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LECTURE 2: Overview and Simple Error Analysis

Opening of the course with the following:

  1. Student survey of interests in scientific computing
  2. Course details
  3. Course plan
  4. Exams and projects
  5. Scientific computing problems
  6. Projects
  7. Email
  8. Basics: Taylor expansion
  9. Four sets of equations
    1. Newton's equations for classical mechanics
    2. Maxwell's equations for E&M
    3. Schrodinger equation for quantum mechanics
    4. Yang-mills equation for QCD

---- Actual Contents:

  1. Scientific Computation: techniques to obtain solutions to mathematical models representing physical situations.
    1. solve equations
    2. data management
    3. visualization
    4. symbolic computing 
2.             Relations of Scientific Computation to other areas:
         Science & engineeringàmathànumerical analysisà programmingà solutions  
  1. Elements of good programs
    1. reliability
    2. robustness
    3. portability
    4. maintainability
    5. Efficiency
  2. Error Estimates:
    1. mean

mean

b.      absolute error

absolute error

c.       relative error

relative error

d.      root mean square error

root mean square error

e.       normalized scientific notation

normalized scientific notation

where n = integer (-, 0, +) and r in [1, 10)

e.g.
  1. Sources of errors
    1. algorithm error---discretization error
    2. machine error---rounding error
    3. unstable: numerical process is unstable if small errors made at one stage get magnified at subsequent stages and seriously degrade the accuracy of the overall calculation
    4. ill condition: relatively small changes in the input data produces large changes in the solution