numerical_methods.systems package

Submodules

numerical_methods.systems.decomposition_LU module

numerical_methods.systems.decomposition_LU.identite(N)[source]
numerical_methods.systems.decomposition_LU.lu_decomposition(A: list[list[float]], b: list[float], n: int) → None[source]

Perform LU decomposition of matrix A and solve the system AX = b.

Parameters:

  • A (list[list[float]]) – Coefficient matrix.

  • b (list[float]) – Right-hand side vector.

  • n (int) – Size of the system.

Returns:

None

numerical_methods.systems.jacobi module

numerical_methods.systems.jacobi.jacobi_method(A: list[list[float]], b: list[float], n: int, num_iterations: int, x_init: list[float]) → list[float][source]

Solve a linear system using the Jacobi iterative method.

Parameters:

  • A (list[list[float]]) – Coefficient matrix of size n x n.

  • b (list[float]) – Right-hand side vector.

  • n (int) – Size of the system (number of equations).

  • num_iterations (int) – Number of iterations to perform.

  • x_init (list[float]) – Initial guess for the solution vector.

Returns:

The approximated solution vector after iterations.

Return type:

list[float]

Example:

>>> A = [[4, -1, 0], [-1, 4, -1], [0, -1, 3]]
>>> b = [15, 10, 10]
>>> x_init = [0, 0, 0]
>>> jacobi_method(A, b, 3, 10, x_init)
[3.75, 4.375, 4.7917]  # Approximated output

numerical_methods.systems.triongulation_partiel module

numerical_methods.systems.triongulation_partiel.partial_pivoting_elimination(A: list[list[float]], b: list[float], n: int) → None[source]

Perform Gaussian elimination with partial pivoting.

Parameters:

  • A (list[list[float]]) – Coefficient matrix of size n x n.

  • b (list[float]) – Right-hand side vector.

  • n (int) – Size of the system.

Returns:

None (modifies A and b in-place).

This function modifies matrix A and vector b to transform A into an upper triangular matrix.

numerical_methods.systems.triongulation_total module

numerical_methods.systems.triongulation_total.full_pivoting_elimination(A: list[list[float]], b: list[float], n: int) → None[source]

Perform Gaussian elimination with full pivoting.

Parameters:

  • A (list[list[float]]) – Coefficient matrix of size n x n.

  • b (list[float]) – Right-hand side vector.

  • n (int) – Size of the system.

Returns:

None (modifies A and b in-place).

Module contents