Polynomial-root solver and root-locus plotter

An analogue method is described for the solution of polynomial equations having complex roots. The method is an adaptation of the known technique of spiral scanning in the complex plane, identified by the variable s, whereby exponentially decreasing oscillatory waveforms are identified with the various powers of s. The required waveforms are generated using conventional linear computing units in such a way that the phase and amplitude relationships are maintained to an accuracy of within 2% throughout each scan. The voltage analogues of the two parts of the polynomial are formed by adding these waveforms in the proportions required by the coefficients. A zero detector generates a short pulse whenever either of these voltages is zero, and this pulse is used to modulate a c.r.t. display. The intersection of the resulting loci of points on the display, being points at which both parts are simultaneously zero, represents the roots of the polynomial. By providing two coefficient-setting potentiometers for each term in the polynomial, the loci of the roots due to, for example, parameter variation in a feedback servomechanism, can be determined. The system described is repetitive at 10 scans/sec and has been used for polynomials of up to the 6th order. For most functions, the accuracy with which the two parts of the root can be determined is within 3% of the maximum value of the scanned variable s.