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Lecture 2

Engineering Computation. Lecture 2. Round-off Error due to Arithmetic Operations. Smearing Occurs when individual terms are larger than summation itself. Consider the exponential series with x = -10 Consider formulas such as:. With 7-decimal-digit accuracy:

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Lecture 2

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  1. EngineeringComputation Lecture 2 E. T. S. I. Caminos, Canales y Puertos

  2. Round-off Error due to Arithmetic Operations Smearing Occurs when individual terms are larger than summation itself. Consider the exponential series with x = -10 Consider formulas such as: • With 7-decimal-digit accuracy: • exact answer = 4.54 10-05 • computed answer = – 6.26 10-05 • (45 terms) wrong sign ! • Largest intermediate terms are: • 9th = –2,755.732 & 10th = 2,755.732 E. T. S. I. Caminos, Canales y Puertos

  3. Truncation Error Error caused by the nature of the numerical technique employed to approximate the solution. Example: Maclaurin series expansion of ex If we use a truncated version of the series: Then the Truncation Error is: E. T. S. I. Caminos, Canales y Puertos

  4. Approximations and Rounding Errors • Precautions: • Sums of large and small numbers: due to equaling the exponent. They are common in sums of infinite series where the individual terms are very small when compared with the accumulated sum. This error can be reduced by summing first the small terms and using double precision. • Cancellation of the subtraction: The subtraction of very similar numbers. • Smearing: The individual terms are larger than the total sum. • Inner products: They are prone to rounding errors. Thus, it is convenient to use double precision in this type of calculations. E. T. S. I. Caminos, Canales y Puertos

  5. Error Propagation • Errors which appear because we are basing current calculations on previous calculations which also incurred some form of error • Stability and Condition Number • Numerically Unstable: Computations which are so sensitive to round-off errors that errors grow uncontrollably during calculations. • Condition: sensitivity to such uncertainty; "well conditioned" vs. "ill conditioned" • Condition Number: measure of the condition; i.e., extent to which uncertainty in x is amplified by ƒ(x) • C.N.  1 ===> "well-conditioned" • C.N. >> 1 ===> "ill-conditioned" E. T. S. I. Caminos, Canales y Puertos

  6. Taylor Series Expansion Basic Idea: Predict the value of a function, ƒ, at a point xi+1 based on the value of the function and all of its derivatives, ƒ, ƒ', ƒ",… at a neighboring point xi Given xi, ƒ(xi), ƒ'(xi), ƒ"(xi), ... ƒn+1(xi), we can predict or approximate ƒ(xi+1) E. T. S. I. Caminos, Canales y Puertos

  7. Taylor Series Expansion General Form: • h = "step size" = xi+1 – xi • Rn = remainder to account for all other terms with xi   xi+1 • = O (hn+1) with x not exactly known "on the order of hn+1 " • Note: f(x) must be a function with n+1 continuous derivatives E. T. S. I. Caminos, Canales y Puertos

  8. Taylor Series Expansion  0th order T.S. approx. (n = 0): f(xi+1) = f(xi) + O (h1)  1st order T.S. approx. (n = 1): f(xi+1) = f(xi) + hf'(xi) + O (h2)  2nd order T.S. approx. (n = 2):  nth order T.S. approximation will be exact for an nth order polynomial E. T. S. I. Caminos, Canales y Puertos

  9. Taylor Series Expansion f(x) f(xi ) Zero order f(xi+1 ) f(xi ) First order f(xi+1 ) f(xi )+f'(xi )h Second order f(xi+1 ) f(xi )+f'(xi )h+ )+f"(xi )h2/2! True f(xi+1 ) x xi xi+1 h E. T. S. I. Caminos, Canales y Puertos

  10. Numerical Differentiation from Taylor Series Expansion Objective: Evaluate the derivatives of function, ƒ(xi), without doing it analytically. When would we want to do this? 1. function is too complicated to differentiate analytically: 2. function is not defined by an equation, i.e., given a set of data points (xi, ƒ(xi)), i=1,…,n i 0 1 2 3 4 xi1.0 3.0 5.0 7.0 9.0 ƒ(xi) 2.3 4.1 5.5 5.7 5.9 E. T. S. I. Caminos, Canales y Puertos

  11. Numerical Differentiation from Taylor Series Expansion • First derivative with backward difference. E. T. S. I. Caminos, Canales y Puertos

  12. Numerical Differentiation from Taylor Series Expansion Backward Difference Approx.:  First Derivative: Letting h = xi - xi-1 first backward difference E. T. S. I. Caminos, Canales y Puertos

  13. Example of 1st Backward FDD Using data below calculate ƒ'(x1) : i 0 1 2 3 4 xi 1.0 3.0 5.0 7.0 9.0 ƒ(xi) 2.3 4.1 5.5 5.7 5.9 First Backward Finite-Divided-Difference at x1: f'(x1)  0.9 { + O (h) } E. T. S. I. Caminos, Canales y Puertos

  14. Backward Difference Approximation Second Derivative: + O([xi-2– xi]3) with h = xi– xi-1 and 2h = xi – xi-2 The 2nd order approximation to ƒ(xi-2) becomes: ƒ(xi-2) = ƒ(xi) – 2hƒ'(xi) + 2h2 ƒ"(xi) +O (h3) [1] 2nd order approximation to ƒ(xi-1): [2] E. T. S. I. Caminos, Canales y Puertos

  15. Backward Difference Approximation Subtracting 2*[2] from [1] yields: f(xi-2) – 2f(xi-1) = –f(xi) + h2f"(xi) + O (h3) Rearranging: h2ƒ"(xi) = f(xi) – 2f(xi-1) + f(xi-2) + O (h3) Second backward difference E. T. S. I. Caminos, Canales y Puertos

  16. Example of 2nd Backward FDD Using data below calculate ƒ"(x2) : i 0 1 2 3 4 xi 1.0 3.0 5.0 7.0 9.0 ƒ(xi) 2.3 4.1 5.5 5.7 5.9 Second Backward Finite-Divided-Difference at x2: f"(5.0)  - 0.1 { + O (h) } E. T. S. I. Caminos, Canales y Puertos

  17. Other Forms of Numerical Differentiation What points are used for each form? Backward: …, ƒ(xi-2), ƒ(xi-1), ƒ(xi), ƒ(xi+1), ƒ(xi+2), … Forward: …, ƒ(xi-2), ƒ(xi-1), ƒ(xi), ƒ(xi+1), ƒ(xi+2), … Centered: …, ƒ(xi-2), ƒ(xi-1), ƒ(xi), ƒ(xi+1), ƒ(xi+2), … E. T. S. I. Caminos, Canales y Puertos

  18. Taylor Series and Truncation errors - Higher order divided differences. - Second finite central divided difference E. T. S. I. Caminos, Canales y Puertos

  19. Other Forms of Numerical Differentiation Forward: Centered: E. T. S. I. Caminos, Canales y Puertos

  20. Taylor Series and Truncation errors • Use of the Taylor series to calculate derivatives. • First derivative with forward difference. E. T. S. I. Caminos, Canales y Puertos

  21. Taylor Series and Truncation errors • First derivative with central differences. E. T. S. I. Caminos, Canales y Puertos

  22. Taylor Series and Truncation errors • Questions: • • Which is a better approximation? • Forward, Centered, or Backward? • • Why? • • When would you use which? • Note: • We also can get higher order forward, centered, and backward difference derivative approximations • [C&C Chapter 23, tabulated in Figs. 23.1-3] E. T. S. I. Caminos, Canales y Puertos

  23. Example Combining Roundoff and Truncation Error Determine h to minimize the total error of a forward finite-divided difference approximation for: • Truncation Error: xi   xi+1 • Round-off Error: with e = machine epsilon. As a result: E. T. S. I. Caminos, Canales y Puertos

  24. Example Combining Roundoff and Truncation Error  Total error  = truncation error + roundoff error  E = | Total Error |  + NOTE: Truncation error decreases as h decreases Round-off error increases as h decreases E. T. S. I. Caminos, Canales y Puertos

  25. Example Combining Roundoff and Truncation Error E. T. S. I. Caminos, Canales y Puertos

  26. Example Combining Roundoff and Truncation Error To minimize total error E with respect to h, set the first derivative to zero: Solve for h and approximate f"() as f"(xi): E. T. S. I. Caminos, Canales y Puertos

  27. Example Combining Roundoff and Truncation Error Linear Application: Determine h that will minimize total error for calculating f’(x) for f(x) =  x at x = 1 Using the first forward-divided-difference approximation with error O(h) and a 5-decimal-digit machine: e = b1-t = 101-5 = 10-4 = 0.0001 f'(x) = ; f"(x) = 0 E. T. S. I. Caminos, Canales y Puertos

  28. Example Combining Roundoff and Truncation Error f(x+h)= {exact: 3.1415} h  (x+h) f(x+h)-f(x) [f(x+h)-f(x)]/h 0 3.1415 0.000001 3.141500 0.00001 3.1416 0.000110 0.0001 3.1419 0.00044.0 0.001 3.1447 0.00323.2 0.01 3.1730 0.0315 3.15 0.1 3. 4557 0.3142 3.142 1 6.2831 3.1416 3.146 Underlined digits are subject to round-off error. They are likely to be in error by ± one or two units. This does not cause much problem when h = 1, but causes large errors in the final result when h < 10-4. E. T. S. I. Caminos, Canales y Puertos

  29. Example Combining Roundoff and Truncation Error Nonlinear Application: Determine h for minimizing the total error for computing f’(x) for ƒ(x) = ex at x = 3 Using the first forward-divided-difference approximation with error O(h) and a 5-decimal-digit machine: e = b1-t = 101-5 = 10-4 = 0.0001 f(x) = f'(x) = f"(x) = ex = 20.0855; or about 0.01 E. T. S. I. Caminos, Canales y Puertos

  30. Example Combining Roundoff and Truncation Error full precision h f(x+h)=ex+h f(x+h)-f(x) [f(x+h)-f(x)][f(x+h)-f(x)] h h 0 20.085 {exact = 20.085} 0.00001 20.085 0.00 20.086 0.0001 20.087 0.00220 20.086 0.001 20.105 0.020 20 20.096 0.01 20.287 0.202 20.2 20.18 0.1 22.198 2.113 21.13 21.12 1 54.598 34.513 34.52334.512 RoundoffTruncation Underlined digits subject to roundoff error. Bold digits in error due to truncation. E. T. S. I. Caminos, Canales y Puertos

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