# library.eigen Submodule¶

## Module Summary¶

Interfaces for the NAG Mark 27.1 eigen Chapter.

eigen - Eigenvalues and Eigenvectors

This module provides functions for various types of matrix eigenvalue problem:

standard eigenvalue problems (finding eigenvalues and eigenvectors of a square matrix );

singular value problems (finding singular values and singular vectors of a rectangular matrix );

generalized eigenvalue problems (finding eigenvalues and eigenvectors of a matrix pencil ).

quadratic eigenvalue problems (finding eigenvalues and eigenvectors of the quadratic ).

Functions are provided for both real and complex data.

The majority of functions for these problems can be found in submodule lapackeig which contains software derived from LAPACK (see Anderson et al. (1999)). However, you should read the F02 Introduction before turning to submodule lapackeig, especially if you are a new user. Submodule sparseig contains functions for large sparse eigenvalue problems, although one such function is also available in this module.

Submodule eigen and submodule lapackeig contain Black Box (or Driver) functions that enable many problems to be solved by a call to a single function, and the decision trees in Decision Trees direct you to the most appropriate functions in submodule eigen and submodule lapackeig. The submodule eigen functions call functions in submodule lapacklin and submodule lapackeig wherever possible to perform the computations, and there are pointers in Decision Trees to the relevant decision trees in submodule lapackeig.

## Functionality Index¶

Black Box functions

complex eigenproblem

selected eigenvalues and eigenvectors: complex_gen_eigsys()

all eigenvalues and optionally eigenvectors, backward

errors and eigenvalue condition numbers: complex_gen_quad()

complex upper triangular matrix

singular values and, optionally, left and/or right singular vectors: complex_triang_svd()

generalized real sparse symmetric-definite eigenproblem

selected eigenvalues and eigenvectors: real_symm_sparse_eigsys()

real eigenproblem

selected eigenvalues and eigenvectors: real_gen_eigsys()

all eigenvalues and optionally eigenvectors, backward

errors and eigenvalue condition numbers: real_gen_quad()

real sparse eigenproblem

selected eigenvalues and eigenvectors: real_gen_sparse_arnoldi()

real sparse symmetric matrix

driver

selected eigenvalues and eigenvectors: real_symm_sparse_arnoldi()

selected eigenvalues and eigenvectors: real_symm_sparse_eigsys()

real upper triangular matrix

singular values and, optionally, left and/or right singular vectors: real_triang_svd()

General Purpose functions (see also submodule sparseig )

real matrix, leading terms SVD: real_gen_partialsvd()

For full information please refer to the NAG Library document

https://www.nag.com/numeric/nl/nagdoc_27.1/flhtml/f02/f02intro.html