linalg

Currently reinforcing my linear algebra and numerical analysis by reimplementing basic, fundamental algorithms in Python. My implementations are tested against numpy and scipy equivalents. Inspired by Alex Nichol's Go repository.

Feel free to read the notes which summarize parts of Justin Solomon's book as well as insights into my thought-process.

Contents

• kahan: kahan summation for adding finite precision floating point numbers.
• gelim: gaussian elimination with naive, partial and full pivoting for solving Ax = b.
• ludecomp: LU, PLU and PLUQ decomposition for solving Ax = b.
• determinant: compute the determinant (or log det) of a square matrix A using PLU factorization.
• inverse: compute the inverse of a square matrix A using PLU factorization.
• cholesky: cholesky decomposition for symmetric positive definite matrices A.
• qrdecomp: QR decomposition of any matrix A using gram-schmidt or householder.
• solve: solve Ax=b using PLU decomposition.
• lstsq: solve least squares using QR decomposition.
• eigen: single and multi eigenvalue finding algorithms, hessenberg factorization and the qr algorithm.
• svd: singular value decomposition SVD of any matrix A.
• optim: iterative linear solvers such as gradient descent and conjugate gradients.

Applications

• imagealign: align a crooked image using least squares.
• deblur: deblur an image by inverting it using conjugate gradients.
• benchmarks: speed comparisons of different decompositions for solving Ax = b.

Resources

• Stanford CS 205A Notes
• Numerical Linear Algebra
• Numerical Recipes: The Art of Scientific Computing

Todos

• Make QR decomposition more efficient for Hessenberg matrices.
• Implement QR decomposition with Givens rotations.