Numerical approximation and solution of equations


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Numerical approximation and solution of equations

  2. 2. General Purpose
  3. 3. Introduction Approximation errors In the numerical analysis, the error between the actual value and the achieved error is called a proxy.
  4. 4. Types of error There are several types of error, but the most common are: Truncation error : Assuming we want to calculate 24 / 7, we know that the outcome of this fraction is 3.428571 ... Truncating to two decimals, is 3.42 only, its expression as would be 171/50 broken, and this, as you can see, is generating an error, which we calculate below. log. step size rounding Fig 1.
  5. 5. <ul><li>Rounding error : Taking the example above, if rounded to two decimals, ie 3.43 only, its expression as 343/100 would be broken, and this will generate an error, as in the previous case, we will calculate later. </li></ul>Model Types Tabla 1.
  6. 6. Calculation error When we obtain an approximate value, regardless of method used, we can calculate the error in two ways: a) Absolute error (AE) : It is the difference between the actual value (RV) and approximate value (VA). Namely EA = | V R - VA | .
  7. 7. b) Relative error : is the percentage difference between the absolute value and real value. Y is calculated as (EA / VR) * 100 = ((| VR  VA |) / VR) * 100 =% error
  8. 8. Approaches Numerical approximation is defined as X * a figure that represents a number whose exact value is X. To the extent that the number X * is closer to the exact value X, is a better approximation of that number Examples: 3.1416 is a numerical approximation of  , 2.7183 is a numerical approximation of e, 1.4142 is a numerical approximation of  2, and 0.333333 is a numerical approximation 1/3
  9. 9. Accuracy yPresicion Presicion refers to the dispersion of the set of values from repeated measurements of a magnitude. The lower the spread the greater the accuracy. A common measure of variability is the standard deviation of measurements and precision can be estimated as a function of it.
  10. 10. Example 1 Several measures are as fired at a target. Accuracy describes the closeness of the arrows at the target center. Arrows that hit closer to the center are considered more accurate. The closer are the steps to an accepted value, the more accurate is a system. The precision, in this example, is the size of the group of arrows. The closer together are the arrows hit the target, the more accurate the system. Note that the fact that the arrows are very near each other is independent of the fact that they are near the center of the target. As such, we can say that accuracy is the degree of repeatability of the result. One could summarize that accuracy is the degree of accuracy, while precision is the degree of reproducibility high accuracy but low precision
  11. 11. Accuracy refers to how close the actual value is the measured value. In statistical terms, accuracy is related to the bias of an estimate. The smaller the bias is a more accurate estimate. When we express the accuracy of a result is expressed by the absolute error is the difference between the experimental value and the true value.
  12. 12. Example 2 An analog clock of hands, moves its minute &quot;just one minute&quot;, but does so in absolute sync with the official schedule or &quot;real&quot; (the target). A second clock uses minute, second, it is even equipped with a measurement system tenths. If we find that your schedule does not coincide fully with the official schedule or real (which remains the goal of every clock), we conclude that the first clock is highly accurate, though not necessary, while the second is highly accurate, although not shown exactly ... at least in our example. high precision but low accuracy
  13. 13. Convergence Convergence is defined as a numerical method ensuring that, when making a &quot;good number&quot; 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 than the other, to approach the desired value, is said to have a faster convergence.
  14. 14. Stability Stability means of a numerical method the level of assurance of convergence and numerical methods is that some do not always converge and, on the other hand, diverge, ie away from the more desired result. To the extent that a numerical method, to a very wide range of possibilities of mathematical modeling, it is safer to converge than the other, 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.
  15. 15. Type of problem to be solved <ul><li>Roots of equations </li></ul><ul><li>Systems of simultaneous linear equations </li></ul><ul><li>Interpolation, differentiation and integration </li></ul><ul><li>Ordinary Differential Equations </li></ul><ul><li>Partial Differential Equations </li></ul>
  16. 16. types of problem Mathematical model Numerical method Team
  17. 17. Problem-solving Software &quot; Program Development: &quot;C&quot; language &quot;Basic&quot; &quot;Fortran&quot; Other. Using mathematical software: &quot;Maple&quot; &quot;MatLab&quot; &quot;Mathcad&quot; &quot;Mathematica.&quot; Managing spreadsheets in PC: Excel Lotus Expedited handling of a graphing calculator
  18. 18. Bibliography c c