The document discusses several numerical methods for finding the roots or zeros of functions, including: 1) The bisection method, which uses interval halving to find roots and converges slowly but reliably. 2) The false position method and fixed point method, which generate sequences of approximations that converge to a solution. 3) Newton's method and the secant method, which use derivatives or secant lines to quickly find better approximations with each iteration. Initial values close to a root are needed for convergence.

1. NORAIMA NAYARITH ZARATE GARCIA
ING. DE PETROLEOS
COD. 2073173

2.  The Taylor series of real or complex function
f (x) is infinitely differentiable in a
neighborhood of a real or complex number is
the power series.

3. In the particular case where a = 0, the series is
also called a McLaren series

4.  The bisection method is a numerical method to
calculate the roots of a polynomial f (x). It is one of
the simplest and most reliable, but not the quickest
method. Suppose that f (x) is continuous.
 Therefore postulate that our function f is
continuous, which is reasonable, because f they
often result from a physical model. Suppose now
that we have an interval [ a, b],with f(a) and f(b)
with different signs, or in other words:
f(a)f(b)<0

5.  It is, therefore, from elemental analysis that the
interval [a,b] will contain at least one root of the
function f . Now suppose that we somehow got an
approximation to this root, for example c , with
a<c<b .You can then evaluate f in c. At present
there are three possibilities:
 f(c) = 0 ;We have found the root, but this is
"infinitely unlikely" to occur.
 f(a)f(c) <0 , We know the root is in the range.
 f(b)f( c) <0 , We know the root is in the range.

6.  So if c not the real root, at least we can update our range and
get a smaller one, which still has the root. If we repeat this
often enough, you can make the range [a,b] smaller and
smaller, and thus we hope that we can obtain a sufficiently
accurate approximation to the root.

7.  Like the bisection method, false position method begins with two
points a0 and b 0 such that f (a 0) and f (b 0) have opposite signs,
which implies the mean value theorem that the function f has a root
in the interval [a0, b 0]. The product method by producing a
sequence of shrinking intervals [ak, bk] containing a root of f.
In the iteration number k, the number.

8.  The fixed point method starts with an initial approach Xo and
Xi+1 generates a sequence of approximations which
converge to the solution of the equation f(x)=0. The function
g is called the iterator function. It can be shown that this
sequence converges provided .

9.  It is an efficient algorithm to find approximations of the zeros
or roots of a real function. It can also be used to find the max
or min of a function, finding the zeros of its first derivative.
 The only way to achieve convergence is to select an initial
value close enough to the desired root. Thus, we must start
the iteration with a value reasonably close to zero.

10.  It is a variation of Newton-Raphson method where instead of
calculating the derivative of the function at the point of study, the
slope is close to the line that connects the function evaluated at the
point of study and at the point of the previous iteration .
 The method is based on obtaining the equation of the line through
the points (xn-1, f (xn-1)) and (xn, f (xn)). In that line drying is
called for cutting the graph of the function.

11.  CHAPRA , STEVEN C. Y CANALE, RAYMOND P.
Numerics Mathods for Engineers. McGraw
Hill 2002.
 Es. Wikipedia. Org/wiki.
 SANTAFE, Elkin R. “Elementos básicos de
modelamiento matemático”.
Clases -universidad de Santander año- 2009.