Roots of polynomials with order 3 to 14.
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
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
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
Changing the origin to (-1.3,0.0) algorithm converges to other real
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
-0.89953 ± j0.873304
-0.03560 ± j1.244722
0.811034 ± j0.903238
0.303097 ± j0.438263
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COPYRIGHT © 2010 Cuthbert Nyack.