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Numerical Analysis Library

Numerical Analysis Library contains the supporting Functionality for Numerical Methods - including Rx 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 Rx To R1 Functions.
  • Measure => Continuous and Discrete Measure Distributions and Variate Evolutions.
  • Numerical => Suite of DROP Numerical Analysis Utilities.

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
  • Successive Over-relaxation
    • Introduction
    • Formulation
    • Convergence
      • Convergence Rate
    • Algorithm
    • Symmetric Successive Over-relaxation
    • Other Applications of the Method
    • References
  • Symmetric Successive Over-relaxation
    • References
  • Tridiagonal matrix algorithm
    • Introduction
    • Derivation
    • Variants
    • References
  • Crank–Nicolson Method
    • Introduction
    • Principle
    • Example: 1D Diffusion
    • Example: 1D Diffusion with advection for steady flow, with multiple channel connections
    • Example: 2D Diffusion
    • Crank–Nicolson for nonlinear problems
    • Application in financial mathematics
    • References
  • Sylvester Equation
    • Existence and uniqueness of the solutions
    • Roth's removal rule
    • Numerical solutions
    • References
  • Bartels–Stewart Algorithm
    • Introduction
    • The Algorithm
      • Computational Cost
      • Simplifications and Special Cases
    • The Hessenberg–Schur algorithm
    • References
  • Triangular Matrix
    • Description
    • Forward and Back Substitution
      • Forward Substitution
    • Properties
    • Special Forms
      • Unitriangular Matrix
      • Strictly Triangular Matrix
      • Atomic Triangular Matrix
        • Lower Block Triangular Matrix
        • Upper Block Triangular Matrix
    • Triangularizability
      • Simultaneous Triangularizability
    • Algebras of Triangular Matrices
      • Borel Subgroups and Borel Subalgebras
      • Examples
    • References
  • QR Decomposition
    • Introduction
    • Cases and Definitions
      • Square Matrix
      • Rectangular Matrix
      • QL, RQ and LQ Decompositions
    • Computing the QR Decomposition
      • Using the Gram–Schmidt Process
        • Example
        • Relation to RQ decomposition
        • Advantages and Disadvantages
      • Using Householder Reflections
        • Example
        • Advantages and Disadvantages
      • Using Givens Rotations
        • Example
        • Advantages and Disadvantages
    • Connection to a Determinant or Product of Eigenvalues
    • Column Pivoting
    • Using for Solution to Linear Inverse Problems
    • Generalizations
    • References
  • Gershgorin Circle Theorem
    • Introduction
    • Statement and Proof
    • Discussion
    • Strengthening of the Theorem
    • Application
    • Example
    • References
  • Condition Number
    • Introduction
    • General Definition in the Context of Error Analysis
    • Matrices
    • Non-linear
      • One Variable
      • Several Variables
    • References
  • Unitary Matrix
    • Introduction
    • Properties
    • Equivalent Conditions
    • Elementary Constructions
      • 2 × 2 Unitary Matrix
    • References
  • Matrix Norm
    • Introduction
    • Preliminaries
    • Matrix Norms induced by Vector Norms
      • Matrix Norms induced by Vector p-Norms
        • p = 1, ∞
        • Spectral Norm (p = 2)
      • Matrix Norms induced by Vector α- and β-norms
      • Properties
      • Square Matrices
    • Consistent and Compatible Norms
    • Entry-wise Matrix Norms
      • L (2,1) and L (p,q) Norms
      • Frobenius Norm
      • Max Norm
    • Schatten Norms
    • Monotone Norms
    • Cut Norms
    • Equivalence of Norms
      • Examples of Norm Equivalence
    • References
  • Spectral Radius
    • Introduction
    • Definition
      • Matrices
      • Bounded Linear Operators
      • Graphs
    • Upper Bounds
      • Upper Bounds on the Spectral Radius of a Matrix
      • Upper Bounds on the Spectral Radius of a Graph
    • Power Sequence
    • Gelfand's Formula
      • Theorem
      • Proof
      • Corollary
    • References

DROP Specifications