Welcome to MTH 308 A/B!! Principles of Numerical Computation / Numerical Analysis and Scientific Computing

Transcription

1 Welcome to MTH 308 A/B!! Principles of Numerical Computation / Numerical Analysis and Scientific Computing

2 Contact Information Instructor : Office : 327, Faculty Building Phone : 7880 or akasha@iitk.ac.in Web : Office Hours : After class Any other times : By (preferred) By appointment

3 Course Outline Introduction to scientific computing Linear equations Nonlinear equations Eigenvalue problems Approximation and interpolation Numerical differentiation and integration

4 Books Scientific Computing: An introductory survey Michael T. Heath Numerical Analysis K.E.Atkinson, J.Stoer, R.Bulirsch, etc.

5 Evaluation Policy Participation (Homework, Quiz, etc.) 10% Challenge Problems (Mini-projects) 40% Mid Semester Exam 20% End Semester Exam 30%

6 Homework Participation Six problem sets -- to be submitted for evaluation. Evaluation policy Quiz One problem will be randomly selected for evaluation. Credit will be based on correctness, clarity of presentation, academic honesty, etc. Collaboration okay; submissions should be individual. Zero or more, in lecture, short surprise quizzes.

7 Challenge Problems Two challenging problems will be given as mini-projects. Involve designing/choosing a solution strategy and implementing it. Evaluation policy 5% for a short report / presentation. 70% credit for correctness 25% for competitive performance with respect to speed Top 25 fastest among correct implementations get credit according to their rank (25% down to 1%).

8 Other ground rules Don t disturb the class. coming late to the class; diverts attention talking, phones, etc. Academic honesty collaborations are okay but don t claim other s work as yours give credit to collaborators wherever necessary no collaborations in quizzes/examinations

9 Opening

10 Opening Problems in scientific computing come from science and engineering. Computational simulation is a representation and emulation of a physical system or a process using a computer.

11 Opening Problems in scientific computing come from science and engineering. Computational simulation is a representation and emulation of a physical system or a process using a computer. Major steps Ø Develop mathematical model expressed by equation of some type Ø Develop algorithms and implement them to solve equations numerically, Ø accurately, and Ø efficiently.

12 Opening Problems in scientific computing come from science and engineering. Computational simulation is a representation and emulation of a physical system or a process using a computer. Major steps Ø Develop mathematical model expressed by equation of some type Ø Develop algorithms and implement them to solve equations numerically, Ø accurately, and Ø efficiently.

13 Question!

14 Question! Compute!

15 Question! Compute! Approximate

16 Question! Compute! Approximate and

17 Question! Compute! Approximate and we get.

18 Question! Compute! Approximate and we get. Approximate

19 Question! Compute! Approximate and we get. Approximate... and

20 Question! Compute! Approximate and we get. Approximate... and... we get.

21 Question! Compute! Approximate and we get. Approximate... and... we get. Matlab returns.

22 Question! Compute! Data Error and Computational Error

23 Question! Compute! Data Error and Computational Error Given and, compute

24 Question! Compute! Data Error and Computational Error Given and, compute

25 Question! Compute! Data Error and Computational Error Given and, compute Total Error

26 Question! Compute! Data Error and Computational Error Given and, compute Total Error

27 Question! Compute! Data Error and Computational Error Given and, compute Total Error

28 Question! Compute! Data Error and Computational Error Given and, compute Total Error Computational Error Propagated Data Error

29 Computational Error

30 Computational Error Computational error (that is, the error made during a computation can be subdivided into Ø truncation (or discretization) error and Ø rounding error.

31 Computational Error Computational error (that is, the error made during a computation can be subdivided into Ø truncation (or discretization) error and Ø rounding error. Truncation Error the difference between true result and the result that would be produced by an algorithm using exact arithmetic.

32 Computational Error Computational error (that is, the error made during a computation can be subdivided into Ø truncation (or discretization) error and Ø rounding error. Rounding Error the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite precision (rounded) arithmetic.

33 Computational Error Computational error (that is, the error made during a computation can be subdivided into Ø truncation (or discretization) error and Ø rounding error. Example

34 Computational Error Computational error (that is, the error made during a computation can be subdivided into Ø truncation (or discretization) error and Ø rounding error. Example by Taylor s theorem,

35 Computational Error Computational error (that is, the error made during a computation can be subdivided into Ø truncation (or discretization) error and Ø rounding error. Example by Taylor s theorem, Thus, the error is bounded by.

36 Computational Error Computational error (that is, the error made during a computation can be subdivided into Ø truncation (or discretization) error and Ø rounding error. Example by Taylor s theorem, Thus, the error is bounded by

37 Computational Error Example by Taylor s theorem, Thus, the error is bounded by

38 Computational Error Example by Taylor s theorem, Thus, the error is bounded by There is a tradeoff between truncation error and rounding error.

39 Computational Error Example by Taylor s theorem, Thus, the error is bounded by There is a tradeoff between truncation error and rounding error. Error is minimized when