condition

condition, a Python 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:

• 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;

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:

test_mat, a Python 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:

1. 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.
2. 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.
3. William Hager,
   Condition Estimates,
   SIAM Journal on Scientific and Statistical Computing,
   Volume 5, Number 2, June 1984, pages 311-316.
4. Nicholas Higham,
   A survey of condition number estimation for triangular matrices,
   SIAM Review,
   Volume 9, Number 4, December 1987, pages 575-596.
5. Diane OLeary,
   Estimating matrix condition numbers,
   SIAM Journal on Scientific and Statistical Computing,
   Volume 1, Number 2, June 1980, pages 205-209.
6. 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:

• condition.py, the source code.
• condition.sh, runs all the tests.
• condition.txt, the output file.