Student lab' Exercises Experiencing Numerical algorithms using Mathematica

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Student lab. Exercises-Experiencing Numerical
algorithms using Mathematica

• A.S. Cvetkovic
• G.V. Milovanovic

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Properties of numerical algorithms

• Numerical stability
• Arithmetic operations in finite precision
• Evaluation of functions in finite precision
• Ill-conditioning of numerical algorithms
• Complexity of numerical algorithms
• Time complexity
• Space complexity

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Why Mathematica

• Number formats in Mathematica
• Exact numbers
• Integers 1, -10, 0
• Rational 2/3, -1/4
• Algebraic ?2, 1?2
• transcendent ?, e

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• Floating Point Format
• IEEE-754, DEC, IBM, MIL-STD-1750
• Double precision 16-decimal digits mantissa
• exponent part at most
• Arbitrary Precision Numbers
• Maximal number
• Minimal number

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Topics covered

• Number representations, arithmetic operations,
  recursive computation
• Computation of transcendental functions
• Construction of orthogonal polynomials
• Linear algebra
• Non-linear equations, polynomial equations
• Interpolation problems
• Approximation problems
• Numerical derivation, integration
• Ordinary differential equations

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Structure of examples

• Start with an obvious algorithm, which is bad
  for numerical purposes, to give motivation
• Explain the reason of the poor behavior
• Present numerical algorithm which solves the
  problem

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Calculation of the Sin function
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How does it look like
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Number representation
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Arithmetic operations
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Transcendental functions
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Linear algebra, linear equations
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Interpolation, Runge example
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Interpolation, Chebyshev