Quick ratio

Henri Padé.

In mathematics a Padé approximant is the "best" approximation of a function by a rational function of given order – under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius who introduced the idea and investigated the features of rational approximations of power series.

The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods in some sense inspired by the Padé theory typically replace them.

Definition

Given a function f and two integers m ≥ 0 and n ≥ 1, the Padé approximant of order [m/n] is the rational function

$R(x)=\frac{\sum _{j=0}^m{a}_j{x}^j}{1+\sum _{k=1}^n{b}_k{x}^k}=\frac{a_0+a_{1}x+a_2{x}^2+\cdots +a_m{x}^m}{1+b_{1}x+b_2{x}^2+\cdots +b_n{x}^n}$

which agrees with f(x) to the highest possible order, which amounts to

$\begin{array}{rcl}f(0)& =& R(0)\\ f^{\prime }(0)& =& R^{\prime }(0)\\ f^{″}(0)& =& R^{″}(0)\\ & ⋮& \\ f^{(m+n)}(0)& =& R^{(m+n)}(0)\end{array}$.

Equivalently, if R(x) is expanded in a Maclaurin series (Taylor series at 0), its first m + n terms would cancel the first m + n terms of f(x), and as such:

$f(x)-R(x)=c_{m+n+1}{x}^{m+n+1}+c_{m+n+2}{x}^{m+n+2}+\cdots$

The Padé approximant is unique for given m and n, that is, the coefficients $a_0,a_1,\dots ,a_m,b_1,\dots ,b_n$ can be uniquely determined. It is for reasons of uniqueness that the zero-th order term at the denominator of R(x) was chosen to be 1, otherwise the numerator and denominator of R(x) would have been unique only up to multiplication by a constant.

The Padé approximant defined above is also denoted as

$[m/n{]}_f(x).$

Computation

For given x, Padé approximants can be computed by Wynn's epsilon algorithm and also other sequence transformations from the partial sums

$T_N(x)=c_0+c_{1}x+c_2{x}^2+\cdots +c_N{x}^N$

of the Taylor series of f, i.e., we have

$c_k=\frac{f^{(k)}(0)}{k!}.$

f can also be a formal power series, and, hence, Padé approximants can also be applied to the summation of divergent series.

One way to compute a Padé approximant is via the extended euclidean algorithm for the polynomial gcd. The relation

$R(x)=P(x)/Q(x)=T_{m+n}(x)\text{ mod }{x}^{m+n+1}$

is equivalent to the existence of some factor K(x) such that

$P(x)=Q(x)T_{m+n}(x)+K(x)x^{m+n+1}$,

which can be interpreted as the Bézout identity of one step in the computation of the extended gcd of the polynomials $T_{m+n}(x)$ and $x^{m+n+1}$.

To recapitulate: to compute the gcd of two polynomials p and q, one computes via long division the remainder sequence

$r_0=p,r_1=q,r_{k-1}=q_k{r}_k+r_{k+1}$,

k =1, 2, 3, ... with $\mathrm{deg}{r}_{k+1}<\mathrm{deg}{r}_k$, until $r_{k+1}=0$. For the Bézout identities of the extended gcd one computes simultaneously the two polynomial sequences

$u_0=1,v_0=0,u_1=0,v_1=1,u_{k+1}=u_{k-1}-q_k{u}_k,v_{k+1}=v_{k-1}-q_k{v}_k$

to obtain in each step the Bézout identity

$r_k(x)=u_k(x)p(x)+v_k(x)q(x)$.

For the [m/n] approximant, one thus carries out the extended euclidean algorithm for

$r_0=x^{m+n+1},r_1=T_{m+n}(x)$

and stops it at the last instant that $v_k$ has degree n or smaller.

Then the polynomials $P=r_k,Q=v_k$ give the [m/n] Padé approximant. If one were to computes all steps of the extended gcd computation, one would obtain an anti-diagonal of the Pade table.

Riemann–Padé zeta function

To study the resummation of a divergent series, say

$\sum _{z=1}^{\mathrm{\infty }}f(z),$

it can be useful to introduce the Padé or simply rational zeta function as

${\zeta }_R(s)=\sum _{z=1}^{\mathrm{\infty }}\frac{R(z)}{z^s},$

where

$R(x)=[m/n{]}_f(x)$

is the Padé approximation of order (m, n) of the function f(x). The zeta regularization value at s = 0 is taken to be the sum of the divergent series.

The functional equation for this Padé zeta function is

$\sum _{j=0}^n{a}_j{\zeta }_R(s-j)=\sum _{j=0}^m{b}_j{\zeta }_0(s-j),$

where a_j and b_j are the coefficients in the Padé approximation. The subscript '0' means that the Padé is of order [0/0] and hence, we have the Riemann zeta function.

DLog Padé method

Padé approximants can be used to extract critical points and exponents of functions. In thermodynamics, if a function f(x) behaves in a non-analytic way near a point x = r like $f(x)\sim {|x-r|}^p$, one calls x = r a critical point and p the associated critical exponent of f. If sufficient terms of the series expansion of f are known, one can approximately extract the critical points and the critical exponents from respectively the poles and residues of the Padé approximants ${[n/n+1]}_g\left(x\right)$ where $g=\frac{f^{\prime }}{f}$.

Generalizations

A Padé approximant approximates a function in one variable. An approximant in two variables is called a Chisholm approximant, in multiple variables a Canterbury approximant (after Graves-Morris at the University of Kent).

See also

• Padé table

Literature

• Baker, G. A., Jr.; and Graves-Morris, P. Padé Approximants. Cambridge U.P., 1996
• Baker, G. A., Jr. Padé approximant, Scholarpedia, 7(6):9756.
• Brezinski, C.; and Redivo Zaglia, M. Extrapolation Methods. Theory and Practice. North-Holland, 1991
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  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
• Frobenius, G.; Ueber Relationen zwischem den Näherungsbrüchen von Potenzreihen, [Journal für die reine und angewandte Mathematik (Crelle's Journal)]. Volume 1881, Issue 90, Pages 1–17
• Gragg, W.B.; The Pade Table and Its Relation to Certain Algorithms of Numerical Analysis [SIAM Review], Vol. 14, No. 1, 1972, pp. 1–62.
• Padé, H.; Sur la répresentation approchée d'une fonction par des fractions rationelles, Thesis, [Ann. \'Ecole Nor. (3), 9, 1892, pp. 1–93 supplement.
• 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

External links

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• Module for Padé Approximation, John H. Mathews California State University, Fullerton
• Padé Approximants, Oleksandr Pavlyk, The Wolfram Demonstrations Project
• A Short Introduction to Padé Approximants, Jerome Soucy Université Laval
• Data Analysis BriefBook: Pade Approximation, Rudolf K. Bock European Laboratory for Particle Physics, CERN
• Sinewave, Scott Dattalo, last accessed 2010-11-11.
• Padé module, Bengt Fornberg, University of Colorade at Boulder, Course-material 2011; last accessed 16.Feb 2012.
• MATLAB function for Pade approximation of models with time delays.