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A basic family of iteration functions for polynomial root finding and its characterizations

## Descriptive

Language
LanguageTerm (authority = ISO 639-3:2007); (type = text)
English
Genre (authority = RULIB-FS)
Other
Genre (authority = marcgt)
technical report
PhysicalDescription
InternetMediaType
application/pdf
Extent
1 online resource (24 pages)
Note (type = special display note)
Technical report lcsr-tr-253
Name (authority = RutgersOrg-School); (type = corporate)
NamePart
School of Arts and Sciences (SAS) (New Brunswick)
Name (authority = RutgersOrg-Department); (type = corporate)
NamePart
Computer Science (New Brunswick)
TypeOfResource
Text
Name (type = personal)
NamePart (type = family)
Kalantari
NamePart (type = given)
Bahman
Affiliation
Computer Science (New Brunswick)
Role
RoleTerm (authority = marcrt); (type = text)
author
Name (type = personal)
NamePart (type = family)
Kalantari
NamePart (type = given)
Iraj
Affiliation
Western Illinois University, Macomb
Role
RoleTerm (authority = marcrt); (type = text)
author
Name (type = personal)
NamePart (type = family)
Zaare-Nahandi
NamePart (type = given)
Rahim
Affiliation
University of Tehran
Role
RoleTerm (authority = marcrt); (type = text)
author
TitleInfo
Title
A basic family of iteration functions for polynomial root finding and its characterizations
Abstract (type = abstract)
Let \$p(x)\$ be a polynomial of degree \$n geq 2\$ with coefficients in a subfield \$K\$ of the complex numbers. For each natural number \$m geq 2\$, let \$L_m(x)\$ be the \$m imes m\$ lower triangular matrix whose diagonal entries are \$p(x)\$ and for each \$j=1, dots, m-1\$, its \$j\$-th subdiagonal entries are \$p^{(j)}(x)/ j!\$. For \$i=1,2\$, let \$L_m^{(i)}(x)\$ be the matrix obtained from \$L_m(x)\$ by deleting its first \$i\$ rows and its last \$i\$ columns. \$L^{(1)}_1(x) equiv 1\$. Then, the function \$B_m(x)=x-p(x)~{{det(L^{(1)}_{m-1}(x))}/ {det(L^{(1)}_{m}(x))}}\$ is a member of \$S(m,m+n-2)\$, where for any \$M geq m\$, \$S(m,M)\$ is the set of all rational iteration functions such that for all roots \$heta\$ of \$p(x)\$ , \$g(x)=heta+ sum_{i=m}^{M} gamma_i(x)(heta-x)^i\$, with \$gamma_i(x)\$'s also rational and well-defined at \$heta\$. Given \$g in S(m,M)\$, and a simple root \$heta\$ of \$p(x)\$, \$g^{(i)}(heta)=0\$, \$i=1, dots, m-1\$, and \$gamma_m(heta)=(-1)^mg^{(m)}(heta)/m!\$. For \$B_m(x)\$ we obtain \$gamma_m(heta)=(-1)^{m} det (L^{(2)}_{m+1}(heta)) /det (L^{(1)}_m(heta))\$. For \$m=2\$ and \$3\$, \$B_m(x)\$ coincides with Newton's and Halley's, respectively. If all roots of \$p(x)\$ are simple, \$B_m(x)\$ is the unique member of \$S(m,m+n-2)\$. By making use of the identity \$0 = sum_{i=0}^n {{p^{(i)}(x)} / {i!}} (heta- x)^i\$, we arrive at two recursive formulas for constructing iteration functions within the \$S(m,M)\$ family. In particular the \$B_m\$'s can be generated using one of these formulas. Moreover, the other formula gives a simple scheme for constructing a family of iteration functions credited to Euler as well as Schr"oder, whose \$m\$-th order member belong to \$S(m,mn)\$, \$m >2\$. The iteration functions within \$S(m,M)\$ can be extended to arbitrary smooth functions \$f\$, with the automatic replacement of \$p^{(j)}\$ with \$f^{(j)}\$ in \$g\$ as well as \$gamma_m(heta)\$.
OriginInfo
DateCreated (encoding = w3cdtf); (keyDate = yes); (qualifier = exact)
1996-06
RelatedItem (type = host)
TitleInfo
Title
Computer Science (New Brunswick)
Identifier (type = local)
rucore21032500001
Location
PhysicalLocation (authority = marcorg); (displayLabel = Rutgers, The State University of New Jersey)
NjNbRU
Identifier (type = doi)
doi:10.7282/t3-hkp5-pp95
Genre (authority = ExL-Esploro)
Technical Documentation
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Open
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## Technical

RULTechMD (ID = TECHNICAL1)
ContentModel
Document
CreatingApplication
Version
1.4
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GPL Ghostscript 9.07
DateCreated (point = start); (encoding = w3cdtf); (qualifier = exact)
2018-06-06T12:37:25
DateCreated (point = start); (encoding = w3cdtf); (qualifier = exact)
2018-06-06T12:37:25
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Version 8.3.13