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Condition number estimation of preconditioned matrices.

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  • 1International Data Centre, the Preparatory Commission for the Comprehensive Nuclear-Test-Ban Treaty Organization, Vienna, Austria. Center for Computational Science and E-systems, Japan Atomic Energy Agency, Ibaraki, Japan.

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A new method accurately estimates condition numbers for preconditioned matrices, outperforming the conventional Lanczos method. This robust, parallelizable approach is crucial for evaluating preconditioners in large-scale scientific computing.

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Area of Science:

  • Numerical Analysis
  • Scientific Computing
  • Linear Algebra

Background:

  • Estimating condition numbers of preconditioned matrices is vital for assessing preconditioner effectiveness, especially in large-scale computations.
  • Traditional methods like the Lanczos connection method struggle with numerical errors and are unsuitable for distributed memory parallel computers.
  • Preconditioning can transform sparse matrices into dense ones, posing challenges for direct condition number estimation.

Purpose of the Study:

  • To introduce a novel, robust, and parallelizable condition number estimation method for preconditioned matrices.
  • To address the limitations of existing methods, particularly the Lanczos connection method, in large-scale and distributed computing environments.
  • To provide a reliable tool for selecting and evaluating effective preconditioners.

Main Methods:

  • Development of a new condition number estimation method based on Hager's method.
  • Feasibility studies involving diagonal scaling and SSOR preconditioners with various matrices (diagonal, tri-diagonal, Pei's matrix).
  • Comparison of the new method's performance against the Lanczos connection method regarding accuracy and applicability.

Main Results:

  • The newly developed method provides accurate condition number estimations with negligible errors.
  • The Lanczos connection method exhibits significant errors (around 10%) even for simple problems.
  • The new method successfully handles complex matrices, including Pei's matrix and those from finite element methods, where the Lanczos method fails.

Conclusions:

  • The proposed method offers a significant improvement over the Lanczos connection method for condition number estimation of preconditioned matrices.
  • The robustness and parallelizability of the new method make it suitable for large-scale scientific computing.
  • This technique is essential for the effective development and application of preconditioners in numerical analysis.