Procrustes analysis

In mathematics, the splitting circle method is a numerical algorithm for the numerical factorization of a polynomial and, ultimately, for finding its complex roots. It was introduced by Arnold Schönhage in his 1982 paper The fundamental theorem of algebra in terms of computational complexity (Technical report, Mathematisches Institut der Universität Tübingen). A revised algorithm was presented by Victor Pan in 1998. An implementation was provided by Xavier Gourdon in 1996 for the Magma and PARI/GP computer algebra systems.

General description

The fundamental idea of the splitting circle method is to use methods of complex analysis, more precisely the residue theorem, to construct factors of polynomials. With those methods it is possible to construct a factor of a given polynomial $p(x)=x^{n}+p_{n-1}x^{n-1}+\cdots +p_{0}$ for any region of the complex plane with a piecewise smooth boundary. Most of those factors will be trivial, that is constant polynomials. Only regions that contain roots of p(x) result in nontrivial factors that have exactly those roots of p(x) as their own roots, preserving multiplicity.

In the numerical realization of this method one uses disks D(c,r) (center c, radius r) in the complex plane as regions. The boundary circle of a disk splits the set of roots of p(x) in two parts, hence the name of the method. To a given disk one computes approximate factors following the analytical theory and refines them using Newton's method. To avoid numerical instability one has to demand that all roots are well separated from the boundary circle of the disk. So to obtain a good splitting circle it should be embedded in a root free annulus A(c,r,R) (center c, inner radius r, outer radius R) with a large relative width R/r.

Repeating this process for the factors found, one finally arrives at an approximative factorization of the polynomial at a required precision. The factors are either linear polynomials representing well isolated zeros or higher order polynomials representing clusters of zeros.

Details of the analytical construction

Newton's identities are a bijective relation between the elementary symmetric polynomials of a tuple of complex numbers and its sums of powers. Therefore, it is possible to compute the coefficients of a polynomial

$p(x)=x^{n}+p_{n-1}x^{n-1}+\cdots +p_{0}=(x-z_{1})\cdots (x-z_{n})$

(or of a factor of it) from the sums of powers of its zeros

$t_{m}=z_{1}^{m}+\cdots +z_{n}^{m}$, $m=0,1,\dots ,n$

by solving the triangular system that is obtained by comparing the powers of u in the following identity of formal power series

$a_{n-1}+2\,a_{n-2}\,u+\cdots +(n-1)\,a_{1}\,u^{n-2}+n\,a_{0}\,u^{n-1}$

$=-(1+a_{n-1}\,u+\cdots +a_{1}\,u^{n-1}+a_{0}\,u^{n})\cdot (t_{1}+t_{2}\,u+t_{3}\,u^{2}+\dots +t_{n}\,u^{n-1}+\cdots ).$

If $G\subset \mathbb {C}$ is a domain with piecewise smooth boundary C and if the zeros of p(x) are pairwise distinct and not on the boundary C, then from the residue theorem of residual calculus one gets

${\frac {1}{2\pi \,i}}\oint _{C}{\frac {p'(z)}{p(z)}}z^{m}\,dz=\sum _{z\in G:\,p(z)=0}{\frac {p'(z)z^{m}}{p'(z)}}=\sum _{z\in G:\,p(z)=0}z^{m}.$

The identity of the left to the right side of this equation also holds for zeros with multiplicities. By using the Newton identities one is able to compute from those sums of powers the factor

$f(x):=\prod _{z\in G:\,p(z)=0}(x-z)$

of p(x) corresponding to the zeros of p(x) inside G. By polynomial division one also obtains the second factor g(x) in p(x) = f(x)g(x).

The commonly used regions are circles in the complex plane. Each circle gives raise to a split of the polynomial p(x) in factors f(x) and g(x). Repeating this procedure on the factors using different circles yields finer and finer factorizations. This recursion stops after a finite number of proper splits with all factors being nontrivial powers of linear polynomials.

The challenge now consists in the conversion of this analytical procedure into a numerical algorithm with good running time. The integration is approximated by a finite sum of a numerical integration method, making use of the fast Fourier transform for the evaluation of the polynomials p(x) and p'(x). The polynomial f(x) that results will only be an approximate factor. To ensure that its zeros are close to the zeros of p inside G and only to those, one must demand that all zeros of p are far away from the boundary C of the region G.

Basic numerical observation

(Schönhage 1982) Let $p\in \mathbb {C} [X]$ be a polynomial of degree n has k zeros inside the circle of radius 1/2 and the remaining n-k zeros outside the circle of radius 2. With N=O(k) large enough, the approximation of the contour integrals using N points results in an approximation $f_{0}$ of the factor f with error

$\|f-f_{0}\|\leq 2^{2k-N}\,nk\,100/98$,

where the norm of a polynomial is the sum of the moduli of its coefficients.

Since the zeros of a polynomial are continuous in its coefficients, one can make the zeros of $f_{0}$ as close as wanted to the zeros of f by choosing N large enough. However, one can improve this approximation faster using a Newton method. Division of p with remainder yields an approximation $g_{0}$ of the remaining factor g. Now

$p-f_{0}g_{0}=(f-f_{0})g_{0}+(g-g_{0})f_{0}+(f-f_{0})(g-g_{0})$,

so discarding the last second order term one has to solve $p-f_{0}g_{0}=f_{0}\Delta g+g_{0}\Delta f$ using any variant of the extended euclidian algorithm to obtain the incremented approximations $f_{1}=f_{0}+\Delta f$ and $g_{1}=g_{0}+\Delta g$. This is repeated until the increments are zero relative to the chosen precision.

Graeffe iteration

The crucial step in this method is to find an annulus of relative width 4 in the complex plane that contains no zeros of p and contains approximately as many zeros of p inside as outside of it. Any annulus of this characteristic can be transformed, by translation and scaling of the polynomial, into the annulus between the radii 1/2 and 2 around the origin. But, not every polynomial admits such a splitting annulus.

To remedy this situation, the Graeffe iteration is applied. It computes a sequence of polynomials

$p_{0}=p,\qquad p_{j+1}(x)=(-1)^{\deg p}p({\sqrt {x}})\,p(-{\sqrt {x}}),$

where the roots of $p_{j}(x)$ are the $2^{j}$-th dyadic powers of the roots of the initial polynomial p. By splitting $p_{j}(x)=e(x)+x\,o(x)$ into even and odd parts, the succeeding polynomial is obtained by purely arithmetic operations as $p_{j+1}(x)=(-1)^{\deg p}(e(x)^{2}-x\,o(x)^{2})$. The ratios of the absolute moduli of the roots increase by the same power $2^{j}$ and thus tend to infinity. Choosing j large enough one finally finds a splitting annulus of relative width 4 around the origin.

The approximate factorization of $p_{j}(x)\approx f_{j}(x)\,g_{j}(x)$ is now to be lifted back to the original polynomial. To this end an alternation of Newton steps and Padé approximations is used. It is easy to check that

${\frac {p_{j-1}(x)}{g_{j}(x^{2})}}\approx {\frac {f_{j-1}(x)}{g_{j-1}(-x)}}$

holds. The polynomials on the left side are known in step j, the polynomials on the right side can be obtained as Padé approximants of the corresponding degrees for the power series expansion of the fraction on the left side.

Finding a good circle

Making use of the Graeffe iteration and any known estimate for the absolute value of the largest root one can find estimates R of this absolute value of any precision. Now one computes estimates for the largest and smallest distances $R_{j}>r_{j}>0$ of any root of p(x) to any of the five center points 0, 2R, −2R, 2Ri, −2Ri and selects the one with the largest ratio $R_{j}/r_{j}$ between the two. By this construction it can be guaranteed that $R_{j}/r_{j}>e^{0{.}3}\approx 1.35$ for at least one center. For such a center there has to be a root-free annulus of relative width $\textstyle e^{0{.}3/n}\approx 1+{\frac {0{.}3}{n}}$. After $\textstyle 3+\log _{2}(n)$ Graeffe iterations, the corresponding annulus of the iterated polynomial has a relative width greater than 11 > 4, as required for the initial splitting described above (see Schönhage (1982)). After $\textstyle 4+\log _{2}(n)+\log _{2}(2+\log _{2}(n))$ Graeffe iterations, the corresponding annulus has a relative width greater than $\textstyle 2^{13{.}8}\cdot n^{6{.}9}>(64\cdot n^{3})^{2}$, allowing a much simplified initial splitting (see Malajovich/Zubelli (1997))

To locate the best root-free annulus one uses a consequence of the Rouché theorem: For k = 1, ..., n − 1 the polynomial equation

$\,0=\sum _{j\neq k}|p_{j}|u^{j}-|p_{k}|u^{k},$

u > 0, has, by Descartes' rule of signs zero or two positive roots $u_{k}<v_{k}$. In the latter case, there are exactly k roots inside the (closed) disk $D(0,u_{k})$ and $A(0,u_{k},v_{k})$ is a root-free (open) annulus.

References

• Schönhage, Arnold (1982): The fundamental theorem of algebra in terms of computational complexity. Preliminary Report, Math. Inst. Univ. Tübingen (1982), 49 pages. (ps.gz)
• 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
• 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
• 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
• 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
• Pan, Victor (1998). Algorithm for Approximating Complex Polynomial Zeros
• Pan, Victor (2002). Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding
• Magma documentation. Real and Complex Fields: Element Operations.