قائمة مواضيع التحليل العددي

تحليل عددي

This is a list of numerical analysis topics, by Wikipedia page.

مواضيع عامة

خوارزمية جمع كاهان Kahan summation algorithm
كيفية تكرارية Iterative method
استكمال ريتشاردسون Richardson extrapolation
تقدير كثيرات الحدود:
- مخطط هورنر Horner scheme
- خوارزمية كلينشاوClenshaw algorithm
- خوارزمية دي كاستيلجاوDe Casteljau's algorithm
تقدير الدوال الخاصة:
- توليد الدوال المثلثية Generating trigonometric tables
- طرق حوسبة الجذور التربيعية Methods of computing square roots
- خوارزمية الجذر النوني n-th root algorithm
- CORDIC
- تقريب لانكزوس Lanczos approximation
طريقة مجموعة المستوى Level set method
Abramowitz and Stegun
Curse of dimensionality
تقارب فائق Superconvergence
Termination
Discretization
- Sequence transformations
Difference quotient

الخطأ

تقريب Approximation
عدد شرطي Condition number
إستقرار رقمي Numerical stability
Well-posed problem
Significant figures
Loss of significance
Propagation of errors resulting from algebraic manipulations
Precision (arithmetic)
Hilbert matrix
Floating point number
Truncation
Round-off error
Discretization error
Approximation error
Percent error

جبر خطي عددي

أنماط المصفوفات التي تظهر في التحليل العددي :

- Sparse matrix
- Circulant matrix
- Pentadiagonal matrix
- Triangular matrix
خوارزميات جداء المصفوفات:
- Strassen algorithm
- Coppersmith-Winograd algorithm
حل جملة معادلات خطية:
- Gaussian elimination
- LU decomposition
  - Crout matrix decomposition
- Block LU decomposition
- LDU decomposition
- Cholesky decomposition
  - Minimum degree algorithm
  - Symbolic Cholesky decomposition
- Conjugate gradient method
خوارزمية القيمة الخاصة Eigenvalue algorithm :
- Power method
- Inverse iteration
- Rayleigh quotient iteration
- Arnoldi iteration
- QR algorithm
- Divide-and-conquer eigenvalue algorithm
تعامد Orthogonalization :
- Gram-Schmidt process
- Householder transformation
- Givens rotation
QR decomposition
Krylov subspace

استيفاء

Interpolation

Nearest Neighbor Interpolation
Polynomial interpolation
- Linear interpolation
- Runge's phenomenon
- Vandermonde matrix
- Chebyshev polynomials
- Chebyshev nodes
- Lebesgue constant (interpolation)
- Different forms for the interpolant:
  - Newton polynomial
    - Divided differences
  - Lagrange polynomial
  - Bernstein polynomial
- Extensions to multiple dimensions:
  - Bilinear interpolation
  - Trilinear interpolation
  - Bicubic interpolation
- Hermite interpolation
- Birkhoff interpolation
Spline interpolation
- Spline
- Cubic Hermite spline
- Hermite spline
- Cardinal spline
- Kochanek-Bartels spline
- Catmull-Rom spline
- de Boor algorithm
- B-spline
  - Truncated power function
- Nonuniform rational B-spline (NURBS)
- Bézier spline
  - Bézier curve
  - de Casteljau's algorithm
- See also: List of numerical computational geometry topics
Slerp
Wavelet
- Continuous wavelet
  - Continuous wavelet transform
Inverse distance weighting
Trigonometric interpolation
- Fast Fourier transform
  - Bluestein's FFT algorithm
  - Bruun's FFT algorithm
  - Cooley-Tukey FFT algorithm
  - Goertzel algorithm
  - Prime-factor FFT algorithm
  - Rader's FFT algorithm
- Sigma approximation
Irrational base discrete weighted transform
Pareto interpolation
Extrapolation
Regression analysis
- Isotonic regression
Approximation theory
- Orders of approximation
- Lebesgue's lemma
- Curve fitting
- Different approximations:
  - Linear approximation
  - Padé approximant
Curve-fitting compaction

إيجاد جذور معادلة خطية

Root-finding algorithm
Bisection method
False position method
Newton's method
Secant method
Müller's method
Inverse quadratic interpolation
Brent's method
Laguerre's method
Shifting nth-root algorithm
Lehmer-Schur algorithm
Wilkinson's polynomial

استمثال

مصطلحات:
- Active set
- Candidate solution
- Global optimum and Local optimum
- Maxima and minima
Continuous optimization
برمجة خطية (أيضا تتعامل مع البرمجة السليمة Integer Programming)
- Simplex algorithm
- Interior point method
  - Mehrotra predictor-corrector method
- Mixed integer programming
- Linear complementarity problem
Quadratic programming
- Linear least squares
- Frank-Wolfe algorithm
Convex optimization
Nonlinear programming
- Descent direction
- Linesearch
  - Backtracking linesearch
  - Wolfe conditions
- Gradient descent
  - Stochastic gradient descent
- Newton's method in optimization
- BFGS method
- Nelder-Mead method
- Golden section search
- Tabu search
- Nonlinear least squares
  - Expectation-maximization algorithm
  - Gauss-Newton algorithm
  - Levenberg-Marquardt algorithm
Combinatorial optimization
Stochastic programming
Dynamic programming
- Bellman equation
Global optimization:
- BRST algorithm
- MCS algorithm
Random optimization algorithms:
- Simulated annealing
- Evolutionary algorithm
  - Differential evolution
- Genetic algorithm, genetic programming
- Particle swarm optimization
  - RPSO
- Stochastic tunneling
- see also the section Monte Carlo method
Infinite-dimensional optimization
- Optimal control
- Shape optimization
Optimal substructure
Theoretical aspects:
- Farkas's lemma
- Karush-Kuhn-Tucker conditions
- Lagrange multipliers
- No-free-lunch theorem, No-Free-Lunch theorems
تطبيقات:
- Automatic label placement
- Cutting stock problem
- Expenditure minimization problem
- Inventory control problem
- Job-shop problem
- Multidisciplinary design optimization
- Paper bag problem

تكامل عددي

Numerical integration

Trapezium rule
Simpson's rule
Newton-Cotes formulas
Gaussian quadrature
Romberg's method
Sparse grid
Numerical differentiation
Euler-Maclaurin formula

معادلات تفاضلية نظامية عددية

معادلات تفاضلية نظامية عددية — الحل العددي للمعادلات التفاضلية النظامية (ODEs)

Euler integration — the most basic method for solving an ODE
Runge-Kutta methods — one of the two main classes of methods for initial value problems
- Midpoint method — a second-order method with two stages
Multistep method — the other main class of methods for initial value problems
Newmark-beta method — a method specifically designed for the solution of problems from classical physics
Verlet integration — another method for problems from classical physics
Geometric integrator — a method that preserves some geometric structure of the equation
- Symplectic integrator — a method for the solution of Hamilton's equations that preserves the symplectic structure
Stiff equation — roughly, an ODE for which the unstable methods needs a very short step size, but stable methods do not.
Shooting method — a method for the solution of boundary value problems
- Illustration of the shooting method

معادلات تفاضلية جزئية عددية

Numerical partial differential equations — the numerical solution of partial differential equations (PDEs)

Methods for the solution of PDEs:
- Finite difference method — based on approximating differential operators with difference operators
  - Discrete Laplace operator — finite-difference approximation of the Laplace operator
  - Crank-Nicolson method — second-order method for heat and related PDEs
- Finite element method, finite element analysis — based on a discretization of the space of solutions
  - Galerkin method — a finite element method in which the residual is orthogonal to the finite element space
  - Rayleigh-Ritz method — a finite element method based on variational principles
  - Direct stiffness method — a particular implementation of the finite element method, often used in structural analysis
- Spectral method — based on the Fourier transformation
  - Pseudo-spectral method
- Boundary element method — based on transforming the PDE to an integral equation on the boundary of the domain
- Analytic element method — similar to the boundary element method, but the integral equation is evaluated analytically
- Finite volume method — based on dividing the domain in many small domains; popular in computational fluid dynamics
- Discrete element method — a method in which the elements can move freely relative to each other
Techniques for improving these methods:
- Multigrid, multigrid method — uses a hierarchy of nested meshes to speed up the methods
- Domain decomposition method — divides the domain in a few subdomains and solves the PDE on these subdomains
- Adaptive mesh refinement — uses the computed solution to refine the mesh only where necessary
Mimetic methods — methods that respect in some sense the structure of the original problem
Numerical diffusion — diffusion introduced by the numerical method, above to that which is naturally present
Numerical resistivity — the same, with resistivity instead of diffusion
Multiphysics — models consisting of various submodels with different physics

طريقة مونتي كارلو

متغيرات طريقة مونتي كارلو:
- Direct simulation Monte Carlo
- Quasi-Monte Carlo method
- Markov chain Monte Carlo
  - Metropolis-Hastings algorithm
  - Gibbs sampling
- Dynamic Monte Carlo method
Box-Muller transform
Low-discrepancy sequence
- Constructions of low-discrepancy sequences
- Illustration of a low-discrepancy sequence
Parallel tempering
Techniques for reducing the variance:
- Control variate
- Importance sampling
- VEGAS algorithm
تطبيقات:
- Metropolis light transport
- Monte Carlo methods in finance
- Quantum Monte Carlo
Also see the list of statistics topics

تطبيقات

Climate model
Computational chemistry
- Coupled cluster
- Density functional theory
- Self-consistent field method
Computational fluid dynamics
- Large eddy simulation
- Smoothed particle hydrodynamics
Computational physics

برمجيات

See also: List of numerical analysis software
Libraries:
- Basic Linear Algebra Subprograms
- dnAnalytics
- EISPACK
- GNU Scientific Library
- LAPACK
- LINPACK
- Netlib
- Numerical Recipes
- SciPy
لغات برمجة :
- bc programming language
- F programming language
- Fortran
- General Algebraic Modeling System
- J programming language
- Numerical Python
- PDL
- R programming language
برامج :
- MATLAB
- GNU Octave
- Scilab
- Rlab
- TK Solver

منطقات تتطلب تصنيف

Courant–Friedrichs–Lewy condition -- Explicit and implicit methods -- Finite element method in structural mechanics -- MISER algorithm -- Newton fractal -- Pseudospectrum -- Uniform theory of diffraction -- Weak formulation -- Radial basis function -- OpenBUGS -- Particle filter -- Braess' paradox -- Convex analysis -- Corner solution -- Discrete optimization -- Goal programming -- Pareto analysis -- Pareto principle -- Partial evaluation -- Subderivative -- Trust region -- Utility maximization problem -- Branch and bound -- Cutting-plane method -- Delayed column generation -- Karmarkar's algorithm -- MTD-f -- Reactive search -- Effective fitness -- Evolution strategy -- Evolutionary art -- Gene expression programming -- Human-based genetic algorithm -- Interactive evolutionary computation -- Interactive genetic algorithm -- Learning classifier system -- Memetic algorithm -- Neuroevolution -- Bairstow's method -- Durand-Kerner method -- Splitting circle method --

Complex wavelet transform / Discrete wavelet transform
Greedy algorithm, Greedy randomized adaptive search procedure, Hill climbing, Local search (optimization), Random-restart hill climbing (currently more about discrete optimization, but may be expanded to cover continuous optimization)
MANIAC II (computer)
Operations research (what is the relation between OR and optimization?)
Secular equation / Secular function
Structural analysis (very applied)

Category:Numerical analysis
Category:Approximation theory
Category:FFT algorithms
Category:Interpolation
Category:Monte Carlo method
Category:Numerical software is excluded, because it is covered by List of numerical analysis software
Category:Numerical linear algebra
Category:Optimization
- Category:Optimization algorithms
  - Category:Evolutionary algorithms
Category:Root-finding algorithms
Category:Splines

Category:Numerical software -- Category:Optimization software -- Category:Combinatorial optimization -- Category:Genetic algorithms -- Category:Genetic programming -- Category:Interactive evolutionary computation