Revision

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

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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

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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

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Part I Truncation Errors

• 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

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Q3

• Let g1(x) be the Taylor series approximation of
  f(x) at p/4.
• Let g2(x) be the Taylor series approximation of
  f(x) at 0.
• 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.

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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.

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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

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Part II Roots Finding

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

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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

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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

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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.

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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

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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

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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

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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