Dr. Hanif Durad 2
Lecture Outline
Computational Physics
Scientific computing
Approximations
Computer Arithmetic
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Computational Physics an Introduction-Computational Physics-Lecture Slides, Slides of Computational Physics
This lecture is for Computational Physics course. It was delivered by Dr. Hanif Durad at Pakistan Institute of Engineering and Applied Sciences, Islamabad (PIEAS). It includes: Computational, Physics, Computer, Arithmetic, Approximations, Visualization, Probabilistic, Approach, Linear, Algebra, Difference, Calculus
Typology: Slides
2011/2012
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Dr. Hanif Durad
2
Lecture Outline Computational Physics Scientific computing Approximations Computer Arithmetic
Dr. Hanif Durad
3
Computational Physics (1/3)
^ Computational physics (CP) is a subfield ofcomputational science. ^ CP is a multidisciplinary subject that combinesaspects of physics, applied mathematics, andcomputer science ^ With the aim of solving realistic physicsproblems
Dr. Hanif Durad
Computational Physics (3/3)
[Landau,P-2]
Physics Problems &(Computational Methods)
^ Deterministic Approach^ ^
Solve the governing equations by analytical ornumerical methods ^ Probabilistic Approach^ ^ Model^ Model the various processes as they actually happenwithout solving the equations
Solution
Visualization
My Part
Scientific Computing
^ What is scientific computing?^ ^
Design and analysis of algorithms for solvingmathematical problems in science and engineeringnumerically ^ Traditionally called numerical analysis ^ Distinguishing features:^ ^
continuous quantities effects of approximations
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Why scientific computing? Simulation of physical phenomena Virtual prototyping of products
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1.1.1 General Strategy
^ Replace difficult problem by easier one havingsame or closely related solution^ ^
Infinite
→^ finite (space, integral sum) ^ Differential
→^ algebraic
^ Nonlinear
→^ linear ^ Complicated
simple
^ Solution obtained may only
approximate
that of
original problem
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[P-3]
1.2.1 Sources of Approximation
^ Before computation:^ ^
modeling empirical measurements ^ During computation:^ ^
previous computations 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 algo.
1.2.2 Absolute Error and RelativeError
^ Absolute error
= approx value
−^ true value
^ Relative error
=(absolute error /true value)
^ Equivalently,^ ^
Approx value = (true value)(1 + rel. error) ^ True value
usually unknown, so
estimate
or^ bound
error rather than compute it exactly Relative error
often taken relative to approximate
value, rather than (unknown) true value
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Data Error and ComputationalError (1/3)
^ Typical problem: compute value of functionfor given argument= true value of input,
= desired result= approximate (inexact) input= approximate function computedTotal error =
= computational error + propagated data error ^ Algorithm has no effect on propagated data error
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f:^
x ( ) f x x f
( )^
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) )
(^
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(^
)^
f^ x^
f^ x
x
f^
f^
f x^
x
x^ f
^ ^
^
^
^
^
Data Error and ComputationalError (3/3) propagated data error
, which does not depend on
the algorithm, might already be present in theinput data; computational error
, which does depend on the
algorithm, might be introduced by thecomputational process into the output data.
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[Adopted from Robert D. Skeel notes CS515,file 1.pdf
→P-13/41 ]
Truncation Error and RoundingError
^ Truncation error : difference between true result (for actual input)and result produced by given algorithm
using exact arithmetic
^ Due to approximations such as truncating infinite series or terminatingiterative sequence before convergence Rounding error : difference between result produced by givenalgorithm using exact arithmetic and result produced by samealgorithm using limited precision arithmetic ^ Due to inexact representation of real numbers and arithmetic operationsupon them Computational error is sum of truncation error and rounding error,but one of these usually dominates
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