UNIVERSIDAD INDUSTRIAL DE SANTANDER

NUMERICAL METHODS

The ever-increasing advances in computer technology has enabled many in science and engineering to apply numerical methods to simulate physical phenomena. Numerical methods are often divided into elementary ones such as finding the root of an equation, integrating a function or solving a linear system of equations to intensive ones like the finite element method. Intensive methods are often needed for the solution of practical problems and they often require the systematic application of a range of elementary methods, often thousands or millions of times over.

In the development of numerical methods, simplifications need to be made to progress towards a solution: for example general functions may need to be approximated by polynomials and computers cannot generally represent numbers exactly anyway. As a result, numerical methods do not usually give the exact answer to a given problem, or they can only tend towards a solution getting closer and closer with each iteration. Numerical methods are generally only useful when they are implemented on computer using a computer programming language .

STEPS OF SOLVING AN ENGINEERING PROBLEM ( Video )

Learn the four steps of solving an engineering problem.

This video teaches you the steps of solving an engineering problem- define the problem, model the problem, solve, and implementation of the solution.

BASIC MATHEMATICAL MODELING

Mechanically, there are a many different ways to construct a model. There are two basic dimensions, however, and these define four classes of models with similar strengths and limitations. First, the underlying processes can be represented in either deterministic or stochastic forms. The difference is analogous to using the mean as a prediction summary versus using the full probability distribution of outcomes. Second, the dynamics over time can be explored either analytically or using computational methods. Analytic, or “closed-form,” solutions isolate the outcome on the left-hand side of an equation, with all of the determinants on the right-hand side, so it is clear how the outcome depends on the inputs. Not all processes can be represented this way. Computational, or “numerical,” solutions must be employed if there are non-trivial feedback loops in the process, so that the outcome ends up on both sides of the equation. Models of this sort are said to be “analytically intractable.” This happens very quickly as simplifying assumptions are relaxed, so most models that attempt to build in realistic heterogeneity need to be solved computationally.

All models divide the population into states (e.g. susceptible and infected) and define the process and rate of movement between those states. Deterministic models are usually built on group aggregates or macro-level states, while stochastic simulation models are usually built to reflect the micro-level states occupied by discrete individual persons. The primary difference between deterministic and stochastic models is how they define the movement between states. Deterministic models define the dynamics using the average rate of transition between states. Stochastic models define the dynamics using the probability that an individual makes the transition from one state to another.

Analytic models of both sorts (deterministic and stochastic) are typically regarded as the ideal, since they reveal a process in terms of simple cause and effect. Many infectious processes are not simple in that way, however, and the assumptions made to gain tractability often come at the cost of ignoring important parts of the process, and thus failure to properly project the outcomes of interest. As computing power has become more widely available, the need for tractability has declined, and computational-deterministic models have become the workhorse of mathematical epidemiology. Their use has led to substantial insight into the population dynamics of HIV and other STIs, as well as a wide range of other infectious diseases. Increasingly, the limitations of deterministic models are leading to the adoption of computational-stochastic or “microsimulation” methods. These methods are better for representing heterogeneities in the transmission process, behavioral or biological, and they are the only way to accurately represent something as simple as a person having multiple ongoing (“concurrent”) partnerships. The advantages of microsimulation are discussed in detail by van Imhoff and Post.(1998) The primary disadvantages are that it requires richer inputs, and may require significantly more computational capacity.

APROXIMATIONS AND ERRORS

Meanings and Example

Significant figures

It is also called significant digits and has been developed to formally designate the reliability of a value numerically. The number of significant figures is the number of digits plus one estimated digit that can be used with confidence. For example:

0.00001845
0.0001845
0.001845

The zeros are not always significant figures as they can be used alone to locate the decimal point. The numbers above have four significant figures. When zeros are included in large numbers, is not clear how many zeros are significant if any.
The concept of significant figures has two important implications for the study of numerical methods .

Accuracy and precision

T

o the number of significant figures representing a quantity 2) extending from the repeated readings of an instrument that measures some property physical .

For example:
When making some shots in a target shooting accuracy refers to the magnitude of the spread of the bullets. Accuracy refers to the approach of a number or measure the true value is supposed to represent. The inaccuracy (known as bias) is defined as a systematic departure from the truth.

Error Definitions

The numerical errors are generated with the use of approximations to represent the operations and quantities math . These errors include truncation, resulting from about a procedure exact mathematical and rounding errors that result from about exact numbers. For both types of errors, the relationship between the exact result and the approximate or true is given by
Estimated value = true value + error
Numerical error is equal to the difference between the true value and approximate value, ie
Eu = true value - approximate value
Example:
Suppose we have to measure the length of a bridge and a rivet, obtained 9999 and 9 cm, respectively. If the values are true 10 000 and 10 cm, calculate: a) wrong and B) the percentage relative error of each case:
Solution: a) the error in the measurement of the bridge is:
Eu = 10 000 - 9999 = 1 cm
And for the rivet is:
Eu = 10-9 = 1 cm
b) The percentage relative error for the bridge is:
Eu = 1 / 10 000 100% = 0.01%
And for the rivet is:
Eu = 1 / 10 100% = 10%
Therefore although both measures have an error of 1 cm, the relative error percentage is much larger rivet. We conclude that it has done a good job as far from the bridge, while the estimate for the fastener leaves much to be desired.
Fractional relative error = error.
True value
The relative error can also be multiplied by 100% to express it as:
Eu = true 100% error
True value
Where Eu denotes the percentage relative error
I) the approximation is larger than the true value (or the previous approach is greater than the current approach), the error is negative, if the approximation is less than the true value, the error is positive. Also in the equations , the denominator may be less than zero, which leads to a negative error.

True Error

ROUNDING ERROR

Rounding error Often, computers cut decimal numbers between e17 and 12th decimal thus introducing a rounding error, for example, the value of "e" is called infinitely 2.718281828 .... If we cut the number 2.71828182 ( 8 significant digits after the decimal point) we are getting or error = E = 2.718281828 -2.71828182 0.000000008 ... However, as we do not consider the number that was cut was greater than 5, then we should have let the number as 2.71828183, in which case the error would only = E = 2.118281828 -2.11828183 -0.000000002 .., that in absolute terms is much smaller than the last. In general, the cutting error of the computers will be much lower than the error introduced by a user, usually cut to a smaller number of significant figures. Example: Depending on the magnitude of the numbers with which you work, the rounding error can have a big impact very small in the final calculation. For example, if we have a product of 502.23 m and a dollar price of U.S. $ 7.52, the total price of U.S. $ 3,776.7696 give us (in Chilean pesos to $ 1 = $ 500 gives us $ 1,888,384 , 8). Now, if we introduce a variation of 0.1% in meters of the product and calculate the total, we get 502.23 * 0.1% = 507, 54, in U.S. $ equivalent of U.S. $ 3,816.7008 (ie, $ 1,908 350.4 Chilean pesos, a difference of $ 19,965.6), which is no less important as a variation of 0.1% in the footage gives a product greater than 1.5% error in the final price.

Rounding error. Almost all real numbers require for its decimal representation of an infinite number of digits. In practice, for management should be considered only a finite number of digits in its representation, we proceed to determination by an appropriate rounding. A typical case as presented by computers, in its report, stored only finite representations of real numbers. In this case we speak of inherent rounding.

EQUATION ROOTS

CLOSED METHODS

A simple method to obtain an approximation to the root of the equation f(x)=0, is to plot the function and see where it crosses the X axis.

The graph shows the existence of several roots, including perhaps a double root around X = 4.2

Reducing the horizontal scale is obtained:

In fact there are two different roots X=4.23 and X=4.26

The Bisection Method

The bisection method, also known as cutting binary or interval partition of Bolzano, is a type of incremental search in which the interval is always divided in half. If the value of the function changes sign on an interval, we evaluate the function value at the midpoint. The root position is determined by placing the midpoint of the subinterval, in which case a change of sign. The process is repeated until a better aproximation.

Error Estimates

An objective criterion to define when a numerical method has to stop, is to estimate the error in a way that does not require prior knowledge of the root. As discussed above, one can calculate the percentage relative error T as follows:

Method of False Position

This is the equation of false position. The estimated value of Xr then to replace either baseline Xi or Xu.

OPEN METHODS

Simple Fixed Point Iteration

Use a simple iteration fixed point to locate the following:

Newton Raphson Method

Secant Method

DIRECT METHODS FOR SOLVING SYSTEMS OF LINEAR EQUATIONS

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Direct methods

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GAUSS JORDAN METHOD

ITERATIVE METHODS FOR THE SOLUTION OF SYSTEM OF LINEAR EQUATIONS

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Iterative methods for the solution

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REFERENCES

http://www.mitecnologico.com/Main/ErrorPorRedondeo

http://www.slideshare.net/nestorbalcazar/mtodos-numricos-03

http://www.monografias.com/trabajos10/menu/menu.shtml

http://www.mty.itesm.mx/etie/deptos/m/ma95-843/lecturas/l843-13.pdf

http://www.monografias.com/trabajos18/sistemas-ecuaciones/sistemas-ecuaciones.shtml