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CE 30125  Lecture 3
p. 3.1
LECTURE 3
DIRECT SOLUTIONS TO LINEAR ALGEBRAIC SYSTEMS  CONTINUED
Illconditioning of Matrices
• There is no clear cut or precise definition of an illconditioned matrix.
Effects of illconditioning
• Roundoff error accrues in the calculations
• Can potentially result in very inaccurate solutions
• Small variation in matrix coefficients causes large variations in the solution
Detection of
illconditioning in a matrix
• An inaccurate solution for
can satisfy an illconditioned matrix quite well!
• Apply back substitution to check for illconditioning
• Solve
through Gauss or other direct method
• Back substitute
• Comparing we find that
X
A
XB
=
X
poor
A
X
B
B
B
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View Full Document CE 30125  Lecture 3
p. 3.2
• Back substitution is
not
a good detection technique.
• The effects of illconditioning are very subtle!
• Examine the inverse of matrix
• If there are elements of
which are many orders of magnitude larger than the orig
inal matrix,
, then
is probably illconditioned
• It is always best to normalize the rows of the original matrix such that the maximum
magnitude is of order 1
• Evaluate
using the same method with which you are solving the system of equa
tions. Now compute
and compare the results to
. If there’s a significant devi
ation, then the presence of serious roundoff exists!
• Compute
using the same method with which you are solving the system of
equations. This is a more severe test of roundoff since it is accumulated both in the
original inversion and the reinversion.
A
A
1
–
A
A
A
1
–
A
1
–
A
I
A
1
–
1
–
CE 30125  Lecture 3
p. 3.3
• Can also evaluate illconditioning by examining the
normalized determinant
. The
matrix may
be illconditioned when:
where
Euclidean Norm of
• If the matrix
is
diagonally dominant
, i.e. the absolute values of the diagonal terms
the sum of the offdiagonal terms for each row, then the matrix is not illcondi
tioned
det
A
a
ij
2
j
1
=
N
i
1
=
N

< 1
A
a
2
j
1
=
N
i
1
=
N
A
a
ii
a
j
1
=
N
i
12
N
=
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p. 3.4
• Effects of illconditioning are most serious in large dense matrices (e.g. especially those
obtained in such problems as curve fitting by least squares)
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This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.
 Fall '08
 Westerink,J

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