def gaussian_elimination(m):
"""
Parameters
----------
m : list of list of floats (matrix)
Returns
-------
list of floats
solution to the system of linear equation
Raises
------
ValueError
no unique solution
"""
# forward elimination
n = height(m)
for i in range(n):
for j in range(i+1, n):
m[j] = [m[j][k] - m[i][k]*m[j][i]/m[i][i] for k in range(n+1)]
if m[n-1][n-1] == 0: raise ValueError('No unique solution')
# backward substitution
x = [0] * n
for i in range(n-1, -1, -1): # for i in reversed(range(n)):
s = sum(m[i][j] * x[j] for j in range(i, n))
x[i] = (m[i][n] - s) / m[i][i]
return x
def gaussian_elimination_with_pivot(m):
"""
Parameters
----------
m : list of list of floats (matrix)
Returns
-------
list of floats
solution to the system of linear equation
Raises
------
ValueError
no unique solution
"""
# forward elimination
n = height(m)
for i in range(n):
pivot(m, n, i)
for j in range(i + 1, n):
m[j] = [
m[j][k] - m[i][k] * m[j][i] / m[i][i] for k in range(n + 1)
]
if m[n - 1][n - 1] == 0: raise ValueError('No unique solution')
# backward substitution
x = [0] * n
for i in range(n - 1, -1, -1):
s = sum(m[i][j] * x[j] for j in range(i, n))
x[i] = (m[i][n] - s) / m[i][i]
return x
def jacobi(m, x0=None, eps=1e-5, max_iteration=100):
n = height(m)
x0 = [0] * n if x0 == None else x0
x1 = [None] * n
for __ in range(max_iteration):
for i in range(n):
s = sum(-m[i][j] * x0[j] for j in range(n) if i != j)
x1[i] = (m[i][n] + s) / m[i][i]
if all(abs(x1[i] - x0[i]) < eps for i in range(n)):
return x1
x0, x1 = x1, x0
raise ValueError('Solution does not converge')
def gauss_seidel(m, x0=None, eps=1e-5, max_iteration=100):
n = height(m)
x0 = [0] * n if x0 == None else x0
x1 = x0[:]
for __ in range(max_iteration):
for i in range(n):
s = sum(-m[i][j] * x1[j] for j in range(n) if i != j)
x1[i] = (m[i][n] + s) / m[i][i]
if all(abs(x1[i] - x0[i]) < eps for i in range(n)):
return x1
x0 = x1[:]
raise ValueError("Solution does not converge")
def jacobi(m, x0=None, eps=1e-5, max_iteration=100):
n = height(m)
x0 = [0] * n if x0 == None else x0
x1 = [None] * n
for __ in range(max_iteration):
for i in range(n):
s = sum(-m[i][j] * x0[j] for j in range(n) if i != j)
x1[i] = (m[i][n] + s) / m[i][i]
if all(abs(x1[i]-x0[i]) < eps for i in range(n)):
return x1
x0, x1 = x1, x0
raise ValueError('Solution does not converge')
