Numerical calculus

The source code for the C# tests in Visual Studio is divided by in 2 parts:

Part 1 contains procedures from DAMath, Boost Math, Boost Odeint and Eigen. The source code for the C# tests in Visual Studio can be found online in the XlCalcNet repository or in the corresponding local XlCalcNet folder in the file B05a_NumericalCalculus.cs.

Part 2 is devoted to numerical integration in ball arithmetic. The source code for the C# tests in Visual Studio can be found online in the XlCalcNet repository or in the corresponding local XlCalcNet folder in the file B05b_FlintNumericalCalculus.cs.

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()