Algorithm 879: EIGENTEST—a test matrix generator for large-scale eigenproblems

ABSTRACT

Eigentest is a package that produces real test matrices with known eigensystems. A test matrix, called an eigenmat, is generated in a factored form, in which the user can specify the eigenvalues and has some control over the condition of the eigenvalues and eigenvectors. An eigenmat A of order n requires only O(n) storage for its representation. Auxiliary programs permit the computation of (A − sI)b, (A − sI)^Tb, (A − sI)⁻¹ b, and (A − sI)^−T b in O(n) operations. A special routine computes specified eigenvectors of an eigenmat and the condition of its eigenvalue. Thus eigenmats are suitable for testing algorithms based on Krylov sequences, as well as others based on matrix-vector products. This article introduces the eigenmat and describes implementations in Fortran 77, Fortran 95, C, and Matlab.

REFERENCES

• 1: Gene H. Golub , Charles F. Van Loan, Matrix computations (3rd ed.), Johns Hopkins University Press, Baltimore, MD, 1996
• 2: Nicholas J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1996
• 3: Söderström, T. and Stewart, G. W. 1975. On the numerical properties of an iterative method for computing the More--Penrose generalized inverse. SIAM J. Numer. Analy. 11, 61--74.
• 4: Stewart, G. W. 1973. Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Rev. 15, 727--764.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Eigenvalues and eigenvectors (direct and iterative methods)

Additional Classification:
  D. Software
  D.2 SOFTWARE ENGINEERING
      D.2.5 Testing and Debugging
          Subjects: Testing tools (e.g., data generators, coverage testing)

General Terms:
Experimentation, Verification

Keywords:
Eigensystem, test matrix generation