Student lab Exercises Experiencing Numerical algorithms using Mathematica
Student lab. Exercises. Experiencing Numerical algorithms using Mathematica A. S. Cvetkovic G. V. Milovanovic
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 •
Why Mathematica • Number formats in Mathematica Exact numbers Integers: 1, -10, 0 Rational: 2/3, -1/4 Algebraic: 2, 1+ 2 transcendent: , e
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
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
Structure of examples 1) Start with an obvious algorithm, which is bad for numerical purposes, to give motivation 2) Explain the reason of the poor behavior 3) Present numerical algorithm which solves the problem
Calculation of the Sin function
How does it look like
Number representation
Arithmetic operations
Transcendental functions
Linear algebra, linear equations
Interpolation, Runge example
Interpolation, Chebyshev
- Slides: 15