OR682Math685CSI700

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OR682/Math685/CSI700

• Lecture 1
• Fall 2000

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Basic Information

• http//iris.gmu.edu/snash/
• Access to Matlab
• Campus computer labs
• Software from www.mathworks.com
• Access to these lecture notes

Matlab Commands diary, help
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Scientific Computing

• Solve and understand mathematical models
• Approximations
• Errors
• Accuracy
• Efficiency
• Continuous versus discrete

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General Approach

• Replace a difficult problem with an easier one
  that has a similar solution
• Complicated ? simple
• Nonlinear ? linear
• Infinite ? finite
• Differential ? algebraic

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Approximation Example

• Area of the earth
• Earth is modeled as a sphere
• Value for radius is measured or estimated
• Value for ? is approximate
• Computations are performed on the computer to
  limited precision

Matlab commands pi,
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Fundamental Sources of Errors

• Input data are inaccurate
• Out of our control
• Computations are inaccurate
• The subject of this course

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Ways of Measuring Error

• Absolute error
• ltApproximate valuegt - lttrue valuegt
• Relative error
• ltAbsolute errorgt / lttrue valuegt
• Blended error
• ltAbsolute errorgt / 1 lttrue valuegt

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Conditioning

• Is the problem good or bad?
• Check if cond (condition number) is large, where

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More on Conditioning

• Let x be the data (with error h), and f (x) be
  the answer

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Conditioning Examples

• Roots of
• Computation of e x
• Computation of sin(x)

Matlab commands syms, pretty, roots
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Forward Backward Error

• Forward error if I make an error x how large
  is the error in the answer f (x)?
• If the problem is ill-conditioned then the error
  in the answer will almost always be large, even
  with a good algorithm.
• Backward error is the answer I computed the
  exact answer to a nearby problem?
• This will be true for a good algorithm.

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Conditioning Stability

• Conditioning applies to problems
• Well conditioned (forward error is small)
• Ill conditioned
• Stability applies to algorithms
• Stable (backward error is small)
• Unstable
• If possible, the algorithm should estimate
  conditioning as it performs the calculation

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Computer Arithmetic

• Floating-point arithmetic
• Machine precision (machine epsilon) relative
  accuracy of computer calculations
• ?mach (Matlab eps), about 10-16
• Overflow largest computer number
• OFL (Matlab realmax), about 10308
• Underflow smallest positive computer number
• UFL (Matlab realmin), about 10-308

Matlab commands eps, realmax, realmin
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Computer Arithmetic (continued)

• Infinity
• Result of computations such as 1/0
• Matlab Inf
• Not a Number
• Result of computations such as 0/0
• Matlab NaN

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Example Summing a Series

• Compute the sum

Matlab function my_sum.m not yet written
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Cancellation

• Cancellation the subtraction of two nearly equal
  numbers
• The result is computed accurately (even on the
  computer)
• Why is this bad?

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Example Cancellation

• Estimate
• Solve

Matlab commands pretty, roots, sqrt, ezplot,
format
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Estimating Derivatives

• Finite-difference approximation to a derivative

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Derivatives (continued)

• How should we choose h?
• Cancellation?
• Accuracy of f (x)?

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Derivatives Choosing h

• Taylor series
• Derivative approximation
• Balance rounding error and truncation error

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Choosing h (continued)

• Rounding error
• Truncation error
• Minimize
• Thus

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For Next Class

• Homework see web site
• Reading
• Heath chapter 1
• Heath chapter 2, sections 2.1, 2.2, 2.3
• Next class on September 13