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Student lab' Exercises Experiencing Numerical algorithms using Mathematica

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Student lab. Exercises- Experiencing Numerical algorithms using Mathematica. A.S. Cvetkovic ... Evaluation of functions in finite precision. Ill-conditioning of ... – PowerPoint PPT presentation

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Title: Student lab' Exercises Experiencing Numerical algorithms using Mathematica


1
Student lab. Exercises-Experiencing Numerical
algorithms using Mathematica
  • A.S. Cvetkovic
  • G.V. Milovanovic

2
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

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

4
  • 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

5
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

6
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

7
Calculation of the Sin function
8
How does it look like
9
Number representation
10
Arithmetic operations
11
Transcendental functions
12
(No Transcript)
13
Linear algebra, linear equations
14
Interpolation, Runge example
15
Interpolation, Chebyshev
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