Numerical Methods Ingles Part 2

8/9/2019 Numerical Methods Ingles Part 2

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5 NUMERICAL METHODS

NUMERICAL APPROXIMATIONS

One of the biggest problems when it comes to do math, is todetermine how accurate and precise and is the data we deliver,because nothing is 100% accurate the only way to deal with thisproblem is to determine how erroneous that our formulation. Tothis end, several techniques have been developed based on thetype of error that a determination.

Significant Figures: the concept of significant figures orsignificant digits has been developed to formally appoints thereliability of a numerical value. The scores of a number are thosethat can be used reliably even usually identify a number of significant figures is a simple procedure in some cases result insome confusion.

Accuracy and precision: the errors in the calculations andmeasurements can be characterized by their accuracy andprecision, accuracy refers to how close the measured or calculated

value of the true value. Accuracy refers to how close they arefrom each other measured or calculated value.


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6 NUMERICAL METHODS

Definition of Errors

Absolute error: defined as the difference between the actual value and the calculated value.

Absolute relative error: defined as the ratio of absoluteerror and the actual value.

Percentage Error: is defined as the percent relative error

Approximate Error: defined as the difference between twosubsequent calculations


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7 NUMERICAL METHODS

Approximate relative error: is defined as the ratio of theapproximate error and calculation n +1

One of the challenges faced by numerical methods is todetermine the error estimates in the absence of knowledge of true values. For example, some numerical methods iterativemethods used to calculate the results. In such methods an

approximation considering the anterior approach. This process isperformed repeatedly or iteratively. To calculate in succession,hoping to get better approximations. In such cases, the error isoften calculated as the equation was. The importance of thiserror is that the absolute error should be a default value asfollows:

Where it is called tolerance, which is preset is simply a measureof acceptable error under which work, there is also a way of defining the acceptable error as a function of correct significantfigures predicted by iteration or calculation:

Where n represents the minimum number of significant figuresthat will be correctly to predict by the formula.

Main causes of error on computers

There are three main types of errors when making mathematical

manipulations on a computer. These occur due to the difficulty of storing figures after the appropriate language translation.


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8 NUMERICAL METHODS

The range of amounts that can be represented is limited, try to save a number too large in terms presents a problembecause sometimes storage space is limited and thereforethe number cannot be causing the overflow error (due tobinary representation of numbers in the computer)

There are only a finite number of quantities that can berepresented within a range; the degree of precision islimited. It is clear that irrational numbers cannot be

represented accurately. Called quantization error and theapproximation is done by two methods.

The interval between the numbers increases as thenumbers grow in magnitude means that the quantizationerrors are proportional to the magnitude of the number tobe represented. Minimum range is defined as:

For court cases and rounding, respectively.


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9 NUMERICAL METHODS

ROOTS OF EQUATIONS

The root of an equation is defined as the point (x) in which thefunction f (x) is zero. To find the points were available formulasor algebraic manipulations, but there were some equations thatcould not be solved analytically. Problem was overcome whenstarting the use of computational tools to find an estimate of these roots.

analytical solution

This equality is satisfied if

numerical solution

It is an impossible equation to solve analytically, but usedifferent algorithms to find a numerical estimate of this.

NUMERICAL APPROACH FOR ROOTS

Numerical approximation is called the estimation of a numberthat is not accurate but serves as this. In this way fails to resolvethe problem of analytical irresolution.


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10 NUMERICAL METHODS

CLOSED METHODS

It is based on the fact that a function changes sign in the vicinity of a root, aspirations take two starting points that have that be oneither side of which is presumably the root of the equation,reducing the interval until you get a reasonable approximation.

Graphical method: it consists in plotting the function and sees where it cuts the axis of the abscissa.


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Advantage

- Is a method that gives the range where the root.- Provides an outline of the behavior of the function which

helps to detect errors in other methods.

Disadvantages

- It is an imprecise method that relies on the observation of the graph.

Note: If the function evaluated at the upper limit and lower limithas the same sign there are no roots or a pair of them and if different is an odd number of them, namely:

No roots or is an even number of them

There are an odd number of roots.


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An example is presented the figure where it is easily seen that theinterval at which is the root

Bisection: Based on the subdivision of the interval where is theroot in various subintervals so that the average interval satisfy a margin of error for the root.

Xr where represents the root for that iteration. While X su and X in the upper and lower limits respectively

The chart above illustrates the method. To define which of thesubintervals is the root using the following criteria

1. then the root is in this subinterval and

is now doing for the next iteration

2. then the root is in the other subinterval

and so for the next iteration.

X in x r x su


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3. then that is the root.

For this method the error of approximation depends on thenumber of iterations, as follows:

Where is the initial interval, n the number of iterations. Viewed another way, if you know the error must then bedetermined the number of iterations.

Advantage

- Do not diverge, this is always tends to the root.

Disadvantages

- You need to know in advance an estimated range for theroots.

- When there are roots tangent to x-axis does not work because the function does not change sign.

X in x su


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- It does not consider the limits which is closer to the root.

False position: it rests on the assumption that the lowestabsolute value of the function within the limits of the range iscloser to the root (which is not always true), thus joining thepoints with a straight line and where short axis was taken on asthe deadline for a new iteration.

The formula for these iterations is:

Advantage

- The process tends more quickly to the root as the previousmethod.


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Disadvantage

- There are functions that do not meet the condition that thelower absolute value of f (x) at the ends of the range is theclosest to the root as shown in the figure below.

You can modify the method so that after a certain number of iterations multiply the value of f (x) are not being moved by a number between zero and one.

Search by increments: the choice is based on a small increase therange and travel through the increase to determine where thefunction changes sign


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Disadvantages

- The determination of the increase is a problem because if is very large you can skip if roots and very little will be longand inefficient.

OPEN METHODS

This iteration method uses only one value of x or two but withthe freedom that they should not therefore enclose the followingmethods is called open.

Disadvantages

- Sometimes the root diverges

Advantage

- When it does converge generally much more efficient thanclosed methods.

Simple iteration method fixed point: this method takes the roleand manipulated so that it is of the form from this gives a starting value and begins to iterate for equality.

Condition:

For this method to converge that must be met:


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Newton-Raphson method: Use the tangent of the position at a point to approximate the root this is best illustrated graphically.

The formula for the approximation is:

Where the error of the iteration i +1 is:

Disadvantage

- The multiple roots are tangent to the x-axis so that f '(x)

becomes zero at point iteration therefore cannot be done.


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- When there are turning points near the root of theapproach diverges from the root.

Secant method: it is a helper method to the Newton-Raphsonand just changes the derivative of a difference.

The formula for this method is:

To become more efficient this method the derivative is redefined:

Where is greater than zero and less than one.

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Numerical Methods Ingles Part 2