CONDITION
Matrix Condition Number Estimation

CONDITION is a Python 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;

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:

TEST_MAT, a Python 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:

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:

• combin.py returns the COMBIN matrix.
• cond_test.py, tests numpy's cond() function.
• condition_hager.py, estimates the matrix L1 condition number using Hager's method.
• condition_linpack.py, estimates the matrix L1 condition number using a method from LINPACK.
• condition_sample1.py, estimates the matrix L1 condition number using sampling.
• conex1.py returns a 4 by 4 LINPACK condition number counterexample matrix.
• conex2.py returns a 3 by 3 LINPACK condition number counterexample matrix.
• conex3.py returns a LINPACK condition number counterexample matrix.
• conex4.py returns a 4 by 4 condition number matrix.
• i4vec_print.py, prints an I4VEC.
• kahan.py returns the KAHAN matrix.
• r8_normal_01.py, returns a unit pseudonormal R8;
• r8_sign.py, returns the sign of an R8.
• r8_uniform_01.py, returns a unit pseudorandom R8.
• r8_uniform_ab.py, returns a scaled pseudorandom R8.
• r8ge.py, functions to apply to a matrix stored in R8GE format.
• r8mat_norm_l1.py, returns the L1 norm of an R8MAT.
• r8mat_print.py, prints an R8MAT.
• r8mat_print_some.py, prints some of an R8MAT.
• r8mat_uniform_01.py, returns a unit pseudorandom R8MAT;
• r8vec_max_abs_index.py, returns the index of the maximum absolute value entry in an R8VEC;
• r8vec_norm.py, computes the L2 norm of an R8VEC.
• r8vec_norm_l1.py, computes the L1 norm of an R8VEC.
• r8vec_normal_01.py, returns an R8VEC of unit pseudonormal values;
• r8vec_print.py, prints an R8VEC.
• r8vec_uniform_01.py, returns an R8VEC of unit pseudouniform values;
• r8vec_uniform_ab.py, returns a scaled pseudorandom R8VEC.
• r8vec_uniform_unit.py, returns a random unit vector.
• timestamp.py returns the YMDHMS date as a timestamp.

Examples and Tests:

• condition_test.py, calls all the tests.
• condition_test.sh, runs all the tests.
• condition_test.txt, the output file.

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