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Topic: Cubic root

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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.
		The cube root function is in some respects not a well-behaved function, or one convenient for the purposes of finding the roots of a cubic equation.
		Therefore the Chebyshev cube root is in fact an analytic function on the whole of the domain D. An alternative construction of the Chebyshev cube root in terms of hypergeometric functions is sketched in the next subsection.
	en.wikipedia.org /wiki/Cubic_equation (2125 words)

	Equation - LoveToKnow 1911 (Site not responding. Last check: 2007-10-23)
		Pure cubic equations are therefore of the form x 3 =r; and hence it appears that a value of the simple power of the unknown quantity may always be found without difficulty, by extracting the cube root of each side of the equation.
		From the theorem that a rational symmetrical function of the roots is expressible in terms of the coefficients, it at once follows that it is possible to determine an equation (of an assignable order) having for its roots the several values of any given (unsymmetrical) function of the roots of the given equation.
		for the cubic the root is a+wb+w 2 c, w an imaginary cube root of unity.
	www.1911encyclopedia.org /Equation (10784 words)

	EQUATION (from Lat. ae... - Online Information article about EQUATION (from Lat. ae...
		The roots of this quadratic are m=+ or 3, and therefore 2y=x, or y=3x.
		Thus we have the biquadratic equation y4+2Py22—84 R.y+P2-4Q=o, one of the roots of which is y= J a+ J b+ A) c, while a, b, c are the roots of the cubic equation z3+Pz2+Qz—R=o.
		for the cubic the root is a+wb+w2c, w an imaginary cube root of unity.
	encyclopedia.jrank.org /EMS_EUD/EQUATION_from_Lat_aequatio_aequ.html (11178 words)

	Quartic equation - Wikipedia, the free encyclopedia
		If only the real rational roots are needed, they can be found (as is true for polynomials of any degree) via trial and error, using Ruffini's rule (so long as all the polynomial coefficients are rational).
		This is correct for both signs of square root, as long as the same sign is taken for both square roots.
		This suggests using a resolvent whose roots may be variously described as a discrete Fourier transform or a Hadamard matrix transform of the roots.
	en.wikipedia.org /wiki/Quartic_equation (1650 words)

	[No title]
		The cubics derived in the algorithms by Descartes, by Neumark and by Ferrari are examined for their stability.
		A root of the cubic is then used to factorize the quartic into quadratics, which may then be solved.
		FERRARI Of the three subsidiary cubics, that from Ferrari's algorithm has two stable combinations of signs of a, b, c and d for the derivation of all of the coefficients of the cubic, p, q, and r.
	www.acooke.org /jara/mllib/cubic.sml (2798 words)

	GNU Scientific Library -- Reference Manual - One dimensional Root-Finding
		The size of this bounded region is reduced, iteratively, until it encloses the root to a desired tolerance.
		Because it is easy to create situations where numerical root finders go awry, it is a bad idea to throw a root finder at a function you do not know much about.
		The root bracketing algorithms described in this section require an initial interval which is guaranteed to contain a root -- if a and b are the endpoints of the interval then f(a) must differ in sign from f(b).
	www.math.umn.edu /systems_guide/gsl-1.3/gsl-ref_31.html (2673 words)

	Search Results for root*
		root) of a square of three square units and of five square units, that these roots are not commensurable in length with the unit length, and he went on in this way, taking all the separate cases up to the root of seventeen square units, at which point, for some reason, he stopped.
		Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction.
		The root of the difficulty is that the proper medium for the accurate expression of physical principles, methods, and conclusions is mathematics: and that physical mathematics, regarded as a language, has an almost untranslatable vocabulary.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=root*&CONTEXT=1 (12226 words)

	Math Forum: Ask Dr. Math FAQ: Cubic Equations
		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.
		Once you have at least one root, the problem of finding the other roots is reduced to solving a quadratic or linear equation.
	mathforum.org /dr.math/faq/faq.cubic.equations2.html (563 words)

	GNU Scientific Library -- Reference Manual: Root Finding Caveats (Site not responding. Last check: 2007-10-23)
		When there are several roots in the search area, the first root to be found will be returned; however it is difficult to predict which of the roots this will be.
		It is not possible to use root-bracketing algorithms on even-multiplicity roots.
		For these algorithms the initial interval must contain a zero-crossing, where the function is negative at one end of the interval and positive at the other end.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_400.html (284 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].
		The values passed into it are the limits x_min, x_max of the interval in which we'll find the root, value returned is the root.
	andromeda.rutgers.edu /~loftin/cs101spr04/sample_programs/cubicroot.cpp (320 words)

	cubic root calculator
		The calculator is designed to solve for the roots of a cubic polynomial with the form:
		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.
	home.att.net /~srschmitt/script_cubic.html (257 words)

	cubic.html
		Problems leading to cubic equations are more rare, perhaps because they are harder to solve.
		Here is a series of problems leading to cubics which occurred prior to Cardano in 1400.
		Omar Khayyam studied the cubic equations in the manner of Archimedes, that is, he solved them by means of conic sections.
	www.ms.uky.edu /~carl/ma330/html/cubic1.html (884 words)

	On the Cubic roots of Unity, and Fermat mod p^k (Site not responding. Last check: 2007-10-23)
		On the Cubic roots of Unity, and Fermat mod p^k
		And a^2 = 1/a, sothat a +1 = -1/a for cubic root 'a' of 1 (mod p^k).....
		In preprint [1] it is shown that all normed FLT mod p^k solutions (case 1 : x,y,z relative prime to p) have the next triplet form : a +1/b = b + 1/c = c + 1/a = -1, with a.b.c=1 (mod p^k), of which the cubic root solution is a special case (a=b=c).
	home.iae.nl /users/benschop/cubic.htm (503 words)

	Quadratic Formula and roots of a cubic
		Furthermore, the formula itself is kind of cumbersome: if the cubic has three real roots, the formula requires the user to calculate cube roots of complex numbers, even though the imaginary parts end up canceling each other out.
		First, every cubic equation ay^3 + by^2 + cy + d = 0 can be reduced to the form x^3 + mx = n divide through by a, then substitute y = x-[b/(3*a)] and expand.
		Let the root found in this manner be denoted r; then (y - r) divides the original cubic, yielding a quadratic which can be solved using the quadratic formula to find the other two roots.
	www.newton.dep.anl.gov /newton/askasci/1995/math/MATH052.HTM (611 words)

	Cubic Formula (Site not responding. Last check: 2007-10-23)
		So the trick is to find one real root, r, then divide the cubic equation by (x-r) to reduce the equation to a quadratic, and then it is easy to find the other two roots.
		The other roots can be found by dividing ax³+bx²+cx+d by (x-r), and solving the resulting quadratic using the quadratic formula.
		Since these three roots are all real, the cubic formula doesn't work without complex numbers, but it does work, and it finds all three roots.
	mcraefamily.com /MathHelp/FactoringCubic1.htm (526 words)

	History of Mathematics--Spring 2005/HWK5
		The number (2 - 11i) is the complex conjugate of (2 + i) and its cubic roots will be the complex conjugates of the cubic root just written down.
		Consider a right triangle having one leg 2 units long, the second leg 11 units long and let θ be the angle between the shorter leg and the hypothenuse.
		Specifically, show that one can find y reducing the right hand side to a square by solving a cubic equation (which was known by then); after this one finds x by extracting a square root.
	www.math.fau.edu /schonbek/HistMath/histmathsp05h7.html (531 words)

	Cubic Equation Solver
		Algebraic theory on polynomials says that there should be three roots for a cubic equation; complex roots, when exist, will appear in complex conjugate pairs.
		If D=0, three real roots of which at least two are equal.
		The objective of this problem is to help you learn how to break a problem into smaller chunks and how to group codes into well-defined chunks (i.e., subroutines).
	www.ee.umd.edu /~nsw/ench250/cubiceq.htm (389 words)

	Pedro Freire - Internet Specialist and Consultant
		random calculations of cubic root of n-bit numbers, is negligible and therefore is not added to this formula (e.g.: for n=16 you get those operations being run less than 0.1% of the time, and the value to be added to this formula would be less than 0.004).
		For other roots, such accuracy is in my opinion pointless since you're dealing with integer numbers after all and precision has gone out the window, but it would make the algorithm perhaps a bit more precise.
		This would mean that the bigger roots are searched for first, so a greater number of values of x are covered in the first few iterations, and on average a root is found sooner.
	www.pedrofreire.com /crea2_en.htm (4886 words)

	Cubic with three real irrational roots.
		Of course, any cubic equation can be solved by the cubic formula, giving the root as a combination of cube roots of an expression involving a square root.
		With the first of those definitions, "any cubic root could be expressed as a something like a nested surd" would be false because two of the cube roots might be complex rather than irrational numbers, the MathWorld definition of "surd".
		With the second of those definitions, "any cubic root could be expressed as a something like a nested surd" is still false because the nested roots might be roots of complex numbers rather than real numbers, the Wikpedia definition of "surd".
	www.physicsforums.com /showthread.php?p=1000511#post1000511 (616 words)

	The "Cubic Formula"
		Solving a cubic equation, on the other hand, was the first major success story of Renaissance mathematics in Italy.
		The other two roots (real or complex) can then be found by polynomial division and the quadratic formula.
		Shortly after the discovery of a method to solve the cubic equation, Lodovico Ferraria (1522-1565), a student of Cardano, found a similar method to solve the quartic equation.
	www.sosmath.com /algebra/factor/fac11/fac11.html (436 words)

	Learn more about Abacus in the online encyclopedia. (Site not responding. Last check: 2007-10-23)
		Unlike the simple counting board used in elementary schools, very efficient suanpan techniques were developed to do multiplication, division, addition, subtraction, square root and cubic root operations at high speed.
		The beads and rods were often lubricated to ensure speed.
		They use an abacus to perform the mathematical functions multiplication, division, addition, subtraction, square root and cubic root.
	www.onlineencyclopedia.org /a/ab/abacus_1.html (937 words)

	PEARSON 30 -- Information (Site not responding. Last check: 2007-10-23)
		means square the displacement in cubic feet (for salt water, divide displacement by 64 to get displacement in cubic feet) and then take the cubic root of that number.
		Then, find the cubic root of that number (i.e., the cubic root of the displacement in cubic feet).
		Finally, divide the beam by the number you just arrived at (i.e., divide the beam by the cubic root of the displacement in cubic feet) to determine the CSF.
	home.comcast.net /~griglack/p30information.html (931 words)

	Calculator Skills
		Powers and roots are another important use of the calculator in chemistry.
		To find the cubic root of 77, enter 77 into the calculator (as the y value), enter 1/3 (as the x value), and depress the [y
		This method should allow you to take the nth root of any number that you may encounter in your studies in chemistry.
	www.chem.vt.edu /chem-dept/long/chemath/S1_CalculatorSkills.html (1685 words)

	18.04 Pictures Home Page
		Here we have plotted the basins of attraction for the three cubic roots of unity under iterations using Newton's method.
		The first picture on the right shows the real part of the cubic root and the second picture shows the imaginary part.
		At each point in the complex plane (except for the origin) there are three possible cubic roots and this gives rise to the "three level surfaces" that show in each of the pictures.
	www-math.mit.edu /18.04/18.04-rrr/Pictures/index.html (2269 words)

	Properties of nuclei (Site not responding. Last check: 2007-10-23)
		The density inside a large nucleus is approximately 1.4 E44 nucleons per cubic meter.
		The radius of a nucleus is proportional to the cubic root of the number of particles,
		From our studies of atomic physics, we saw that the radius of the innermost Bohr orbit around a nucleus of charge Z was (5.3 E-11 m)/Z. So for the heaviest nucleus the radius of the inner-most electronic orbit is 100 times larger than the radius of the nucleus.
	www.pa.msu.edu /courses/1997spring/PHY232/lectures/nuclear/properties.html (127 words)

	[No title] (Site not responding. Last check: 2007-10-23)
		(/ rat (* 2 a)))) (defun cubic (a b c d) (cond ((minusp a) (cubic (- a) (- b) (- c) (- d))) ((zerop a) (quadratic b c d)) ;a = 0.
		((zerop e) (cons 0 (cubic a b c d))) ;e = 0.
		(- c) (- (* b d) (* 4 a e)) (- (* 4 a c e) (+ (* a d d) (* b b e))))))))) (defun quartic-aux (a b c d e s) ;s is root of resolvent.
	www.engin.umd.umich.edu /CIS/course.des/cis479/winston.lisp/32mathem.lsp (786 words)

	Visual Basic Complex Math (Site not responding. Last check: 2007-10-23)
		"Evaluate The Cubic Expression" Evaluates the expression A X^3 + B X^2 + C X + D where A, B, C, D, and X are real and/or complex numbers.
		"Root Of Cubic Equation" Returns one of the three roots of the cubic equation.
		"Root Of Quadratic Equation" Returns one of the three roots of the cubic equation.
	www.entisoft.com /ESTools/MathComplex.HTML (575 words)

	Problem A: R U Kidding Mr. Feynman
		Feynman lost in operations such as addition and multiplication but he won in cubic roots.
		To calculate square root of 17, as Feynman has an excelent memory, he knows 'all' perfect squares (as well cubes), he knows that 44 = 16 then he just use the method above and calculate 4+1/8 that equals 4.125 (not very bad as square root* of 17= 4.123...
		For each line of input print the value of the cubic root approximated by the method explained above.
	acm.uva.es /p/v105/10509.html (370 words)

	MIT OpenCourseWare \| Mathematics \| 18.04 Complex Variables with Applications, Fall 1999 \| Study Materials \| Riemann ...
		These are given by the principal value of the square root and its negative (the principal value of the square root is defined in exactly the same fashion as the principal value of the cubic root was defined earlier).
		What this all means is that: the Riemann Surface for the square root is an object in four dimensional space.
		In the pictures we have color coded the surface: at the (inevitable in three dimensions) crossings each of the two sheets that cross have a clearly distinct coloring (this is the best we could do with our 4-D coloring pens broken).
	ocw.mit.edu /OcwWeb/Mathematics/18-04Complex-Variables-with-ApplicationsFall1999/StudyMaterials/detail/Riemann-Surfaces--The-Square-Root.htm (363 words)

	The Geometry of the Cubic Formula
		Tartaglia's first step was to depress the cubic by shifting the graph of the cubic horizontally by the quantity b/3a.
		Our goal is to find one real root; the other two real roots can then be found by polynomial division and the quadratic formula.
		There will never be an algebraic improvement of the cubic formula, which avoids the usage of complex numbers.
	www.sosmath.com /algebra/factor/fac111/fac111.html (739 words)

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