Parlett The Symmetric Eigenvalue Problem Pdf
Book Details
- Title: The Symmetric Eigenvalue Problem
- Author: Beresford N. Parlett
- Publisher: Society for Industrial and Applied Mathematics (SIAM)
- Series: Classics in Applied Mathematics
- Year: Originally 1980 (SIAM Classics edition 1998)
A Taste of Parlett’s Style: The Perturbation Approach
To illustrate why Parlett’s text is so valuable, consider the problem of computing eigenvectors of nearly multiple eigenvalues. Standard textbooks say “the eigenvectors become ill-conditioned.” Parlett says:
“When eigenvalues cluster, the eigenvectors are not individually meaningful; only their invariant subspace is well-determined. Any rotation of an orthonormal basis for that subspace is also a valid eigenbasis.”
He then introduces the canonical angles between subspaces (the sin(Θ) metric) to measure how close two invariant subspaces are. This geometric viewpoint directly informs algorithms: if you only need the subspace (e.g., for PCA), you can stop early without computing individual eigenvectors.
No other book on symmetric eigenvalues gives such a clear geometric and numerical treatment of subspaces. parlett the symmetric eigenvalue problem pdf
4. Writing Style and Pedagogy
Parlett is a gifted writer. His style can be described as "rigorous but conversational."
- Clarity: He avoids unnecessary jargon but does not shy away from complex proofs.
- Historical Context: He frequently acknowledges the historical development of the algorithms, giving credit to pioneers like Wilkinson, Givens, and Householder.
- Examples: The book uses small, tractable examples to illustrate where naive methods fail and why sophisticated methods succeed.
2. Core Philosophy: Theory in Service of Computation
Parlett’s central thesis is that to compute eigenvalues efficiently and accurately, one must understand the underlying mathematical structure. Unlike generic linear algebra texts that list algorithms as recipes, Parlett explains why algorithms work by leveraging the deep properties of symmetric matrices.
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. Book Details
1. Overview
Title: The Symmetric Eigenvalue Problem
Author: Beresford N. Parlett
Series: Classics in Applied Mathematics (SIAM)
Original Publication: 1980 (SIAM edition 1998)
This book is a definitive, rigorous, and practical treatment of numerical methods for computing eigenvalues and eigenvectors of symmetric (and Hermitian) matrices. It is widely considered the canonical reference in the field, bridging pure linear algebra, numerical analysis, and software implementation.
Who Is Beresford Parlett?
Beresford N. Parlett is a towering figure in numerical analysis, having spent most of his career at the University of California, Berkeley. His work spans error analysis, iterative methods, and particularly the Lanczos algorithm. Parlett was not an algorithm inventor in the commercial sense (like Golub or Wilkinson, whom he frequently cites), but rather a synthesizer and critic. He ferrets out hidden assumptions, exposes numerical pitfalls, and provides unifying mathematical frameworks. A Taste of Parlett’s Style: The Perturbation Approach
Parlett’s writing style is distinctive: dense, witty, and unapologetically mathematical. He warns readers early: “No pain, no gain.” This is not a cookbook; it is an intellectual journey.
5. Numerical Stability and Accuracy
- Orthogonality and loss thereof: iterative and inverse-iteration methods can lose orthogonality; justify reorthogonalization when computing clustered eigenvalues.
- Relative vs absolute accuracy:
- MRRR emphasizes relative accuracy for small eigenvalues.
- For well-conditioned problems, standard QR provides good absolute accuracy.
- Shifts and deflation: proper shifts accelerate convergence; careful deflation improves both speed and numerical stability.
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.
If you belong to the first group, be prepared to work through the exercises. Many are labeled “Research problem”—Parlett expects you to discover open questions.
