Roots Of Equations

ROOTS OF EQUATIONS

Let y = f (x). The values of x which makes y = 0 are called the roots of the equation. The fundamental theorem of algebra states that every polynomial of degree n has n roots. In the case of real estate, it must correspond to x values that make the feature cut the x-axis, as shown in the figure.

CLOSED METHODS:

• Bisection Method: Start with a function f(x) and two values of x (a and b) such that f(a) and f(b) have opposite signs. These values of a and b may be the final interval of an incremental search method with a relitavely large step size.

THE PURPOSE OF SUCH METHOD IS DIVIDING THE RANGE IN SUCCESSIVE ITERATIONS INVESTIGATING THE CHANGE SIGN.

Procedure:

• Evaluate f(x) at the midpoint of the interval, at xmid = a+b / 2
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Limitations:

• False Position Method:
  • It's a root-finding algorithm that combins features of the bisection method and the secant method.
  If f (x) is real and continuous in the interval between xL and xU f (xL), f (xU) have opposite signs, there is at least one root in the interval under study.

It's a more rational method than the bisection method. It considers the proximity to one of the points.

It is based on graphical display.

Example:

Consider finding the root of f(x) = e^-x(3.2 sin(x) - 0.5 cos(x)) on the interval [3,4], ε=0,001

Thus after the sixth iteration, step is 0.0009 and therefore b = 3.2969 to be our approximation of the root.

OPEN METHODS:

• Fixed Point Method: The method starts with this concept f(x) = 0 ---> x = g(x) It predicts the value of the function based on the value of the same function. Xi+1 = g(xi).

• Newton-Raphson Method: This method is the most used for the determination of roots. A good approach is where the derivative cuts zero, and it should exist a near point.

In the Newton-Raphson method, the root is not bracketed. In fact, only one initial guess of the root is needed to get the iterative process started to find the root of an equation. The method hence falls in the category of open methods. Convergence in open methods is not guaranteed but if the method does converge, it does so much faster than the bracketing methods.

The Newton-Raphson method is based on the principle that if the initial guess of the root of f(x)=0 is at xi, then if one draws the tangent to the curve at f(xi), the point xi+1 where the tangent crosses the x-axis is an improved estimate of the root.

Procedure:

Drawdowns of the method:

• Secant Method: This is a derivation of the Newton - Raphson method. This method needs two points of departure and has and advantage: it doesn't need the derivative of f(x).

The above equation is called the secant method. This method now requires two initial guesses, but unlike the bisection method, the two initial guesses do not need to bracket the root of the equation. The secant method is an open method and may or may not converge. However, when secant method converges, it will typically converge faster than the bisection method and slower than the Newton-Raphson method.

Source: Cálculo avanzado, raices de ecuaciones. Cristian Bay. 2007

http://www.frsr.utn.edu.ar/documentos/pdf/raices_2_civil.pdf

MULTIPLE ROOTS

A multiple root corresponds to a point where a function is tangent to the x axis. For example, a double root results from:

f(x) = (x-3)(x-1)(x-1)

Multiplying terms, this polynomial looks like:

f(x) = X^3 - 5X^2 + 7X - 3

In the next figure we see how the function tangentially touches the x-axis at the point where a double root exists.

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