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# Methods of calculate roots of equations

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### Methods of calculate roots of equations

1. 1. NORAIMA NAYARITH ZARATE GARCIA ING. DE PETROLEOS COD. 2073173
2. 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. 3. In the particular case where a = 0, the series is also called a McLaren series
4. 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. 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. 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. 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. 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. 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. 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. 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.