def gauss(system: sympy.Matrix, symbol_list: list):
"""
Performs gauss elimination with partial pivoting on a system of
linear equations.
:param system: system of linear equations.
:param symbol_list: list of symbols used in the equations.
:return: a [n, 1] matrix containing result.
"""
output = Output()
output.title = "Gaussian-Elimination"
system = system.as_mutable()
n = system.shape[0]
begin = timeit.default_timer()
# iterate over columns
for i in range(0, n):
# find maximum magnitude and index in this column
max_ind = _get_max_elem(system, i)
# swap current row with the row found to have the maximum element
system.row_swap(max_ind, i)
# forward elimination, iterate over remaining rows and eliminate
for j in range(i + 1, n):
_eliminate(system, i, j)
# perform back substitution.
end = timeit.default_timer()
output.execution_time = abs(end - begin)
output.dataframes.append(
equations_util.create_equ_sys_df(symbol_list, _back_sub(system)))
return output
def gauss_jordan(system: sympy.Matrix, symbol_list):
"""
Performs gauss jordan elimination with partial pivoting on a system of
linear equations.
:param system: system of linear equations.
:param symbol_list: list of symbols used in the equations.
:return: a [n, 1] matrix (vector) containing result.
"""
system = system.as_mutable()
n = system.shape[0]
output = Output()
output.title = "Gauss Jordan"
begin = timeit.default_timer()
# iterate over rows
for i in range(0, n):
# find maximum magnitude and index in this column
max_ind = _get_max_elem(system, i)
# swap current row with the row found to have the maximum element
system.row_swap(max_ind, i)
# normalize current row
system.row_op(i, lambda u, v: u / system[i, i])
# forward elimination, iterate over remaining rows and eliminate
for j in range(i + 1, n):
_eliminate(system, i, j)
# forward elimination, iterate over previous rows and eliminate
for j in range(i - 1, -1, -1):
_eliminate(system, i, j)
# return last column
end = timeit.default_timer()
output.execution_time = abs(end - begin)
output.dataframes.append(
equations_util.create_equ_sys_df(
symbol_list, sympy.Matrix(system.col(system.shape[0]))))
return output
def gauss_seidel(A: sympy.Matrix,
symbols: list,
b=None,
max_iter=100,
max_err=1e-5,
x=None):
"""Gauss-Seidel Iterative Method for Solving A System of Linear Equations:
takes a system of linear equations and returns an approximate solution
for the system using Gauss-Seidel approximation.
Keyword arguments:
A: sympy.Matrix -- The augmented matrix representing the system if b = None
else the coefficients matrix. A is an [n, n] matrix.
symbols: list of sympy.Symbol representing the variables' names.
b: sympy.Matrix -- The r.h.s matrix of the system. b is an n-dimensional
vector.
max_iter: int -- The maximum number of iterations to perform.
max_err: float -- The maximum allowed error.
x: sympy.Matrix -- The initial value for the variables. x is an n-dimensional
vector.
return:
1) The n-dimensional vector x containing the final approximate solution.
2) The [n, number_of_iterations] matrix x_hist containing the values
of x during each iteration.
3) The numpy array err_hist containing the values of the error during each iteration.
"""
A = A.as_mutable()
n = len(symbols)
output = Output()
output.title = "Gauss-Seidel"
if b is None:
A, b = [A[:, :-1], A[:, -1]]
if x is None:
x = sympy.Matrix.zeros(n, 1)
x_prev = x[:, :]
err_hist = [float('NaN')]
x_hist = sympy.Matrix(x)
begin = timeit.default_timer()
for _ in range(0, max_iter):
for i in range(0, n):
xi_new = b[i]
for j in range(0, n):
if i != j:
xi_new -= A[i, j] * x[j]
x[i] = xi_new / A[i, i]
x_hist = x_hist.row_join(x)
diff = (x - x_prev).applyfunc(abs)
err = numpy.amax(numpy.array(diff).astype(numpy.float64))
err_hist.append(err)
x_prev = x[:, :]
if err < max_err:
break
end = timeit.default_timer()
output.execution_time = abs(end - begin)
output.roots = numpy.array(x[:]).astype(numpy.float64)
output.errors = numpy.append(output.errors, err)
output.dataframes.append(create_dataframe_part2(x_hist, err_hist, symbols))
return output
def lu_decomp(system: sympy.Matrix, symbol_list):
system = system.as_mutable()
output = Output()
output.title = "LU Decomposition"
begin = timeit.default_timer()
n = system.shape[0]
a = system[:, :n]
b = system[:, n]
indexMap = numpy.array(range(n), dtype=numpy.int)
a, indexMap = _decompose(a, indexMap)
y = _forward_sub(a, b, indexMap)
output.dataframes.append(
equations_util.create_equ_sys_df(symbol_list,
_back_sub(a.row_join(y), indexMap)))
end = timeit.default_timer()
output.execution_time = abs(end - begin)
return output
def _init_output(output: Output, method_name: str, f, f_bound):
output.roots = []
output.errors = []
output.title = method_name
output.function = f
output.boundary_function = f_bound