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Welcome to MTH 308 A/B !!
Principles of Numerical Computation
/Numerical Analysis and
Scientific Computing
Contact Information
Akash Anand
• Instructor : Akash Anand
• Office : 327, Faculty Building• Phone : 7880 or 0512-259-7880
• Email : [email protected]
• Web : www.home.iitk.ac.in/~akasha
• Office Hours : – After class
– Any other times : • By email (preferred)
• By appointment
Course Outline
Akash Anand
• Introduction to scientific computing• Linear equations
• Nonlinear equations• Eigenvalue problems• Approximation and interpolation• Numerical differentiation and
integration
Books
Akash Anand
• Scientific Computing: An introductory survey
– Michael T. Heath
• Numerical Analysis
– K.E.Atkinson,
– J.Stoer, R.Bulirsch, etc.
Evaluation Policy
Akash Anand
• Participation
(Homework, Quiz, etc.) 10%
• Challenge Problems
(Mini-projects) 40%
• Mid Semester Exam 20%
• End Semester Exam 30%
Participation
Akash Anand
• Homework
– Six problem sets -- to be submitted for evaluation.
– Evaluation policy
• 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.
• Quiz
– Zero or more, in lecture, short surprise quizzes.
Challenge Problems
Akash Anand
• 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%).
Other ground rules
Akash Anand
• 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
Opening …
Akash Anand
Opening …
Akash Anand
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.
Opening …
Akash Anand
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, andØ efficiently.
Opening …
Akash Anand
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, andØ efficiently.
Question !
Akash Anand
Question !
Akash Anand
• Compute !
Question !
Akash Anand
• Compute !– Approximate
Question !
Akash Anand
• Compute !– Approximate …– … and
Question !
Akash Anand
• Compute !– Approximate …– … and– … we get .
Question !
Akash Anand
• Compute !– Approximate …– … and– … we get .
– Approximate …
Question !
Akash Anand
• Compute !– Approximate …– … and– … we get .
– Approximate …– ... and
Question !
Akash Anand
• Compute !– Approximate …– … and– … we get .
– Approximate …– ... and– ... we get .
Question !
Akash Anand
• Compute !– Approximate …– … and– … we get .
– Approximate …– ... and– ... we get .
– Matlab returns .
Question !
Akash Anand
• Compute !
• Data Error and Computational Error
Question !
Akash Anand
• Compute !
• Data Error and Computational Error– Given and , compute
Question !
Akash Anand
• Compute !
• Data Error and Computational Error– Given and , compute …
Question !
Akash Anand
• Compute !
• Data Error and Computational Error– Given and , compute …
– Total Error
Question !
Akash Anand
• Compute !
• Data Error and Computational Error– Given and , compute …
– Total Error
Question !
Akash Anand
• Compute !
• Data Error and Computational Error– Given and , compute …
– Total Error
Question !
Akash Anand
• Compute !
• Data Error and Computational Error– Given and , compute …
– Total Error Computational Error
Propagated Data Error
Computational Error
Akash Anand
Computational Error
Akash Anand
Computational error (that is, the error made during a computation can be subdivided into Ø truncation (or discretization) error and Ø rounding error.
Computational Error
Akash Anand
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.
Computational Error
Akash Anand
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.
Computational Error
Akash Anand
Computational error (that is, the error made during a computation can be subdivided into Ø truncation (or discretization) error and Ø rounding error.
Example
Computational Error
Akash Anand
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,
Computational Error
Akash Anand
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 .
Computational Error
Akash Anand
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
Computational Error
Akash Anand
Example
… by Taylor’s theorem,
Thus, the error is bounded by
Computational Error
Akash Anand
Example
… by Taylor’s theorem,
Thus, the error is bounded by
There is a tradeoff between truncation error and rounding error.
Computational Error
Akash Anand
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
