Python/matrix/nth_fibonacci_using_matrix_exponentiation.py

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"""
Implementation of finding nth fibonacci number using matrix exponentiation.
Time Complexity is about O(log(n)*8), where 8 is the complexity of matrix
multiplication of size 2 by 2.
And on the other hand complexity of bruteforce solution is O(n).
As we know
f[n] = f[n-1] + f[n-1]
Converting to matrix,
[f(n),f(n-1)] = [[1,1],[1,0]] * [f(n-1),f(n-2)]
-> [f(n),f(n-1)] = [[1,1],[1,0]]^2 * [f(n-2),f(n-3)]
...
...
-> [f(n),f(n-1)] = [[1,1],[1,0]]^(n-1) * [f(1),f(0)]
So we just need the n times multiplication of the matrix [1,1],[1,0]].
We can decrease the n times multiplication by following the divide and conquer approach.
"""
2019-10-05 05:14:13 +00:00
def multiply(matrix_a: list[list[int]], matrix_b: list[list[int]]) -> list[list[int]]:
matrix_c = []
n = len(matrix_a)
for i in range(n):
list_1 = []
for j in range(n):
val = 0
for k in range(n):
val = val + matrix_a[i][k] * matrix_b[k][j]
list_1.append(val)
matrix_c.append(list_1)
return matrix_c
def identity(n: int) -> list[list[int]]:
return [[int(row == column) for column in range(n)] for row in range(n)]
def nth_fibonacci_matrix(n: int) -> int:
"""
>>> nth_fibonacci_matrix(100)
354224848179261915075
>>> nth_fibonacci_matrix(-100)
-100
"""
if n <= 1:
return n
res_matrix = identity(2)
fibonacci_matrix = [[1, 1], [1, 0]]
n = n - 1
while n > 0:
if n % 2 == 1:
res_matrix = multiply(res_matrix, fibonacci_matrix)
fibonacci_matrix = multiply(fibonacci_matrix, fibonacci_matrix)
n = int(n / 2)
return res_matrix[0][0]
def nth_fibonacci_bruteforce(n: int) -> int:
"""
>>> nth_fibonacci_bruteforce(100)
354224848179261915075
>>> nth_fibonacci_bruteforce(-100)
-100
"""
if n <= 1:
return n
fib0 = 0
fib1 = 1
Add flake8 pluin flake8 bugbear to pre-commit (#7132) * ci(pre-commit): Add ``flake8-builtins`` additional dependency to ``pre-commit`` (#7104) * refactor: Fix ``flake8-builtins`` (#7104) * fix(lru_cache): Fix naming conventions in docstrings (#7104) * ci(pre-commit): Order additional dependencies alphabetically (#7104) * fix(lfu_cache): Correct function name in docstring (#7104) * Update strings/snake_case_to_camel_pascal_case.py Co-authored-by: Christian Clauss <cclauss@me.com> * Update data_structures/stacks/next_greater_element.py Co-authored-by: Christian Clauss <cclauss@me.com> * Update digital_image_processing/index_calculation.py Co-authored-by: Christian Clauss <cclauss@me.com> * Update graphs/prim.py Co-authored-by: Christian Clauss <cclauss@me.com> * Update hashes/djb2.py Co-authored-by: Christian Clauss <cclauss@me.com> * refactor: Rename `_builtin` to `builtin_` ( #7104) * fix: Rename all instances (#7104) * refactor: Update variable names (#7104) * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * ci: Create ``tox.ini`` and ignore ``A003`` (#7123) * revert: Remove function name changes (#7104) * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci * Rename tox.ini to .flake8 * Update data_structures/heap/heap.py Co-authored-by: Dhruv Manilawala <dhruvmanila@gmail.com> * refactor: Rename `next_` to `next_item` (#7104) * ci(pre-commit): Add `flake8` plugin `flake8-bugbear` (#7127) * refactor: Follow `flake8-bugbear` plugin (#7127) * fix: Correct `knapsack` code (#7127) * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci Co-authored-by: Christian Clauss <cclauss@me.com> Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com> Co-authored-by: Dhruv Manilawala <dhruvmanila@gmail.com>
2022-10-13 16:03:06 +00:00
for _ in range(2, n + 1):
fib0, fib1 = fib1, fib0 + fib1
return fib1
def main() -> None:
for ordinal in "0th 1st 2nd 3rd 10th 100th 1000th".split():
n = int("".join(c for c in ordinal if c in "0123456789")) # 1000th --> 1000
print(
f"{ordinal} fibonacci number using matrix exponentiation is "
f"{nth_fibonacci_matrix(n)} and using bruteforce is "
f"{nth_fibonacci_bruteforce(n)}\n"
)
# from timeit import timeit
# print(timeit("nth_fibonacci_matrix(1000000)",
# "from main import nth_fibonacci_matrix", number=5))
# print(timeit("nth_fibonacci_bruteforce(1000000)",
# "from main import nth_fibonacci_bruteforce", number=5))
# 2.3342058970001744
# 57.256506615000035
if __name__ == "__main__":
import doctest
doctest.testmod()
main()