Transcript Stability and Sensitivity of Certain Computational Methods

Stability and Sensitivity of Certain
Computational
Methods: A Symbolic-Numeric
Treatment with
Mathematica
Ali YAZICI
Software Engineering Department
Atilim University
Ankara, Turkey 06836
[email protected]
Introduction
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Computer Algebra Systems (CAS) such as
Mathematica, Maple and Matlab offer so many
different capabilities, which can be used as an aid
in Computer Assisted Instruction.
The use of Mathematica (especially its symbolic
power) in teaching some of the numerical
methods is shown to be effective ([1], [2],[3],
[4]).
In this study, symbolic and numerical properties
of stability and sensitivity of some basic
computational procedures are given using
Mathematica.
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Definitions
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Sensitivity: The question how much does a
function change under perturbations of its
arguments is of importance in numerical
computations.
Stability: An algorithm, is numerically stable if an
error, whatever the reason is, does not grow
unexpectedly larger during the calculation.
In this work, Mathematica notebooks have been
prepared to demonstrate the issues of sensitivity
and stability problems for some introductory
numerical computations.
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Sensitivity of Polynomials
Obviously, the zeros of p(x) are 1,2,...,10 and are simple.
However, it can be shown that a small change in the
coefficient of x9 from -55 to -55 + eps for very small values
of eps may change some of the real roots of p(x) to be
complex (Wilkinson).
The roots of the perturbed polynomial are simply
calculated (Mathematica) as:
{x->1}, {x->2}, {x->2.9998}, {x->4:0061},
{x-> 4.9396},{x-> 10.194}, {x->6.3473-0.37935i},
{x->6.3473 + 0.37935i},{x->8.5828-0.69933i},
{x->8.5828 + 0.69933i}
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Sensitivity animated
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Analysis
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By differentiating the equation
p(x,eps) = 0 with respect to eps, one
gets [10]
This equation can be used to compute
the measure of sensitivity against the
roots 1,2,3,...,9, and 10.
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Analysis by Mathematica
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...Mathematica
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The input commands 1-3 below computes
the sensitivity constants by invoking the
module above.
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Stability of Gram-Schmidt &QR
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Let A = (a1, a2, ... , an) be a mxn
matrix with real values with m ¸ n
and rank(A) = n. The CGS process
produces an orthogonal basis Q =
(q1, q2, ... , qn) of span(A), such
that A = QR where R is an upper
triangular matrix of order n.
This factorization can be utilized to
determine the eigenvalues of A.
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CGS Algorithm
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Stability Measures
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The accuracy of the QR factorization
based on the CGS process can be
simply measured by computing the
norm ||A-QR||
The measure of orthogonalization
can be obtained similarly by
computing the norm ||QTQ -I||
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Mathematica Implementation of
CGS
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Running CGS
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To display the instability numerically,
Hilbert’s matrix (highly ill-conditioned) of
order n is chosen.
Mathematica commands to produce the
Hilbert matrix of order 5 and running the
CGS program
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...Running CGS
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Results
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Divided Differences
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Mathematica in action
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Propagation of initial error
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Conclusions
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This study is about using one of the well-known CAS,
namely, Mathematica to demonstrate rather
complicated concepts just mentioned in a numerical
computing course taught to engineering
undergraduates.
The code segments produced in this study have been
successfully used and class tested in the Numerical
Methods courses taught at the undergraduate level.
The learner with some intermediate knowledge of
Mathematica can easily modify the codes at his/her
wish and perform additional experiments to understand
the concepts of stability, sensitivity and error analysis
in scientific problems.
This study can be expanded to cover the stability of
other (more sophisticated) computational methods. A
GUI based interactive tool is also necessary to provide
an educational environment for computational
problems of similar nature.
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