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Topic: Multiple roots of a polynomial

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	Complex number - Wikipedia, the free encyclopedia
		Multiplication with i corresponds to a counter clockwise rotation by 90 degrees (π / 2 radians).
		A root of the polynomial p is a complex number z such that p(z) = 0.
		The earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid.
	en.wikipedia.org /wiki/Complex_number (2775 words)

	Complex number (Site not responding. Last check: 2007-10-13)
		Complex numbers were first introduced in connection with explicit formulas for the roots of cubic polynomials.
		Multiplication with i corresponds to a counter clockwise rotation by 90 degrees.
		) is rooted in the ambiguity in choice of i (-1 has two square roots); note, however, that conjugate is not differentiable (see holomorphic).
	hallencyclopedia.com /Complex_number (2794 words)

	The factor theorem. The rational root theorem.
		Determine the polynomial whose roots are −1, 1, 2, and sketch its graph.
		Determine the polynomial with integer coefficients whose roots are −½, −2, −2, and sketch the graph.
		This Integer Root Theorem is an instance of the more general Rational Root Theorem: If the rational number r/s is a root of a polynomial whose coefficients are integers, then the integer r is a factor of the constant term, and the integer s is a factor of the leading coefficient.
	www.themathpage.com /aPreCalc/factor-theorem.htm (1496 words)

	roots (Site not responding. Last check: 2007-10-13)
		ROOTS employs a relatively little known algorithm described in the computer mathematics literature several years ago.
		This algorithm iteratively seeks all the roots simultaneously.
		Accuracy is usually within two digits of the precision of the BASIC employed in addition to displaying the calculated solutions, the program also shows the results of substituting those values into the original polynomial convergence failure is highly unlikely and, if it occurs, it is generally limited to special multiple (M-2) roots.
	people.becon.org /~echoscan/24-04.htm (160 words)

	Complex number - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-13)
		The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots.
		= −1, i.e., i is a square root of −1.
		Every complex number can be represented in the form x + iy, where x and y are real numbers called the real part and the imaginary part of the complex number respectively.
	encyclopedia.learnthis.info /c/co/complex_number.html (2277 words)

	Multiple roots
		3 is a root of multiplicity 4, and −1 is a root of multiplicity 5.
		−2 is a root of multiplicity 2, and 1 is a root of multiplicity 3.
		This polynomial is of the 5th degree, which is odd.
	www.themathpage.com /aPreCalc/point-inflection.htm (406 words)

	RootOf -- the set of roots of a polynomial (Site not responding. Last check: 2007-10-13)
		a polynomial, an arithmetical expression representing a polynomial in
		Since it is generally impossible to represent the roots of a polynomial in terms of radicals,
		For reducible polynomials, the result may be a multiple of the correct minimal polynomial.
	www.sciface.com /STATIC/DOC25/eng/stdlib/RootOf.shtml (401 words)

	LAB #5: Polynomials and Newton's Method (Site not responding. Last check: 2007-10-13)
		The polynomial p2(x) is well defined at x = r1 by invoking continuity, and has all the roots of p(x), except for r1.
		That is, if 3.5 is a root with multiplicity 4, we would have found that value 4 separate times.
		Then the polynomial p2(x) is defined by the coefficients b and the remainder rem is the value p(r1).
	www.math.pitt.edu /~troy/math2070/lab_05.html (2341 words)

	Solving Polynomial Equations
		A polynomial of degree n will have n roots, some of which may be multiple roots.
		When the polynomial is arranged in standard form, a variation in sign occurs when the sign of a coefficient is different from the sign of the preceding coefficient.
		−2 is not a root of the equation f(x)=0.
	oakroadsystems.com /math/polysol.htm (3916 words)

	Polynomial Division and Factoring
		In the previous section, we discussed a technique for sketching polynomials that depended on our ability to find all the factors/roots of a polynomial.
		Whenever the outputs of a polynomial changes from a negative to positive value, or vice versa, there must be a root between the corresponding inputs.
		Use the Rational Root Theorem to make a list of all possible candidates for rational roots; that is, take all combinations ±b/a, where a and b are integer factors of the coefficients of the highest and lowest degree terms, respectively, and write them from smallest to largest.
	campus.northpark.edu /math/PreCalculus/Algebraic/Polynomial/Factoring (2897 words)

	Publications of the SPACES team
		The multiplication by a constant problem consists in generating code to perform a multiplication by an integer constant, using elementary operations, such as left shifts (multiplications by powers of two), additions and subtractions.
		Let f1,andhellip;,fk be k multivariate polynomials which have a finite number of common zeros in the algebraic closure of the ground field, counting the common zeros at infinity.
		When all the polynomials have the same degree, the complexity of this algorithm is polynomial relative to the generic number of solutions.
	www-calfor.lip6.fr /~safey/Spaces/publications.html (13078 words)

	ABSTRACT ALGEBRA ON LINE: Galois Theory
		To study solvability by radicals of a polynomial equation f(x) = 0, we let K be the field generated by the coefficients of f(x), and let F be a splitting field for f(x) over K. Galois considered permutations of the roots that leave the coefficient field fixed.
		When we say that a polynomial equation is solvable by radicals, we mean that the solutions can be obtained from the coefficients in a finite sequence of steps, each of which may involve addition, subtraction, multiplication, division, or taking nth roots.
		There exists a polynomial of degree 5 with rational coefficients that is not solvable by radicals.
	www.math.niu.edu /~beachy/aaol/galois.html (1898 words)

	A Level:Polynomial Zeros
		The program may be used as an investigative tool, for example, to study how changes in the coefficients of a given polynomial affect its zeros or to investigate zeros of families of polynomials of a given form.
		As a quick check, row 1 of A should contain the coefficients of the last entered polynomial starting with the constant term, row 2 should contain the coefficients of the derivative and there should be at least three rows.
		This option allows the user to read in the polynomial coefficients from matrix E. It is particularly useful if the polynomial has a high degree with only a few non-zero coefficients (i.e.
	www.infj.ulst.ac.uk /NI-Maths/A-Level/Programs/polyzero/polyzero.htm (710 words)

	LAB #5: Polynomials and Newton's Method
		This is the method by which you can simplify a polynomial fraction, in that beloved method of partial fractions from calculus.
		Your polynomial coefficient vector is actually a matrix, with 1 row and n columns.
		Moreover, it could be that the polynomial has only one real root, with multiplicity greater than 1.
	www.csit.fsu.edu /~burkardt/math2070/lab_05.html (2130 words)

	Resultants and Discriminants.
		Thus the discriminant of p is zero precisely when p has multiple roots.
		And we see that if a is any root of the polynomial s, then r has a multiple root (given explicitly by setting the last factor to zero).
		is the polynomial whose roots are the multiple roots of the original polynomial.
	www.apmaths.uwo.ca /~rcorless/AM563/NOTES/Feb_28_96/node4.html (385 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Date: 11 Jan 1999 04:07:04 GMT Keywords: Numerical root-finding appropriate with multiple roots The problem with using the gcd of the polynomial and its derivative is that this relies on the coefficients being integer/rational.
		but the bad message is that computing the gcd is terribly unstable under roundoff, so besides some trivial cases computing the gcd numerically and obtaining the multiple roots from it will fail.
		Of course, any multiple root will inherently be illconditioned against perturbations and if one needs high precision then roundoff free computation of the gcd is mandatory.
	www.math.niu.edu /~rusin/known-math/99/jenkins (283 words)

	[No title]
		The # individual roots are obtained as > r[1]; r[2]; # To find basis of eigenvectors corresponding to r[1] do: > N:= nullspace(evalm(A -r[1]* diag(1,1))); # We see that the basis in N consists of single vector [1,1].
		However it might # happen that some roots of the characteristic polynomal are multiple # roots.
		For instance, the polynomial p(t)=(1-t)(3-t)(3-t), has the # root t=1 of multiplicity 1 and the root 3 of the multiplicity 2.
	www.math.utah.edu /~kapovich/TEA/2002F/proj3.txt (1198 words)

	Complex Roots of Polynomials
		Polynomial root computation by means of the LR algorithm.
		Root distribution of a polynomial in subregions of the complex plane
		Inclusion of multiple polynomial roots in complex rectangular arithmetic.
	math.fullerton.edu /mathews/c2003/PolyRootComplexBib/Links/PolyRootComplexBib_lnk_3.html (1267 words)

	[20040901] ERROR-CORRECTING CODES AND FINITE FIELDS
		Characterization of the minimal polynomial and the set (ideal) of polynomials with x as a root.
		Construction of a field contining a root of a given polynomial.
		Calculation of the minimum polynomial of B using the Frobenius automorphism.
	reussir.ticmundi.com /0-19-269067-1.html (716 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		If you want to accurately determine the zeroes of a polynomial having real coefficients, ROOTS is the program for the task.
		In addition to displaying the calculated solutions, the program also shows the results of substituting those values into the original polynomial.
		Convergence failure is highly unlikely and if it occurs, it is generally limited to special multiple (M-2) roots.
	www.dynacompsoftware.com /ebay/91-252.HTM (102 words)

	Types of roots (Site not responding. Last check: 2007-10-13)
		where x is a single variable which can have multiple values (roots) that satisfy this equation.
		In this case, no intersection with the x axis of the cartesian coordinate system occurs since all the roots are located in the complex plane.
		The roots of an nth-degree polynomial such as Eq.
	uranus.eng.auth.gr /lessons/1/2.html (306 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		ROOTS(C) computes the roots of the polynomial whose coefficients are the elements of the vector C. If C has N+1 components, the polynomial is C(1)*X^N +...
		If there are no multiple roots, B(s) R(1) R(2) R(n) ---- = -------- + -------- +...
		The residues are returned in the column vector R, the pole locations in column vector P, and the direct terms in row vector K. The number of poles is n = length(A)-1 = length(R) = length(P).
	www.eg.bucknell.edu /~kozick/elec32098/residue.diary (168 words)

	Symbolic Computation With Eigenvalues
		For some special problems, efficiency or insight can be gained from working with symbolic representations of the eigenvalues, perhaps using the characteristic polynomial as an aid to symbolic manipulation.
		This polynomial will have multiple roots when its discriminant with respect to x is zero (for a polynomial of degree three this is a precise characterization --- see [6] for more details).
		We thus see that this very small perturbation changes the eigenvalues of A (which are 1, 2, and 3) into a set with a multiple root.
	www.apmaths.uwo.ca /~rcorless/frames/AM563/NOTES/Jan_31_96/node26.html (611 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		In the theory of equations, he is particularlh known for the transformation that converts an nth degree polynomial equation in x into an nth degree polynomial equation in y in which the coefficients of y^(n-1) and y^(n-2) are both zero
		Found a Tschinhausen transformation that converts an nth degree polynomial equation in x into an nth degree polynomial equation in y in which the coefficients of y^(n-1), y^(n-2), y^(n-3) are all zero which became important to the transcendental solution of the quintic equation by means of elliptic functions
		Gave an ingenious rule for finding multiple roots of a polynomial that is equivalent to our present method by finding the roots of the highest common factor and its derivative
	faculty.salisbury.edu /~hwaustin/homword.htm (1403 words)

	Abstract of traces paper (Site not responding. Last check: 2007-10-13)
		One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components.
		We exploit the reduction to the univariate root finding problem as a way to sample the polynomial more efficiently, certify the decomposition with linear traces, and apply interpolation techniques to construct the irreducible factors.
		With a random combination of differentials we lower multiplicities and reduce to the regular case.
	www.math.uic.edu /~jan/Articles/factorabs.html (182 words)

	PlanetMath: fundamental theorem of algebra result
		You may wait a while and refresh this page, or click here to try again.
		Cross-references: induction, division, fundamental theorem of algebra, multiple, roots, degree, polynomial, theorem
		This is version 4 of fundamental theorem of algebra result, born on 2004-05-11, modified 2004-11-05.
	planetmath.org /encyclopedia/FundamentalTheoremOfAlgebraResult.html (134 words)

	mailing-list: Re: Return on investment of stocks/mutual funds (Site not responding. Last check: 2007-10-13)
		You basically take the n flows, use them as the factors associated with a polynomial in x, of degree n, where: 0 = Flow1 * x^0 + Flow2 * x^2 + Flow3 * x^3 +...
		The thorny problem is that for n>2 there may be multiple possible rates of return, corresponding to there being multiple roots for the polynomial.
		At any rate, it's all pretty well-understood, and it should not be tough to compute an IRR given a well-defined set of cash flows.
	www.gnucash.org /gnucash-devel/June-2000/msg00433.php3 (377 words)

	Glossary of research economics
		The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence.
		The condition number is computed from the characteristic roots or eigenvalues of the matrix.
		If the largest characteristic root is denoted L and the smallest characteristic root is S (both being presumed to be positive here, that is, the matrix being diagnosed is presumed to be positive definite), then the condition number is:
	econterms.com /econtent.html (14743 words)

	Computational aspects of the Jordan canonical form - Beelen, Van Dooren (ResearchIndex) (Site not responding. Last check: 2007-10-13)
		seen as a polynomial matrix equals n r n l n f n1 min(m; n) From a numerical viewpoint, the computation of the KCF (2) is untractable
		2 Multiple roots of a polynomial From common definitions, a point x 2 C is an m...
		On Ill-Conditioned Eigenvalues, Multiple Roots of Polynomials, and..
	citeseer.ist.psu.edu /beelen90computational.html (575 words)

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