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Assignment 4: matrix factorization and linear equation

In this assignment we want to solve the linear equation:

$$ \mathbf{A}\boldsymbol{x} = \boldsymbol{b} $$

where $\mathbf{A}$ is a $N \times N$ matrix, $\boldsymbol{x}$ and $\boldsymbol{b}$ are $N \times 1$ vectors.

With the same matrix $\mathbf{A}$, we want to solve multiple different $\boldsymbol{x}$ with different $\boldsymbol{b}$. To do that, we firstly do a LU-factorization:

$$ \mathbf{A} = \mathbf{P} \mathbf{L} \mathbf{U} $$

where $\mathbf{P}$ is the row-permutation matrix, $\mathbf{L}$ lower triangular matrix and $\mathbf{U}$ is the upper triangular matrix.

Once we have the factorization of $\mathbf{A}$, we can use the factorization to solve multiple equations of $\boldsymbol{x}$ and $\boldsymbol{b}$.

In this assignment, we use the object-oriented programming pattern. Given the input matrix in the constructor, we decompose it and store the factorization and permutation into class members. The class can have a method solve to solve different equations of $\boldsymbol{x}$ and $\boldsymbol{b}$.

Class skeleton

We use the following class skeleton to implement:

import numpy as np


class LUSolver:
    def __init__(self, input_matrix: np.ndarray):
        """
        Constructor of the class. It takes the input matrix and decompose it into LU factorization.
        Store the factorization and permutation into class members.
        """
        pass

    def solve(self, b: np.ndarray) -> np.ndarray:
        """
        Solve the linear equation with the input matrix and the given vector b.
        """
        pass

Implementation

To implement the class, you need two scipy functions: scipy.linalg.lu_factor and scipy.linalg.lu_solve. Have a look at their documentations about how to use them.

Error checking

You should check the following conditions in the constructor. If a condition is not satisfied, you should raise an error with a meaningful message.

  • input_matrix should be an instance of np.ndarray.
  • input_matrix should be a square matrix.
  • input_matrix should have the dtype of np.float64.
  • The size of input_matrix should be at least 1.
  • input_matrix should not contain any inf.
  • input_matrix should not contain any nan.
  • input_matrix should not be singular.

You should check the following conditions in the solve method. If a condition is not satisfied, you should raise an error with a meaningful message.

  • b should be an instance of np.ndarray.
  • b should be a one-dimensional vector.
  • b should have the dtype of np.float64.
  • The size of b should be the same as the size of the input matrix.
  • b should not contain any inf.
  • b should not contain any nan.

Testing

Same as Assignment 2, you should test the class with happy flow and all error conditions. In the happy flow, try to test with manually created input in the test script, and pre-defined input in a json file.