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The characteristic roots (roots of the characteristic equation) also provide qualitative information about the behavior of the variable whose evolution is described by the dynamic equation. For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is negative.
A quadratic with two real roots, for example, will have exactly two angles that satisfy the above conditions. For complex roots, one must also find a series of similar triangles, but with the vertices of the root path displaced from the polynomial path by a distance equal to the imaginary part of the root. In this case, the root path will not ...
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjugate a − bi is also a root of P. [1]
Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
More generally, if an equation P(x) = 0 of prime degree p with rational coefficients is solvable in radicals, then one can define an auxiliary equation Q(y) = 0 of degree p – 1, also with rational coefficients, such that each root of P is the sum of p-th roots of the roots of Q.
Let f(z) be a polynomial (with complex coefficients) of degree n with no roots on the imaginary axis (i.e. the line z = ic where i is the imaginary unit and c is a real number).Let us define real polynomials P 0 (y) and P 1 (y) by f(iy) = P 0 (y) + iP 1 (y), respectively the real and imaginary parts of f on the imaginary line.
For simple roots, this results immediately from the implicit function theorem. This is true also for multiple roots, but some care is needed for the proof. A small change of coefficients may induce a dramatic change of the roots, including the change of a real root into a complex root with a rather large imaginary part (see Wilkinson's polynomial).
Before continuing to the roots of (), it might be necessary to numerically improve the accuracy of the root approximations for (), for instance by Newton's method. Graeffe's method works best for polynomials with simple real roots, though it can be adapted for polynomials with complex roots and coefficients, and roots with higher multiplicity.
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