What's inside NM Dev

01

Linear Algebra

• vector
• vector space
• various matrix representations
  • bi-diagonal
  • diagonal
  • tri-diagonal
  • Givens
  • Hilbert
  • lower/upper triangular
  • permutation
• sparse vector representations
• sparse matrix representations
  • CSR
  • DOK
  • LIL
• iterative sparse matrix solver
  • stationary: Jacobi, Gauss-Seidel, SOR, SSOR
  • non-stationary: Steepest Descent, BiCG, BiCGStabl, CGNE, CGNR, CG, CGS, GCR, GMRES, MinRes, QMR
  • pre-conditioner support: Jacobi, SSOR, customized
• matrix elementary operations
• Householder transformation
• matrix inverse
• matrix measures: determinant rank trace max min
• power of matrix
• matrix pseudoinverse
• matrix bi-diagonalization
• matrix tri-diagonalization
• Cholesky decomposition
• Doolittle factorization
• Eigen factorization
• Gauss-Jordan elimination
• SVD factorization (for asymmetric matrix)
• Gram-Schmidt factorization
• Hessenberg factorization
• LDL decomposition
• LU decomposition
• QR decomposition

02

Statistics

This is a collection of advanced statistics algorithms.

• cointegration
• covariance selection
• factor analysis
• hidden Markov model (Rabiner)
• hidden Markov model (mixture models: Beta, Binomial, Exponential, Gamma, Log-Normal, Normal, Poisson)
• hypothesis testing
• Kalman filter
• LASSO (Least Absolute Shrinkage and Selection Operator)
• linear regression
• principal component analysis
• stochastic process simulation
• time series analysis

03

Digital Signal Processing

This is a collection of the algorithms for digital signal processing.

• Fast Fourier Transform (FFT)

04

Calculus

• finite difference
• differentiation using Ridders’ method
• derivaties of functions for: Beta Beta Regularized Erf Gamma Gaussian
• multivariate differentiation: (bordered) Hessian gradient Jacobian
• Riemann integration
• Riemann integration using substitution rules: Double-Exponential rule Exponential rule Power law singularity Map to the Standard interval
• integration using Romberg’s method
• integration using Newton-Cotes formula Simpson’s rule Trapezoidal rule Midpoint rule
• integration using Gaussian quadrature Legendre polynomials Chebyshev polynomials Laguerre polynomials Hermite polynomials

05

Time Series Analysis

• various moving average filters
• univariate/multivariate time series analysis
  • ARIMA modeling, simulation, fitting, prediction
  • GARCH modeling, simulation, fitting, prediction
  • Auto-Covariance computation

06

Extreme Value Theory

This is a collection of algorithms for Extreme Value Theory.

• univariate distributions
  • Gumbel
  • Frechet
  • Weibull
  • Generalized EV (GEVD)
  • Generalized Pareto (GPD)
  • maxima, minima, order statistics
• bivariate distributions
  • asymmetric logic
  • asymmetric mixed
  • asymmetric negative logistic
  • bilogistic
  • Coles-Tawn
  • Husler-Reiss logistic
  • negative bilogistic
  • negative logistic
• return level and return period
• extremal index estimation Ferro-Seegers algorithm
• cluster analysis
• extreme value markov chain
• extreme value time series, i.e., MARMA(p,q) model
• fitting GEVD by maximum likelihood estimation
• fitting GPD by Peaks-over-Threshold (POT) method
• fitting Gumbel type by Average Conditional Exceedance Rate (ACER) method

07

Optimization

This is a collection of the advanced optimization algorithms.

• linear programming (LP)
• quadratic programming (QP)
• second order conic programming (SOCP)
• semi-definite programming (SDP)
• sequential quadratic programming (SQP)
• mixed linear integer programming (MIP)
• genetic algorithms (GA)
• differential evolution optimization (DE)

08

Differential Equation

This is a collection of numerical algorithms to solve ordinary and partial differential equation problems.

• Ordinary Differential Equation (ODE) solvers for initial value problem (IVP)
  • Euler’s method
  • Runge Kutta 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th and 10th order
  • Runge-Kutta-Fehlberg (RKF45) with adaptive step-size control
  • Adams-Bashforth-Moulton predictor-corrector multi-step method, 1st to 5th order
  • solvers based on Richardson extrapolation
  • Burlisch-Stoer extrapolation
  • semi-implicit extrapolation (suitable for stiff systems)
  • first order system of ODEs: conversion from high order ODE to first order ODE system
• Partial Differential Equation (PDE) solvers: finite difference methods
  • elliptic problem: iterative central difference method (for Poisson’s equation)
  • 1D hyperbolic problem: explicit central difference method (for 1D wave equation)
  • 2D hyperbolic problem: explicit central difference method (for 2D wave equation)
  • 1D parabolic problem: Crank-Nicolson method (for 1D heat or diffusion equation)
  • 2D parabolic problem: alternating direction implicit (ADI) method (for 2D heat or diffusion equation)