I'm iterating 4 million times (for a project). This is taking forever to do. I was wondering how I can go faster.
numbers = [0,1]
evenNumbers = []
y = 0
l = 0
for x in range (1,4000000):
l = numbers[x-1] + numbers[x]
numbers.append(l)
for k in numbers:
if k % 2 ==0:
evenNumbers.append(k)
for n in evenNumbers:
y += n
print(y)
This is going to be very slow regardless due to the how big the numbers are getting, but you can speed it up significantly by just not storing all the intermediate values:
m, n = 0, 1
y = 0
for _ in range(1, 4000000):
m, n = n, m + n
if n % 2 == 0:
y += n
print(y)
You should just compare the time it takes for each function here to complete, as they are the three different ways that most would approach iteration.
import time
def foreach(arr):
for i in range(len(arr)):
print(arr[i])
def forin(arr):
for i in arr:
print(i)
def whileloop(arr):
i = 0
while i < len(arr):
print(arr[i])
i += 1
def main():
arr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
start = time.time()
foreach(arr)
end = time.time()
print("foreach: ", end - start)
start = time.time()
forin(arr)
end = time.time()
print("forin: ", end - start)
start = time.time()
whileloop(arr)
end = time.time()
print("whileloop: ", end - start)
if __name__ == "__main__":
main()
Im not very sure if I will translate the assignment correctly, but the bottom line is that I need to consider the function f(i) = min dist(i, S), where S is number set of length k and dist(i, S) = sum(j <= S)(a (index i) - a(index j)), where a integer array.
I wrote the following code to accomplish this task:
n, k = (map(int, input().split()))
arr = list(map(int, input().split()))
sorted_arr = arr.copy()
sorted_arr.sort()
dists = []
returned = []
ss = 0
indexed = []
pop1 = None
pop2 = None
for i in arr:
index = sorted_arr.index(i)
index += indexed.count(i)
indexed.append(i)
dists = []
if (index == 0):
ss = sorted_arr[1:k+1]
elif (index == len(arr) - 1):
sorted_arr.reverse()
ss = sorted_arr[1:k+1]
else:
if index - k < 0:
pop1 = 0
elif index + k > n - 1:
pop2 = None
else:
pop1 = index - k
pop2 = index + k + 1
ss = sorted_arr[pop1:index] + sorted_arr[index + 1: pop2]
for ind in ss:
dists.append(int(abs(i - ind)))
dists.sort()
returned.append(str(sum(dists[:k])))
print(" ".join(returned))
But I need to speed up its execution time significantly.
I try to write an algorithm in python, and i dont understand why its out of range.I am realy dont understand where is the problem
This is my code
B = [1, 2, 3, 4]
n = 4
def b(m, i):
if m % 2 == 0 and m > 2:
if i < m - 1:
return 1
else:
return m - 2
else:
return m - 1
def PERM(m):
if m == 1:
print(P)
else:
for i in range(0, m):
PERM(m - 1)
if i < m:
k = P[m]
P[m] = P[b(m, i)]
P[b(m, i)] = k
PERM(n)
in for i in range(0, m):, the loop is iterting from 0 to 4 which is total of 5 index so better you change the n = 4 to n = 3, hope this help :)
In factoring methods based on the quadratic sieve, finding the left null space of a binary matrix (values computed mod 2) is a crucial step. (This is also the null space of the transpose.) Does numpy or scipy have tools to do this quickly?
For reference, here is my current code:
# Row-reduce binary matrix
def binary_rr(m):
rows, cols = m.shape
l = 0
for k in range(min(rows, cols)):
print(k)
if l >= cols: break
# Swap with pivot if m[k,l] is 0
if m[k,l] == 0:
found_pivot = False
while not found_pivot:
if l >= cols: break
for i in range(k+1, rows):
if m[i,l]:
m[[i,k]] = m[[k,i]] # Swap rows
found_pivot = True
break
if not found_pivot: l += 1
if l >= cols: break # No more rows
# For rows below pivot, subtract row
for i in range(k+1, rows):
if m[i,l]: m[i] ^= m[k]
l += 1
return m
It is pretty much a straightforward implementation of Gaussian elimination, but since it's written in python it is very slow.
qwr, I found a very fast gaussian elimination routine that finishes so qiuckly that the slow point is the Quadratic Sieving or SIQS Sieving step. The gaussian elimination functions were taken from skollmans factorise.py at https://raw.githubusercontent.com/skollmann/PyFactorise/master/factorise.py
I'll soon be working on a SIQS/GNFS implementation from scratch, and hope to write something super quick for python with multithreading and possiblly cython. In the meantime, if you want something that compiles C (Alpertons ECM Engine) but uses python, you can use: https://github.com/oppressionslayer/primalitytest/ which requires you to cd into calculators directory and run make before importing p2ecm with from sfactorint import p2ecm. With that you can factorise 60 digit numbers in a few seconds.
# Requires sympy and numpy to be installed
# Adjust B and I accordingly. Set for 32 length number
# Usage:
# N=1009732533765251*1896182711927299
# factorise(N, 5000, 25000000) # Takes about 45-60 seconds on a newer computer
# N=1009732533765251*581120948477
# Linear Algebra Step finishes in 1 second, if that
# N=factorise(N, 5000, 2500000) # Takes about 5 seconds on a newer computer
# #Out[1]: 581120948477
import math
import numpy as np
from sympy import isprime
#
# siqs_ functions are the Gaussian Elimination routines right from
# skollmans factorise.py. It is the fastest Gaussian Elimination that i have
# found in python
#
def siqs_factor_from_square(n, square_indices, smooth_relations):
"""Given one of the solutions returned by siqs_solve_matrix_opt,
return the factor f determined by f = gcd(a - b, n), where
a, b are calculated from the solution such that a*a = b*b (mod n).
Return f, a factor of n (possibly a trivial one).
"""
sqrt1, sqrt2 = siqs_calc_sqrts(square_indices, smooth_relations)
assert (sqrt1 * sqrt1) % n == (sqrt2 * sqrt2) % n
return math.gcd(abs(sqrt1 - sqrt2), n)
def siqs_find_factors(n, perfect_squares, smooth_relations):
"""Perform the last step of the Self-Initialising Quadratic Field.
Given the solutions returned by siqs_solve_matrix_opt, attempt to
identify a number of (not necessarily prime) factors of n, and
return them.
"""
factors = []
rem = n
non_prime_factors = set()
prime_factors = set()
for square_indices in perfect_squares:
fact = siqs_factor_from_square(n, square_indices, smooth_relations)
if fact != 1 and fact != rem:
if isprime(fact):
if fact not in prime_factors:
print ("SIQS: Prime factor found: %d" % fact)
prime_factors.add(fact)
while rem % fact == 0:
factors.append(fact)
rem //= fact
if rem == 1:
break
if isprime(rem):
factors.append(rem)
rem = 1
break
else:
if fact not in non_prime_factors:
print ("SIQS: Non-prime factor found: %d" % fact)
non_prime_factors.add(fact)
if rem != 1 and non_prime_factors:
non_prime_factors.add(rem)
for fact in sorted(siqs_find_more_factors_gcd(non_prime_factors)):
while fact != 1 and rem % fact == 0:
print ("SIQS: Prime factor found: %d" % fact)
factors.append(fact)
rem //= fact
if rem == 1 or sfactorint_isprime(rem):
break
if rem != 1:
factors.append(rem)
return factors
def add_column_opt(M_opt, tgt, src):
"""For a matrix produced by siqs_build_matrix_opt, add the column
src to the column target (mod 2).
"""
M_opt[tgt] ^= M_opt[src]
def find_pivot_column_opt(M_opt, j):
"""For a matrix produced by siqs_build_matrix_opt, return the row of
the first non-zero entry in column j, or None if no such row exists.
"""
if M_opt[j] == 0:
return None
return lars_last_powers_of_two_trailing(M_opt[j] + 1)
def siqs_build_matrix_opt(M):
"""Convert the given matrix M of 0s and 1s into a list of numbers m
that correspond to the columns of the matrix.
The j-th number encodes the j-th column of matrix M in binary:
The i-th bit of m[i] is equal to M[i][j].
"""
m = len(M[0])
cols_binary = [""] * m
for mi in M:
for j, mij in enumerate(mi):
cols_binary[j] += "1" if mij else "0"
return [int(cols_bin[::-1], 2) for cols_bin in cols_binary], len(M), m
def siqs_solve_matrix_opt(M_opt, n, m):
"""
Perform the linear algebra step of the SIQS. Perform fast
Gaussian elimination to determine pairs of perfect squares mod n.
Use the optimisations described in [1].
[1] Koç, Çetin K., and Sarath N. Arachchige. 'A Fast Algorithm for
Gaussian Elimination over GF (2) and its Implementation on the
GAPP.' Journal of Parallel and Distributed Computing 13.1
(1991): 118-122.
"""
row_is_marked = [False] * n
pivots = [-1] * m
for j in range(m):
i = find_pivot_column_opt(M_opt, j)
if i is not None:
pivots[j] = i
row_is_marked[i] = True
for k in range(m):
if k != j and (M_opt[k] >> i) & 1: # test M[i][k] == 1
add_column_opt(M_opt, k, j)
perf_squares = []
for i in range(n):
if not row_is_marked[i]:
perfect_sq_indices = [i]
for j in range(m):
if (M_opt[j] >> i) & 1: # test M[i][j] == 1
perfect_sq_indices.append(pivots[j])
perf_squares.append(perfect_sq_indices)
return perf_squares
def sqrt_int(N):
Nsqrt = math.isqrt(N)
assert Nsqrt * Nsqrt == N
return Nsqrt
def siqs_calc_sqrts(square_indices, smooth_relations):
"""Given on of the solutions returned by siqs_solve_matrix_opt and
the corresponding smooth relations, calculate the pair [a, b], such
that a^2 = b^2 (mod n).
"""
res = [1, 1]
for idx in square_indices:
res[0] *= smooth_relations[idx][0]
res[1] *= smooth_relations[idx][1]
res[1] = sqrt_int(res[1])
return res
def quad_residue(a,n):
l=1
q=(n-1)//2
x = q**l
if x==0:
return 1
a =a%n
z=1
while x!= 0:
if x%2==0:
a=(a **2) % n
x//= 2
else:
x-=1
z=(z*a) % n
return z
def STonelli(n, p):
assert quad_residue(n, p) == 1, "not a square (mod p)"
q = p - 1
s = 0
while q % 2 == 0:
q //= 2
s += 1
if s == 1:
r = pow(n, (p + 1) // 4, p)
return r,p-r
for z in range(2, p):
#print(quad_residue(z, p))
if p - 1 == quad_residue(z, p):
break
c = pow(z, q, p)
r = pow(n, (q + 1) // 2, p)
t = pow(n, q, p)
m = s
t2 = 0
while (t - 1) % p != 0:
t2 = (t * t) % p
for i in range(1, m):
if (t2 - 1) % p == 0:
break
t2 = (t2 * t2) % p
b = pow(c, 1 << (m - i - 1), p)
r = (r * b) % p
c = (b * b) % p
t = (t * c) % p
m = i
return (r,p-r)
def build_smooth_relations(smooth_base, root_base):
smooth_relations = []
for xx in range(len(smooth_base)):
smooth_relations.append((root_base[xx], smooth_base[xx], xx))
return smooth_relations
def strailing(N):
return N>>lars_last_powers_of_two_trailing(N)
def lars_last_powers_of_two_trailing(N):
p,y=1,2
orign = N
#if orign < 17: N = N%16
N = N&15
if N == 1:
if ((orign -1) & (orign -2)) == 0: return orign.bit_length()-1
while orign&y == 0:
p+=1
y<<=1
return p
if N in [3, 7, 11, 15]: return 1
if N in [5, 13]: return 2
if N == 9: return 3
return 0
def build_matrix(factor_base, smooth_base):
factor_base = factor_base.copy()
factor_base.insert(0, 2)
sparse_matrix = []
col = 0
for xx in smooth_base:
sparse_matrix.append([])
for fx in factor_base:
count = 0
factor_found = False
while xx % fx == 0:
factor_found = True
xx=xx//fx
count+=1
if count % 2 == 0:
sparse_matrix[col].append(0)
continue
else:
if factor_found == True:
sparse_matrix[col].append(1)
else:
sparse_matrix[col].append(0)
col+=1
return np.transpose(sparse_matrix)
def get_mod_congruence(root, N, withstats=False):
r = root - N
if withstats==True:
print(f"{root} ≡ {r} mod {N}")
return r
def primes_sieve2(limit):
a = np.ones(limit, dtype=bool)
a[0] = a[1] = False
for (i, isprime) in enumerate(a):
if isprime:
yield i
for n in range(i*i, limit, i):
a[n] = False
def remove_singletons(XX):
no_singletons = []
for xx in XX:
if len(xx) != 1:
no_singletons.append(xx)
return no_singletons
def fb_sm(N, B, I):
factor_base, sieve_base, sieve_list, smooth_base, root_base = [], [], [], [], []
primes = list(primes_sieve2(B))
i,root=-1,math.isqrt(N)
for x in primes[1:]:
if quad_residue(N, x) == 1:
factor_base.append(x)
for x in range(I):
xx = get_mod_congruence((root+x)**2, N)
sieve_list.append(xx)
if xx % 2 == 0:
xx = strailing(xx+1) # using lars_last_modulus_powers_of_two(xx) bit trick
sieve_base.append(xx)
for p in factor_base:
residues = STonelli(N, p)
for r in residues:
for i in range((r-root) % p, len(sieve_list), p):
while sieve_base[i] % p == 0:
sieve_base[i] //= p
for o in range(len(sieve_list)):
# This is set to 350, which is only good for numbers
# of len < 32. Modify
# to be more dynamic for larger numbers.
if len(smooth_base) >= 350:
break
if sieve_base[o] == 1:
smooth_base.append(sieve_list[o])
root_base.append(root+o)
return factor_base, smooth_base, root_base
def isSquare(hm):
cr=math.isqrt(hm)
if cr*cr == hm:
return True
return False
def find_square(smooth_base):
for x in smooth_base:
if isSquare(x):
return (True, smooth_base.index(x))
else:
return (False, -1)
t_matrix=[]
primes=list(primes_sieve2(1000000))
def factorise(N, B=10000, I=10000000):
global primes, t_matrix
if isprime(N):
return N
for xx in primes:
if N%xx == 0:
return xx
factor_base, smooth_base, root_base = fb_sm(N,B,I)
issquare, t_matrix = find_square(smooth_base)
if issquare == True:
return math.gcd(math.isqrt(smooth_base[t_matrix])+get_mod_congruence(root_base[t_matrix], N), N)
t_matrix = build_matrix(factor_base, smooth_base)
smooth_relations = build_smooth_relations(smooth_base, root_base)
M_opt, M_n, M_m = siqs_build_matrix_opt(np.transpose(t_matrix))
perfect_squares = remove_singletons(siqs_solve_matrix_opt(M_opt, M_n, M_m))
factors = siqs_find_factors(N, perfect_squares, smooth_relations)
return factors
I'm trying to use this http://www.irma-international.org/viewtitle/41011/ algorithm to invert a nxn matrix.
I ran the function on this matrix
[[1.0, -0.5],
[-0.4444444444444444, 1.0]]
and got the output
[[ 1.36734694, 0.64285714]
[ 0.57142857, 1.28571429]]
the correct output is meant to be
[[ 1.28571429, 0.64285714]
[ 0.57142857, 1.28571429]]
My function:
def inverse(m):
n = len(m)
P = -1
D = 1
mI = m
while True:
P += 1
if m[P][P] == 0:
raise Exception("Not Invertible")
else:
D = D * m[P][P]
for j in range(n):
if j != P:
mI[P][j] = m[P][j] / m[P][P]
for i in range(n):
if i != P:
mI[i][P] = -m[i][P] / m[P][P]
for i in range(n):
for j in range(n):
if i != P and j != P:
mI[i][j] = m[i][j] + (m[P][j] * mI[i][P])
mI[P][P] = 1 / m[P][P]
if P == n - 1: # All elements have been looped through
break
return mI
Where am I making my mistake?
https://repl.it/repls/PowerfulOriginalOpensoundsystem
Output
inverse: [[Decimal('1.285714285714285693893862813'),
Decimal('0.6428571428571428469469314065')],
[Decimal('0.5714285714285713877877256260'),
Decimal('1.285714285714285693893862813')]]
numpy: [[ 1.28571429 0.64285714] [ 0.57142857 1.28571429]]
from decimal import Decimal
import numpy as np
def inverse(m):
m = [[Decimal(n) for n in a] for a in m]
n = len(m)
P = -1
D = Decimal(1)
mI = [[Decimal(0) for n in a] for a in m]
while True:
P += 1
if m[P][P] == 0:
raise Exception("Not Invertible")
else:
D = D * m[P][P]
for j in range(n):
if j != P:
mI[P][j] = m[P][j] / m[P][P]
for i in range(n):
if i != P:
mI[i][P] = -m[i][P] / m[P][P]
for i in range(n):
for j in range(n):
if i != P and j != P:
mI[i][j] = m[i][j] + (m[P][j] * mI[i][P])
mI[P][P] = 1 / m[P][P]
m = [[Decimal(n) for n in a] for a in mI]
mI = [[Decimal(0) for n in a] for a in m]
if P == n - 1: # All elements have been looped through
break
return m
m = [[1.0, -0.5],
[-0.4444444444444444, 1.0]]
print(inverse(m))
print(np.linalg.inv(np.array(m)))
My thought process:
At first, I thought you might have lurking floating point roundoff errors. This turned out to not be true. That's what the Decimal jazz is for.
Your bug is here
mI = m # this just creates a pointer that points to the SAME list as m
and here
for i in range(n):
for j in range(n):
if i != P and j != P:
mI[i][j] = m[i][j] + (m[P][j] * mI[i][P])
mI[P][P] = 1 / m[P][P]
# you are not copying mI to m for the next iteration
# you are also not zeroing mI
if P == n - 1: # All elements have been looped through
break
return mI
In adherence to the algorithm, every iteration creates a NEW a' matrix, it does not continue to modify the same old a. I inferred this to mean that in the loop invariant, a becomes a'. Works for your test case, turns out to be true.