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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Numerical Approach

It is a systematic technique whose results are approximations of the true value that assumes the variable of interest, the consistent repetition of the art, to which the called "iterations" is what you closer and closer to the desired value.

Examples:

• 3.1416 is a numerical approximation of π
• 2.7183 is a numerical approximation of e
• 1.4142 is a numerical approximation of the square root of 2
• 0.333333 is a numerical approximation of 1 / 3

Significant Figures

The measurements are normally done through instruments, such as a speedometer to measure the speed of a car, or an odometer to measure the distance covered.

The number of significant figures is the number of digits t, which can be used with confidence to measure a variable, for example: 3 significant figures on the speedometer and 7 on the odometer.

The management of significant figures can develop criteria to detect how accurate are the results and to assess levels of accuracy and precision with which they are expressed some numbers such as pi, or root of 2.

Accuracy And Precition

Precision:
Refers to the number of significant figures representing a quantity.

Accuracy:
It refers to the approach of a number or measure the numerical value is supposed to represent.

The numerical methods should provide sufficiently accurate and precise solutions. The term "error" is used to represent both the inaccuracy and to measure the uncertainty in the predictions.

Covergence And Stability

Convergence:

Convergence is defined as a numerical method ensuring that, when making a "good number" of iterations, the approximations obtained eventually move closer and closer to the true value sought.

To the extent that a numerical method requires fewer iterations to get close to another desired value, is said to have a faster convergence.

Stability:

Stability means of a numerical method the level of assurance of convergence. Some never numerical methods and instead converge, diverge, that is, away from the more desired result.

To the extent that a numerical method, to a very wide range of possibilities of mathematical modeling, cnverja is safer than another, is said to have greater stability.

It is common to find methods that converge quickly, but they are very unstable and, in contrast, very stable models, but slow convergence.

Error In Numerical Methods

There are two types of error that are common in numerical calcula

tions:

Roundoff error:

Is due to the fact that floating

point numbers are represented by finite precision.

Truncation error:

Are related to the method of approach to be used because they generally face an infinite series of terms, will tend to cut the number of terms, introducing an error at that time, not to use the series complete (assumed to be exact). In one itation is understood as the error by not following iteration and continue moving towards the solution.

Source: Error in numerical methods, Brian D. Storey

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¿What is a model?

"The object of empirical study exists in the tangible world, or empirical research, as researchers call it. In most research projects, one of the first goals is to create a theoretical picture of the empirical object of study in the conceptual world of thought and theory. Scientists often use the model name of this picture of the object of study, as seen in the diagram on the right. In the early stages of a research project the model often exists only as an idea in the mind of the researcher, but soon he will want to put it on paper or computer, too."

"Representation of reality by means of abstractions.The models focus on important parts of a system (at least, that she is interested in a specific model type), downplaying otras.Los models are created using modeling tools."

Fuente: http://www2.uiah.fi/projekti/metodi/25b.htm

Mathematical Model:

"Representation of reality by means of abstractions.The models focus on important parts of a system (at least, that she is interested in a specific model type), downplaying otras.Los models are created using modeling tools."

It is a kind of scientific model that uses a mathematical formula to express relationships, substantive propositions of facts, variables, parameters, entities and relationships between variables and / or entities and operations.

These models are used to analyze the behavior of complex systems in situations that are difficult to observe in reality.

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