Numerical solvers for linear systems, polynomial roots, and matrix equations.
| Algorithm | Description |
|---|---|
| Gaussian Elimination | Direct solver for dense linear systems using partial pivoting |
| Levinson-Durbin | Fast solver for Toeplitz linear systems exploiting structural symmetry |
| Durand-Kerner | Simultaneous iterative root-finder for polynomials |
| Discrete Algebraic Riccati Equation | Iterative solver for the DARE arising in LQR and Kalman filter design |
| Runge-Kutta ODE Integrators | Fixed-step RK4 and adaptive Dormand-Prince RK45 for ODE integration |
| Spectral Radius | Dominant eigenvalue magnitude and discrete-time Schur stability margin |
| Condition Number | 1-norm condition number estimate via column-wise inverse solve |
| QR Decomposition | Householder factorization and Givens streaming row update for least-squares solves |
| LU Decomposition | PA = LU factorization with partial pivoting for general dense linear systems |
| Lyapunov / Sylvester Solvers | Sylvester AX+XB=C and continuous/discrete Lyapunov solvers via Kronecker vectorisation |
| Singular Value Decomposition | Golub-Kahan bidiagonalization and implicit-shift QR for SVD, pseudo-inverse, rank, and condition number |
| Jacobi Eigenvalue Solver | Cyclic Jacobi rotations for the full symmetric eigenvalue/eigenvector problem |
| Symplectic Integrators | Semi-implicit Euler and velocity Verlet for second-order systems — bounded energy error on conservative plants |
| Damped Pseudo-Inverse | Damped (Tikhonov) least-squares inverse and exact null-space projector from one SVD — tall or wide matrices |
| Active-Set QP (Goldfarb–Idnani) | Dense strictly convex QP with equality and inequality constraints via the dual active-set method — no feasible start needed |