1. Revision

2. 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

3. 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)?
0.1
0.1 x 102
The value of 1/1024
Answer: (B) 0.1

4. 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

5. 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

6. 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 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.

7. 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.

8. 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

9. 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

10. 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

11. 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

12. 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.

13. 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

14. 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

15. 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

16. 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