fib_recursive(n: int) -> list[int]:
"""
Calculates the first n (0-indexed) Fibonacci numbers using recursion
>>> fib_recursive(0)
[0]
>>> fib_recursive(1)
[0, 1]
>>> fib_recursive(5)
[0, 1, 1, 2, 3, 5]
>>> fib_recursive(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
>>> fib_recursive(-1)
Traceback (most recent call last):
...
ValueError: n is negative
"""
def fib_recursive_term(i: int) -> int:
"""
Calculates the i-th (0-indexed) Fibonacci number using recursion
>>> fib_recursive_term(0)
0
>>> fib_recursive_term(1)
1
>>> fib_recursive_term(5)
5
>>> fib_recursive_term(10)
55
>>> fib_recursive_term(-1)
Traceback (most recent call last):
...
ValueError: n is negative
"""
if i < 0:
raise ValueError("n is negative")
if i < 2:
return i
return fib_recursive_term(i - 1) + fib_recursive_term(i - 2)
if n < 0:
raise ValueError("n is negative")
return [fib_recursive_term(i) for i in range(n + 1)]
def fib_recursive_cached(n: int) -> list[int]:
"""
Calculates the first n (0-indexed) Fibonacci numbers using recursion
>>> fib_recursive_cached(0)
[0]
>>> fib_recursive_cached(1)
[0, 1]
>>> fib_recursive_cached(5)
[0, 1, 1, 2, 3, 5]
>>> fib_recursive_cached(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
>>> fib_recursive_cached(-1)
Traceback (most recent call last):
...
ValueError: n is negative
"""
@functools.cache
def fib_recursive_term(i: int) -> int:
"""
Calculates the i-th (0-indexed) Fibonacci number using recursion
"""
if i < 0:
raise ValueError("n is negative")
if i < 2:
return i
return fib_recursive_term(i - 1) + fib_recursive_term(i - 2)
if n < 0:
raise ValueError("n is negative")
return [fib_recursive_term(i) for i in range(n + 1)]
def fib_memoization(n: int) -> list[int]:
"""
Calculates the first n (0-indexed) Fibonacci numbers using memoization
>>> fib_memoization(0)
[0]
>>> fib_memoization(1)
[0, 1]
>>> fib_memoization(5)
[0, 1, 1, 2, 3, 5]
>>> fib_memoization(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
>>> fib_memoization(-1)
Traceback (most recent call last):
...
ValueError: n is negative
"""
if n < 0:
raise ValueError("n is negative")
# Cache must be outside recursive function
# other it will reset every time it calls itself.
cache: dict[int, int] = {0: 0, 1: 1, 2: 1} # Prefilled cache
def rec_fn_memoized(num: int) -> int:
if num in cache:
return cache[num]
value = rec_fn_memoized(num - 1) + rec_fn_memoized(num - 2)
cache[num] = value
return value
return [rec_fn_memoized(i) for i in range(n + 1)]
def fib_binet(n: int) -> list[int]:
"""
Calculates the first n (0-indexed) Fibonacci numbers using a simplified form
of Binet's formula:
https://en.m.wikipedia.org/wiki/Fibonacci_number#Computation_by_rounding
NOTE 1: this function diverges from fib_iterative at around n = 71, likely
due to compounding floating-point arithmetic errors
NOTE 2: this function doesn't accept n >= 1475 because it overflows
thereafter due to the size limitations of Python floats
>>> fib_binet(0)
[0]
>>> fib_binet(1)
[0, 1]
>>> fib_binet(5)
[0, 1, 1, 2, 3, 5]
>>> fib_binet(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
>>> fib_binet(-1)
Traceback (most recent call last):
...
ValueError: n is negative
>>> fib_binet(1475)
Traceback (most recent call last):
...
ValueError: n is too large
"""
if n < 0:
raise ValueError("n is negative")
if n >= 1475:
raise ValueError("n is too large")
sqrt_5 = sqrt(5)
phi = (1 + sqrt_5) / 2
return [round(phi**i / sqrt_5) for i in range(n + 1)]
def matrix_pow_np(m: ndarray, power: int) -> ndarray:
"""
Raises a matrix to the power of 'power' using binary exponentiation.
Args:
m: Matrix as a numpy array.
power: The power to which the matrix is to be raised.
Returns:
The matrix raised to the power.
Raises:
ValueError: If power is negative.
>>> m = np.array([[1, 1], [1, 0]], dtype=int)
>>> matrix_pow_np(m, 0) # Identity matrix when raised to the power of 0
array([[1, 0],
[0, 1]])
>>> matrix_pow_np(m, 1) # Same matrix when raised to the power of 1
array([[1, 1],
[1, 0]])
>>> matrix_pow_np(m, 5)
array([[8, 5],
[5, 3]])
>>> matrix_pow_np(m, -1)
Traceback (most recent call last):
...
ValueError: power is negative
"""
result = np.array([[1, 0], [0, 1]], dtype=int) # Identity Matrix
base = m
if power < 0: # Negative power is not allowed
raise ValueError("power is negative")
while power:
if power % 2 == 1:
result = np.dot(result, base)
base = np.dot(base, base)
power //= 2
return result
def fib_matrix_np(n: int) -> int:
"""
Calculates the n-th Fibonacci number using matrix exponentiation.
https://www.nayuki.io/page/fast-fibonacci-algorithms#:~:text=
Summary:%20The%20two%20fast%20Fibonacci%20algorithms%20are%20matrix
Args:
n: Fibonacci sequence index
Returns:
The n-th Fibonacci number.
Raises:
ValueError: If n is negative.
>>> fib_matrix_np(0)
0
>>> fib_matrix_np(1)
1
>>> fib_matrix_np(5)
5
>>> fib_matrix_np(10)
55
>>> fib_matrix_np(-1)
Traceback (most recent call last):
...
ValueError: n is negative
"""
if n < 0:
raise ValueError("n is negative")
if n == 0:
return 0
m = np.array([[1, 1], [1, 0]], dtype=int)
result = matrix_pow_np(m, n - 1)
return int(result[0, 0])
if __name__ == "__main__":
from doctest import testmod
testmod()
# Time on an M1 MacBook Pro -- Fastest to slowest
num = 30
time_func(fib_iterative_yield, num) # 0.0012 ms
time_func(fib_iterative, num) # 0.0031 ms
time_func(fib_binet, num) # 0.0062 ms
time_func(fib_memoization, num) # 0.0100 ms
time_func(fib_recursive_cached, num) # 0.0153 ms
time_func(fib_recursive, num) # 257.0910 ms
time_func(fib_matrix_np, num) # 0.0000 ms
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