Solution of Cubic and Quartic for Partial fraction Expansion.
Cuthbert Nyack
If it is required to find the inverse Laplace Transform of a
function which has a cubic or quartic in its denominator, then
the roots of the cubic or quartic must be found.
Analytical methods are available for the solution of cubic and quartic equations and the applet below can be used to find the roots of a cubic or quartic in s using these analytical techniques.
In the applet each coefficient is set by 4 scrollbars. eg a2 is set by
scrollbars 6, 7, 8 and 9. 6 is for coarse adjustment, 7 for fine adjustment, 8 for finer adjustment and 9 adds a power of 10.
The gif image below shows the roots of the cubic equation in s
s3 + 2.1 s2 + 50.02 s + 86.7 = 0 are given by
-1.7545665 and -0.17271673 ± j7.02738094.
The gif image below shows the roots of the quartic equation in s
s4 + 4.67 s3 + 0.1995 s2 + 4.67467 s + 5.9699 = 0 are given by
-0.8327028, -4.7782705, and 0.470486697 ± j1.13094633.
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COPYRIGHT © 2010 Cuthbert Nyack.