Numerical approximation


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Numerical approximation

  2. 2. The concept of significant figures has been developed to designate formally the reliability of a numerical value. The significant figures of a number are those that can be used reliably, e.g, used to determine the accuracy of the measurement.
  3. 3. The measurement of fluid volume in the graduated cylinder is between 38 and 39 cm³. We believe that it is 38.4 cm or 38.2 cm³. As modified, a move the last digit is estimated and therefore uncertain. The measure of this volume has three significant figures.
  4. 4. The concept of significant figures has two important implications for the study of numerical methods. 1 The numerical methods obtain approximate results. Therefore, we must develop criteria to specify how accurate are the results obtained. A way to do it is in terms of significant figures. For example, we can say that the approach is acceptable to four significant figures.
  5. 5. 2 Although certain quantities such as π or e represent specific numbers, they can’t to express exactly with a finite number of digits. For example: π = 3.14159265358973238462643... To infinity. The computers hold only a finite number of significant figures, these numbers can never be represented accurately.
  6. 6. 1 All nonzero digits are significant: 1,284 g  4 significant figures 1,2 g  2 significant figures With zeroes, the situation is particularly: Zeroes placed before other digits are 2 not significant. 0.046  2 significant digits.
  7. 7. 3 Zeroes placed between other digits are always significant. 4009 kg  4 significant digits. Zeroes placed after other digits 4 but behind a decimal point are significant. 7,90  3 significant digits.
  8. 8. 5 Zeroes at the end of a number are significant only if they are behind a decimal point. Otherwise, it is impossible to tell if they are significant. For example, in the number 8200, it is not clear if the zeroes are significant or not. The number of significant digits in 8200 is at least two, but could be three or four. To avoid uncertainty, use scientific notation to place 8.200 x 103  4 S.F significant zeroes behind a decimal point: 8.20 x 103  3 S.F 8.2 x 103  2 S.F
  9. 9. In math operations, the significant number its in answer should equal to the least number of significant digits in any one of the numbers being multiplied, divided etc. Example: 5.67 J ( 3 S.F) + 1.1 J (2 S.F) 0.9378 J (4 S.F) -------------------- (2 S.F) 7.7 J
  10. 10. It's called rounding to the process of eliminating non-significant digits of a number. The rules are the following: 1 If the digit removed is greater than 5, the previous digit increases by one. E.g: 8.236 → 8.24 If the digit removed is less than 5, the previous digit 2 is not modified. E.g: 8.231 → 8.23 If the digit removed is 5 followed by a different 3 number than 0, the previous digit increases by one. E.g: 8.2353→8.24 If the digit removed is 5 followed by 0 looks to the 4 next that follows, if it is odd increase or if it’s pair remains unchanged. E.g: (1) 8.23503→8.24 (2) 8.23502→8.23
  11. 11. 1. Writing more digits in an answer (intermediate or final) than justified by the number of digits in the data. 2. Rounding-off, say, to two digits in an intermediate answer, and then writing three digits in the final answer.
  12. 12. ACCURACY refers to how close is d measured or calculated value to the true value. PRECISION refers to how close is an measured or calculated individual value the others. with respect to b THE INACCURACY OR BIAS is c defined as a systematic departure from the truth. THE VAGUENESS OR UNCERTAINTY, refers to the magnitude of the spread of values. a
  13. 13. The numerical methods should be sufficiently accurate or no bias to satisfy the requirements of a particular engineering problem.
  14. 14. Rounding error Truncation error Occurs when the numbers has Represents the difference between a limit of significant figures an exact mathematical formulation which are used to represent of a problem and the approximation exact numbers. given by a numerical method. For the types of errors, the relationship between the exact or true result and the approximate is given by: True value = Approximation + error
  15. 15. • True or Absolute Error It is equal to the difference between the true value and approximate value • Relative Error It is the quotient (division) between the absolute error and the true value. If you multiply by 100 to obtain the true percentage relative error. True Value - Approximation True Value - Approximation Relative Error= True Value or t = x100 True Value
  16. 16. For numerical methods, the true value will only be known when the functions can be solved analytically. Thus, in real life to know the true value early, it is difficult. In these cases, normalizing the error is an alternative to have the best possible estimate of true value: Approximate Error a= x100 Approximate Value Some numerical methods use an iterative method to calculate results. In such cases, the error is calculated as the difference between the previous and the current approach. Therefore, the percentage relative error is given by: Current approach - Anterior approach a= x100 Current approach
  17. 17. In essence, the Taylor series provides a means to predict the value of a function at a point in terms of the function value and its derivatives at another point. In particular, the theorem states that any smooth function can be Aproximación de la función exponencial approximated with a Fuente: polynomial. ns/6/64/Taylorspolynomialexbig.svg
  18. 18. To the Taylor’s series construction makes use of approximations, what allows us to understand more about them. Initially requires a first term which is a zero-order approximation (f value at the new point is equal to the value in the previous point) If (xi ) is next to (xi+1),then F(xi) soon will be equal to F(xi+1):
  19. 19. To achieve greater approach adds one more term to the series; this is an order 1 approximation, which generates an adjustment for straight lines. To make the Taylor´s series expansion and to gain better approach generalizes the series for all functions, as follows:
  20. 20. Similarly can be added terms and obtain the Taylor series of order n: f ''(a ) f ( n )(a ) f( x)  f(a )  f '(a )( x  a )  ( x  a )2  ...  ( x  a )n  Rn 2! n! Where Rn term is included, to notice the term of n +1 to infinity. : x ( x  t )n ( n1) Rn   f (t )dt a n! “With the Taylor’ series we can estimate the truncation errors”