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[[Image:Rungesphenomenon.png|right|thumb|300px|The red curve is {{red|the Runge function}}.<br/> The blue curve is {{blue|a 5th-order interpolating polynomial}} (using six equally-spaced interpolating points).<br/>
The green curve is {{green|a 9th-order interpolating polynomial}} (using ten equally-spaced interpolating points).<br/>At the interpolating points, the error between the function and the interpolating polynomial is (by definition) zero. Between the interpolating points (especially in the region close to the endpoints 1 and −1), the error between the function and the interpolating polynomial gets worse for higher-order polynomials.]]
 
In the [[mathematics|mathematical]] field of [[numerical analysis]], '''Runge's phenomenon''' is a problem of oscillation at the edges of an interval that occurs when using [[polynomial interpolation]] with polynomials of high degree over a set of equispaced interpolation points. It was discovered by [[Carl David Tolmé Runge]] when exploring the behavior of errors when using polynomial interpolation to approximate certain functions.<ref>{{ Citation
| first = Carl
| last = Runge
| author-link = Carl David Tolmé Runge
| year = 1901
| title = Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten
| journal = Zeitschrift für Mathematik und Physik
| volume = 46
| pages = 224–243
| postscript = . }} available at [http://www.archive.org/details/zeitschriftfrma12runggoog www.archive.org]</ref>
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 every [[continuous function]] ''f(x)'' defined on an [[interval (mathematics)|interval]] [''a'',''b''] can be [[uniform convergence|uniformly approximated]] as closely as desired by a [[polynomial]] function ''P''<sub>''n''</sub>(''x'') of sufficiently large degree &le;&nbsp;''n'', i.e.,
:<math>\lim_{n \rightarrow \infty} \left( \max_{a \leq x \leq b} | f(x) -P_n(x)| \right) = 0.</math>
[[Interpolation]] at equidistant points is a natural and well-known approach to construct approximating polynomials.
Runge's phenomenon demonstrates, however, that [[interpolation]] can easily result in divergent approximations.
 
==Problem==
 
Consider the [[function (mathematics)|function]]:
 
:<math>f(x) = \frac{1}{1+25x^2}.\,</math>
 
Runge found that if this function is [[interpolation|interpolated]] at equidistant points ''x''<sub>''i''</sub> between &minus;1 and 1 such that:
 
:<math>x_i = \frac{2i}{n} - 1,\quad i \in \left\{ 0, 1, \dots, n \right\}</math>
 
with a [[polynomial]] ''P''<sub>''n''</sub>(''x'') of degree &le;&nbsp;''n'', the resulting interpolation oscillates toward the end of the interval, i.e. close to &minus;1 and 1. It can even be proven that the interpolation error increases (without bound) when the degree of the polynomial is increased:
 
:<math>\lim_{n \rightarrow \infty} \left( \max_{-1 \leq x \leq 1} | f(x) -P_n(x)| \right) = +\infty.</math>
 
This shows that high-degree polynomial interpolation at equidistant points can be troublesome.
 
===Reason===
The error between the generating function and the interpolating polynomial of order ''n'' is given by
:<math>f(x) - P_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \prod_{i=1}^{n+1} (x-x_i) </math>
for some <math>\xi</math> in (&minus;1,&nbsp;1). Thus,
:<math>\max_{-1 \leq x \leq 1} |f(x) - P_n(x)| \leq
\max_{-1 \leq x \leq 1} \frac{|f^{(n+1)}(x)|}{(n+1)!}
\max_{-1 \leq x \leq 1} \prod_{i=0}^n |x-x_i|. </math>
 
For the case of the Runge function, interpolated at equidistant points,
each of the two multipliers in the upper bound for the approximation error grows to infinity with ''n.''  
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 to the problem==
 
===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 [−1,1]) given by the formula<ref>{{Citation | last1=Berrut | first1=Jean-Paul | last2=Trefethen | first2=Lloyd N. | author2-link=Lloyd N. Trefethen | title=Barycentric Lagrange interpolation | doi=10.1137/S0036144502417715 | year=2004 | journal=SIAM Review | issn=1095-7200 | volume=46 | pages=501–517 | issue=3}}</ref>
<math>
1/\sqrt{1-x^2}
</math>.
A standard example of such a set of nodes is [[Chebyshev nodes]], for which the maximum error in approximating the Runge function is guaranteed to diminish with increasing polynomial order. The phenomenon demonstrates that high degree polynomials are generally unsuitable for interpolation with equidistant nodes.
 
===Use of piecewise polynomials===
The problem can be avoided by using [[spline curve]]s 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 <math>2n</math> instead of <math>n+1</math>), and fit an interpolating polynomial whose first (or second) derivative has minimal <math>L^2</math> norm.
 
===Least squares fitting===
Another method is fitting a polynomial of lower degree using the method of [[least squares]]. Generally, when using m equidistant points, if <math>N<2\sqrt{m}</math> then least squares approximation <math>P_N(x)</math> is well-conditioned.<ref>{{Citation | last1=Dahlquist | first1=Germund | last2=Björk | first2=Åke | author1-link=Germund Dahlquist | title=Numerical Methods | year=1974 | isbn=0-13-627315-7 | chapter=4.3.4. Equidistant Interpolation and the Runge Phenomenon | pages=101–103}}</ref>
 
==Related statements from the [[approximation theory]]==
For every predefined table of interpolation nodes there is a continuous function for which the interpolation process on those nodes diverges. For every continuous function there is a table of nodes on which the interpolation process converges. Chebyshev interpolation (i.e., on [[Chebyshev nodes]]) converges uniformly for every absolutely continuous function.
 
==See also==
* Compare with the [[Gibbs phenomenon]] for sinusoidal basis functions
* [[Taylor series]]
* [[Chebyshev nodes]]
* [[Stone–Weierstrass theorem]]
 
==References==
<references />
 
[[Category:Interpolation]]
[[Category:Continuous mappings]]
 
[[de:Polynominterpolation#Runges Phänomen]]

Revision as of 11:49, 25 July 2013

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At the interpolating points, the error between the function and the interpolating polynomial is (by definition) zero. Between the interpolating points (especially in the region close to the endpoints 1 and −1), the error between the function and the interpolating polynomial gets worse for higher-order polynomials.

In the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points. It was discovered by Carl David Tolmé Runge when exploring the behavior of errors when using polynomial interpolation to approximate certain functions.[1] 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 every continuous function f(x) defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function Pn(x) of sufficiently large degree ≤ n, i.e.,

Interpolation at equidistant points is a natural and well-known approach to construct approximating polynomials. Runge's phenomenon demonstrates, however, that interpolation can easily result in divergent approximations.

Problem

Consider the function:

Runge found that if this function is interpolated at equidistant points xi between −1 and 1 such that:

with a polynomial Pn(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:

This shows that high-degree polynomial interpolation at equidistant points can be troublesome.

Reason

The error between the generating function and the interpolating polynomial of order n is given by

for some in (−1, 1). Thus,

For the case of the Runge function, interpolated at equidistant points, each of the two multipliers in the upper bound for the approximation error grows to infinity with n. 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 to the problem

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 [−1,1]) given by the formula[2] . A standard example of such a set of nodes is Chebyshev nodes, for which the maximum error in approximating the Runge function is guaranteed to diminish with increasing polynomial order. The phenomenon demonstrates that high degree polynomials are generally unsuitable for interpolation with equidistant nodes.

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 instead of ), and fit an interpolating polynomial whose first (or second) derivative has minimal norm.

Least squares fitting

Another method is fitting a polynomial of lower degree using the method of least squares. Generally, when using m equidistant points, if then least squares approximation is well-conditioned.[3]

Related statements from the approximation theory

For every predefined table of interpolation nodes there is a continuous function for which the interpolation process on those nodes diverges. For every continuous function there is a table of nodes on which the interpolation process converges. Chebyshev interpolation (i.e., on Chebyshev nodes) converges uniformly for every absolutely continuous function.

See also

References

  1. Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

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    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 available at www.archive.org
  2. Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  3. Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

de:Polynominterpolation#Runges Phänomen