NumericalMathematics

Numerical Mathematics

This task view on numerical mathematics lists R packages and functions that are useful for solving numerical problems in linear algebra and analysis. It shows that R is a viable computing environment for implementing and applying numerical methods, also outside the realm of statistics.

The task view will not cover differential equations, optimization problems and solvers, or packages and functions operating on times series, because all these topics are treated extensively in the corresponding task views DifferentialEquations, Optimization, and TimeSeries. All these task views together will provide a good selection of what is available in R for the area of numerical mathematics. The HighPerformanceComputing task view with its many links for parallel computing may also be of interest.

The task view has been created to provide an overview of the topic. If some packages are missing or certain topics in numerical math should be treated in more detail, please let the maintainer know.

Numerical Linear Algebra

As statistics is based to a large extent on linear algebra, many numerical linear algebra routines are present in R, and some only implicitly. Examples of explicitly available functions are vector and matrix operations, matrix (QR) decompositions, solving linear equations, eigenvalues/-vectors, singular value decomposition, or least-squares approximation.

  • The recommended package Matrix provides classes and methods for dense and sparse matrices and operations on them, for example Cholesky and Schur decomposition, matrix exponential, or norms and conditional numbers for sparse matrices.
  • Recommended package MASS adds generalized (Penrose) inverses and null spaces of matrices.
  • expm computes the exponential, logarithm, and square root of square matrices, but also powers of matrices or the Frechet derivative. expm() is to be preferred to the function with the same name in Matrix.
  • SparseM provides classes and methods for sparse matrices and for solving linear and least-squares problems in sparse linear algebra
  • Package rmumps provides a wrapper for the MUMPS library, solving large linear systems of equations applying a parallel sparse direct solver
  • svd provides R bindings to state-of-the-art implementations of singular value decomposition (SVD) and eigenvalue/eigenvector computations. Package ssvd will obtain sparse SVDs using an iterative thresholding method, while irlba will compute approximate singular values/vectors of large matrices.
  • The packages geigen and QZ compute generalized eigenvalues and -vectors for pairs of matrices, and QZ (generalized Schur) decompositions.
  • eigeninv generates matrices with a given set of eigenvalues ('inverse eigenvalue problem').
  • Package rARPACK, a wrapper for the ARPACK library, is typically used to compute only a few eigenvalues/vectors, e.g., a small number of largest eigenvalues.
  • Package RSpectra interfaces the 'Spectra' library for large-scale eigenvalue decomposition and SVD problems.
  • optR uses elementary methods of linear algebra (Gauss, LU, CGM, Cholesky) to solve linear systems.
  • matrixcalc contains a collection of functions for matrix calculations, special matrices, and tests for matrix properties, e.g., (semi-)positive definiteness.
  • Package onion contains routines for manipulating quaternions and octonians (normed division algebras over the real numbers); quaternions can be useful for handling rotations in three-dimensional space.
  • Packages RcppArmadillo and RcppEigen enable the integration of the C++ template libraries 'Armadillo' resp. 'Eigen' for linear algebra applications written in C++ and integrated in R using Rcpp for performance and ease of use.

Special Functions

Many special mathematical functions are present in R, especially logarithms and exponentials, trigonometric and hyperbolic functions, or Bessel and Gamma functions. Many more special functions are available in contributed packages.

  • Package gsl provides an interface to the 'GNU Scientific Library' that contains implementations of many special functions, for example the Airy and Bessel functions, elliptic and exponential integrals, the hypergeometric function, Lambert's W function, and many more.
  • Airy and Bessel functions, for real and complex numbers, are also computed in package Bessel, with approximations for large arguments.
  • Package pracma includes special functions, such as error functions and inverses, incomplete and complex gamma function, exponential and logarithmic integrals, Fresnel integrals, the polygamma and the Riemann zeta function.
  • appell computes Gauss' 2F1 and Appell's F1 hypergeometric functions for complex parameters and arguments quite accurately.
  • The hypergeometric (and generalized hypergeometric) function, is computed in hypergeo, including transformation formulas and special values of the parameters.
  • Elliptic and modular functions are available in package elliptic, including the Weierstrass P function and Jacobi's theta functions. There are tools for visualizing complex functions.
  • Package lamW implements both real-valued branches of the Lambert W function (using Rcpp).

Polynomials

Function polyroot() in base R determines all zeros of a polynomial, based on the Jenkins-Traub algorithm. Linear regression function lm() can perform polynomial fitting when using poly() in the model formula (with option raw = TRUE).

  • Packages polynom and PolynomF provide similar functionality for manipulating univariate polynomials, like evaluating polynomials (Horner scheme), differentiating or integrating them, or solving polynomials, i.e. finding all roots (based on an eigenvalue computation).
  • Package MonoPoly fits univariate polynomials to given data, applying different algorithms.
  • For multivariate polynomials, package multipol provides various tools to manipulate and combine these polynomials of several variables.
  • Package mpoly facilitates symbolic manipulations on multivariate polynomials, including basic differential calculus operations on polynomials, plus some Groebner basis calculations.
  • Package orthopolynom consists of a collection of functions to construct orthogonal polynomials and their recurrence relations, among them Chebyshev, Hermite, and Legendre polynomials, as well as spherical and ultraspherical polynomials. There are functions to operate on these polynomials.

Differentiation and Integration

D() and deriv() in base R compute derivatives of simple expressions symbolically. Function integrate() implements an approach for numerically integrating univariate functions in R. It applies adaptive Gauss-Kronrod quadrature and can handle singularities and unbounded domains to a certain extent.

  • Package Deriv provides an extended solution for symbolic differentiation in R; the user can add custom derivative rules, and the output for a function will be an executable function again.
  • numDeriv sets the standard for numerical differentiation in R, providing numerical gradients, Jacobians, and Hessians, computed by simple finite differences, Richardson extrapolation, or the highly accurate complex step approach.
  • Package pracma contains functions for computing numerical derivatives, including Richardson extrapolation or complex step. fderiv() computes numerical derivatives of higher orders. pracma has several routines for numerical integration: adaptive Lobatto quadrature, Romberg integration, Newton-Cotes formulas, Clenshaw-Curtis quadrature rules. integral2() integrates functions in two dimensions, also for domains characterized by polar coordinates or with variable interval limits.
  • Package gaussquad contains a collection of functions to perform Gaussian quadrature, among them Chebyshev, Hermite, Laguerre, and Legendre quadrature rules, explicitly returning nodes and weights in each case. Function gaussquad() in package statmod does a similar job.
  • Package fastGHQuad provides a fast Rcpp-based implementation of (adaptive) Gauss-Hermite quadrature.
  • Adaptive multivariate integration over hyper-rectangles in n-dimensional space is available in package cubature as function adaptIntegrate(), based on a C library of the same name. The integrand functions can even be multi-valued.
  • Multi-dimensional numerical integration is also covered in package R2Cuba, a wrapper around the C library Cuba. With vegas() it includes an approach to Monte Carlo integration based on importance sampling.
  • mvQuad provides methods for generating multivariate grids that can be used for multivariate integration. These grids will be based on different quadrature rules such as Newton-Cotes or Gauss quadrature formulas.
  • Package SparseGrid provides another approach to multivariate integration in high-dimensional spaces. It creates sparse n-dimensional grids that can be used as with quadrature rules.
  • Package SphericalCubature employs cubature to integrate functions over unit spheres and balls in n-dimensional space; SimplicialCubature provides methods to integrate functions over m-dimensional simplices in n-dimensional space. Both packages comprise exact methods for polynomials.
  • Package polyCub holds some routines for numerical integration over polygonal domains in two dimensions.
  • Package Pade calculates the numerator and denominator coefficients of the Pade approximation, given the Taylor series coefficients of sufficient length.
  • madness allows to compute derivatives of multivariate functions, defined by matrix operations, via forward differentiation and the chain rule.
  • features extracts features from functional data, such as first and second derivatives, or curvature at critical points, while RootsExtremaInflections finds roots, extrema and inflection points of curves defined by discrete points.

Interpolation and Approximation

Base R provides functions approx() for constant and linear interpolation, and spline() for cubic (Hermite) spline interpolation, while smooth.spline() performs cubic spline approximation. Base package splines creates periodic interpolation splines in function periodicSpline().

  • Interpolation of irregularly spaced data is possible with the akima package: aspline() for univariate data, bicubic() or interp() for data on a 2D rectangular domain. (This package is distributed under ACM license and not available for commercial use.)
  • Package signal contains several filters to smooth discrete data, notably interp1() for linear, spline, and cubic interpolation, pchip() for piecewise cubic Hermite interpolation, and sgolay() for Savitzky-Golay smoothing.
  • Package pracma provides barycentric Lagrange interpolation (in 1 and 2 dimensions) in barylag() resp. barylag2d(), 1-dim. akima in akimaInterp(), and interpolation and approximation of data with rational functions, i.e. in the presence of singularities, in ratinterp() and rationalfit().
  • sinterp() in package stinepack realizes interpolation based on piecewise rational functions by applying Stineman's algorithm. The interpolating function will be monoton in regions where the specified points change monotonically.
  • Schumaker() in package schumaker implements shape-preserving splines, guaranteed to be monotonic resp. concave or convex if the data is monotonic, concave, or convex.
  • Package conicfit provides several (geometric and algebraic) algorithms for fitting circles, ellipses, and conics in general.

Root Finding

uniroot(), implementing the Brent-Decker algorithm, is the basic routine in R to find roots of univariate functions. There are implementations of the bisection algorithm in several contributed packages. For root finding with higher precision there is function unirootR() in the multi-precision package Rmpfr. And for finding roots of multivariate functions see the following two packages:

  • For solving nonlinear systems of equations the BB package provides (non-monotone) Barzilai-Borwein spectral methods in sane(), including a derivative-free variant in dfsane(), and multi-start features with sensitivity analysis.
  • Package nleqslv solves nonlinear systems of equations using alternatively the Broyden or Newton method, supported by strategies such as line searches or trust regions.
  • ktsolve defines a common interface for solving a set of equations with BB or nleqslv.

Discrete Mathematics and Number Theory

Not so many functions are available for computational number theory. Note that integers in double precision can be represented exactly up to 2^53 - 1, above that limit a multi-precision package such as gmp is needed, see below.

  • Package numbers provides functions for factorization, prime numbers, twin primes, primitive roots, modular inverses, extended GCD, etc. Included are some number-theoretic functions like divisor functions or Euler's Phi function.
  • contfrac contains various utilities for evaluating continued fractions and partial convergents.
  • The partitions package enumerates additive partitions of integers, including restricted and unequal partitions.
  • permutations treats permutations as invertible functions of finite sets and includes several mathematical operations on them.
  • Package combinat generates all permutations or all combinations of a certain length of a set of elements (i.e. a vector); it also computes binomial coefficients.
  • magic creates and investigates magical squares.

Multi-Precision Arithmetic and Symbolic Mathematics

  • Multiple precision arithmetic is available in R through package gmp that interfaces to the GMP C library. Examples are factorization of integers, a probabilistic prime number test, or operations on big rationals -- for which linear systems of equations can be solved.
  • Multiple precision floating point operations and functions are provided through package Rmpfr using the MPFR and GMP libraries. Special numbers and some special functions are included, as well as routines for root finding, integration, and optimization in arbitrary precision.
  • Brobdingnag handles very large numbers by holding their logarithm plus a flag indicating their sign. (An excellent vignette explains how this is done using S4 methods.)
  • Package rSymPy accesses the symbolic algebra system 'SymPy' (written in Python) from R. It supports arbitrary precision computations, linear algebra and calculus, solving equations, discrete mathematics, and much more.
  • Package Ryacas interfaces the computer algebra system 'Yacas'. It supports symbolic and arbitrary precision computations in calculus and linear algebra.

MATLAB, Octave, and Python Interfaces

Interfaces to numerical computation software such as MATLAB (commercial) or Octave (free) will be important when solving difficult numerical problems. (Please note that the commercial programs SAS and Mathematica do have facilities to call R functions.)

  • The matlab emulation package contains about 30 simple functions, replicating MATLAB functions, using the respective MATLAB names and being implemented in pure R.
  • Package R.matlab provides tools to read and write MAT files, which is the MATLAB data format. It also enables a one-directional interface with a MATLAB process, sending and retrieving objects through a TCP/IP connection.
  • Package RcppOctave provides an interface to Octave (a MATLAB clone). It enables calling Octave functions, passing variables between R and Octave, and browsing Octave documentation.
  • Python, through its modules 'NumPy', 'SciPy', 'Matplotlib', 'SymPy', and 'pandas', has elaborate and efficient numerical and graphical tools available. R package rPython permits calls from R to Python, while RPy (with Python module 'rpy2') interfaces R from Python.
  • 'pyRserve' is a Python module for connecting Python to an R process running Rserve as an RPC gateway. This R process can run on a remote machine, variable access and function calls will be delegated through the network.

View on CRAN

7 months ago

Hans W. Borchers

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