قائمة مواضيع التحليل العددي

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 تحليل عددي

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

 مواضيع عامة

• خوارزمية جمع كاهان Kahan summation algorithm
• طريقة تكرارية Iterative method
• استكمال ريتشاردسون Richardson extrapolation
• تقدير كثيرات الحدود:
  • مخطط هورنر Horner scheme
  • خوارزمية كلينشاو Clenshaw algorithm
  • خوارزمية دي كاستيلجاو De 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 الصحيحة 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

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 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

 مقالات 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