This repo was created as an extra credit project for my MATH 2331: Linear Algebra class at Northeastern University. The file numerical.py contains functions which implement the following features:
- Vector operations, such as adding, subtracting, and scaling vectors
- L1-norm, L2-norm, and L∞-norm of vectors
- Absolute and relative errors of vectors
- Matrix-vector and matrix-matrix products
- Row and column operations on matrices
- Transpose of matrices
- Inverting and computing the sign of permutations
- Simple partial pivoting, scaled partial pivoting, and full pivoting
- Gaussian elimination
- Back-substitution
- Solving systems of equations using elimination
- LU decomposition
- Forward-substitution
- Solving systems of equations using the LU decomposition
- Finding the image, kernel, and inverse of a matrix given the LU decomposition
- Determining if a matrix is singular given the LU decomposition
- Finding the exact determinant of an integer-valued matrix using permutations
- Finding the determinant of a matrix given the LU decomposition
- QR decomposition using Householder reflections
- Finding the least-squares solution of a system of equations using QR decomposition
- Calculating the best-fit coefficients of a regression model using the linear least squares estimator
- Schur decomposition using a naive form of the QR algorithm
- Finding the eigendecomposition given the Schur decomposition
- Finding the roots of a monic polynomial by finding the Schur decomposition of the companion matrix
Finally, the numerical_test.py file contains several unit tests which ensure that all of these functions work as intended.