condition
condition,
a MATLAB code which
implements methods for computing or estimating the condition number of a matrix.
Let ||*|| be a matrix norm, let A be an invertible matrix, and inv(A) the inverse of A.
The condition number of A with respect to the norm ||*|| is defined to be
kappa(A) = ||A|| * ||inv(A)||
If A is not invertible, the condition number is taken to be infinity.
Facts about the condition number include:
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1 <= kappa(A) for all matrices A.
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1 = kappa(I), where I is the identity matrix.
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for the L2 matrix norm, the condition number of any orthogonal matrix is 1.
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for the L2 matrix norm, the condition number is the ratio of the maximum
to minimum singular values;
MATLAB includes a function cond() which computes the condition number of
a matrix, with respect to a particular matrix norm:
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cond(a,1) uses the L1 matrix norm;
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cond(a,2) uses the L2 matrix norm;
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cond(a,inf) uses the Loo matrix norm;
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cond(a,'fro') uses the Frobenius matrix norm;
Licensing:
The computer code and data files described and made available on this
web page are distributed under
the MIT license
Languages:
condition is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version and
a Python version.
Related Data and Programs:
condition_test
test_mat,
a MATLAB code which
defines test matrices for which some of the determinant, eigenvalues, inverse,
null vectors, P*L*U factorization or linear system solution are already known.
Reference:
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Alan Cline, Cleve Moler, Pete Stewart, James Wilkinson,
An estimate for the Condition Number of a Matrix,
Technical Report TM-310,
Argonne National Laboratory, 1977.
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Alan Cline, Russell Rew,
A set of counterexamples to three condition number estimators,
SIAM Journal on Scientific and Statistical Computing,
Volume 4, Number 4, December 1983, pages 602-611.
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William Hager,
Condition Estimates,
SIAM Journal on Scientific and Statistical Computing,
Volume 5, Number 2, June 1984, pages 311-316.
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Nicholas Higham,
A survey of condition number estimation for triangular matrices,
SIAM Review,
Volume 9, Number 4, December 1987, pages 575-596.
-
Diane OLeary,
Estimating matrix condition numbers,
SIAM Journal on Scientific and Statistical Computing,
Volume 1, Number 2, June 1980, pages 205-209.
-
Pete Stewart,
Efficient Generation of Random Orthogonal Matrices With an Application
to Condition Estimators,
SIAM Journal on Numerical Analysis,
Volume 17, Number 3, June 1980, pages 403-409.
Source Code:
Last revised on 13 December 2018.