328 lines
10 KiB
JavaScript
328 lines
10 KiB
JavaScript
|
import _extends from "@babel/runtime/helpers/extends";
|
|||
|
|
import { factory } from '../../utils/factory.js';
|
|||
|
|
import { format } from '../../utils/string.js';
|
|||
|
|
import { createComplexEigs } from './eigs/complexEigs.js';
|
|||
|
|
import { createRealSymmetric } from './eigs/realSymmetric.js';
|
|||
|
|
import { typeOf, isNumber, isBigNumber, isComplex, isFraction } from '../../utils/is.js';
|
|||
|
|
var name = 'eigs';
|
|||
|
|
|
|||
|
|
// The absolute state of math.js's dependency system:
|
|||
|
|
var dependencies = ['config', 'typed', 'matrix', 'addScalar', 'equal', 'subtract', 'abs', 'atan', 'cos', 'sin', 'multiplyScalar', 'divideScalar', 'inv', 'bignumber', 'multiply', 'add', 'larger', 'column', 'flatten', 'number', 'complex', 'sqrt', 'diag', 'size', 'reshape', 'qr', 'usolve', 'usolveAll', 'im', 're', 'smaller', 'matrixFromColumns', 'dot'];
|
|||
|
|
export var createEigs = /* #__PURE__ */factory(name, dependencies, _ref => {
|
|||
|
|
var {
|
|||
|
|
config,
|
|||
|
|
typed,
|
|||
|
|
matrix,
|
|||
|
|
addScalar,
|
|||
|
|
subtract,
|
|||
|
|
equal,
|
|||
|
|
abs,
|
|||
|
|
atan,
|
|||
|
|
cos,
|
|||
|
|
sin,
|
|||
|
|
multiplyScalar,
|
|||
|
|
divideScalar,
|
|||
|
|
inv,
|
|||
|
|
bignumber,
|
|||
|
|
multiply,
|
|||
|
|
add,
|
|||
|
|
larger,
|
|||
|
|
column,
|
|||
|
|
flatten,
|
|||
|
|
number,
|
|||
|
|
complex,
|
|||
|
|
sqrt,
|
|||
|
|
diag,
|
|||
|
|
size,
|
|||
|
|
reshape,
|
|||
|
|
qr,
|
|||
|
|
usolve,
|
|||
|
|
usolveAll,
|
|||
|
|
im,
|
|||
|
|
re,
|
|||
|
|
smaller,
|
|||
|
|
matrixFromColumns,
|
|||
|
|
dot
|
|||
|
|
} = _ref;
|
|||
|
|
var doRealSymmetric = createRealSymmetric({
|
|||
|
|
config,
|
|||
|
|
addScalar,
|
|||
|
|
subtract,
|
|||
|
|
column,
|
|||
|
|
flatten,
|
|||
|
|
equal,
|
|||
|
|
abs,
|
|||
|
|
atan,
|
|||
|
|
cos,
|
|||
|
|
sin,
|
|||
|
|
multiplyScalar,
|
|||
|
|
inv,
|
|||
|
|
bignumber,
|
|||
|
|
complex,
|
|||
|
|
multiply,
|
|||
|
|
add
|
|||
|
|
});
|
|||
|
|
var doComplexEigs = createComplexEigs({
|
|||
|
|
config,
|
|||
|
|
addScalar,
|
|||
|
|
subtract,
|
|||
|
|
multiply,
|
|||
|
|
multiplyScalar,
|
|||
|
|
flatten,
|
|||
|
|
divideScalar,
|
|||
|
|
sqrt,
|
|||
|
|
abs,
|
|||
|
|
bignumber,
|
|||
|
|
diag,
|
|||
|
|
size,
|
|||
|
|
reshape,
|
|||
|
|
qr,
|
|||
|
|
inv,
|
|||
|
|
usolve,
|
|||
|
|
usolveAll,
|
|||
|
|
equal,
|
|||
|
|
complex,
|
|||
|
|
larger,
|
|||
|
|
smaller,
|
|||
|
|
matrixFromColumns,
|
|||
|
|
dot
|
|||
|
|
});
|
|||
|
|
|
|||
|
|
/**
|
|||
|
|
* 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})
|
|||
|
|
*
|
|||
|
|
* 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})
|
|||
|
|
*
|
|||
|
|
* See also:
|
|||
|
|
*
|
|||
|
|
* inv
|
|||
|
|
*
|
|||
|
|
* @param {Array | Matrix} x Matrix to be diagonalized
|
|||
|
|
*
|
|||
|
|
* @param {number | BigNumber | OptsObject} [opts] Object with keys `precision`, defaulting to config.relTol, and `eigenvectors`, defaulting to true and specifying whether to compute eigenvectors. If just a number, specifies precision.
|
|||
|
|
* @return {{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.
|
|||
|
|
*
|
|||
|
|
*/
|
|||
|
|
return typed('eigs', {
|
|||
|
|
// The conversion to matrix in the first two implementations,
|
|||
|
|
// just to convert back to an array right away in
|
|||
|
|
// computeValuesAndVectors, is unfortunate, and should perhaps be
|
|||
|
|
// streamlined. It is done because the Matrix object carries some
|
|||
|
|
// type information about its entries, and so constructing the matrix
|
|||
|
|
// is a roundabout way of doing type detection.
|
|||
|
|
Array: function Array(x) {
|
|||
|
|
return doEigs(matrix(x));
|
|||
|
|
},
|
|||
|
|
'Array, number|BigNumber': function Array_numberBigNumber(x, prec) {
|
|||
|
|
return doEigs(matrix(x), {
|
|||
|
|
precision: prec
|
|||
|
|
});
|
|||
|
|
},
|
|||
|
|
'Array, Object'(x, opts) {
|
|||
|
|
return doEigs(matrix(x), opts);
|
|||
|
|
},
|
|||
|
|
Matrix: function Matrix(mat) {
|
|||
|
|
return doEigs(mat, {
|
|||
|
|
matricize: true
|
|||
|
|
});
|
|||
|
|
},
|
|||
|
|
'Matrix, number|BigNumber': function Matrix_numberBigNumber(mat, prec) {
|
|||
|
|
return doEigs(mat, {
|
|||
|
|
precision: prec,
|
|||
|
|
matricize: true
|
|||
|
|
});
|
|||
|
|
},
|
|||
|
|
'Matrix, Object': function Matrix_Object(mat, opts) {
|
|||
|
|
var useOpts = {
|
|||
|
|
matricize: true
|
|||
|
|
};
|
|||
|
|
_extends(useOpts, opts);
|
|||
|
|
return doEigs(mat, useOpts);
|
|||
|
|
}
|
|||
|
|
});
|
|||
|
|
function doEigs(mat) {
|
|||
|
|
var _opts$precision;
|
|||
|
|
var opts = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {};
|
|||
|
|
var computeVectors = 'eigenvectors' in opts ? opts.eigenvectors : true;
|
|||
|
|
var prec = (_opts$precision = opts.precision) !== null && _opts$precision !== void 0 ? _opts$precision : config.relTol;
|
|||
|
|
var result = computeValuesAndVectors(mat, prec, computeVectors);
|
|||
|
|
if (opts.matricize) {
|
|||
|
|
result.values = matrix(result.values);
|
|||
|
|
if (computeVectors) {
|
|||
|
|
result.eigenvectors = result.eigenvectors.map(_ref2 => {
|
|||
|
|
var {
|
|||
|
|
value,
|
|||
|
|
vector
|
|||
|
|
} = _ref2;
|
|||
|
|
return {
|
|||
|
|
value,
|
|||
|
|
vector: matrix(vector)
|
|||
|
|
};
|
|||
|
|
});
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
if (computeVectors) {
|
|||
|
|
Object.defineProperty(result, 'vectors', {
|
|||
|
|
enumerable: false,
|
|||
|
|
// to make sure that the eigenvectors can still be
|
|||
|
|
// converted to string.
|
|||
|
|
get: () => {
|
|||
|
|
throw new Error('eigs(M).vectors replaced with eigs(M).eigenvectors');
|
|||
|
|
}
|
|||
|
|
});
|
|||
|
|
}
|
|||
|
|
return result;
|
|||
|
|
}
|
|||
|
|
function computeValuesAndVectors(mat, prec, computeVectors) {
|
|||
|
|
var arr = mat.toArray(); // NOTE: arr is guaranteed to be unaliased
|
|||
|
|
// and so safe to modify in place
|
|||
|
|
var asize = mat.size();
|
|||
|
|
if (asize.length !== 2 || asize[0] !== asize[1]) {
|
|||
|
|
throw new RangeError("Matrix must be square (size: ".concat(format(asize), ")"));
|
|||
|
|
}
|
|||
|
|
var N = asize[0];
|
|||
|
|
if (isReal(arr, N, prec)) {
|
|||
|
|
coerceReal(arr, N); // modifies arr by side effect
|
|||
|
|
|
|||
|
|
if (isSymmetric(arr, N, prec)) {
|
|||
|
|
var _type = coerceTypes(mat, arr, N); // modifies arr by side effect
|
|||
|
|
return doRealSymmetric(arr, N, prec, _type, computeVectors);
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
var type = coerceTypes(mat, arr, N); // modifies arr by side effect
|
|||
|
|
return doComplexEigs(arr, N, prec, type, computeVectors);
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
/** @return {boolean} */
|
|||
|
|
function isSymmetric(arr, N, prec) {
|
|||
|
|
for (var i = 0; i < N; i++) {
|
|||
|
|
for (var j = i; j < N; j++) {
|
|||
|
|
// TODO proper comparison of bignum and frac
|
|||
|
|
if (larger(bignumber(abs(subtract(arr[i][j], arr[j][i]))), prec)) {
|
|||
|
|
return false;
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
return true;
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
/** @return {boolean} */
|
|||
|
|
function isReal(arr, N, prec) {
|
|||
|
|
for (var i = 0; i < N; i++) {
|
|||
|
|
for (var j = 0; j < N; j++) {
|
|||
|
|
// TODO proper comparison of bignum and frac
|
|||
|
|
if (larger(bignumber(abs(im(arr[i][j]))), prec)) {
|
|||
|
|
return false;
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
return true;
|
|||
|
|
}
|
|||
|
|
function coerceReal(arr, N) {
|
|||
|
|
for (var i = 0; i < N; i++) {
|
|||
|
|
for (var j = 0; j < N; j++) {
|
|||
|
|
arr[i][j] = re(arr[i][j]);
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
/** @return {'number' | 'BigNumber' | 'Complex'} */
|
|||
|
|
function coerceTypes(mat, arr, N) {
|
|||
|
|
/** @type {string} */
|
|||
|
|
var type = mat.datatype();
|
|||
|
|
if (type === 'number' || type === 'BigNumber' || type === 'Complex') {
|
|||
|
|
return type;
|
|||
|
|
}
|
|||
|
|
var hasNumber = false;
|
|||
|
|
var hasBig = false;
|
|||
|
|
var hasComplex = false;
|
|||
|
|
for (var i = 0; i < N; i++) {
|
|||
|
|
for (var j = 0; j < N; j++) {
|
|||
|
|
var el = arr[i][j];
|
|||
|
|
if (isNumber(el) || isFraction(el)) {
|
|||
|
|
hasNumber = true;
|
|||
|
|
} else if (isBigNumber(el)) {
|
|||
|
|
hasBig = true;
|
|||
|
|
} else if (isComplex(el)) {
|
|||
|
|
hasComplex = true;
|
|||
|
|
} else {
|
|||
|
|
throw TypeError('Unsupported type in Matrix: ' + typeOf(el));
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
if (hasBig && hasComplex) {
|
|||
|
|
console.warn('Complex BigNumbers not supported, this operation will lose precission.');
|
|||
|
|
}
|
|||
|
|
if (hasComplex) {
|
|||
|
|
for (var _i = 0; _i < N; _i++) {
|
|||
|
|
for (var _j = 0; _j < N; _j++) {
|
|||
|
|
arr[_i][_j] = complex(arr[_i][_j]);
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
return 'Complex';
|
|||
|
|
}
|
|||
|
|
if (hasBig) {
|
|||
|
|
for (var _i2 = 0; _i2 < N; _i2++) {
|
|||
|
|
for (var _j2 = 0; _j2 < N; _j2++) {
|
|||
|
|
arr[_i2][_j2] = bignumber(arr[_i2][_j2]);
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
return 'BigNumber';
|
|||
|
|
}
|
|||
|
|
if (hasNumber) {
|
|||
|
|
for (var _i3 = 0; _i3 < N; _i3++) {
|
|||
|
|
for (var _j3 = 0; _j3 < N; _j3++) {
|
|||
|
|
arr[_i3][_j3] = number(arr[_i3][_j3]);
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
return 'number';
|
|||
|
|
} else {
|
|||
|
|
throw TypeError('Matrix contains unsupported types only.');
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
});
|