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Numerical Methods
林文杰 Wen-Chieh LinAssistant Professor
Department of Computer Science andInstitute of Multimedia Engineering
Numerical Methods © Wen-Chieh Lin 2
Introduction
AdministrationAbout meCourse overview
Numerical Methods © Wen-Chieh Lin 3
AdministrationCourse web
http://caig.cs.nctu.edu.tw/course/NM
Office hoursEC707, Wed 3:30—5:00 PM
TA:劉政旻
[email protected] 512, ext. 54787
黃怡錚[email protected], ext. 56675
Numerical Methods © Wen-Chieh Lin 4
Textbook Applied Numerical Analysis, 7th ed. (2004), by
Gerald & Wheatley, Addison-Wesley.
Numerical Methods © Wen-Chieh Lin 5
References
Applied Numerical Methods with MATLAB forEngineers and Scientists, (2005), by S. C.Chapra, McGraw-Hill.
Scientific Computing: An Introductory Survey,2nd Ed. (2002), by Michael Heath, McGraw-Hill
Numerical Recipes in C++, 2nd Ed. (2002) byWilliam H. Press, Saul A. Teukolsky, WilliamT. Vetterling, Brian P. Flannery, CambridgeUniversity Press
Numerical Methods © Wen-Chieh Lin 6
Perquisites and Grading
PerquisitesCalculusLinear algebra
Assignments (40%)Written homeworkMATLAB programmingQuiz (depending on class attendance)
Midterm exam (30%) Final exam (30%)
Numerical Methods © Wen-Chieh Lin 7
Late and Cheating Policies
Turn in assignments on the class of the duedate
Penalty of 10% of the value of theassignment/day
0 points for any cheating on assignments ortest
Numerical Methods © Wen-Chieh Lin 8
About me
Ph.D. in Robotics, School of CS, CMUTexture synthesis, replacement and trackingPhysics-based animation
M.S. & B.S. in Control Eng., NCTU Join NCTU in March 2006
Numerical Methods © Wen-Chieh Lin 9
Near-regular TextureAnalysis and Manipulation
Numerical Methods © Wen-Chieh Lin 10
Dynamic Near-regular TextureTracking and Replacement
visibility map
Numerical Methods © Wen-Chieh Lin 11
Application: Superimposition with Occlusion
Numerical Methods © Wen-Chieh Lin 12
Dynamic Texture Replacement—underwater texture
Numerical Methods © Wen-Chieh Lin 13
Physics-based Animation
Numerical Methods © Wen-Chieh Lin 14
What is this course about?
This is not a course to teach you to code. This is a course to teach you computer
algorithms for analyzing and solving scienceand engineering problems in numerical ways.
Numerical Methods © Wen-Chieh Lin 15
Numerical Analysis/Methods
What is numerical analysis/method?Analysis and design of algorithms for numerically
solving mathematical problems in science andengineering
Why do we care about numerical analysis?Simulation of real-world phenomena and eventsVirtual prototyping of engineering designs
Numerical Methods © Wen-Chieh Lin 16
Analysis vs. Numerical Analysis
Consider solvingAnalytically, we know that is a root of the
equationNumerically, how do we find the root of the
equation using a computer program?Computer can only do arithmetic operationsDesign a procedure consisting of only
arithmetic operations to find the root
22 x2
Numerical Methods © Wen-Chieh Lin 17
Numerically Solving
99515625.14125.1
030625.2425.1
1025.245.1
96.14.1
25.25.1
2
2
2
2
2
Too large
Too small
A bit closer
Close
Pretty close
22 x
This is actually a root finding methodcalled “Bisection”!
Numerical Methods © Wen-Chieh Lin 18
Course OverviewApproximation and errors Solving nonlinear equations
Solving sets of equations
Numerical Methods © Wen-Chieh Lin 19
Course Overview (cont.) Interpolation and curve
fittingFind intermediate values
from a table of dataFit curves to data If the curve passes all
data points, we call itinterpolation.
Interpolation
Numerical Methods © Wen-Chieh Lin 20
Course Overview (cont.)Approximation of functions
with polynomials or ratio of polynomials
Numerical differentiation and integration approximate derivative values of a function approximate definite integral, even when no
analytical form exists
Numerical solution of ordinary differentialequations
Optimization
Numerical Methods © Wen-Chieh Lin 21
Applications of Numerical Methods
Computer graphics—root finding,interpolation, curve fitting, optimization, ODEsolver, PDE solver, finite element methodPhysics-based animationGeometry modeling
Computer vision—optimization, curve fitting,linear equationsStereo visionShape from shading
Numerical Methods © Wen-Chieh Lin 22
Applications of Numerical Methods (cont.)
Machine learning—curve fitting, linearequations, function approximationPattern recognitionNeural network
Simulation for prototyping—ODE solver, PDEsolver, optimization, numerical integration,interpolation, finite element methodCircuit designMechanical designCAD/CAM
Numerical Methods © Wen-Chieh Lin 23
Examples in Physics-based Animation
Generate motion based on physical laws (e.g.,Newton’s laws, Fluid dynamics)
Simulated physical phenomena gravity momentum collision friction fluid flow (liquid, gas, turbulence) flexibility, elasticity fracture
Numerical Methods © Wen-Chieh Lin 24
Simulated Flames
Duc Quang Nguyen, Ronald Fedkiw and Henrik Wann Jensen,SIGGRAPH 2003
Numerical Methods © Wen-Chieh Lin 25
Simulated Water
Nick Foster and Ronald Fedkiw, SIGGRAPH 2001
Numerical Methods © Wen-Chieh Lin 26
Simulated Cloth
Kwang-Jin Choi and Hyeong-Seok Ko, SIGGRAPH 2002
Numerical Methods © Wen-Chieh Lin 27
Simulated Deformable Object
Doug James & Dinesh Pai, SIGGRAPH 2002
Numerical Methods © Wen-Chieh Lin 28
Questions?