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NumericalMathematics
Numerical Mathematics
Hans W. Borchers, Robin Hankin, Serguei Sokol
2024-07-27

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 r view("DifferentialEquations"), r view("Optimization"), and r view("TimeSeries"). All these task views together will provide a good selection of what is available in R for the area of numerical mathematics. The r view("HighPerformanceComputing") task view with its many links for parallel computing may also be of interest.

Contents

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 contact the maintainer, either via e-mail or by submitting an issue or pull request in the GitHub repository linked above.

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 r pkg("Matrix", priority = "core") 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 r pkg("MASS") adds generalized (Penrose) inverses and null spaces of matrices.
  • r pkg("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 r pkg("Matrix").
  • r pkg("QZ") provides generalized eigenvalues and eigenvectors using a generalized Schur decomposition (QZ decomposition). It is able to compute the exponential of a matrix quite efficiently.
  • r pkg("EigenR") with complex Linear Algebra functions for determinant, rank, inverse, pseudo-inverse, kernel and image, QR decomposition, Cholesky decomposition, (complex) Schur decomposition, and Hessenberg decomposition.
  • r pkg("geigen") calculates Generalized Eigenvalues, the Generalized Schur Decomposition and the Generalized Singular Value Decomposition of a pair of (complex) matrices.
  • r pkg("SparseM") provides classes and methods for sparse matrices and for solving linear and least-squares problems in sparse linear algebra
  • r pkg("rmumps") provides a wrapper for the MUMPS library, solving large linear systems of equations applying a sparse direct solver.
  • r pkg("sanic") supports routines for solving (dense and sparse) large systems of linear equations; direct and iterative solvers from the Eigen C++ library are made available, including Cholesky, LU, QR, and Krylov subspace methods.
  • r pkg("Rlinsolve") is a collection of iterative solvers for sparse linear system of equations; stationary iterative solvers such as Jacobi or Gauss-Seidel, as well as nonstationary (Krylov subspace) methods are provided.
  • r pkg("svd") provides R bindings to state-of-the-art implementations of singular value decomposition (SVD) and eigenvalue/eigenvector computations. Package r pkg("irlba") will compute approximate singular values/vectors of large matrices.
  • Package r pkg("PRIMME") interfaces PRIMME, a C library for computing eigenvalues and eigenvectors of real symmetric or complex Hermitian matrices. It will find largest, smallest, or interior eigen-/singular values and will apply preconditioning to accelerate convergence.
  • Package r pkg("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 r pkg("RSpectra") interfaces the 'Spectra' library for large-scale eigenvalue decomposition and SVD problems.
  • r pkg("optR") uses elementary methods of linear algebra (Gauss, LU, CGM, Cholesky) to solve linear systems.
  • Package r pkg("mbend") for bending non-positive-definite (symmetric) matrices to positive-definiteness, using weighted and unweighted methods.
  • r pkg("matrixcalc") contains a collection of functions for matrix calculations, special matrices, and tests for matrix properties, e.g., (semi-)positive definiteness; mainly used for teaching and research purposes
  • r pkg("matlib") contains a collection of matrix functions for teaching and learning matrix linear algebra as used in multivariate statistical methods; mainly for tutorial purposes in learning matrix algebra ideas using R.
  • Package r pkg("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 r pkg("RcppArmadillo") and r pkg("RcppEigen") enable the integration of the C++ template libraries 'Armadillo' resp. 'Eigen' for linear algebra applications written in C++ and integrated in R using r pkg("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 r pkg("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. r pkg("RcppGSL") provides an easy-to-use interface between 'GSL' data structures and R, using concepts from 'Rcpp'.
  • Airy and Bessel functions, for real and complex numbers, are also computed in package r pkg("Bessel"), with approximations for large arguments.
  • Package r pkg("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 Dirichlet and Riemann zeta functions.
  • The hypergeometric (and generalized hypergeometric) function, is computed in r pkg("hypergeo"), including transformation formulas and special values of the parameters.
  • Elliptic and modular functions are available in package r pkg("elliptic"), e.g., the Weierstrass P function and Jacobi's theta functions. There are tools for visualizing complex functions.
  • r pkg("jacobi") evaluates Jacobi theta and related functions: Weierstrass elliptic functions, Weierstrass sigma and zeta function, Klein j-function, Dedekind eta function, lambda modular function, Jacobi elliptic functions, Neville theta functions, Eisenstein series. Complex values of the variable are supported.
  • r pkg("Carlson") evaluates Carlson elliptic and incomplete elliptic integrals (with compex arguments).
  • Package r pkg("expint") wraps C-functions from the GNU Scientific Library to calculate exponential integrals and the incomplete Gamma function, including negative values for its first argument.
  • r pkg("fourierin") computes Fourier integrals of functions of one and two variables using the Fast Fourier Transform.
  • r pkg("logOfGamma") uses approximations to compute the natural logarithms of the Gamma function for large values.
  • Package r pkg("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 r pkg("PolynomF", priority = "core") (recommended) and r pkg("polynom") provide similar functionality for manipulating univariate polynomials, like evaluating polynomials (Horner scheme), or finding their roots. 'PolynomF' generates orthogonal polynomials and provides graphical display features.
  • r pkg("polyMatrix") (based on 'polynom') implements basic matrix operations and provides thus an infrastructure for the manipulation of polynomial matrices.
  • Package r pkg("MonoPoly") fits univariate polynomials to given data, applying different algorithms.
  • For multivariate polynomials, package r pkg("multipol") provides various tools to manipulate and combine these polynomials of several variables.
  • Package r pkg("mpoly") facilitates symbolic manipulations on multivariate polynomials, including basic differential calculus operations on polynomials.
  • r pkg("mvp") enables fast manipulation of symbolic multivariate polynomials, using print and coercion methods from the 'mpoly' package, but offers speed improvements.
  • Package r pkg("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.
  • Symbolic calculation and evaluation of the Jack polynomials, zonal and Schur polynomials are available in package r pkg("jack").
  • The Free Algebra in R package r pkg("freealg") handles multivariate polynomials with non-commuting indeterminates.
  • Package r pkg("spray") provides functionality for sparse multidimensional arrays, interpreted as multivariate polynomials.
  • Package r pkg("qspray") provides symbolic calculation and evaluation of multivariate polynomials with rational coefficients, plus some Groebner basis calculations.
  • Package r pkg("minimaxApprox") implements the algorithm of Remez (1962) for polynomial minimax approximation and that of Cody et al. (1968) for rational minimax approximations.

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 r pkg("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.
  • r pkg("numDeriv", priority = "core") 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 r pkg("dual") achieves automatic differentiation (for univariate functions) by employing dual numbers; for a mathematical function its value and its exact first derivative are returned.
  • Package r github("Non-Contradiction/autodiffr") (on Github) provides an R wrapper for the Julia packages ForwardDiff.jl and ReverseDiff.jl to do automatic differentiation for native R functions.
  • r pkg("pracma") contains functions for computing numerical derivatives, including Richardson extrapolation or complex step. fderiv() computes numerical derivatives of higher orders. r pkg("pracma") also 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 polar coordinates or domains with variable interval limits.
  • r pkg("cubature", priority = "core") is a package for adaptive multivariate integration over hypercubes in n-dimensional space, based on the C-library 'cubature', resp. for deterministic and Monte-Carlo integration based on library 'Cuba'. Function 'cubintegrate()' wraps all the integration methods provided.
  • Package r pkg("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 r pkg("statmod") does a similar job.
  • r pkg("GramQuad") allows for numerical integration based on Gram polynomials.
  • Package r pkg("fastGHQuad") provides a fast r pkg("Rcpp")-based implementation of (adaptive) Gauss-Hermite quadrature.
  • r pkg("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 r pkg("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 r pkg("SphericalCubature") employs r pkg("cubature") to integrate functions over unit spheres and balls in n-dimensional space; r pkg("SimplicialCubature") provides methods to integrate functions over m-dimensional simplices in n-dimensional space. Both packages comprise exact methods for polynomials.
  • Package r pkg("polyCub") holds some routines for numerical integration over polygonal domains in two dimensions.
  • Package r pkg("Pade") calculates the numerator and denominator coefficients of the Pade approximation, given the Taylor series coefficients of sufficient length.
  • r pkg("calculus") provides efficient functions for high-dimensional numerical and symbolic calculus, including accurate higher-order derivatives, Taylor series expansion, differential operators, and Monte-Carlo integration in orthogonal coordinate systems.
  • r pkg("features") extracts features from functional data, such as first and second derivatives, or curvature at critical points, while r pkg("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 r pkg("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 r pkg("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 r pkg("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().
  • The r pkg("interp") package provides bivariate data interpolation on regular and irregular grids, either linear or using splines. Currently the piecewise linear interpolation part is implemented. (It is intended to provide a free replacement for the ACM licensed akima::interp and tripack::tri.mesh functions.)
  • Package r pkg("splines2") provides basis matrices of B-splines, M-splines, I-splines, convex splines (C-splines), periodic splines, natural cubic splines, generalized Bernstein polynomials, and their integrals (except C-splines) and derivatives by closed-form recursive formulas.
  • r pkg("bspline") uses B-splines for creating functions interpolating and smooting 1D data. fitsmbsp() can optimize knot positions and impose monotonicity and positivity constraints. Produced functions can be differentiated with dbsp() or integrated with ibsp().
  • r pkg("tripack") for triangulation of irregularly spaced data is a constrained two-dimensional Delaunay triangulation package providing both triangulation and generation of Voronoi mosaics of irregular spaced data.
  • sinterp() in package r pkg("stinepack") realizes interpolation based on piecewise rational functions by applying Stineman's algorithm. The interpolating function will be monotone in regions where the specified points change monotonically.
  • Schumaker() in package r pkg("schumaker") implements shape-preserving splines, guaranteed to be monotonic resp. concave or convex if the data is monotonic, concave, or convex.
  • r pkg("ADPF") uses least-squares polynomial regression and statistical testing to improve Savitzky-Golay smoothing.
  • Package r pkg("conicfit") provides several (geometric and algebraic) algorithms for fitting circles, ellipses, and conics in general.

Root Finding and Fixed Points

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 r pkg("Rmpfr"). For finding roots of univariate and multivariate functions see the following packages:

  • Package r pkg("itp") implements the Interpolate, Truncate, Project (ITP) root-finding algorithm. The user provides a univariate (1-dim.) function and the endpoints of an interval where the function values have different signs.
  • Package r pkg("rootSolve") includes function multiroot() for finding roots of systems of nonlinear (and linear) equations; it also contains an extension uniroot.all() that attempts to find all zeros of a univariate function in an intervall (excepting quadratic zeros).
  • For solving nonlinear systems of equations the r pkg("BB") package provides Barzilai-Borwein spectral methods in sane(), including a derivative-free variant in dfsane(), and multi-start features with sensitivity analysis.
  • Package r pkg("nleqslv") solves nonlinear systems of equations using alternatively the Broyden or Newton method, supported by strategies such as line searches or trust regions.
  • r pkg("ktsolve") defines a common interface for solving a set of equations with BB or nleqslv.
  • r pkg("FixedPoint") provides algorithms for finding fixed point vectors of functions, including Anderson acceleration, epsilon extrapolation methods, or minimal polynomial methods .
  • Package r pkg("daarem") implements the DAAREM method for accelerating the convergence of any smooth, monotone, slow fixed point iteration.
  • Algorithms for accelerating the convergence of slow, monotone sequences from smooth contraction mappings such as the expectation-maximization (EM) algorithm are provided in packages r pkg("SQUAREM") resp. r pkg("turboEM").

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 r pkg("gmp") is needed, see below.

  • Package r pkg("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.
  • r pkg("contfrac") contains various utilities for evaluating continued fractions and partial convergents. r pkg("contFracR")Converts numbers to continued fractions and back again. A solver for Pell's Equation is provided.
  • r pkg("magic") creates and investigates magical squares and hypercubes, including functions for the manipulation and analysis of arbitrarily dimensioned arrays.
  • Package r pkg("freegroup") provides functionality for manipulating elements of a free group including juxtaposition, inversion, multiplication by a scalar, power operations, and Tietze forms.
  • The r pkg("partitions") package enumerates additive partitions of integers, including restricted and unequal partitions.
  • r pkg("permutations") treats permutations as invertible functions of finite sets and includes several mathematical operations on them.
  • Package r pkg("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.
  • Package r pkg("arrangements") provides generators and iterators for permutations, combinations and partitions. The iterators allow users to generate arrangements in a fast and memory efficient manner. Permutations and combinations can be drawn with/without replacement and support multisets.
  • Package r github("xoopR/set6") (on Github) implements (as R6 classes) many forms of mathematical sets (sets, tuples, intervals) and allows for standard operations on them (unions, products, differences, etc.).
  • r pkg("RcppAlgos") provides flexible functions for generating combinations or permutations of a vector with or without constraints; the extension package r pkg("RcppBigIntAlgos") features a quadratic sieve algorithm for completely factoring large integers.
  • Package r pkg("Zseq") generates well-known integer sequences; the 'gmp' package is adopted for computing with arbitrarily large numbers. Every function has on its help page a hyperlink to the corresponding entry in the On-Line Encyclopedia of Integer Sequences ( OEIS ).
  • Package r pkg("primes") provides quite fast (Rcpp) functions for identifying and generating prime numbers. And r pkg("primefactr") uses prime factorization for computations such as reducing ratios of large factorials.
  • Package r pkg("frab") provides methods to "add" two tables with the free Abelian group as the underlying structure.

Multiprecision and Symbolic Calculations

  • Multiple precision arithmetic is available in R through package r pkg("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 r pkg("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.
  • r pkg("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.)
  • r pkg("VeryLargeIntegers") implements a multi-precision library that allows to store and manage arbitrarily big integers; it includes probabilistic primality tests and factorization algorithms.
  • r pkg("bignum") is a package for arbitrary-precision integer and floating-point numbers of 50 decimal digits of precision. The package utilizes the 'Boost.Multiprecision' C++ library and is specifically designed to work with the 'tidyverse' collection of R packages.
  • Package r pkg("Ryacas") interfaces the computer algebra system 'Yacas'; it supports symbolic and arbitrary precision computations in calculus and linear algebra.
  • Package r pkg("caracas") (based on 'reticulate') accesses the symbolic algebra system 'SymPy'; supported are symbolic operations in linear algebra and calculus, such as eigenvalues, derivatives, integrals, limits, etc., computing special functions, or solving systems of equations.
  • Package r pkg("symengine") provides an interface to 'SymEngine', a C++ library for fast symbolic calculations, such as manipulating mathematical expressions, finding exact derivatives, performing symbolic matrix computations, or solving ordinary differential equations (numerically).
  • Package r pkg("rim") provides an interface to the free and powerful computer algebra system Maxima. Results can be output in LaTeX or MathML, and 2D and 3D plots will be displayed directly. Maxima code chunks can be included in 'RMarkdown' documents.
  • Package r pkg("m2r") provides a persistent interface to Macauley2, an extended software program supporting research in algebraic geometry and commutative algebra. Macauley2 has to be installed independently, otherwise a Macauley2 process in the cloud will be instantiated.
  • There are several packages for working with algebras over the real numbers. r pkg("clifford") provides a suite of routines for arbitrary dimensional Clifford algebras and discusses special cases such as Lorentz transforms or quaternion multiplication. Package r pkg("weyl") provides functionality for Weyl algebras, which have applications in quantum mechanics. Package r pkg("stokes") works with the algebra of diffrential $k$-forms as used in exterior calculus (these packages all use r pkg("disordR") discipline). Package r pkg("jordan") provides functionality for working with Jordan algebras, which are commutative but non-associative algebras that obey the Jordan identity $(xy)xx=x(yxx)$.

Python and SAGE Interfaces

Python, through its modules 'NumPy', 'SciPy', 'Matplotlib', 'SymPy', and 'pandas', has elaborate and efficient numerical and graphical tools available.

  • r pkg("reticulate") is an R interface to Python modules, classes, and functions. When calling Python in R data types are automatically converted to their equivalent Python types; when values are returned from Python to R they are converted back to R types. This package from the RStudio team is a kind of standard for calling Python from R.
  • r pkg("feather") provides bindings to read and write feather files, a lightweight binary data store designed for maximum speed. This storage format can also be accessed in Python, Julia, or Scala.
  • 'pyRserve' is a Python module for connecting Python to an R process running r pkg("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.
  • r pkg("XRPython") (and 'XRJulia') are based on John Chambers' r pkg("XR") package and his "Extending R" book and allow for a structured integration of R with Python resp. Julia.

SageMath is an open source mathematics system based on Python, allowing to run R functions, but also providing access to systems like Maxima, GAP, FLINT, and many more math programs. SageMath can be freely used through a Web interface at CoCalc .

MATLAB Octave Julia and other Interfaces

Interfaces to numerical computation software such as MATLAB (commercial) or Octave (free) will be important when solving difficult numerical problems. Unfortunately, at the moment there is no package allowing to call Octave functions from within R.

  • The r pkg("matlab") emulation package contains about 30 simple functions, replicating MATLAB functions, using the respective MATLAB names and being implemented in pure R.
  • Packages r pkg("rmatio") and r pkg("R.matlab") provides tools to read and write MAT files (the MATLAB data format) for versions 4 and 5. 'R.matlab' also enables a one-directional interface with a MATLAB v6 process, sending and retrieving objects through a TCP connection.

Julia is "a high-level, high-performance dynamic programming language for numerical computing", which makes it interesting for optimization problems and other demanding scientific computations in R.

  • r pkg("JuliaCall") provides seamless integration between R and Julia; the user can call Julia functions just like any R function, and R functions can be called in the Julia environment, both with reasonable automatic type conversion. Notes on Julia Call provides an introduction of how to apply Julia functions with 'JuliaCall'.
  • r pkg("JuliaConnectoR") provides a functionally oriented interface for integrating Julia with R; imported Julia functions can be called as R functions; data structures are converted automatically.
  • Package r pkg("XRJulia") provides an interface from R to computations in the Julia language, based on the interface structure described in the book "Extending R" by John M. Chambers.

Java Math functions can be employed through the 'rjava' or 'rscala' interfaces. Then package r pkg("commonsMath") allows calling Java JAR files of the Apache Commons Mathematics Library, a specialized library for all aspects of numerics, optimization, and differential equations.

Please note that commercial programs such as MATLAB, Maple, or Mathematica have facilities to call R functions.

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