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Topic: Cubic polynomial

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	Cubic equation - Wikipedia, the free encyclopedia
		In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power.
		Cubic equations were first discovered by Jaina mathematicians in ancient India sometime between 400 BC and 200 CE.
		Every cubic equation with real coefficients has at least one solution x among the real numbers; this is a consequence of the intermediate value theorem.
	en.wikipedia.org /wiki/Cubic_equation (2125 words)

	Spline interpolation - Wikipedia, the free encyclopedia
		Spline interpolation is preferred over polynomial interpolation because the interpolation error can be made small even when using low degree polynomials for the spline.
		For the n cubic polynomials comprising S, this means to determine these polynomials, we need to determine 4n conditions (since for one polynomial of degree three, there are four conditions on choosing the curve).
		The natural cubic spline is approximately the same curve as created by the spline device.
	en.wikipedia.org /wiki/Spline_interpolation (564 words)

	Graphing Polynomial Functions - Cubic and Quartic E quations
		, a polynomial, a quartic equation, of degree 4.
		That is to say, the third degree polynomial had three roots, and the fourth degree polynomials had four roots, some were at an inflection point and some were a double root.
		The 3d degree polynomials, F1 and F2, were similar in that as x became smaller and smaller, the function steadily decreased or (became more negative), and as x increased beyond the rightmost root, the function steadily increased.
	www.theoldpro.net /math/graphingpolynomials (978 words)

	Cubic Spline Interpolation
		We seek to fit a cubic polynomial on the interval [1, 2] and another cubic polynomial on the interval [2, 3].
		The cubic spline, along with the three points upon which it is based, is shown in Exhibit 1.
		Taylor series expansion In calculus, a power series obtained as a limit of Taylor polynomials that may approximate or equal the function from which it is constructed.
	www.riskglossary.com /articles/cubic_spline.htm (371 words)

	Mathcad Library
		Cubic splines are a common method of approximating a curve.
		A cubic polynomial is completely determined if its value and the value of its derivative is specified at two distinct points.
		This property of cubic polynomials is used for creating "good-looking", that is, continuous, smooth and well behaved interpolation curves.
	www.mathcad.com /library/LibraryContent/MathML/complex_splines.htm (1486 words)

	Egwald Mathematics - Linear Algebra: Polynomials and Polynomial Roots
		Any polynomial of degree n with real coefficients can be factored into the product of quadratic polynomials with real coefficients, times one linear polynomial with real coefficients if n is an odd number.
		The derivative of a polynomial is a polynomial of degree 1 less than the degree of the original polynomial.
		is a root of the polynomial p(x) of degree n, ie p(x
	www.egwald.com /linearalgebra/polynomials.php (1117 words)

	Puzzle 232. Primes and Cubic polynomials
		Regarding the primes produced by each polynomial we report the total primes (T), their subdivision into negative (N) and positive (P) ones, and also the quantity of distinct (D) primes found in the total primes.
		In the first polynomial of Goetgheluck one prime is repeated three times (T-D=2).In the second polynomial of Goetgheluck two primes is repeated (each of them is repeated two times (T-D=2(2-1)).
		This polynomial is linked with a polynomial I discovered on January 2006 (I spoke about this polynomial in a mail to S.M Ruiz, February 23, 2006).
	www.primepuzzles.net /puzzles/puzz_232.htm (1507 words)

	cs261-lect15 (Site not responding. Last check: 2007-10-23)
		This polynomial can have n-1 relative maxima and mimima and the graph can wiggle in order to pass through the points.
		The disadvantage of this is that the polynomial for the whole interval is not smooth at the points (i.e.
		This is a fairly simple procedure and the resulting curve created by joining all the cubic polynomials together is smooth.
	www2.cs.uregina.ca /~norma/cs261-3.4.html (724 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		In this paper we address this problem in the case of an isochronous annulus of periodic orbits (all orbits have the same constant period), and the unperturbed system is explicitly linearizable by a birational transformation of Darboux form, i.e.
		Unlike the Kukles isochrone \cite{10}, it admits a polynomial linearizing transformation that preserves the polynomial nature of the perturbation one-form, allowing the use of the relative cohomology decomposition.
		Cubic Hamiltonian Isochrones\endheading \smallskip Assuming the degenerate singularity on the $y-$axis without loss of generality, a cubic Hamiltonian system may be written as $$ \aligned \dot x=& -y-a_1x^2-2a_2xy-3a_3y^2-a_4x^3-2a_5x^2y\\ \dot y=&x+3a_6x^2+2a_1xy+a_2y^2+4a_7x^3+3a_4x^2y+2b_5xy^2, \endaligned \tag{$\Cal H_3$}$$ with Hamiltonian function $$ H(x,y)=\frac{x^2+y^2}{2}+a_6x^3+a_1x^2y+a_2xy^2+a_3y^3+a_7x^4+a_4x^3y+a_5x^2y^2.
	www.univie.ac.at /EMIS/journals/EJDE/Volumes/Volumes/1999/35/toni-tex (3024 words)

	[No title]
		Returning to the concept of interpolation, an alternative to using a single polynomial to interpolate a set of data points is to use a spline.
		Polynomials are one type of function used to interpolate data.
		This is a cubic spline because it satisfies all the conditions for a cubic spline (it also happens to be a polynomial, too, although that won't be true in general).
	www.cs.wisc.edu /~deppeler/cs310/notes/InterpApprox/InterpApproxNotes.html?class=MathML (3913 words)

	Cubic Spline Interpolation
		The cubic spline interpolation is a piecewise continuous curve, passing through each of the values in the table.
		Cubic splines are popular because they are easy to implement and produce a curve that appears to be seamless.
		Cubic splines avoid this problem, but they are only piecewise continuous, meaning that a sufficiently high derivative (third) is discontinous.
	www.physics.utah.edu /~detar/phycs6720/handouts/cubic_spline/cubic_spline/node1.html (258 words)

	[No title]
		Eckert BEZIER POLYNOMIAL CURVES The parametric equations for a 2-D cubic polynomial curve are: x = axt^3 + bxt^2 + cxt + dx 0 <= t <= 1 y = ayt^3 + byt^2 + cyt + dy The shape of the curve is determined by the polynomial coefficients: (ax,bx,cx,dx, ay,by,cy,dy).
		In the case of uniform cubic Bezier polynomials, the 4 X 4 matrix is called the Bezier geometry matrix.
		Other kinds of polynomial curves will have their polynomial coefficients given by a similar equation--only the matrix elements of the constant 4 X 4 geometry matrix will change.
	www.cs.binghamton.edu /~reckert/460/bezier.htm (860 words)

	Math Forum - Ask Dr. Math
		Show that every cubic polynomial function y=ax^3+bx^2+cx+d can be expressed in the form y=a(x-h)^3+m(x-h)+k for some h,k; thus the graph of every cubic polynomial has a point of symmetry.
		This is our standard answer for the question "What are the general solutions to cubic and quartic polynomial equations?" This actually gives a description of how to solve the cubic, but we don't describe how to do the quartic.
		Contained in the method for solving the cubic equation is a transformation (a variable substitution, which is essentially the step you ask for too - substitute x-h for x) that will eliminate the squared term.
	mathforum.org /library/drmath/view/53514.html (573 words)

	Hermite Polynomial
		Hermite polynomials were studied by the French Mathematician
		(1822-1901), and are referred to as a "clamped cubic," where "clamped" refers to the slope at the endpoints being fixed.
		The cubic Hermite polynomial is a generalization of both the Taylor polynomial and Lagrange polynomial, and it is referred to as an "osculating polynomial." Hermite polynomials can be generalized to higher degrees by requiring that the use of more nodes
	math.fullerton.edu /mathews/n2003/HermitePolyMod.html (206 words)

	Polynomial Interpolations
		To be more precise, assume that we have found a polynomial that interpolates to the n first data points p0,..., pn-1, and also a polynomial that interpolates to the n last data points p1,..., pn.
		Polynomial interpolation is not restricted to interpolation to point data: one can also interpolate to other information, such as derivative data.
		The objective is to find a cubic polynomial curve p that interpolates to these data: p(0) = p0, p'(0) = m0, p'(1) = m1, p(1) = p1, where the prime denotes differentiation.
	www.math.hmc.edu /faculty/gu/math142/mellon/Application_to_CAGD/Interpolations_and_Blossoms/Polynomial_Interpolation.html (717 words)

	Legendre Polynomials
		The differential equation allows us to apply the polynomials to problems arising in mathematics and physics, among which is the important problem of the solution of Laplace's equation and spherical harmonics.
		The polynomials can also be found by solving the differential equation by determining the coefficients of a power series substituted in the equation.
		When differentiated n times, it becomes a polynomial of order n consisting of either all odd or all even powers of x, as n is odd or even.
	www.du.edu /~jcalvert/math/legendre.htm (1164 words)

	Piecewise Cubic Interpolation (Site not responding. Last check: 2007-10-23)
		There are sufficiently many degrees of freedom in choosing the cubic polynomials that the resulting piecewise cubic not only can be made continuous, but it can also have a continuous first derivative (i.e., Hermite cubic) and continuous second derivative (i.e., cubic spline).
		A periodic cubic spline is most appropriate for periodic data, but for nonperiodic data the first and second derivatives can still be equated at the endpoints, and this strategy is implemented here.
		Similarly, a monotonic Hermite cubic interpolant is most appropriate for monotonic data, but for nonmonotonic data, the interpolant can be forced to be monotonic on each subinterval in which the data are montonic, as implemented here.
	www.cse.uiuc.edu /eot/modules/interpolation/pcwcubic (377 words)

	Polynomial Functions
		The degree of the polynomial is the largest exponent of x which appears in the polynomial -- it is also the subscript on the leading term.
		A degree 1 polynomial is a linear function, a degree 2 polynomial is a quadratic function, a degree 3 polynomial a cubic, a degree 4 a quartic, and so on.
		All polynomial functions grow without bound in a positive or negative direction for large and large negative x.
	oregonstate.edu /instruct/mth251/cq/FieldGuide/polynomial/lesson.html (329 words)

	Indirect Electron Tunneling - Analysis
		The cubic spline method fits cubic polynomials to between consecutive data points, called a "zone" for this discussion.
		The left-most and right-most second derivatives which need to assumed are actually computed by fitting a cubic polynomial to 4 data points at either edge of the domain.
		For example, if 2 averages are specified it completes a cubic spline interpolation on every other data point and averages the results with the cubic spline on the remaining points.
	mxp.physics.umn.edu /s98/projects/menz/analysis.htm (1212 words)

	Math Forum: Ask Dr. Math FAQ: Cubic Equations
		Here is another method of solving a cubic polynomial equation submitted independently by Paul A. Torres and Robert A. Warren.
		It is based on the idea of "completing the cube," by arranging matters so that three of the four terms are three of the four terms of a perfect cube.
		These are the roots of the cubic equation that were sought.
	mathforum.org /dr.math/faq/faq.cubic.equations2.html (563 words)

	cubic root calculator
		The program is operated by entering the coefficients for the cubic polynomial to be solved, selecting the rounding option desired, and then pressing the Calculate button.
		In this case, the cubic polynomial can be reduced to a quadratic which is easily solved.
		A cubic polynomial can have three real zeros, or one real zero and one pair of complex zeros.
	home.att.net /~srschmitt/script_cubic.html (257 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		In this case, the binary search is done in terms of an interval [x_min,x_max] in which there is a root of the cubic polynomial xxx + axx + b*x + c.
		By checking the sign of the polynomial at the midpoint x_mid of the interval, we can check whether the root is in the first half of the interval [x_min,x_mid] or in the second half [x_mid,x_max].
		This is always possible for a CUBIC polynomial (or indeed any polynomial of odd degree).
	andromeda.rutgers.edu /~loftin/cs101spr04/sample_programs/cubicroot.cpp (320 words)

	Interpolation with Polynomials and Splines (Site not responding. Last check: 2007-10-23)
		A cubic spline is a piecewise cubic polynomial such that the function, its derivative and its second derivative are continuous at the interpolation nodes.
		It is the smoothest of all possible interpolating curves in the sense that it minimizes the integral of the square of the second derivative.
		For the cubic spline, however, the changes rapidly decay away from the perturbed node.
	www.wam.umd.edu /~petersd/interp.html (198 words)

	What are OMDI Math Tools (Site not responding. Last check: 2007-10-23)
		Cubic interpolation is like linear interpolation except that it attempts to fit to a curve.
		Polynomial tools include a Newtonian root finder, some methods for evaluation of a function or derivative from arrays of coefficients, and a polynomial fitting routine.
		Fitting a polynomial has already been described, however you can fit many non-linear equations that are linearizable using the least squares method.
	www.octavian.com /Layout.jsp?product=xl_math&type=document (3375 words)

	PlanetMath: cubic formula
		This is version 6 of cubic formula, born on 2002-01-06, modified 2005-03-05.
		I also wrote a program to check their answers for a variety of coefficients, a, b, and c.
		We looked at the graph of the cubic equation and found that it should have 1 real and 2 complex solutions.
	planetmath.org /encyclopedia/CubicEquation.html (245 words)

	[No title]
		The formula for solving cubic equations is much more complex than the formula for solving quadratic equations, so we're not going to use the cubic formula.
		Usually, these cubic polynomials are cooked up so that they have roots that are integers with small absolute value (or at least one root that is).
		Since this is a factor of the cubic also, we are now interested in the roots of this quadratic polynomial.
	www.gomath.com /messageboard/index.php?message=45002 (726 words)

	[No title]
		$f$ is to be approximated by a piecewise cubic polynomial $g(x)$ with the properties that $g(x_i)=f(x_i)$, $g'(x_i)=f'(x_i)$, and $g(x)$ is a cubic polynomial $P_i(x)$ on each interval $[x_i,x_{i+1}]$, where $x_0
		A spline of degree $m$ with nodes $x_0polynomial of degree $\le m$ in $(-\infty,x_0)$, $(x_0,x_1)$, $\ldots$, $(x_{n-1},x_n)$, $(x_n,\infty)$.
		A natural spline of degree $2k+1$ is a spline of degree $2k+1$ which is a polynomial of degree $\le k$ in $(-\infty,x_0)$ and $(x_n,\infty)$.
	ei.cs.vt.edu /~cs3414/S97/spline.txt (1023 words)

	flipcode - Polynomial Root-Finder
		By definition, the eigenvalues of a matrix are the roots of the characteristic polynomial of the matrix.
		This seems pretty difficult as the coefficients of the characteristic polynomial depend on the entries of the matrix in a rather nontrivial way; for instance, one of the coefficients will be the determinant of the matrix.
		Recall that the discriminant of a cubic polynomial is positive when there is one simple root; it is negative when there are three simple roots; it is zero when there is a non-simple root.
	www.flipcode.com /cgi-bin/fcarticles.cgi?show=64168 (2855 words)

	CADull's Website, Mathematics (Interpolating with Splines) (Site not responding. Last check: 2007-10-23)
		That is, use a cubic polynomial, which has four unknowns, to join the points.
		They have been joined with cubic polynomials that have a gradient of zero at each of the points.
		That is, a single cubic curve is defined over the two intervals at both the beginning and the end of the function.
	www.cadull.com /old/maths/splines.html (2302 words)

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