Numerical calculus

Numerical calculus#

Numerical calculus

Introduction
- Delegates, lambda expressions, callback functions: an example
- Usage in Python
- Usage in CSharp
DAMath: Numerical Rootfinding and Minimization
- Quadratic Equation, Real Coefficients
  - ctx.squadx()
- Cubic Equation, Real Coefficients
  - ctx.cubsolve()
- Brent’s algorithm for finding a local minimum
  - math53.localmin()
- Modified Brent’s algorithm for finding a local minimum
  - math53.mbrent()
- Rootfinding: Brent/Dekker algorithm
  - math53.zbrent()
- Rootfinding: Simplified Brent/Dekker algorithm
  - math53.zeroin()
Boost/Math: Root Finding and Minimization Algorithms
- Root Finding Without Derivatives (TOM 748 algorithm)
  - ctxboost.BracketRoot()
- Root Finding With Derivatives: Newton-Raphson
  - ctxboost.NewtonRaphson()
- Root Finding With Derivatives: Halley
  - ctxboost.Halley()
- Root Finding With Derivatives: Schröder
  - ctxboost.Schroeder()
- Locating Function Minima using Brent’s algorithm
  - ctx.BrentMinimum()
Mpmath: Rootfinding and optimization
- General root-finding interface
  - ctx.findroot()
- Bisection algorithm
  - ctx.bisection()
- Secant algorithm
  - ctx.secant()
- Illinois, Pegasus, Anderson algorithms
  - ctx.illinois()
- Muller algorithm
  - ctx.muller()
- Ridder algorithm
  - ctx.ridder()
- Brent algorithm (to be implemented from Scipy brent)
  - ctx.brent()
- Newton algorithm
  - ctx.newton_c()
- Newton-Steffenson algorithm (ANewton)
  - ctx.newton_steffenson_c()
- Modified Newton algorithm (MNewton)
  - ctx.mnewton_c()
- Halley algorithm
  - ctx.halley_c()
- Roots of a vector function (MDNewton, to be implemented)
  - ctx.md_newton_c()
- Levenberg-Marquardt algorithm
  - ctx.levenberg_marquardt_c()
- BFGS algorithm.
  - ctx.bfgs_c()
- L-BFGS algorithm.
  - ctx.lbfgs_c()
- Conjugate gradient.
  - ctx.conj_grad_c()
DAMath: Numerical Quadrature
- General error codes for Amath integration functions, type 1
- General error codes for Amath integration functions, type 2
- Global adaptive quadrature by Forsythe, Malcolm, Moler (quanc8)
  - ctx.quanc8()
- 21-point Gauss-Kronrod rule, finite interval (qags)
  - ctx.qags()
- 15-point Gauss-Kronrod rule, infinite interval (qagi)
  - math53.qagi()
- Cauchy principal value, finite interval (qawc)
  - math53.qawc()
- Double Exponential (DE) transformation, finite interval (intde)
  - math53.intde()
- DE transformation, infinite interval, no oscillatory factor (intdei)
  - math53.intdei()
- DE transformation, infinite interval, oscillatory factor (intdeo)
  - math53.intdeo()
Boost/Math: Numerical integration
- Trapezoidal Quadrature
  - ctxboost.Trapezoidal()
- Gauss-Legendre quadrature
  - ctxboost.GaussLegendre()
- Gauss-Kronrod Quadrature
  - ctxboost.GaussKronrod()
- Double-exponential quadrature: tanh_sinh
  - ctxboost.TanhSinh()
- Double-exponential quadrature: sinh_sinh
  - ctxboost.SinhSinh()
- Double-exponential quadrature: exp_sinh
  - ctxboost.ExpSinh()
- Fourier Integral, Cosine
  - ctxboost.OouraCos()
- Fourier Integral, Sine
  - ctxboost.OouraSin()
- Calculating the pdf from the characteristic function
  - ctx.pdf_from_cf_boost()
- Calculating the cdf from the characteristic function
  - ctx.cdf_from_cf_continuous_boost()
Mpmath: Numerical integration
- General quadrature interface
  - ctx.quad()
- Quadrature with subdivision
  - ctx.quadsubdiv()
- Quadrature of oscillatory functions (Filon)
  - ctx.quadosc()
- Verified numerical integration (Okayama)
  - quad_verified()
- Specialized Gauss quadrature rules
  - ctx.gauss_quad_rules()
Mpmath: Numerical inverse Laplace transform
- General inverse Laplace transform interface
  - ctx.invertlaplace()
- Talbot method
  - ctx.invlaptalbot()
- Stehfest method
  - ctx.invlapstehfest()
- de Hoog, Knight, and Stokes method
  - ctx.invlapdehoog()
- Cohen method
  - ctx.invlapcohen()
Boost/Odeint: Ordinary differential equations
- Runge-Kutta 4 method, constant stepper
  - ctxboost.RungeKutta4Const()
- Cash-Karp method, constant stepper
  - ctxboost.CashKarpConst()
- Dormand-Prince 5 method, constant stepper
  - ctxboost.DormandPrince5Const()
- Fehlberg 78 method, constant stepper
  - ctxboost.Fehlberg78Const()
- Adams-Bashforth-Moulton, constant stepper
  - ctxboost.AdamsBashforthMoultonConst()
- Cash-Karp method, adaptive stepper
  - ctxboost.CashKarpAdaptive()
- Dormand-Prince 5 method, adaptive stepper
  - ctxboost.DormandPrince5Adaptive()
- Fehlberg 78 method, adaptive stepper
  - ctxboost.Fehlberg78Adaptive()
- Bulirsch-Stoer method, adaptive stepper
  - ctxboost.BulirschStoerAdaptive()
- Dormand-Prince 5 method, dense output stepper
  - ctxboost.DormandPrince5DenseOutput()
- Bulirsch-Stoer method, dense output stepper
  - ctxboost.BulirschStoerDenseOutput()
Mpmath: Numerical differentiation
- First derivative, using the complex-step derivative approximation
  - ctx.deriv1_c()
- Second derivative, using the complex-step derivative approximation
  - ctx.deriv2_c()
- Gradient, using the complex-step derivative approximation
  - ctx.gradient_c()
- Jacobi matrix, using the complex-step derivative approximation
  - ctx.jacobi_c()
- Nth numerical (partial) derivative, using finite differences or numerical quadrature
  - ctx.diff()
- Function object which evaluates the nth derivative of a given function
  - ctx.diffun()
- Forward difference, based on a given sequence
  - ctx.difference()
- Generating a sequence of derivatives
  - ctx.diffs()
- Composition of derivatives
  - ctx.diffs_prod()
- Composition of exponential of derivatives
  - ctx.diffs_exp()
- Fractional derivatives / differintegration
  - ctx.differint()
- Taylor series
  - ctx.taylor()
- Solving an ODE using high-order Taylor series
  - ctx.odefun()
Mpmath: Asymptotic expansions
- Edgeworth expansion: general approximation of the pdf, cdf and sf
  - ctx.edgeworth()
- Cornish-Fisher expansion: general approximation of the qtf and isf
  - ctx.cornish_fisher()
- Sheppard correction, cumulants
  - ctx.sheppard()
- The Sheppard correction, cumulant generating function
  - ctx.sheppard_per_cgf()
- Luggannini-Rice expansion: general approximation of the pdf, cdf, and sf
  - ctx.luggannini_rice()
- Jensen expansion: general approximation of the qtf and isf
  - ctx.jensen_inv()
- Box-Davis expansion: general approximation of the pdf, cdf and sf
  - ctx.box_davis_expansion()
- Box-Davis expansion: general approximation of the qtf and isf (for \(\omega_1 = 0\))
  - ctx.box_davis_expansion_inv()
Mpmath: Function approximation
- Pade approximation
  - ctx.pade()
- Chebyshev approximation
  - ctx.chebyfit()
- Fourier series
  - ctx.fourier()
- Fourier series evaluation
  - ctx.fourierval()
Mpmath: Sums, products, limits and extrapolation
- Summation of infinite series
  - ctx.nsum()
- Summation using the Euler-Maclaurin formula
  - ctx.sumem()
- Summation using the Abel-Plana formula
  - ctx.sumap()
- Products
  - ctx.nprod()
- Limits, general
  - ctx.limit()
- Richardson extrapolation
  - ctx.richardson()
- Shanks extrapolation
  - ctx.shanks()
- Levin extrapolation
  - ctx.levin()
- Cohan alternating extrapolation
  - ctx.cohen_alt()
Mpmath: Number identification
- Constant recognition
  - ctx.pslq()
- Algebraic identification
  - ctx.findpoly()
- Integer relations (PSLQ)
  - ctx.identify()
Mpmath: Polynomials
- Polynomial evaluation
  - ctx.polyval()
- Polynomial roots
  - ctx.polyroots()
Eigen: Polynomials
- RootsToMonicPolynomial
  - matA.RootsToMonicPolynomial()
- PolyEval
  - matA.PolyEval()
- PolyEvalComplex
  - matA.PolyEvalComplex()
- PolynomialSolver
  - matA.PolynomialSolver()
Eigen/MinPack: non linear optimization
- Multidimensional Rootfinding: Powell Hybrid
  - ctxboost.PowellHybrid()
- Nonlinear LeastSquares: Levenberg-Marquardt
  - ctxboost.LevenbergMarquardt()
Eigen/CppOptLib: multidimensional optimization
- Nelder-Mead Solver
  - ctxboost.NelderMead()
- CMAes solver
  - ctxboost.CMAes()
- BFGS Solver
  - ctxboost.BFGS()
- L-BFGS Solver
  - ctxboost.LBFGS()
- Gradient descent solver
  - ctxboost.GradientDescent()
- Conjugate gradient descent solver
  - ctxboost.ConjugateGradientDescent()
- Newton descent solver (description needs correction)
  - ctxboost.NewtonDescent()
Flint/Functions for polynomials
- Product of polynomials
  - poly.Times()
- Quotient of polynomials
  - poly.Div()
- Quotient and remainder of polynomials
  - poly.Divrem()
- Taylor shift of polynomials
  - poly.TaylorShift()
- Composition of polynomials
  - poly.Compose()
- Evaluation of polynomials, same type
  - poly.Evaluate()
- Evaluation of polynomials, complex number
  - poly.EvaluatePolyCplx()
- Polynomial from roots
  - poly.ProductRoots()
- Polynomial from roots, complex number
  - poly.ProductRootsCplx()
- Multipoint evaluation for polynomials
  - poly.EvaluateVec()
- Multipoint evaluation for polynomials, fast
  - poly.EvaluateVecFast()
- Interpolation for polynomials, Newton
  - poly.InterpolateNewton()
- Interpolation for polynomials, Lagrange
  - poly.InterpolateFast()
- Differentiation for polynomials
  - poly.Derivative()
- Integral of polynomials
  - poly.Integral()
- Roots of a polynomial, floating point input
  - poly.FindRoots()
- Root finding, integer input
  - poly.FmpzComplexRoots()
- Examples: poly_roots.c
- Polynomial square-free (integers)
  - poly.IsSquarefree()
- Convert to monic polynomial (rational)
  - poly.MakeMonic()
- Test for monic polynomial (rational)
  - poly.IsMonic()
- Test for square-free polynomial (rational)
  - poly.IsSquarefree2()
Flint/Power series and Taylor arithmetic
- Taylor Arithmetic
- Borel transform of power series
  - poly.BorelTransform()
- Inverse Borel transform of power series
  - poly.InvBorelTransform()
- Sum of power series
  - poly.PlusSeries()
- Difference of power series
  - poly.MinusSeries()
- Product of power series
  - poly.TimesSeries()
- Quotient of power series
  - poly.DivSeries()
- Inverse of power series
  - poly.InvSeries()
- Composition of power series
  - poly.ComposeSeries()
- Reversion of power series
  - poly.RevertSeries()
Flint/Verified numerical differentiation
- Finding “all” real roots within a finite interval
  - Apr.RealRoots()
- Taylor Polynomial: rigorous error bounds of \(f(x)-T_n(x)\)
- Chebyshev interpolation polynomials: rigorous error bounds of \(|f(x) - P_n(x)|\)
- Trapezoidal rule and related quadratures: approximate error bounds
Flint/Verified numerical integration
- Complex Tanh-Sinh integration (non-adaptive)
  - aflintc.DE_Integration()
- Complex Gauss-Legendre integration (non-adaptive)
  - aflintc.GLIntegration1()
- Complex Gauss-Legendre integration (adaptive)
  - aflintc.GLIntegration()