Revision

Part I: Errors Floating-point number representations Round-off and chopping errors Overflow and underflow Absolute vs. Relative Errors Errors propagation through arithmetic operations Subtractive Cancellation

Q1 Which of these decimal numbers cannot be represented exactly using the IEEE 754 standard for double precision floating point numbers (1 sign bit, 11 bits exponent and 52 bits mantissa)? 1234567890 0.1 0.1 x 102 The value of 1/1024 -123.25 Answer: (B) 0.1

Q2 What are the general approaches we can take to minimize errors in the calculated results? Avoid adding huge number to small number Avoid subtracting numbers that are close Minimize the number of arithmetic operations involved

Part I: Truncation Errors Taylor's Series Approximation Derive Taylor's series Understand the characteristics of Taylor's Series approximation Estimate truncation errors using the remainder term Estimating truncation errors by Alternating Series, Geometry series, Integration

Q3 Let g1(x) be the Taylor series approximation of f(x) at π/4. Suppose we want to approximate f(0.5) with error less than 10-6. Which Taylor series approximation is more efficient (involves less arithmetic operations)? g1(x) is better than g2(x) for all f(x) g2(x) is better than g1(x) for all f(x) Depends on what f(x) is. Answer: C [g1(x) is better than g2(x) for most f(x)]. A counter example would be f(x) = sin(x) or f(x) = cos(x) in which the alternative terms are zeroes.

Q4 Estimate the truncation errors for both series if we only include the first 10 terms of the series. Answer: For S1, infinity. For S2, 1/11.

Part II: Roots Finding Closed or Bracketing methods How to select initial interval? How to select the sub-interval for subsequent iterations? Bisection vs. False-Positioning methods in terms of performance Closed vs. Open methods Performance Convergence Multiple roots

Part II: Roots Finding Fixed point iteration Newton Raphson method Convergence Analysis Newton Raphson method Deriving the updating formula Selecting initial point Pitfalls Convergence Rate (Single/multiple roots) Secant vs. Newton Raphson

Part II: Roots Finding Convergent Rate Definition Estimation e.g.: Suppose the approximated errors in each iteration are 0.1, -0.05, 0.001, 10-4, 10-8, 10-16, 10-32, 0, 0, 0, … What is the convergent rate of the corresponding method? Modified Newton's methods for multiple roots

Part III: Systems of Equations Gauss Elimination Forward Elimination and its complexity Back substitution and its complexity Effect of pivoting and scaling LU Decomposition Algorithm LU Decomposition vs. Gauss Elimination Similarity Advantages (Disadvantages?) Complexity

Part III: Systems of Equations Error Analysis Ill-Condition system What is an ill-condition function? Iterative Methods (Gauss-Seidel, Jacobi) What are they good for? Convergent criteria Special Form of matrices [Optional] Tri-diagonal, symmetric, etc.

Part IV: Optimization Classification of Optimization Problems Single or multiple variables Linear or non-linear Constrained or unconstrained 1-D unconstrained optimization Golden-Section Search What is so special about this method? Quadratic Interpolation How fast does it converges? Newton's Method

Part IV: Optimization Multi-Dimensional Unconstrained Optimization Non-gradient or direct methods Gradient methods What is gradient? What is Hessian matrices? How to check if a point is a local maxima/minima or saddle point? Linear Programming Simplex Method

Part V: Curve Fitting Linear Least-Square Regression Why is it called "Linear"? When to use Regression and when to use interpolation? How to construct polynomials? Newton form Lagrange form

Part V: Curve Fitting What is spline interpolation? Spline interpolation vs. polynomial interpolation Linear vs. Quadratic vs. Cubic Splines What are the conditions used to determine the spline functions? Does the order of the data points matter in Regression Polynomial interpolation Spline interpolation