Function eigs #
Compute eigenvalues and optionally eigenvectors of a square matrix.
The eigenvalues are sorted by their absolute value, ascending, and
returned as a vector in the values property of the returned project.
An eigenvalue with algebraic multiplicity k will be listed k times, so
that the returned values vector always has length equal to the size
of the input matrix.
The eigenvectors property of the return value provides the eigenvectors.
It is an array of plain objects: the value property of each gives the
associated eigenvalue, and the vector property gives the eigenvector
itself. Note that the same value property will occur as many times in
the list provided by eigenvectors as the geometric multiplicity of
that value.
If the algorithm fails to converge, it will throw an error –
in that case, however, you may still find useful information
in err.values and err.vectors.
Note that the ‘precision’ option does not directly specify the accuracy of the returned eigenvalues. Rather, it determines how small an entry of the iterative approximations to an upper triangular matrix must be in order to be considered zero. The actual accuracy of the returned eigenvalues may be greater or less than the precision, depending on the conditioning of the matrix and how far apart or close the actual eigenvalues are. Note that currently, relatively simple, “traditional” methods of eigenvalue computation are being used; this is not a modern, high-precision eigenvalue computation. That said, it should typically produce fairly reasonable results.
Syntax #
math.eigs(x, [prec])
math.eigs(x, {options})
Parameters #
| Parameter | Type | Description |
|---|---|---|
x |
Array | Matrix | Matrix to be diagonalized |
opts |
number | BigNumber | OptsObject | Object with keys precision, defaulting to config.epsilon, and eigenvectors, defaulting to true and specifying whether to compute eigenvectors. If just a number, specifies precision. |
Returns #
| Type | Description |
|---|---|
| {values: Array | Matrix, eigenvectors?: Array<EVobj>}} Object containing an array of eigenvalues and an array of {value: number | BigNumber, vector: Array | Matrix | objects. The eigenvectors property is undefined if eigenvectors were not requested. |
Throws #
Type | Description —- | ———–
Examples #
const { eigs, multiply, column, transpose, matrixFromColumns } = math
const H = [[5, 2.3], [2.3, 1]]
const ans = eigs(H) // returns {values: [E1,E2...sorted], eigenvectors: [{value: E1, vector: v2}, {value: e, vector: v2}, ...]
const E = ans.values
const V = ans.eigenvectors
multiply(H, V[0].vector)) // returns multiply(E[0], V[0].vector))
const U = matrixFromColumns(...V.map(obj => obj.vector))
const UTxHxU = multiply(transpose(U), H, U) // diagonalizes H if possible
E[0] == UTxHxU[0][0] // returns true always
// Compute only approximate eigenvalues:
const {values} = eigs(H, {eigenvectors: false, precision: 1e-6})
