CONDITION
Matrix Condition Number Estimation
CONDITION,
a C library 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:

1 <= kappa(A) for all matrices A.

1 = kappa(I), where I is the identity matrix.

for the L2 matrix norm, the condition number of any orthogonal matrix is 1.

for the L2 matrix norm, the condition number is the ratio of the maximum
to minimum singular values;
The CONDITION library needs access to a copy of the R8LIB library.
Licensing:
The computer code and data files described and made available on this
web page are distributed under
the GNU LGPL 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
LINPACK_D,
a C library which
solves linear systems using double precision real arithmetic;
R8LIB,
a C library which
contains many utility routines using double precision real (R8) arithmetic.
TEST_MAT,
a C library 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:

Alan Cline, Cleve Moler, Pete Stewart, James Wilkinson,
An estimate for the Condition Number of a Matrix,
Technical Report TM310,
Argonne National Laboratory, 1977.

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 602611.

William Hager,
Condition Estimates,
SIAM Journal on Scientific and Statistical Computing,
Volume 5, Number 2, June 1984, pages 311316.

Nicholas Higham,
A survey of condition number estimation for triangular matrices,
SIAM Review,
Volume 9, Number 4, December 1987, pages 575596.

Diane OLeary,
Estimating matrix condition numbers,
SIAM Journal on Scientific and Statistical Computing,
Volume 1, Number 2, June 1980, pages 205209.

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 403409.
Source Code:
Last revised on 15 July 2019.