Synthetic Division Algorithm

If the nonlinear equation being solved is of the polynomial form, each real root (located by one of the methods already discussed) can be removed from the polynomial by synthetic division, thus reducing the degree of the polynomial to (n - 1). Each successive application of the synthetic division algorithm will reduce the degree of the polynomial further, until all real roots have been located. 

A simple computational algorithm for synthetic division has been given by Lapidus [4], Consider the fourth-degree polynomial 

In order to determine the coefficients (b) of the third-degree polynomial first multiply out Eq. (1.49) and rearrange in descending power of x 

Equating Eqs. (1.48) and (1.50), the coefficients of like powers of x must be equal to each other, that is, 

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In order to determine whether the system is stable or unstable, the two polynomials are combined, as shown in the Method of Solution, using as the multiplier of the polynomial from the numerator of the transfer function. Function NRsdivision (which uses the Newton-Raphson method with synthetic division algorithm) or function roots (which uses the eigenvalue algorithm) is called to calculate the roots of the overall polynomial function and the sign of all roots is checked for positive real parts. A flag named stbl indicates that the system is stable (all negative roots stbl = 1) or unstable (positive root stbl = 0). 

The most effective way of finding the roots of nonlinear equations is to devise iterative algorithms that start at an initial estimate of a root and converge to the exact value of the desired root in a finite number of. steps. Once a root is located, it may be removed by synthetic division if the equation is of the polynomial form. Otherwise, convergence on the same root may be avoided by initiating the search for subsequent roots in different region of the feasible space. 

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