It solves the standard eigenvalue problem ( ) and the generalized problem (
Specifically Level 3 BLAS, which performs matrix-matrix operations to maximize data reuse in cache.
This overview details the history, structure, and enduring legacy of the library, the definitive collection of Fortran subroutines for solving matrix eigenvalue problems. The Genesis of Numerical Reliability Matrix Eigensystem Routines — EISPACK Guide
Despite being technologically superseded, the EISPACK Guide remains a foundational text for numerical analysts. It established the standards for , including the use of "check-results" and rigorous error analysis. The logic embedded in its Fortran IV code continues to serve as the "gold standard" for verifying the correctness of new numerical libraries across all modern programming languages.
In response, the NATS project (National Activity to Test Software), involving Argonne National Laboratory and various universities, began translating and refining these algorithms. The result was , a milestone in software engineering that prioritized numerical stability, documentation, and systematic testing over simple execution speed. Scope and Mathematical Coverage It solves the standard eigenvalue problem ( )
One of EISPACK's greatest innovations was the introduction of . While the library contains dozens of low-level "building block" routines—such as TRED1 for Householder reduction or IMTQL1 for the implicit QL algorithm—the drivers (like RG for general real matrices or RS for real symmetric matrices) simplified the user experience. A single call to a driver would handle the necessary transformations, the eigenvalue extraction, and the back-transformations of eigenvectors. Numerical Stability and the QR Algorithm
EISPACK was designed to be a "pathway" system. Users would select a specific path of subroutines based on the characteristics of their matrix and the specific data required: It established the standards for , including the
Should we focus on the for calling these routines, or would you prefer a comparison of execution speeds between EISPACK and its successor, LAPACK?