CONDITION
Matrix Condition Number Estimation

CONDITION is a MATLAB 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;

MATLAB includes a function cond() which computes the condition number of a matrix, with respect to a particular matrix norm:

• cond(a,1) uses the L1 matrix norm;
• cond(a,2) uses the L2 matrix norm;
• cond(a,inf) uses the Loo matrix norm;
• 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 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 MATLAB library which solves linear systems using double precision real arithmetic;

NAPACK, a FORTRAN77 library which includes many routines for applied numerical linear algebra tasks, including the matrix condition number, by William Hager.

TEST_MAT, a MATLAB 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.m returns the COMBIN matrix.
• combin_inverse.m returns the inverse of the COMBIN matrix A.
• condition_hager.m, estimates the matrix L1 condition number using Hager's method.
• condition_linpack.m, estimates the matrix L1 condition number using a method from LINPACK.
• condition_sample1.m, estimates the matrix L1 condition number using sampling.
• conex1.m returns a 4 by 4 LINPACK condition number counterexample matrix.
• conex1_inverse.m returns the inverse of the CONEX1 matrix.
• conex2.m returns a 3 by 3 LINPACK condition number counterexample matrix.
• conex2_inverse.m returns the inverse of the CONEX2 matrix.
• conex3.m returns a LINPACK condition number counterexample matrix.
• conex3_inverse.m returns the inverse of the CONEX3 matrix.
• conex4.m returns a 4 by 4 condition number matrix.
• conex4_inverse.m returns the inverse of the CONEX4 matrix.
• kahan.m returns the KAHAN matrix.
• kahan_inverse.m returns the inverse of the KAHAN matrix.
• r8_sign.m, returns the sign of an R8.
• r8ge_fa.m, factors a matrix stored in general storage.
• r8mat_uniform_01.m, returns a unit pseudorandom R8MAT;
• r8vec_max_abs_index.m, returns the index of the maximum absolute value entry in an R8VEC;
• r8vec_normal_01.m, returns an R8VEC of unit pseudonormal values;
• r8vec_uniform_01.m, returns an R8VEC of unit pseudouniform values;
• r8vec_uniform_unit.m, returns a random unit vector.
• timestamp.m returns the YMDHMS date as a timestamp.