Polynomial MATLAB roots

The following experiments validate our assessment of troubles with Newton or bisection root finders for multiple roots. First we use the bisection method based MATLAB root finder f zero, followed by a simple Newton iteration code, both times using the chosen polynomial p x) of degree 9 in its extended form (1.6). 

The tutorial demonstrates how MATLAB is used to generate root loeus diagrams, and lienee how to design eontrol systems in the. v-plane. Examples given in Chapter 5 are used to illustrate the MATLAB eommands. The roots of the eharaeteristie equation (or any polynomial) ean be found using the roots eommand. 

We first take the painful route just so we better understand the results from MATLAB. If we have to do the chore by hand, we much prefer the completing the perfect square method in Example 2.8. Even without MATLAB, we can easily find that the roots of the polynomial s2 + 4s +13 are -2 3j, and F(s) can be written as the sum of... 

The complete Routh array analysis allows us to find, for example, the number of poles on the imaginary axis. Since BIBO stability requires that all poles lie in the left-hand plane, we will not bother with these details (which are still in many control texts). Consider the fact that we can calculate easily the exact roots of a polynomial with MATLAB, we use the Routh criterion to the extent that it serves its purpose.1 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simplified Routh-Hurwitz recipe without proof follows. 

MATLAB s polynomial-roots finder roots does not handle repeated or clustered roots very well, but otherwise it is the best 0(n3) root finder available. Note that an operations count of 0(nP) for an algorithm signifies that the algorithm performs K n additions and multiplications (for some algorithm specific constants K and j, but depending on n) to obtain its output from n input data. Most of the polynomial-root finders of the last century unfortunately were even slower 0(n4) algorithms and all in all much too slow and inaccurate. 

To illustrate we first verify the identical behavior of the MATLAB QR based polynomial-root finder roots and MATLAB s QR based matrix eigenvalue finder eig for p s companion matrix P = C(p) First we define p by its coefficient vector in MATLAB s workspace, then we invoke the MATLAB polynomial-root finder roots, followed by its matrix eigenvalue finder eig on the companion matrix ofp. Finally we display the companion matrix P of p. As an example we use p(x) = x3 — 2.x2 + 4 here and represent p by its coefficient vector [1 -2 0 4] in the following line of MATLAB commands. 

MATLAB has a built-in root finder for scalar equations f(x) = 0 in one real variable x that are in standard form. The built-in MATLAB function is fzero. The use of fzero hinges on a user-defined function, such as the function f inside the following fzero tester, called fzerotryl, that we apply to our previously studied third degree polynomial. 

The first call of fzero inside fzerotryl takes 24 iterations to arrive at the real root x = -1.1304 of our trial polynomial p(x) x — 2x2 + 4 when starting at x0 = —2, while the second call converges after 9 iterations when looking for real roots of p inside the interval [—2, —1]. Please look up help fzero to learn more about this MATLAB function and how it was used. 

MATLAB s 0(n3) polynomial-root finder roots, used for the same polynomialp, encounters different problems and computes 4 complex conjugate root pairs instead. These lie on a small radius circle around the ninefold root 2. As input for roots, we represent our polynomial (x — 2)9 of degree 9 in extended form by its coefficient vector [1 -18. .. 2304 -512],... 

Only the earlier mentioned faster 0(n2) polynomial-root finder pzero discovers the ninefold real root 2 of p correctly see the Resources appendix for a quote of the literature for pzero and the folder pzero on our CD for the actual MATLAB code of pzero. 

In Examples 11.6 to 11.8, the characteristic equations were either first- or second-order, and thus we could find the roots analytically. For higher-order polynomials, this is not possible, and numerical root-finding techniques (Chapra and Canale, 2010), also available in MATLAB and Mathematica, must be employed. An attractive alternative, the Routh stability criterion, is available to evaluate stability without requiring calculation of the roots of the characteristic equation. 

Example 1.2 Finding a Root of an nth-Degree Polynomial by Newton-Raphson Method Applied to the Soave-Redlich-Kwong Equation of State. Develop a MATLAB function to calculate a root of a polynomial equation by Newton-Raphson method. Calculate the specific volume of a pure gas, at a given temperature and pressure, by using the Soave-Redlich-Kwong equation of state... 

MATLAB has its own function, roots.m, for calculating all the roots of a polynomial equation of the form in Eq. (1.11). This function accomplishes the task of finding the roots of the polynomial equation [Eq. (1.11)] by first converting the polynomial to the companion matrix A shown in Eq. (1.56), It then uses the built-in function eig.m, which calculates the eigenvalues of a matrix, to evaluate the eigenvalues of the companion matrix, which are also the roots of the polynomial Eq. (1.11) ... 

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