Eigenvalues and Singular Values :: Sparse Matrices (Mathematics)

Mathematics

Eigenvalues and Singular Values

Two functions are available which compute a few specified eigenvalues or singular values. svds is based on eigs which uses ARPACK [6].

Functions to Compute a Few Eigenvalues or Singular Values
Function
Description

eigs
Few eigenvalues

svds
Few singular values

**Functions to Compute a Few Eigenvalues or Singular Values**
Function	Description
`eigs`	Few eigenvalues
`svds`	Few singular values

These functions are most frequently used with sparse matrices, but they can be used with full matrices or even with linear operators defined by M-files.

The statement

```
[V,lambda] = eigs(A,k,sigma)
```

finds the k eigenvalues and corresponding eigenvectors of the matrix A which are nearest the "shift" sigma. If sigma is omitted, the eigenvalues largest in magnitude are found. If sigma is zero, the eigenvalues smallest in magnitude are found. A second matrix, B, may be included for the generalized eigenvalue problem

The statement

```
[U,S,V] = svds(A,k)
```

finds the k largest singular values of A and

```
[U,S,V] = svds(A,k,0)
```

finds the k smallest singular values.

For example, the statements

```
L = numgrid('L',65);
A = delsq(L);
```

set up the five-point Laplacian difference operator on a 65-by-65 grid in an L-shaped, two-dimensional domain. The statements

```
size(A)
nnz(A)
```

show that A is a matrix of order 2945 with 14,473 nonzero elements.

The statement

```
[v,d] = eigs(A,1,0);
```

computes the smallest eigenvalue and eigenvector. Finally,

L(L>0) = full(v(L(L>0)));
x = -1:1/32:1;
contour(x,x,L,15)
axis square

distributes the components of the eigenvector over the appropriate grid points and produces a contour plot of the result.

contour plot of the result

The numerical techniques used in eigs and svds are described in [6].

Simultaneous Linear Equations Selected Bibliography