NumericalAnalysisLibrary.md

Numerical Analysis Library

Numerical Analysis Library contains the supporting Functionality for Numerical Methods - including R^x Solvers, Linear Algebra, and Statistical Measure Distributions.

Documentation

Document	Link
Technical Specification	Latest Previous
User Guide
API	Javadoc

Component Projects

• Function => Implementation and Solvers for a Suite of R^x To R¹ Functions.
  • { Home | Project }
• Measure => Continuous and Discrete Measure Distributions and Variate Evolutions.
  • { Home | Project }
• Numerical => Suite of DROP Numerical Analysis Utilities.
  • { Home | Project }

Coverage

• Introduction
  • Framework Glossary
  • Document Layout
• Framework
• Search Initialization
  • Bracketing
  • Objective Function Failure
  • Bracketing Start Initialization
  • Open Search Initialization
  • Search/Bracketing Initializer Heuristic Customization
• Numerical Challenges in Search
• Variate Iteration
• Open Search Method: Newton Method
• Closed Search Methods
  • Secant
  • Bracketing Iterative Search
  • Univariate Iterator Primitive: Bisection
  • Univariate Iterator Primitive: False Position
  • Univariate Iterator Primitive: Inverse Quadratic
  • Univariate Iterator Primitive: Ridder
  • Univariate compound iterator: Brent and Zheng
• Polynomial Root Search
• Meta-heuristics
  • Introduction
  • Properties and Classification
  • Meta-heuristics Techniques
  • Meta-heuristics Techniques in Combinatorial Problems
  • Key Meta-heuristics Historical Milestones
  • References
• Multivariate Distribution
  • Parallels between Vector Calculus and Statistical Distribution Analysis
• Linear Systems Analysis and Transformation
  • Matrix Transforms
  • Systems of Linear Equations
  • Orthogonalization
  • Gaussian Elimination
• Rayleigh Quotient Iteration
  • Introduction
  • The Algorithm
  • References
• Power Iteration
  • Introduction
  • The Method
  • Analysis
  • Applications
  • References
• Sylvester's Formula
  • Overview
  • Conditions
  • Generalization
  • References
• Numerical Integration
  • Introduction and Overview
  • Reasons for Numerical Integration
  • Methods for One-Dimensional Integrals
  • Quadrature Rules Based on Interpolating Functions
  • Generalized Mid-Point Rule Formulation
  • Adaptive Algorithms
  • Extrapolation Methods
  • A Priori Conservative Error Estimation
  • Integrals Over Infinite Intervals
  • Multi-dimensional Integrals
  • Monte Carlo
  • Sparse Grids
  • Bayesian Quadrature
  • Connections to Differential Equations
  • References
• Gaussian Quadrature
  • Introduction and Overview
  • Gauss-Legendre Quadrature
  • Change of Interval
  • Other Forms
  • Fundamental Theorem
  • General Formula for the Weights
  • Proof that the Weights are Positive
  • Computation of Gaussian Quadrature Rules
  • Recurrence Relation
  • The Golub-Welsch Algorithm
  • Error Estimates
  • Gauss-Kronrod Rules
  • Gauss-Lobatto Rules
  • References
• Gauss-Kronrod Quadrature
  • Introduction and Overview
  • Description
  • Example
  • Implementation
  • References
• Gamma Distribution
  • Overview
  • Gamma Measures - Central Distribution Table
  • Definitions
  • Characterization Using Shape 𝜶 and Rate 𝜷
  • Characterization using Shape 𝒌 and Scale 𝜽
  • Properties – Skewness
  • Properties – Median Calculation
  • Properties – Summation
  • Properties – Scaling
  • Properties – Exponential Family
  • Properties – Logarithmic Expectation and Variance
  • Properties – Information Entropy
  • Properties – Kullback-Liebler Divergence
  • Properties – Laplace Transform
  • Related Distributions – General
  • Properties – Compound Gamma
  • Statistical Inference – Maximum Likelihood Parameter Estimation
  • Closed-Form Estimators
  • Bayesian Minimum Mean-Squared Error
  • Bayesian Inference Conjugate Prior
  • Occurrence and Applications
  • Computational Methods – Generating Gamma Distributed Random Variables
  • References
• Chi Square Distribution
  • Overview
  • Definition
  • Introduction
  • Probability Density Function
  • Cumulative Distribution Function
  • Additivity
  • Sample Mean
  • Entropy
  • Non-central Moments
  • Asymptotic Properties
  • Cumulants
  • Relation to Other Distributions
  • Generalizations
  • Linear Combinations
  • Non-Central Chi-Squared Distribution
  • Generalized Chi-Squared Distribution
  • Gamma, Exponential, and Related Distribution
  • Occurrence and Applications
  • Table of χ^2 Values vs. p-Values
  • Summary Expressions
  • References
• Non-central Chi-Square Distribution
  • Overview
  • Background
  • Non-central Chi-Square Distribution Table
  • Definition
  • Properties – Moment Generating Function
  • Properties – Moments
  • Cumulative Distribution Function
  • Approximation – including for Quantiles
  • Derivation of the PDF
  • Related Distributions
  • Transformations
  • Use in Tolerance Intervals
  • References
• Exponential Distribution
  • Overview
  • Definitions
    • Probability Density Function
    • Cumulative Distribution Function
    • Alternative Parameterization
  • Properties
    • Mean, Variance, Moments, and Median
    • Memorylessness Property of Exponential Random Variable
    • Quantiles
    • Conditional Value at Risk (Expected Shortfall)
    • Buffered Probability of Exceedance (bPOE)
    • Kullback-Leibler Divergence
    • Maximum Entropy Distribution
    • Distribution of the Minimum of Exponential Random Variables
    • Joint Moments of i.i.d. Exponential Order Statistics
    • Sum of Two Independent Exponential Random Variables
  • Related Distributions
  • Statistical Inference
    • Parameter Estimation
    • Fisher Information
    • Confidence Intervals
    • Bayesian Inference
  • Occurrence and Applications
    • Occurrence of Events
    • Prediction
  • Random Variate Generation
  • References
• Householder Transformation
  • Overview
  • Definition
    • Transformation
    • Householder Matrix
    • Properties
  • Applications
    • Numerical Linear Algebra
      • QR Decomposition
      • Tridiagonalization
      • Examples
  • Computational and Theoretical Relationship to other Unitary Transformations
  • References
• The Householder Transformation in Numerical Linear Algebra
  • Abstract
  • Linear Algebra
    • Geometric Meanings of the Determinant and Matrix Norm
    • Computation of Determinants
    • Computation of Matrix Inverses
    • Error Propagation
  • Gaussian Elimination
    • Row Reduction using Gaussian Elimination
    • Gaussian Elimination without Pivoting
    • Gaussian Elimination with Pivoting
  • Householder Transformations
    • Geometric Construction
    • Construction with Specified Source and Destination
    • Properties of Q, Algebraically Obtained
    • Properties of Q, Geometrically Obtained
    • Repeated Householders for Upper-Triangularization
    • Householders for Column-zeroing
    • Computation of Determinants
    • Computation of Matrix Inverses
    • Rotation Matrices
  • References
• Hilbert Space
  • Overview
  • Definition and Illustration
    • Motivating Example: Euclidean Vector Space
    • Definition
    • Second Example: Sequence Spaces
  • Examples
    • Lebesgue Spaces
    • Sobolev Spaces
    • Spaces of Holomorphic Functions
      • Hardy Spaces
      • Bergman Spaces
  • Applications
    • Sturm-Liouville Theory
    • Partial Differential Equations
    • Ergodic Theory
    • Fourier Analysis
    • Quantum Mechanics
    • Color Perception
  • Properties
    • Pythagorean Identity
    • Parallelogram Identity and Polarization
    • Best Approximation
    • Duality
    • Weakly-convergent Sequences
    • Banach Space Properties
  • Operators on Hilbert Spaces
    • Bounded Operators
    • Unbounded Operators
  • Constructions
    • Direct Sums
    • Tensor Products
  • Orthonormal Bases
    • Sequence Spaces
    • Bessel's Inequality and Parseval's Formula
    • Hilbert Dimension
    • Separable Spaces
  • Orthogonal Complements and Projections
  • Spectral Theory
  • References
• Positive-definite Kernel
  • Definition
    • Some General Properties
    • Examples of Positive-definite Kernels
  • Connection with Reproducing Kernel Hilbert Spaces and Feature Maps
  • Kernels and Distances
  • Some Applications
    • Kernels in Machine Learning
    • Kernels in Probabilistic Models
    • Numerical Solution of Partial Differential Equations
    • Other Applications
  • References
• Reproducing Kernel Hilbert Space
  • Overview
  • Definition
  • Example
  • Moore-Aronszajn Theorem
  • Integral Operator's and Mercer's Theorem
  • Featur Maps
  • Properties
  • Common Examples
    • Bilinear Kernels
    • Polynomial Kernels
    • Radial Basis Function Kernels
      • Gaussian or Squared Exponential Kernel
      • Laplacian Kernel
    • Bergman Kernels
  • Extension to Vector-valued Functions
  • Connection between RKHS and ReLU Function
  • References
• Representer Theorem
  • Overview
  • Formal Statement
  • Generalizations
  • Applications
  • References
• Kernel Methods
  • Overview
  • Motivation and Informal Explanation
  • Mathematics: The Kernel Trick
  • Applications
  • References

DROP Specifications

• Main => https://lakshmidrip.github.io/DROP/
• Wiki => https://github.com/lakshmiDRIP/DROP/wiki
• GitHub => https://github.com/lakshmiDRIP/DROP
• Repo Layout Taxonomy => https://lakshmidrip.github.io/DROP/Taxonomy.md
• Javadoc => https://lakshmidrip.github.io/DROP/Javadoc/index.html
• Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
• Release Versions => https://lakshmidrip.github.io/DROP/version.html
• Community Credits => https://lakshmidrip.github.io/DROP/credits.html
• Issues Catalog => https://github.com/lakshmiDRIP/DROP/issues