Roots of polynomials with order 3 to 14.

Cuthbert Nyack

The applet below can be used to find the roots of polynomials with order 3 to 14 using Laguerre's algorithm for finding roots of polynomials. In the applet Laguerre's algorithm uses the origin as the starting point and the different roots are found by moving the origin to the vicinity of the root. When the function is plotted, the roots are located at the starting points of the blue/red boundaries.
Scrollbar (0) sets the order of the polynomial.
Scrollbar (1) suppresses plotting of the function to enable quicker setting of the coefficients.
Scrollbars (2) to (43) can be used to set the coefficients.
Scrollbars (44) and (45) sets the origin (47) can be used to scale the origin.
Scrollbars (47) and (48) sets the width and height (49) can be used to scale the width and height.
If the root is not accurate enough, then the number of iterations can be increased by scrollbar(50). Alternatively the origin may be changed. The algorithm may occasionally fail to converge. Usually this problem can be resolved by moving the origin around. Identical roots may sometimes occur, these can be identified by 2 boundaries starting at the same point.

Image below shows the applet set to solve the 12th order polynomial:-
z¹² + 2z¹¹ + 3z¹⁰ + 4z⁹ + 5z⁸ + 6z⁷ + 7z⁶ + 8z⁵ + 4z⁴ + 2z³ + z² + 0.5z + 0.25 = 0

Image below shows a plot of the function. Starting from -0.1,0.0 the algorithm converges to -0.25745 - j0.514837

Changing the origin to (-0.5,0.0) algorithm converges to real root -0.59081

Changing the origin to (-1.3,0.0) algorithm converges to other real root -1.25226.

Changing the origin to (-1.0,0.7) algorithm converges to complex root -0.89953 + j0.873304

Continuing we find the 12 roots are:-
-0.25745 ± j0.514837
-0.59081
-1.25226
-0.89953 ± j0.873304
-0.03560 ± j1.244722
0.811034 ± j0.903238
0.303097 ± j0.438263

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