Lills method

Lills method (after Eduard Lill ) is a graphical procedure for the determination of the zeros of a polynomial .

For a given polynomial, starting from a common point, two polygons with and sections are constructed. If they end at a common point, the negative tangent of their intersection angle at the common starting point is a zero of the polynomial. ${\ displaystyle n}$ ${\ displaystyle n-1}$

Polynomial and the corresponding polygonal lines with found zero

Traverses and zeros for the polynomial Note that if a straight line of the second polygon intersects the corresponding straight line of a segment from the first polygon at its end point, the second polygon does not turn at a right angle, but continues straight ahead. This is because the perpendicular to be constructed here intersects the next line at the same point and the perpendicular to be constructed there becomes an extension of the original straight line.

{\ displaystyle f (x) = 4x ^ {3} + 2x ^ {2} -2x-1}

The polygon with segments is constructed first and results from the coefficients of the polynomial . First, assume that all coefficients are positive. Starting from any starting point, the first section of the route runs to the right by length units, the next then turns left at a right angle and is length units long. At its end, the next section turns left again and this process is continued for all coefficients. A four-cycle of right, upward, left, downward is thus obtained, which assigns one of these four directions to each coefficient. If a coefficient is now negative, one moves against the direction assigned to the coefficient by the cycle. ${\ displaystyle n}$ ${\ displaystyle f (x) = a_ {n} x ^ {n} + a_ {n-1} x ^ {n-1} \ ldots + a_ {1} x + a_ {0}}$ ${\ displaystyle a_ {n}}$ ${\ displaystyle a_ {n-1}}$

Polynomial and the associated polygonal lines without a zero found

The second traverse is constructed based on a starting angle and the first traverse. A straight line is selected in such a way that it forms the specified starting angle with the first section of the first polygon . Then you intersect this straight line with the straight line on which the second section of the first polygon lies. This point of intersection forms the first section of the second polygon with the starting point. At this point of intersection you now set up the vertical and calculate its point of intersection with the straight line on which the third section of the first polygon is located, thus obtaining the second section of the second polygon. You continue like this until you have reached the last section of the first polygon. If you hit its end point there, i.e. the intersection of the last perpendicular of the second polygonal line intersects the straight line on which the last section of the first polygonal line lies, exactly at its end point, you have found a zero and its value corresponds to the negative Tangent of the angle at the starting point. If you do not hit the end point of the last line of the first polygon, you have not found a zero point and you therefore construct a new second polygon with a different starting angle. By cleverly varying the starting angle, all zeros can theoretically be determined. ${\ displaystyle \ alpha}$

If you apply Lills method in a slightly modified form to a normalized quadratic function, you get a derivation of the Carlyle circle (see there).

literature

Dan Kalman: Uncommon Mathematical Excursions: Polynomia and Related Realms . AMS, 2009, ISBN 978-0-88385-341-2 , pp. 13-22
Rainer Kaenders (Hrsg.), Reinhard Schmidt (Hrsg.): Understanding more mathematics with GeoGebra . Springer Spectrum, 2nd edition, 2014, ISBN 9783658042226 , pp. 71–75
Thomas C. Hull: Solving Cubics With Creases: The Work of Beloch and Lill . American Mathematical Monthly, April 2011, pp. 307-315 ( online copy )
Eduard Lill: Résolution graphique des équations numériques de tous les degrés à une seule inconnue, et description d'un instrument inventé dans ce but . Nouvelles Annales de mathématiques (2), Vol. 6, 1867, pp. 359–362 ( online copy )
Eduard Lill: Résolution graphique des équations algébriques qui ont des racines imaginaires . Nouvelles Annales de mathématiques (2), Vol. 7, 1868, pp. 363–367 ( online copy )

Web links

Commons : Lill's method - collection of images, videos and audio files