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ECIV 301

ECIV 301. Programming & Graphics Numerical Methods for Engineers REVIEW III. Topics. Regression Analysis Linear Regression Linearized Regression Polynomial Regression Numerical Integration Newton Cotes Trapezoidal Rule Simpson Rules Gaussian Quadrature. Topics.

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ECIV 301

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  1. ECIV 301 Programming & Graphics Numerical Methods for Engineers REVIEW III

  2. Topics • Regression Analysis • Linear Regression • Linearized Regression • Polynomial Regression • Numerical Integration • Newton Cotes • Trapezoidal Rule • Simpson Rules • Gaussian Quadrature

  3. Topics • Numerical Differentiation • Finite Difference Forms • ODE – Initial Value Problems • Runge Kutta Methods • ODE – Boundary Value Problems • Finite Difference Method

  4. what value of y corresponds to x=0.935? Regression Often we are faced with the problem…

  5. e.g. Best Fit ? Curve Fitting Question 2: Is it possible to find a simple and convenient formula that represents dataapproximately ? Approximation

  6. Experimental Measurements Stress Strain

  7. BEST FIT CRITERIA Error at each Point y Stress Strain

  8. Best Fit => Minimize Error Best Strategy

  9. Objective: What are the values of ao and a1 that minimize ? Best Fit => Minimize Error

  10. Least Square Approximation In our case Since xi and yi are known from given data

  11. Least Square Approximation

  12. Least Square Approximation 2 Eqtns 2 Unknowns

  13. Least Square Approximation

  14. Example

  15. Quantification of Error Average

  16. Quantification of Error Average

  17. Quantification of Error Average

  18. Quantification of Error Standard Deviation Shows Spread Around mean Value

  19. Quantification of Error

  20. Quantification of Error “Standard Deviation” for Linear Regression

  21. Quantification of Error Better Representation Less Spread

  22. Quantification of Error Coefficient of Determination Correlation Coefficient

  23. Linearized Regression The Exponential Equation

  24. Linearized Regression The Power Equation

  25. Linearized Regression The Saturation-Growth-Rate Equation

  26. Polynomial Regression A Parabola is Preferable

  27. Polynomial Regression Minimize

  28. Polynomial Regression

  29. Polynomial Regression 3 Eqtns 3 Unknowns

  30. Polynomial Regression Use any of the Methods we Learned

  31. Polynomial Regression With a0, a1, a2 known the Total Error Standard Error Coefficient of Determination

  32. Polynomial Regression For Polynomial of Order m Standard Error Coefficient of Determination

  33. Numerical Integration & Differentiation

  34. Motivation

  35. Motivation

  36. Motivation

  37. AREA BETWEEN a AND b Motivation

  38. Motivation

  39. Motivation

  40. Motivation

  41. Calculate Derivative Motivation Given

  42. Motivation Given Calculate

  43. Think as Engineers!

  44. In Summary INTERPOLATE

  45. In Summary Newton-Cotes Formulas Replace a complicated function or tabulated data with an approximating function that is easy to integrate

  46. In Summary Also by piecewise approximation

  47. Closed/Open Forms CLOSED OPEN

  48. Trapezoidal Rule Linear Interpolation

  49. Trapezoidal Rule Multiple Application

  50. Trapezoidal Rule Multiple Application

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