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Scientific Computing: An Introductory Survey讲义
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Scientific Computing: An Introductory Survey讲义
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Scientific Computing
Approximations
Computer Arithmetic
Scientific Computing: An Introductory Survey
Chapter 1 – Scientific Computing
Prof. Michael T. Heath
Department of Computer Science
University of Illinois at Urbana-Champaign
Copyright
c
2002. Reproduction permitted
for noncommercial, educational use only.
Michael T. Heath Scientific Computing 1 / 46
Scientific Computing
Approximations
Computer Arithmetic
Outline
1
Scientific Computing
2
Approximations
3
Computer Arithmetic
Michael T. Heath Scientific Computing 2 / 46
Scientific Computing
Approximations
Computer Arithmetic
Introduction
Computational Problems
General Strategy
Scientific Computing
What is scientific computing?
Design and analysis of algorithms for numerically solving
mathematical problems in science and engineering
Traditionally called numerical analysis
Distinguishing features of scientific computing
Deals with continuous quantities
Considers effects of approximations
Why scientific computing?
Simulation of natural phenomena
Virtual prototyping of engineering designs
Michael T. Heath Scientific Computing 3 / 46
Scientific Computing
Approximations
Computer Arithmetic
Introduction
Computational Problems
General Strategy
Well-Posed Problems
Problem is well-posed if solution
exists
is unique
depends continuously on problem data
Otherwise, problem is ill-posed
Even if problem is well posed, solution may still be
sensitive to input data
Computational algorithm should not make sensitivity worse
Michael T. Heath Scientific Computing 4 / 46
Scientific Computing
Approximations
Computer Arithmetic
Introduction
Computational Problems
General Strategy
General Strategy
Replace difficult problem by easier one having same or
closely related solution
infinite → finite
differential → algebraic
nonlinear → linear
complicated → simple
Solution obtained may only approximate that of original
problem
Michael T. Heath Scientific Computing 5 / 46
Scientific Computing
Approximations
Computer Arithmetic
Sources of Approximation
Error Analysis
Sensitivity and Conditioning
Sources of Approximation
Before computation
modeling
empirical measurements
previous computations
During computation
truncation or discretization
rounding
Accuracy of final result reflects all these
Uncertainty in input may be amplified by problem
Perturbations during computation may be amplified by
algorithm
Michael T. Heath Scientific Computing 6 / 46
Scientific Computing
Approximations
Computer Arithmetic
Sources of Approximation
Error Analysis
Sensitivity and Conditioning
Example: Approximations
Computing surface area of Earth using formula A = 4πr
2
involves several approximations
Earth is modeled as sphere, idealizing its true shape
Value for radius is based on empirical measurements and
previous computations
Value for π requires truncating infinite process
Values for input data and results of arithmetic operations
are rounded in computer
Michael T. Heath Scientific Computing 7 / 46
Scientific Computing
Approximations
Computer Arithmetic
Sources of Approximation
Error Analysis
Sensitivity and Conditioning
Absolute Error and Relative Error
Absolute error : approximate value − true value
Relative error :
absolute error
true value
Equivalently, approx value = (tr ue value) × (1 + rel error)
True value usually unknown, so we estimate or bound
error rather than compute it exactly
Relative error often taken relative to approximate value,
rather than (unknown) true value
Michael T. Heath Scientific Computing 8 / 46
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