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'''
Various Linear Algebra Functions
Author: Igor van Loo
'''
[docs]def gauss_jordan_elimination(matrix, augmentedpart = None):
'''
Performs `Gauss Jordan Elimination on the given matrix <https://en.wikipedia.org/wiki/Gaussian_elimination>`_
:param matrix: Matrix to perform Algoithm on
:param augmentedpart: Optional argument, will attach the augmented part onto the matrix and then perform the algorithm
:returns: True if algorithm was successful, false otherwise
.. code-block:: python
matrix = [[2, 1, -1],
[-3, -1, 2],
[-2, 1, 2]]
print(gauss_jordan_elimination(matrix)) #True
.. note::
This function simply performs the Gauss Jordan Algorithm, it is used with with solve and inverse to solve Ax = b, and the find
the inverse of a matrix
'''
if type(matrix) != list:
return "matrix must be a list"
if augmentedpart != None:
matrix = concatenate(matrix, augmentedpart)
m, n = len(matrix), len(matrix[0])
h = 0
k = 0
while h < m and k < n:
i_max = argmax([abs(matrix[i][k]) for i in range(h, m)]) + h
if matrix[i_max][k] == 0:
k += 1
else:
if h != i_max:
temp_row = matrix[i_max]
matrix[i_max] = matrix[h]
matrix[h] = temp_row
for i in range(h + 1, m):
f = matrix[i][k]/matrix[h][k]
matrix[i][k] = 0
for j in range(k + 1, n):
matrix[i][j] -= f*matrix[h][j]
h += 1
k += 1
for y in range(m-1, -1, -1):
t = matrix[y][y]
if abs(t) < pow(10, -10):
return False
for z in range(0,y):
for x in range(n-1, y-1, -1):
matrix[z][x] -= matrix[y][x] * matrix[z][y] / t
matrix[z][x] = round(matrix[z][x], 4)
matrix[y][y] /= t
matrix[y][y] = round(matrix[y][y], 4)
for x in range(m, n):
matrix[y][x] /= t
matrix[y][x] = round(matrix[y][x], 4)
return True
[docs]def solve(M, b):
'''
Finds the solution, x, to the equation Mx = b
:param M: A Matrix
:param b: A Vector
:returns: The Vector x, or an error message
.. code-block:: python
matrix = [[2, 1, -1],
[-3, -1, 2],
[-2, 1, 2]]
b = [[8],
[-11],
[-3]]
print(solve(matrix, b)) #[[2.0], [3.0], [-1.0]]
'''
if type(M) != list:
return "M must be a list"
m, n = len(M), len(M[0])
if len(b[0]) > 1:
return "b must be a vector"
if m != len(b):
return "Impossible to solve"
if gauss_jordan_elimination(M, b):
return [M[x][n:] for x in range(m)]
else:
return "No solution, or infinite solutions"
[docs]def inverse(matrix):
'''
Finds the inverse of the given matrix by performing Gauss Jordan Elimination
:param matrix: Matrix to be inverted
:returns: Inverted matrix, or an error message
.. code-block:: python
matrix = [[1, -1, 0],
[-8, 9, -1],
[-9, 0, 10]]
print(inverse(matrix)) #[[90.0, 10.0, 1.0], [89.0, 10.0, 1.0], [81.0, 9.0, 1.0]]
'''
if type(matrix) != list:
return "matrix must be a list"
m, n = len(matrix), len(matrix[0])
if m != n:
return "Must be a square matrix"
if gauss_jordan_elimination(matrix, identity(m)):
return [matrix[x][m:] for x in range(m)]
else:
return "Matrix is not invertible"
[docs]def determinant(matrix):
'''
Finds the determinant of a matrix
:param matrix: Matrix
:returns: det(Matrix)
.. code-block:: python
matrix = [[7, -1, 0],
[-8, 9, -1],
[-9, 0, 10]]
print(determinant(matrix)) #541.0
'''
if type(matrix) != list:
return "matrix must be a list"
m, n = len(matrix), len(matrix[0])
if m != n:
return "Must be a square matrix"
h = 0
k = 0
det = 1
while h < m and k < n:
i_max = argmax([abs(matrix[i][k]) for i in range(h, m)]) + h
if matrix[i_max][k] == 0:
k += 1
else:
if h != i_max:
temp_row = matrix[i_max]
matrix[i_max] = matrix[h]
matrix[h] = temp_row
det *= -1
for i in range(h + 1, m):
f = matrix[i][k]/matrix[h][k]
matrix[i][k] = 0
for j in range(k + 1, n):
matrix[i][j] -= f*matrix[h][j]
h += 1
k += 1
for i in range(m):
det *= matrix[i][i]
return round(det, 4)
[docs]def matrix_addition(A, B, subtract = False):
'''
Performs matrix addition/subtraction on matrix A
:param A: First Matrix
:param B: Second Matrix
:param subtract: If True, will perform matrix subtraction
:returns: A + B
.. code-block:: python
matrix = [[1, 0, 0],
[0, 1, 0],
[0, 0, 1]]
print(matrix_addition(matrix, matrix)) #[[2, 0, 0], [0, 2, 0], [0, 0, 2]]
print(matrix_addition(matrix, matrix, True)) #[[0, 0, 0], [0, 0, 0], [0, 0, 0]]
'''
if type(A) != list or type(B) != list or type(subtract) != bool:
return "A and B must be lists"
m1, n1 = len(A), len(A[0])
m2, n2 = len(B), len(B[0])
if m1 != m2 or n1 != n2:
return "Dimensions are unequal"
else:
for x in range(m1):
for y in range(n1):
if subtract:
A[x][y] -= B[x][y]
else:
A[x][y] += B[x][y]
return A
[docs]def identity(l, val = 1):
'''
Generats an Square Identity Matrix
:param l: Size of matrix
:param val: diagonal values, default is 1
:returns: val * identity matrix of size l
.. code-block:: python
print(identity(2)) #[[1, 0], [0, 1]]
print(identity(2, 5)) #[[5, 0], [0, 5]]
'''
if type(l) != int or type(val) != int:
return "l and val must be integers"
matrix = []
for x in range(l):
row = [0]*l
row[x] = val
matrix.append(row)
return matrix
[docs]def concatenate(A, B):
'''
Concatenates 2 matrices, A and B
:param A: First Matrix
:param B: Second Matrix
:returns: A concatenated with B
.. code-block:: python
A = [[1, 0],
[0, 1]]
print(concatenate(A, A)) #[[1, 0, 1, 0], [0, 1, 0, 1]]
.. note::
This is a helper function for inverse and solve
'''
if type(A) != list or type(B) != list:
return "A and B must be lists"
m1 = len(A)
m2 = len(B)
if m1 != m2:
return "Cannot concatenate horizontally"
for row in range(m1):
A[row] += B[row]
return A
[docs]def argmax(alist):
'''
Finds the `argmax <https://en.wikipedia.org/wiki/Arg_max>`_ of a list
:param alist: A list
:returns: the arg max of the list
.. code-block:: python
print(argmax([1, 3, 2])) #1
'''
if type(alist) != list:
return "Input must be a list"
f = lambda i: alist[i]
return max(range(len(alist)), key = f)
[docs]def fillmatrix(size, val = 0):
'''
Fills a matrix with a value
:param size: A tuple, denoted (width, height) of matrix
:param val: Value to fill matrix with, default is 0
:returns: A matrix
.. code-block:: python
print(fillmatrix((2, 2))) #[[0, 0], [0, 0]]
print(fillmatrix((2, 3))) #[[0, 0, 0], [0, 0, 0]]
'''
if type(size) != tuple:
return "Size must be a tuple"
return [[val]*size[1] for _ in range(size[0])]
[docs]def matrix_mul(A, B):
'''
Matrix multplication of 2 matrices, A and B
:param A: First Matrix
:param B: Second Matrix
:returns: Matrix AB
.. code-block:: python
A = [[2, 0],
[0, 2]]
B = [[1, 2],
[3, 1]]
print(matrix_mul(A, B)) #[[2, 4], [6, 2]]
'''
if type(A) != list or type(B) != list:
return "A and B must be lists"
m1, n1 = len(A), len(A[0])
m2, n2 = len(B), len(B[0])
if n1 != m2:
return "Cannot multiply matrices"
matrix = fillmatrix((m1, n2))
for row in range(m1):
for col in range(n2):
for elt in range(len(B)):
matrix[row][col] += A[row][elt] * B[elt][col]
return matrix
[docs]def matrix_pow(A, n):
'''
Matrix exponentiation. Uses the `Exponentiation by squaring <https://en.wikipedia.org/wiki/Exponentiation_by_squaring>`_ method
:param A: Matrix
:param n: exponenent
:returns: Matrix A^n
.. code-block:: python
A = [[1, 1],
[1, 0]]
print(matrix_pow(A, 10)) #[[89, 55], [55, 34]]
.. note::
Seem familiar? These are fibonacci numbers!
This is nearly identical to my fibonacci generation function as it uses the same method,
however the fibonacci is slightly more optimized due to it's properties
'''
A_res = A
for bit in bin(n)[3:]:
A_res = matrix_mul(A_res, A_res)
if bit == "1":
A_res = matrix_mul(A_res, A)
return A_res