CS3220 - Introduction to Scientific Computing Summer 2008 Jonathan - - PowerPoint PPT Presentation

CS3220 - Introduction to Scientific Computing

Summer 2008 Jonathan Kaldor

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Overview

• Course Mechanics
• What is scientific computing anyways?
• MATLAB refresher

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Course Mechanics

• See syllabus
• Homeworks
• Exams
• Grading
• TA: Jesse Simons

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Scientific Computing

• a.k.a. numerical analysis, a.k.a. numerical

methods...

• Different things to different people

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Scientific Computing

• Pedantic definition: use of computers to

support other sciences

• Analysis of data
• Model Generation
• ... but not entirely

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Scientific Computing

• Another definition: intersection of analytical

math with applied computer science

• How to solve (or approximate) answers to

problems in

• linear algebra
• calculus
• differential equations

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Scientific Computing

• In reality, combination of the two
• Given a problem (maybe, but not always,

motivated by other scientific fields)...

• ... determine the methods needed to

solve the problem...

• ... and understand how to implement the

solutions efficiently on a computer

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Scientific Computing

• “But don’t we already know how to solve

those problems?”

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Why it is important

• Yes, but the computer adds its own

peculiarities

• Floating point numbers introduce

additional error

• Running time important

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Floating Point Numbers

• In math, numbers have as much precision as

needed

• On computers, numbers have a finite

precision

• e.g. π versus 3.141592
• Leads to accumulation of error
• 1e-10 + 1e10 = ???

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Floating Point Numbers

• In math, zero is well-defined and distinct
• In floating point, imprecision leads to

uncertainty

• Result of computation is 1e-15 ... is it

actually zero?

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Floating Point Numbers (a real life example)

• As part of research, needed to solve cubic

equation

• detect collisions between two triangles as

they move over time

• Quadratic formula for quadratics, no

closed-form solution for quartics

• For cubics, though...

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Floating Point Numbers (a real life example)

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Floating Point Numbers (a real life example)

• Implemented closed

form solution for cubics...

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Floating Point Numbers (a real life example)

• Implemented closed

form solution for cubics...

• ... and it worked!...

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Floating Point Numbers (a real life example)

• Implemented closed

form solution for cubics...

• ... and it worked!...
• ... well, almost all of the

time

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Floating Point Numbers (a real life example)

• So what happened?
• Method was unstable - accumulation of

floating point error in method meant answers were not exact

• ... and it was just large enough to
• ccasionally miss

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Conditioning

• Another example: in linear algebra, n x n

matrices are either singular (not invertible)

• r non-singular (invertible)
• In scientific computing, third option: ill-

conditioned (nearly-singular)

• matrix is technically non-singular, but

close enough that inverse cannot be computed with good accuracy

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This Course

• Study types of problems, methods of

solution

• Emphasis on:
• Problem recognition
• How each method works
• How things can go wrong

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This Course

• Additional goals:
• Efficient code
• Being able to discuss sources of error
• MATLAB plotting tools
• Identifying easy / hard problems

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Course Topics (List)

• Linear Systems
• Linear Fitting
• Interpolation
• Root finding / optimization
• Differential Equations

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Course Topics (Visually)

[see http://www.graphics.cornell.edu/~jmkaldor/caf-scarf-2-h264-960-24.mov ]

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