Numpy matrix power/exponent with modulo?
Question:
Is it possible to use numpy’s linalg.matrix_power with a modulo so the elements don’t grow larger than a certain value?
Answers:
What’s wrong with the obvious approach?
E.g.
import numpy as np
x = np.arange(100).reshape(10,10)
y = np.linalg.matrix_power(x, 2) % 50
In order to prevent overflow, you can use the fact that you get the same result if you first take the modulo of each of your input numbers; in fact:
(M**k) mod p = ([M mod p]**k) mod p,
for a matrix M. This comes from the following two fundamental identities, which are valid for integers x and y (and a positive power p):
(x+y) mod p = ([x mod p]+[y mod p]) mod p # All additions can be done on numbers *modulo p*
(x*y) mod p = ([x mod p]*[y mod p]) mod p # All multiplications can be done on numbers *modulo p*
The same identities hold for matrices as well, since matrix addition and multiplication can be expressed through scalar addition and multiplication. With this, you only exponentiate small numbers (n mod p is generally much smaller than n) and are much less likely to get overflows. In NumPy, you would therefore simply do
((arr % p)**k) % p
in order to get (arr**k) mod p.
If this is still not enough (i.e., if there is a risk that [n mod p]**k causes overflow despite n mod p being small), you can break up the exponentiation into multiple exponentiations. The fundamental identities above yield
(n**[a+b]) mod p = ([{n mod p}**a mod p] * [{n mod p}**b mod p]) mod p
and
(n**[a*b]) mod p = ([n mod p]**a mod p)**b mod p.
Thus, you can break up power k as a+b+… or a*b*… or any combination thereof. The identities above allow you to perform only exponentiations of small numbers by small numbers, which greatly lowers the risk of integer overflows.
Using the implementation from Numpy:
https://github.com/numpy/numpy/blob/master/numpy/matrixlib/defmatrix.py#L98
I adapted it by adding a modulo term. HOWEVER, there is a bug, in that if an overflow occurs, no OverflowError or any other sort of exception is raised. From that point on, the solution will be wrong. There is a bug report here.
Here is the code. Use with care:
from numpy.core.numeric import concatenate, isscalar, binary_repr, identity, asanyarray, dot
from numpy.core.numerictypes import issubdtype
def matrix_power(M, n, mod_val):
# Implementation shadows numpy's matrix_power, but with modulo included
M = asanyarray(M)
if len(M.shape) != 2 or M.shape[0] != M.shape[1]:
raise ValueError("input must be a square array")
if not issubdtype(type(n), int):
raise TypeError("exponent must be an integer")
from numpy.linalg import inv
if n==0:
M = M.copy()
M[:] = identity(M.shape[0])
return M
elif n<0:
M = inv(M)
n *= -1
result = M % mod_val
if n <= 3:
for _ in range(n-1):
result = dot(result, M) % mod_val
return result
# binary decompositon to reduce the number of matrix
# multiplications for n > 3
beta = binary_repr(n)
Z, q, t = M, 0, len(beta)
while beta[t-q-1] == '0':
Z = dot(Z, Z) % mod_val
q += 1
result = Z
for k in range(q+1, t):
Z = dot(Z, Z) % mod_val
if beta[t-k-1] == '1':
result = dot(result, Z) % mod_val
return result % mod_val
I had overflow issues with all the previous solutions, so I had to write an algorithm that accounts for overflows after every single integer multiplication. This is how I did it:
def matrix_power_mod(x, n, modulus):
x = np.asanyarray(x)
if len(x.shape) != 2:
raise ValueError("input must be a matrix")
if x.shape[0] != x.shape[1]:
raise ValueError("input must be a square matrix")
if not isinstance(n, int):
raise ValueError("power must be an integer")
if n < 0:
x = np.linalg.inv(x)
n = -n
if n == 0:
return np.identity(x.shape[0], dtype=x.dtype)
y = None
while n > 1:
if n % 2 == 1:
y = _matrix_mul_mod_opt(x, y, modulus=modulus)
x = _matrix_mul_mod(x, x, modulus=modulus)
n = n // 2
return _matrix_mul_mod_opt(x, y, modulus=modulus)
def matrix_mul_mod(a, b, modulus):
if len(a.shape) != 2:
raise ValueError("input a must be a matrix")
if len(b.shape) != 2:
raise ValueError("input b must be a matrix")
if a.shape[1] != a.shape[0]:
raise ValueError("input a and b must have compatible shape for multiplication")
return _matrix_mul_mod(a, b, modulus=modulus)
def _matrix_mul_mod_opt(a, b, modulus):
if b is None:
return a
return _matrix_mul_mod(a, b, modulus=modulus)
def _matrix_mul_mod(a, b, modulus):
r = np.zeros((a.shape[0], b.shape[1]), dtype=a.dtype)
bT = b.T
for rowindex in range(r.shape[0]):
x = (a[rowindex, :] * bT) % modulus
x = np.sum(x, 1) % modulus
r[rowindex, :] = x
return r
What worked for me was changing Tamas Hegedus code in a single line, to specify dtype=object. This is needed, since Numpy does not support large integers as python does. Using dtype=object forces numpy to support large integers and avoid silent overflow errors.
Tested with exponent 10^(15) and modulus about 10^(12).
def matrix_power_mod(x, n, modulus):
x = np.asanyarray(x)
if len(x.shape) != 2:
raise ValueError("input must be a matrix")
if x.shape[0] != x.shape[1]:
raise ValueError("input must be a square matrix")
if not isinstance(n, int):
raise ValueError("power must be an integer")
if n < 0:
x = np.linalg.inv(x)
n = -n
if n == 0:
return np.identity(x.shape[0], dtype=x.dtype)
y = None
while n > 1:
if n % 2 == 1:
y = _matrix_mul_mod_opt(x, y, modulus=modulus)
x = _matrix_mul_mod(x, x, modulus=modulus)
n = n // 2
return _matrix_mul_mod_opt(x, y, modulus=modulus)
def matrix_mul_mod(a, b, modulus):
if len(a.shape) != 2:
raise ValueError("input a must be a matrix")
if len(b.shape) != 2:
raise ValueError("input b must be a matrix")
if a.shape[1] != a.shape[0]:
raise ValueError("input a and b must have compatible shape for multiplication")
return _matrix_mul_mod(a, b, modulus=modulus)
def _matrix_mul_mod_opt(a, b, modulus):
if b is None:
return a
return _matrix_mul_mod(a, b, modulus=modulus)
def _matrix_mul_mod(a, b, modulus):
r = np.zeros((a.shape[0], b.shape[1]), dtype=object)
bT = b.T
for rowindex in range(r.shape[0]):
x = (a[rowindex, :] * bT) % modulus
x = np.sum(x, 1) % modulus
r[rowindex, :] = x
return r
Is it possible to use numpy’s linalg.matrix_power with a modulo so the elements don’t grow larger than a certain value?
What’s wrong with the obvious approach?
E.g.
import numpy as np
x = np.arange(100).reshape(10,10)
y = np.linalg.matrix_power(x, 2) % 50
In order to prevent overflow, you can use the fact that you get the same result if you first take the modulo of each of your input numbers; in fact:
(M**k) mod p = ([M mod p]**k) mod p,
for a matrix M. This comes from the following two fundamental identities, which are valid for integers x and y (and a positive power p):
(x+y) mod p = ([x mod p]+[y mod p]) mod p # All additions can be done on numbers *modulo p*
(x*y) mod p = ([x mod p]*[y mod p]) mod p # All multiplications can be done on numbers *modulo p*
The same identities hold for matrices as well, since matrix addition and multiplication can be expressed through scalar addition and multiplication. With this, you only exponentiate small numbers (n mod p is generally much smaller than n) and are much less likely to get overflows. In NumPy, you would therefore simply do
((arr % p)**k) % p
in order to get (arr**k) mod p.
If this is still not enough (i.e., if there is a risk that [n mod p]**k causes overflow despite n mod p being small), you can break up the exponentiation into multiple exponentiations. The fundamental identities above yield
(n**[a+b]) mod p = ([{n mod p}**a mod p] * [{n mod p}**b mod p]) mod p
and
(n**[a*b]) mod p = ([n mod p]**a mod p)**b mod p.
Thus, you can break up power k as a+b+… or a*b*… or any combination thereof. The identities above allow you to perform only exponentiations of small numbers by small numbers, which greatly lowers the risk of integer overflows.
Using the implementation from Numpy:
https://github.com/numpy/numpy/blob/master/numpy/matrixlib/defmatrix.py#L98
I adapted it by adding a modulo term. HOWEVER, there is a bug, in that if an overflow occurs, no OverflowError or any other sort of exception is raised. From that point on, the solution will be wrong. There is a bug report here.
Here is the code. Use with care:
from numpy.core.numeric import concatenate, isscalar, binary_repr, identity, asanyarray, dot
from numpy.core.numerictypes import issubdtype
def matrix_power(M, n, mod_val):
# Implementation shadows numpy's matrix_power, but with modulo included
M = asanyarray(M)
if len(M.shape) != 2 or M.shape[0] != M.shape[1]:
raise ValueError("input must be a square array")
if not issubdtype(type(n), int):
raise TypeError("exponent must be an integer")
from numpy.linalg import inv
if n==0:
M = M.copy()
M[:] = identity(M.shape[0])
return M
elif n<0:
M = inv(M)
n *= -1
result = M % mod_val
if n <= 3:
for _ in range(n-1):
result = dot(result, M) % mod_val
return result
# binary decompositon to reduce the number of matrix
# multiplications for n > 3
beta = binary_repr(n)
Z, q, t = M, 0, len(beta)
while beta[t-q-1] == '0':
Z = dot(Z, Z) % mod_val
q += 1
result = Z
for k in range(q+1, t):
Z = dot(Z, Z) % mod_val
if beta[t-k-1] == '1':
result = dot(result, Z) % mod_val
return result % mod_val
I had overflow issues with all the previous solutions, so I had to write an algorithm that accounts for overflows after every single integer multiplication. This is how I did it:
def matrix_power_mod(x, n, modulus):
x = np.asanyarray(x)
if len(x.shape) != 2:
raise ValueError("input must be a matrix")
if x.shape[0] != x.shape[1]:
raise ValueError("input must be a square matrix")
if not isinstance(n, int):
raise ValueError("power must be an integer")
if n < 0:
x = np.linalg.inv(x)
n = -n
if n == 0:
return np.identity(x.shape[0], dtype=x.dtype)
y = None
while n > 1:
if n % 2 == 1:
y = _matrix_mul_mod_opt(x, y, modulus=modulus)
x = _matrix_mul_mod(x, x, modulus=modulus)
n = n // 2
return _matrix_mul_mod_opt(x, y, modulus=modulus)
def matrix_mul_mod(a, b, modulus):
if len(a.shape) != 2:
raise ValueError("input a must be a matrix")
if len(b.shape) != 2:
raise ValueError("input b must be a matrix")
if a.shape[1] != a.shape[0]:
raise ValueError("input a and b must have compatible shape for multiplication")
return _matrix_mul_mod(a, b, modulus=modulus)
def _matrix_mul_mod_opt(a, b, modulus):
if b is None:
return a
return _matrix_mul_mod(a, b, modulus=modulus)
def _matrix_mul_mod(a, b, modulus):
r = np.zeros((a.shape[0], b.shape[1]), dtype=a.dtype)
bT = b.T
for rowindex in range(r.shape[0]):
x = (a[rowindex, :] * bT) % modulus
x = np.sum(x, 1) % modulus
r[rowindex, :] = x
return r
What worked for me was changing Tamas Hegedus code in a single line, to specify dtype=object. This is needed, since Numpy does not support large integers as python does. Using dtype=object forces numpy to support large integers and avoid silent overflow errors.
Tested with exponent 10^(15) and modulus about 10^(12).
def matrix_power_mod(x, n, modulus):
x = np.asanyarray(x)
if len(x.shape) != 2:
raise ValueError("input must be a matrix")
if x.shape[0] != x.shape[1]:
raise ValueError("input must be a square matrix")
if not isinstance(n, int):
raise ValueError("power must be an integer")
if n < 0:
x = np.linalg.inv(x)
n = -n
if n == 0:
return np.identity(x.shape[0], dtype=x.dtype)
y = None
while n > 1:
if n % 2 == 1:
y = _matrix_mul_mod_opt(x, y, modulus=modulus)
x = _matrix_mul_mod(x, x, modulus=modulus)
n = n // 2
return _matrix_mul_mod_opt(x, y, modulus=modulus)
def matrix_mul_mod(a, b, modulus):
if len(a.shape) != 2:
raise ValueError("input a must be a matrix")
if len(b.shape) != 2:
raise ValueError("input b must be a matrix")
if a.shape[1] != a.shape[0]:
raise ValueError("input a and b must have compatible shape for multiplication")
return _matrix_mul_mod(a, b, modulus=modulus)
def _matrix_mul_mod_opt(a, b, modulus):
if b is None:
return a
return _matrix_mul_mod(a, b, modulus=modulus)
def _matrix_mul_mod(a, b, modulus):
r = np.zeros((a.shape[0], b.shape[1]), dtype=object)
bT = b.T
for rowindex in range(r.shape[0]):
x = (a[rowindex, :] * bT) % modulus
x = np.sum(x, 1) % modulus
r[rowindex, :] = x
return r