Ambiguity function

In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials. In other words, Laguerre's method can be used to solve numerically the equation

$p(x)=0$

for a given polynomial p. One of the most useful properties of this method is that it is, from extensive empirical study, very close to being a "sure-fire" method, meaning that it is almost guaranteed to always converge to some root of the polynomial, no matter what initial guess is chosen. This method is named in honour of Edmond Laguerre, a French mathematician.

Definition

The algorithm of the Laguerre method to find one root of a polynomial p is:

• Choose an initial guess $x_0$
• For k = 0, 1, 2, …
  • Calculate $G=\frac{p^{\prime }(x_k)}{p(x_k)}$
  • Calculate $H=G^2-\frac{p^{″}(x_k)}{p(x_k)}$
  • Calculate $a=\frac{n}{G\pm \sqrt{(n-1)(nH-G^2)}}$, where the sign is chosen to give the denominator with the larger absolute value, to avoid loss of significance as iteration proceeds.
  • Set $x_{k+1}=x_k-a$
• Repeat until a is small enough or if the maximum number of iterations has been reached.

If a root has been found, the corresponding linear factor can be removed from p. This deflation step reduces the degree of the polynomial by one, so that eventually, approximations for all roots of p can be found. Note however that deflation can lead to approximate factors that differ significantly from the corresponding exact factors. This error is least if the roots are found in the order of increasing magnitude.

Derivation

The fundamental theorem of algebra states that every nth degree polynomial p can be written in the form

$p(x)=C(x-x_1)(x-x_2)\cdots (x-x_n),$

such that $x_k$ where $(k=1,2,...,n)$ are the roots of the polynomial. If we take the natural logarithm of both sides, we find that

$\ln |p(x)|=\ln \left|C\right|+\ln |x-x_1|+\ln |x-x_2|+\cdots +\ln |x-x_n|.$

Denote the derivative by

$G=\frac{d}{dx}\ln |p(x)|=\frac{1}{x-x_1}+\frac{1}{x-x_2}+\cdots +\frac{1}{x-x_n},$

and the negated second derivative by

$H=-\frac{d^2}{d{x}^2}\ln |p(x)|=\frac{1}{(x-x_1{)}^2}+\frac{1}{(x-x_2{)}^2}+\cdots +\frac{1}{(x-x_n{)}^2}.$

We then make what Acton calls a 'drastic set of assumptions', that the root we are looking for, say, $x_1$ is a certain distance away from our guess $x$, and all the other roots are clustered together some distance away. If we denote these distances by

$a=x-x_1$

and

$b=x-x_i,i=2,3,\dots ,n$

then our equation for G may be written as

$G=\frac{1}{a}+\frac{n-1}{b}$

and the expression for H becomes

$H=\frac{1}{a^2}+\frac{n-1}{b^2}.$

Solving these equations for a, we find that

$a=\frac{n}{G\pm \sqrt{(n-1)(nH-G^2)}}$,

where the square root of a complex number is chosen to produce larger absolute value of the denominator, or equivalently, to satisfy:

$\mathrm{\Re }(\bar{G}\sqrt{(n-1)(nH-G^2)})>0$,

where $\mathrm{\Re }$ denotes real part of a complex number, and $\bar{G}$ is a complex conjugation of $G$; or

$a=\frac{p(x)}{p^{\prime }(x)}\cdot {(\frac{1}{n}+\frac{n-1}{n}\sqrt{1-\frac{n}{n-1}\frac{p(x)p^{″}(x)}{p^{\prime }(x{)}^2}})}^{-1}$,

where the square root of a complex number is chosen to have a non-negative real part.

For small values of p(x) this formula differs from the offset of the third order Halley's method by an error of $O(p(x{)}^3)$, so convergence close to a root will be cubic as well.

Note that, even if the 'drastic set of assumptions' does not work for some particular polynomial P, P can be transformed into a related polynomial Q for which the assumptions are correct, e.g. by shifting the origin towards a suitable complex number w, $Q(z)=P(z-w)$, to give distinct roots distinct magnitudes if necessary (which it will be if some roots are complex conjugates), and then repeatedly applying the root squaring transformation used in Graeffe's method enough times to make the smaller roots significantly smaller than the largest root (and so, clustered in comparison); the Graeffe's method approximation can be used to start the new iteration for Laguerre's method. An approximate root for P may then be obtained straightforwardly from that for Q.

Properties

If x is a simple root of the polynomial p, then Laguerre's method converges cubically whenever the initial guess x₀ is close enough to the root x. On the other hand, if x is a multiple root then the convergence is only linear. This is obtained with the penalty of calculating values for the polynomial and its first and second derivatives at each stage of the iteration.

A major advantage of Laguerre's method is that it is almost guaranteed to converge to some root of the polynomial no matter where the initial approximation is chosen. This is in contrast to other methods such as the Newton–Raphson method which may fail to converge for poorly chosen initial guesses. It may even converge to a complex root of the polynomial, because of the square root being taken in the calculation of a above may be of a negative number. This may be considered an advantage or a liability depending on the application to which the method is being used. Empirical evidence has shown that convergence failure is extremely rare, making this a good candidate for a general purpose polynomial root finding algorithm. However, given the fairly limited theoretical understanding of the algorithm, many numerical analysts are hesitant to use it as such, and prefer better understood methods such as the Jenkins–Traub algorithm, for which more solid theory has been developed. Nevertheless, the algorithm is fairly simple to use compared to these other "sure-fire" methods, easy enough to be used by hand or with the aid of a pocket calculator when an automatic computer is unavailable. The speed at which the method converges means that one is only very rarely required to compute more than a few iterations to get high accuracy.

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534