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In the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
, Runge's phenomenon () is a problem of oscillation at the edges of an interval that occurs when using
polynomial interpolation In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no ...
with polynomials of high degree over a set of equispaced interpolation points. It was discovered by
Carl David Tolmé Runge Carl David Tolmé Runge (; 30 August 1856 – 3 January 1927) was a German mathematician, physicist, and spectroscopist. He was co-developer and co- eponym of the Runge–Kutta method (German pronunciation: ), in the field of what is today know ...
(1901) when exploring the behavior of errors when using polynomial interpolation to approximate certain functions. The discovery was important because it shows that going to higher degrees does not always improve accuracy. The phenomenon is similar to the Gibbs phenomenon in Fourier series approximations.


Introduction

The Weierstrass approximation theorem states that for every
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
''f''(''x'') defined on an interval 'a'',''b'' there exists a set of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
functions ''P''''n''(''x'') for ''n''=0, 1, 2, …, each of degree at most ''n'', that approximates ''f''(''x'') with
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
over 'a'',''b''as ''n'' tends to infinity, that is, :\lim_ \left( \sup_ \left, f(x) -P_n(x)\ \right) = 0. Consider the case where one desires to
interpolate In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has ...
through ''n''+1 equispaced points of a function ''f''(''x'') using the ''n''-degree polynomial ''P''''n''(''x'') that passes through those points. Naturally, one might expect from Weierstrass' theorem that using more points would lead to a more accurate reconstruction of ''f''(''x''). However, this ''particular'' set of polynomial functions ''P''''n''(''x'') is not guaranteed to have the property of uniform convergence; the theorem only states that a set of polynomial functions exists, without providing a general method of finding one. The ''P''''n''(''x'') produced in this manner may in fact diverge away from ''f''(''x'') as ''n'' increases; this typically occurs in an oscillating pattern that magnifies near the ends of the interpolation points. The discovery of this phenomenon is attributed to Runge.


Problem

Consider the Runge function :f(x) = \frac\, (a scaled version of the Witch of Agnesi). Runge found that if this function is interpolated at equidistant points ''x''''i'' between −1 and 1 such that: :x_i = \frac - 1,\quad i \in \left\ with a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
''P''''n''(''x'') of degree ≤ ''n'', the resulting interpolation oscillates toward the end of the interval, i.e. close to −1 and 1. It can even be proven that the interpolation error increases (without bound) when the degree of the polynomial is increased: :\lim_ \left( \sup_ , f(x) -P_n(x), \right) = \infty. This shows that high-degree polynomial interpolation at equidistant points can be troublesome.


Reason

Runge's phenomenon is the consequence of two properties of this problem. * The magnitude of the ''n''-th order derivatives of this particular function grows quickly when ''n'' increases. * The equidistance between points leads to a Lebesgue constant that increases quickly when ''n'' increases. The phenomenon is graphically obvious because both properties combine to increase the magnitude of the oscillations. The error between the generating function and the interpolating polynomial of order ''n'' is given by :f(x) - P_n(x) = \frac \prod_^ (x - x_i) for some \xi in (−1, 1). Thus, : \max_ , f(x) - P_n(x), \leq \max_ \frac \max_ \prod_^n , x - x_i, . Denote by w_n(x) the nodal function :w_n(x) = (x - x_0)(x - x_1)\cdots(x - x_n) and let W_n be the maximum of the magnitude of the w_n function: :W_n=\max_ , w_n(x), . It is elementary to prove that with equidistant nodes :W_n \leq n!h^ where h=2/n is the step size. Moreover, assume that the (''n''+1)-th derivative of f is bounded, i.e. :\max_ , f^(x), \leq M_. Therefore, :\max_ , f(x) - P_(x), \leq M_ \frac. But the magnitude of the (''n''+1)-th derivative of Runge's function increases when ''n'' increases, since M_\leq(n+1)!5^. The consequence is that the resulting upper bound, (10/n)^n!, tends to infinity when ''n'' tends to infinity. Although often used to explain the Runge phenomenon, the fact that the upper bound of the error goes to infinity does not necessarily imply, of course, that the error itself also diverges with ''n.''


Mitigations


Change of interpolation points

The oscillation can be minimized by using nodes that are distributed more densely towards the edges of the interval, specifically, with asymptotic density (on the interval ) given by the formula 1 / \sqrt. A standard example of such a set of nodes is
Chebyshev nodes In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial m ...
, for which the maximum error in approximating the Runge function is guaranteed to diminish with increasing polynomial order.


S-Runge algorithm without resampling

When equidistant samples must be used because resampling on well-behaved sets of nodes is not feasible, the S-Runge algorithm can be considered. In this approach, the original set of nodes is mapped on the set of
Chebyshev nodes In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial m ...
, providing a stable polynomial reconstruction. The peculiarity of this method is that there is no need of resampling at the mapped nodes, which are also called '' fake nodes''. A Python implementation of this procedure can be foun
here


Use of piecewise polynomials

The problem can be avoided by using spline curves which are piecewise polynomials. When trying to decrease the interpolation error one can increase the number of polynomial pieces which are used to construct the spline instead of increasing the degree of the polynomials used.


Constrained minimization

One can also fit a polynomial of higher degree (for instance, with n points use a polynomial of order N = n^2 instead of n + 1), and fit an interpolating polynomial whose first (or second) derivative has minimal L^2 norm. A similar approach is to minimize a constrained version of the L^p distance between the polynomial's m-th derivative and the mean value of its m-th derivative. Explicitly, to minimize : V_p = \int_a^b \left, \frac - \frac \int_a^b \frac \mathrmz\^p \mathrm x - \sum_^n \lambda_i \, \left(P_N(x_i) - f(x_i)\right), where N \ge n - 1 and m < N , with respect to the polynomial coefficients and the
Lagrange multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
s, \lambda_i. When N = n - 1, the constraint equations generated by the Lagrange multipliers reduce P_N(x) to the minimum polynomial that passes through all n points. At the opposite end, \lim_ P_N(x) will approach a form very similar to a piecewise polynomials approximation. When m=1, in particular, \lim_ P_N(x) approaches the linear piecewise polynomials, i.e. connecting the interpolation points with straight lines. The role played by p in the process of minimizing V_p is to control the importance of the size of the fluctuations away from the mean value. The larger p is, the more large fluctuations are penalized compared to small ones. The greatest advantage of the Euclidean norm, p=2, is that it allows for analytic solutions and it guarantees that V_p will only have a single minimum. When p\neq 2 there can be multiple minima in V_p, making it difficult to ensure that a particular minimum found will be the
global minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
instead of a local one.


Least squares fitting

Another method is fitting a polynomial of lower degree using the method of
least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the re ...
. Generally, when using m equidistant points, if N < 2\sqrt then least squares approximation P_N(x) is well-conditioned.


Bernstein polynomial

Using
Bernstein polynomial In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials. The idea is named after Sergei Natanovich Bernstein. A numerically stable way to evaluate pol ...
s, one can uniformly approximate every continuous function in a closed interval, although this method is rather computationally expensive.


External fake constraints interpolation

This method proposes to optimally stack a dense distribution of constraints of the type on nodes positioned externally near the endpoints of each side of the interpolation interval, where P"(x) is the second derivative of the interpolation polynomial. Those constraints are called External Fake Constraints as they do not belong to the interpolation interval and they do not match the behaviour of the Runge function. The method has demonstrated that it has a better interpolation performance than Piecewise polynomials (splines) to mitigate the Runge phenomenon.


Related statements from the

approximation theory In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' wil ...

For every predefined table of interpolation nodes there is a continuous function for which the sequence of interpolation polynomials on those nodes diverges. For every continuous function there is a table of nodes on which the interpolation process converges. {{Citation needed, date=March 2014 Chebyshev interpolation (i.e., on
Chebyshev nodes In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial m ...
) converges uniformly for every absolutely continuous function.


See also

*
Chebyshev nodes In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial m ...
* Compare with the Gibbs phenomenon for sinusoidal basis functions *
Occam's razor Occam's razor, Ockham's razor, or Ocham's razor ( la, novacula Occami), also known as the principle of parsimony or the law of parsimony ( la, lex parsimoniae), is the problem-solving principle that "entities should not be multiplied beyond neces ...
argues for simpler models *
Schwarz lantern In mathematics, the Schwarz lantern is a polyhedron, polyhedral approximation to a cylinder, used as a Pathological (mathematics), pathological example of the difficulty of defining the area of a smooth (curved) surface as the Limit (mathemati ...
, another example of failure of convergence *
Stone–Weierstrass theorem In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the ...
*
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...


References

Interpolation Theory of continuous functions Numerical artefacts de:Polynominterpolation#Runges Phänomen