Parlett The Symmetric Eigenvalue Problem Pdf

Review: The Symmetric Eigenvalue Problem by B. N. Parlett

1. Overview

Title: The Symmetric Eigenvalue Problem
Author: Beresford N. Parlett
Series: Classics in Applied Mathematics (SIAM)
Original Publication: 1980 (SIAM edition 1998)

Not recommended for:

  • Beginners looking for a first introduction (start with Strang or Trefethen & Bau).
  • Practitioners who only need to call eig() in MATLAB or Python (the theory is overkill).
  • Those allergic to dense notation and theorem-proof structure.

Large-Scale Problems: Detailed treatment of the Lanczos algorithm and Krylov subspace methods, which are essential for huge, sparse matrices where computing all eigenvalues is computationally impossible. parlett the symmetric eigenvalue problem pdf

He focuses heavily on the Spectral Theorem and the concept of orthogonal transformations. The book treats the symmetric eigenvalue problem not as a subset of the general problem, but as a distinct and elegant field where real eigenvalues and orthogonal eigenvectors allow for much more robust methods than in the non-symmetric case. Review: The Symmetric Eigenvalue Problem by B

  • Option A: accumulate Q explicitly (cost O(n^2) memory/time) if eigenvectors needed.
  • Option B: store compact reflectors (vectors and scalars) and apply later to backtransform.

Error Analysis and Stability
A standout feature is the thorough treatment of backward stability, rounding errors, and practical convergence criteria. Parlett bridges pure analysis and computational reality better than most textbooks. Beginners looking for a first introduction (start with

  1. Introduction: The book begins with an introduction to the symmetric eigenvalue problem, including its definition, properties, and applications.
  2. Theoretical background: The second chapter provides a review of the theoretical background, including the properties of symmetric matrices and the definition of eigenvalues and eigenvectors.
  3. The QR algorithm: The third chapter covers the QR algorithm, a popular method for computing the eigenvalues and eigenvectors of a symmetric matrix.
  4. The divide-and-conquer algorithm: The fourth chapter discusses the divide-and-conquer algorithm, a fast and efficient method for solving the symmetric eigenvalue problem.
  5. The Jacobi algorithm: The fifth chapter covers the Jacobi algorithm, a classical method for solving the symmetric eigenvalue problem.
  • Parlett, B. N. (1980). The symmetric eigenvalue problem. Prentice Hall.
  • Parlett, B. N. (1998). The symmetric eigenvalue problem. SIAM.

The Symmetric Eigenvalue Problem | SIAM Publications Library