Review After Midterm

Review (after the Mid-term) _______________________________________________________________________ _ 1. What is linear regression? 2. What is the least square fitting? 3. What is the least square fitting of a line? 4. What is the major difference between error measure Sr and St? 5. What is the linearization of a non-linear relationship? 6. What is polynomial regression? 7. What is the least square fitting of a polynomial? 8. What is multiple linear regression? 9. What is the least square fitting of a hyper-plane? 10. What is the general linear squares fitting? _______________________________________________________________________ _ 11. What is polynomial interpolation? 12. What is quadratic interpolation? 13. What is Newton’s Polynomial Interpolation? What’s its advantage? 14. What people use to estimate the errors of Newton’s Interpolation Polynomial? 15. What is Lagrange Interpolation Polynomial? 16. Inverse interpolation. 17. What’s the general idea of splines? 18. Quadratic spline. 19. Cubic spline. 20. Cheney and Kincaid Method of Cubic Spline _______________________________________________________________________ _ 21. What is the main motivation of Fourier approximation? 22. What’s the least squares fitting of a sinusoid?

23. How least-squares fitting of a sinusoid is simplified by equi-space sampling? 24. What is a harmonic function? 25. What is the least-squares fitting of a general sinusoidal function (harmonic)? 26. What is the continuous Fourier series representation of a periodical function? _______________________________________________________________________ _ 27. What is Newton-Cotes formulas for integration? 28. The trapezoidal rule. 29. Error estimate of trapezoidal rule. 30. Multiple applications of trapezoidal rule. 31. Error estimate of multiple applications of trapezoidal rule. 32. Simpson’s 1/3 rule. 33. Error estimate of Simpson’s 1/3 rule. 34. Multiple applications of Simpson’s 1/3 rule. 35. Simpson’s 3/8 rule. 36. Error estimate of Simpson’s 3/8 rule. 37. Multiple integrals. _______________________________________________________________________ _ 38. Richardson’s extrapolation of integration. What are its benefits? 39. The Romberg integration algorithm. 40. The motivation behind the Gauss Quadrature. 41. What is the Two-point Gauss-Legendre formula? 42. Arbitrary integral range for the two-point Gauss-Legendre formula. 43. Higher-point Gauss-Legendre formula. _______________________________________________________________________ _ 44. What is the key point using more sample points in numerical differentiation?

45. Forward, backward, and centered finite differences. 46. Richardson’s extrapolation. _______________________________________________________________________ _ 47. What is an ODE? 48. Higher order ODE and 1st order ODE, how to convert. 49. The Euler’s method, its error estimate. 50. The Heun’s method, its iterative nature. 51. Heun’s method and the trapezoidal rule. 52. The midpoint method. 53. The general Runge-Kutta method. 54. The 2nd order RK method – how are the coefficients determined? 55. The 4th order RK method. 56. System of 1st order ODEs. _______________________________________________________________________ _ 57. What is the main characteristic of a stiff ODE? 58. Difference between the implicit Euler’s method and the normal Euler’s method. 59. What is the general idea of multistep method? 60. Non-self-start Heun’s method. 61. Idea of using integration for solving ODE. 62. The open Newton-Cotes formulas; why is it called “open”? 63. The closed Newton-Cotes formulas; why is it called “closed”? 64. 2nd-order Adams-Bashforth Formulas; open or closed? 65. 2nd-order Adams-Moulton Formulas; open or closed? _______________________________________________________________________ _ 66. Boundary-Value Vs. Initial-Value ODE.

67. The shooting method for boundary-value ODE. 68. The finite difference approach in boundary-value ODEs. 69. What is the Power’s method? 70. How to use Power’s method to compute the smallest eigen value. _______________________________________________________________________ _ 71. What is a linear PDE? 72. What is an elliptic PDE? 73. The general idea of finite difference method for solving an elliptic PDE. 74. Liebmann’s method. 75. How to handle derivative boundary conditions in finite difference method. 76. How to handle irregular boundaries in finite difference method. 77. What is the control-volume approach? What are its differences with the point-wise approach? 78. What is a parabolic equation? What’s its main difference with an elliptic equation? 79. The explicit (finite difference) method for solving a parabolic PDE. What does “explicit” mean? 80. Convergence and stability of the explicit method. 81. The simple implicit method. Why it is called “implicit”? 82. The Crank-Nicolson method. Why it is an improvement to the simple implicit method? 83. Parabolic equations in two spatial dimensions. 84. The explicit method for two spatial dimensional parabolic PDE. 85. The Crank-Nicolson method for two spatial dimensional parabolic PDEs. 86. The Alternating-direction implicit method. Its major difference with the CrankNicolson method.