# Numerical Analysis

**Bounds for the Roots of Polynomials:** Let A = (a_{ij}) be an n * n matrix.

If Au = λu, then λ and u are called the eigenvalue and eigenvector of A,
respectively. The

eigenvalues of A are the roots of the characteristic polynomial

The eigenvectors are the solutions to the Homogeneous system

If A is symmetric, i.e., A^{t} = A, then all the eigenvalues of A are real. Let

be the eigenvalues of A, then

Our fist theorem is known as the Gerschgorin's Disks Theorem.

**Theorem 1.** Let A = (a_{ij} be an n * n
matrix. For j = 1, 2, ... , n, define

Let D_{j} (a_{jj} , r_{j}) be the
disk of radius r_{j} with the center at the point (0, a_{ij}) of
the complex plane. Then

all the eigenvalues of the matrix A are contained within the union of the D_{j}
's. Thus

contains all the eigenvalues of A.

**Remark.** Since A and A^{t} have the same set
of eigenvalues, we may use Theorem 1. for

both A and A^{t} and get the best neighborhood
for the eigenvalues of A.

Consider now the polynomial of degree n

The polynomial P is said to be monic, if the leading
coefficient a_{0} equals one. Clearly,

the matrix

is monic. To this monic polynomial we associate an n * n matrix

The matrix C_{P} is called the Companion Matrix of
P(x).

**Theorem 2.** x_{0} is a root of p(x) if and
only if x_{0} is an eigenvalue of the matrix C_{P}.

Corollary. Consider a monic polynomial P(x) of degree n. Then

( i) all the roots of P(x) is contained within , where

(ii) if {x_{1}, x_{2}, ..., x_{n}}
are the n roots of P(x), then

**Proof.** By using C_{P}, the above theorems,
the Remark and the fact that trace(C_{P}) = -a_{1},

one may readily prove the corollary.

**Rational Roots: **Although a real polynomial may have
complex roots, but there

is a well known theorem concerning the rational roots of polynomial with integer
coefficients.

**Theorem 3.** Let be
a polynomial with integer coefficients.

If p/q is a rational root of P(x), then a_{n} = pr and a_{0} =
qs.

**Nested Form:** Consider the following polynomial of
degree n

The following form of P(x) is called the nested form of P(x):

Finally, we present a root finding tool known as Horner's method or Synthetic division.

**Horner's method (Synthetic Division):** Consider the
polynomial:

The following chart shows how to evaluate