Roots of a polynomial

Descartes Rule of Signs The number of positive real roots of a polynomial equation with real coefficients either is equal to the number V of its variations in sign or is less than i by a positive even integer. The number of negative roots of P(x) = 0 either is equal to the number of variations of sign of P - ) or is less than that number by a positive even integer. 

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. 

The Routh method can be used to find out if there are any roots of a polynomial in the RHP. It can be applied to either closedloop or openloop systems by using the appropnate characteristic equation. 

Nowadays, numerical analysts give the following global ready advice when any applied problem leads to trying to find roots of a polynomial to go back to before the problem was expressed in polynomial form . There was likely a linear model at an earlier stage of the problem development whose eigen information could be extracted much more reliably from the respective matrix than the roots can be from the unfortunate polynomial. 

Adapt f zerotry2. m on p. 30 to various other polynomials of your choice. Use polynomials of degrees less than 7 that have some multiple roots, as well as no multiple roots. What happens to the complex roots of a polynomial under fzero ... 

One of the weaknesses of the linear prediction method is the need to find the M (complex) roots of a polynomial. In the presence of noise, it is difficult to separate the true signal-related roots from the spurious roots due to the noise. A rather different approach to the signal identification problem was proposed by Kung,122 and introduced into NMR by Barkhuijsen and colleagues.123 Despite its origins in state-space theory, many of the processing steps involved are markedly similar to those of the linear prediction method. Useful reviews are those by de Beer and colleagues.20 23... 

For stability at a rest point one wishes to show that the eigenvalues of the linearization lie in the left half of the complex plane. There is a totally general result, the Routh-Hurwitz criterion, that can determine this. It is an algorithm for determining the signs of the real parts of the zeros of a polynomial. Since the eigenvalues of a matrix A are the roots of a polynomial... 

In this chapter you ll learn how to use graphical and numerical methods to solve chemical problems. The methods described range from the simple (finding the roots of a polynomial from a graph of the function) to the complex (using matrix methods to solve sets of simultaneous equations). Excel does most of the work, though, by virtue of its graphing capabilities and built-in numerical analysis capabilities. 

Sometimes a chemical problem can be reduced algebraically, by pencil and paper, to a polynomial expression for which the solution to the problem is one of the roots of the polynomial. Almost everyone remembers the quadratic formula for the roots of a quadratic equation, but finding the roots of a more complicated polynomial is more difficult. We begin by describing three methods for finding the real roots of a polynomial. 

The graphical method can give us only an approximate idea of the roots of a polynomial. To obtain a more accurate numerical result, the roots of 1/ = F(a ) can easily be obtained by trial and error, finding the values of x that make the function y equal to zero. 

Goal Seek.xls illustrates the use of Goal Seek... in the Tools menu to find the roots of a polynomial. It also shows, in hidden rows 8-25, how to find a root by successive approximations. 

NewtRaph.xls illustrates the use of the Newton-Raphson method to find the roots of a polynomial. 

The set of roots of a polynomial with real coefficients can include conjugate pairs of complex numbers. Thus eigenvalues can be complex, appearing in conjugate pairs. When this happens the corresponding column eigenvectors also form a conjugate pair, as do the rows. 

This procedure transforms the ill-conditioned problem of finding the roots of a polynomial into the well-conditioned problem of finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix. As shown by Wilf (1962), the N weights can then be calculated as Wa = OToV ai where tpai is the first component of the ath eigenvector (pa of the Jacobi matrix. 

The fundamental theorem of algebra states that a polynomial of degree n has n number of roots, although some of the roots may be complex numbers. If a root of a polynomial of degree n is known to be, say r, then a polynomial of degree n - 1 is given by... 

Quadratic formula The roots of a polynomial of degree n = 2 are given by the quadratic formula... 

Cardano s formula The roots of a polynomial of degree n = 3 are given by the cubic formula. The number of real roots of a cubic equation depends on the sign of the polynomial discriminant... 

Circular.xls shows how to use an intentional circular reference with the Newton-Raphson method to find the roots of a polynomial. A "Circular Reference" error message will be displayed upon opening the workbook. 

As we have seen, if is a root of a polynomial equation, then z is also a root. Recall that for real numbers, absolute value refers to the magnitude of a number, independent of its sign. Thus, 3.14 = — 3.14 = 3.14. We can also write - 3.14 = - 3.14. The absolute value of a complex number z, also called its magnitude or modulus, is likewise written as z - It is defined by... 

It is efficient at finding the roots of a polynomial with degree higher than 3. 

Two particularly useful functions are those for finding roots root(f(x),x) and polyrootsiy). Root will find one root of a fnnction if one exists but requires an initial guess of X as well as the function as inputs. Polyroots will find all of the roots of a polynomial using the coefficients of the polynomial in increasing order of x. Please see Figure 5.9 for an example. 

Roots of a polynomial. Example 4.6 shows that a first guess of x = 2 leads to the root x = 3 for the equation y = x- - 8x + IS. Toward what root does the Newton method converge if your first guess is x = 6 ... 

Coulson attended Clifton College in Bristol where his "wise and cunning mathematics master" H. C. Beaven, a Balliol graduate, invited him to a lecture at the University of Bristol by Selig Brodetsky, a rather well-known applied mathematician. Years later, Coulson remembered how flattered he was to be asked to a lecture at the university and also how impressed he was by what he heard. The speaker explained how a specific root-squaring process could be used to find the roots of a polynomial equation. The event marked him for life. 

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... 

This program solves the problem posed in Example 1.2. % It calculates the real gas specific volume from the % SRK equation of state using the Newton-Raphson method % for calculating the roots of a polynomial. 

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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