ml-matrix
Version:
Matrix manipulation and computation library
784 lines (688 loc) • 21.4 kB
JavaScript
;
const Matrix = require('../matrix');
const util = require('./util');
const hypotenuse = util.hypotenuse;
const getFilled2DArray = util.getFilled2DArray;
const defaultOptions = {
assumeSymmetric: false
};
// https://github.com/lutzroeder/Mapack/blob/master/Source/EigenvalueDecomposition.cs
function EigenvalueDecomposition(matrix, options) {
options = Object.assign({}, defaultOptions, options);
if (!(this instanceof EigenvalueDecomposition)) {
return new EigenvalueDecomposition(matrix, options);
}
matrix = Matrix.checkMatrix(matrix);
if (!matrix.isSquare()) {
throw new Error('Matrix is not a square matrix');
}
var n = matrix.columns,
V = getFilled2DArray(n, n, 0),
d = new Array(n),
e = new Array(n),
value = matrix,
i, j;
var isSymmetric = false;
if (options.assumeSymmetric) {
isSymmetric = true;
} else {
isSymmetric = matrix.isSymmetric();
}
if (isSymmetric) {
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
V[i][j] = value.get(i, j);
}
}
tred2(n, e, d, V);
tql2(n, e, d, V);
}
else {
var H = getFilled2DArray(n, n, 0),
ort = new Array(n);
for (j = 0; j < n; j++) {
for (i = 0; i < n; i++) {
H[i][j] = value.get(i, j);
}
}
orthes(n, H, ort, V);
hqr2(n, e, d, V, H);
}
this.n = n;
this.e = e;
this.d = d;
this.V = V;
}
EigenvalueDecomposition.prototype = {
get realEigenvalues() {
return this.d;
},
get imaginaryEigenvalues() {
return this.e;
},
get eigenvectorMatrix() {
if (!Matrix.isMatrix(this.V)) {
this.V = new Matrix(this.V);
}
return this.V;
},
get diagonalMatrix() {
var n = this.n,
e = this.e,
d = this.d,
X = new Matrix(n, n),
i, j;
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
X[i][j] = 0;
}
X[i][i] = d[i];
if (e[i] > 0) {
X[i][i + 1] = e[i];
}
else if (e[i] < 0) {
X[i][i - 1] = e[i];
}
}
return X;
}
};
function tred2(n, e, d, V) {
var f, g, h, i, j, k,
hh, scale;
for (j = 0; j < n; j++) {
d[j] = V[n - 1][j];
}
for (i = n - 1; i > 0; i--) {
scale = 0;
h = 0;
for (k = 0; k < i; k++) {
scale = scale + Math.abs(d[k]);
}
if (scale === 0) {
e[i] = d[i - 1];
for (j = 0; j < i; j++) {
d[j] = V[i - 1][j];
V[i][j] = 0;
V[j][i] = 0;
}
} else {
for (k = 0; k < i; k++) {
d[k] /= scale;
h += d[k] * d[k];
}
f = d[i - 1];
g = Math.sqrt(h);
if (f > 0) {
g = -g;
}
e[i] = scale * g;
h = h - f * g;
d[i - 1] = f - g;
for (j = 0; j < i; j++) {
e[j] = 0;
}
for (j = 0; j < i; j++) {
f = d[j];
V[j][i] = f;
g = e[j] + V[j][j] * f;
for (k = j + 1; k <= i - 1; k++) {
g += V[k][j] * d[k];
e[k] += V[k][j] * f;
}
e[j] = g;
}
f = 0;
for (j = 0; j < i; j++) {
e[j] /= h;
f += e[j] * d[j];
}
hh = f / (h + h);
for (j = 0; j < i; j++) {
e[j] -= hh * d[j];
}
for (j = 0; j < i; j++) {
f = d[j];
g = e[j];
for (k = j; k <= i - 1; k++) {
V[k][j] -= (f * e[k] + g * d[k]);
}
d[j] = V[i - 1][j];
V[i][j] = 0;
}
}
d[i] = h;
}
for (i = 0; i < n - 1; i++) {
V[n - 1][i] = V[i][i];
V[i][i] = 1;
h = d[i + 1];
if (h !== 0) {
for (k = 0; k <= i; k++) {
d[k] = V[k][i + 1] / h;
}
for (j = 0; j <= i; j++) {
g = 0;
for (k = 0; k <= i; k++) {
g += V[k][i + 1] * V[k][j];
}
for (k = 0; k <= i; k++) {
V[k][j] -= g * d[k];
}
}
}
for (k = 0; k <= i; k++) {
V[k][i + 1] = 0;
}
}
for (j = 0; j < n; j++) {
d[j] = V[n - 1][j];
V[n - 1][j] = 0;
}
V[n - 1][n - 1] = 1;
e[0] = 0;
}
function tql2(n, e, d, V) {
var g, h, i, j, k, l, m, p, r,
dl1, c, c2, c3, el1, s, s2,
iter;
for (i = 1; i < n; i++) {
e[i - 1] = e[i];
}
e[n - 1] = 0;
var f = 0,
tst1 = 0,
eps = Math.pow(2, -52);
for (l = 0; l < n; l++) {
tst1 = Math.max(tst1, Math.abs(d[l]) + Math.abs(e[l]));
m = l;
while (m < n) {
if (Math.abs(e[m]) <= eps * tst1) {
break;
}
m++;
}
if (m > l) {
iter = 0;
do {
iter = iter + 1;
g = d[l];
p = (d[l + 1] - g) / (2 * e[l]);
r = hypotenuse(p, 1);
if (p < 0) {
r = -r;
}
d[l] = e[l] / (p + r);
d[l + 1] = e[l] * (p + r);
dl1 = d[l + 1];
h = g - d[l];
for (i = l + 2; i < n; i++) {
d[i] -= h;
}
f = f + h;
p = d[m];
c = 1;
c2 = c;
c3 = c;
el1 = e[l + 1];
s = 0;
s2 = 0;
for (i = m - 1; i >= l; i--) {
c3 = c2;
c2 = c;
s2 = s;
g = c * e[i];
h = c * p;
r = hypotenuse(p, e[i]);
e[i + 1] = s * r;
s = e[i] / r;
c = p / r;
p = c * d[i] - s * g;
d[i + 1] = h + s * (c * g + s * d[i]);
for (k = 0; k < n; k++) {
h = V[k][i + 1];
V[k][i + 1] = s * V[k][i] + c * h;
V[k][i] = c * V[k][i] - s * h;
}
}
p = -s * s2 * c3 * el1 * e[l] / dl1;
e[l] = s * p;
d[l] = c * p;
}
while (Math.abs(e[l]) > eps * tst1);
}
d[l] = d[l] + f;
e[l] = 0;
}
for (i = 0; i < n - 1; i++) {
k = i;
p = d[i];
for (j = i + 1; j < n; j++) {
if (d[j] < p) {
k = j;
p = d[j];
}
}
if (k !== i) {
d[k] = d[i];
d[i] = p;
for (j = 0; j < n; j++) {
p = V[j][i];
V[j][i] = V[j][k];
V[j][k] = p;
}
}
}
}
function orthes(n, H, ort, V) {
var low = 0,
high = n - 1,
f, g, h, i, j, m,
scale;
for (m = low + 1; m <= high - 1; m++) {
scale = 0;
for (i = m; i <= high; i++) {
scale = scale + Math.abs(H[i][m - 1]);
}
if (scale !== 0) {
h = 0;
for (i = high; i >= m; i--) {
ort[i] = H[i][m - 1] / scale;
h += ort[i] * ort[i];
}
g = Math.sqrt(h);
if (ort[m] > 0) {
g = -g;
}
h = h - ort[m] * g;
ort[m] = ort[m] - g;
for (j = m; j < n; j++) {
f = 0;
for (i = high; i >= m; i--) {
f += ort[i] * H[i][j];
}
f = f / h;
for (i = m; i <= high; i++) {
H[i][j] -= f * ort[i];
}
}
for (i = 0; i <= high; i++) {
f = 0;
for (j = high; j >= m; j--) {
f += ort[j] * H[i][j];
}
f = f / h;
for (j = m; j <= high; j++) {
H[i][j] -= f * ort[j];
}
}
ort[m] = scale * ort[m];
H[m][m - 1] = scale * g;
}
}
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
V[i][j] = (i === j ? 1 : 0);
}
}
for (m = high - 1; m >= low + 1; m--) {
if (H[m][m - 1] !== 0) {
for (i = m + 1; i <= high; i++) {
ort[i] = H[i][m - 1];
}
for (j = m; j <= high; j++) {
g = 0;
for (i = m; i <= high; i++) {
g += ort[i] * V[i][j];
}
g = (g / ort[m]) / H[m][m - 1];
for (i = m; i <= high; i++) {
V[i][j] += g * ort[i];
}
}
}
}
}
function hqr2(nn, e, d, V, H) {
var n = nn - 1,
low = 0,
high = nn - 1,
eps = Math.pow(2, -52),
exshift = 0,
norm = 0,
p = 0,
q = 0,
r = 0,
s = 0,
z = 0,
iter = 0,
i, j, k, l, m, t, w, x, y,
ra, sa, vr, vi,
notlast, cdivres;
for (i = 0; i < nn; i++) {
if (i < low || i > high) {
d[i] = H[i][i];
e[i] = 0;
}
for (j = Math.max(i - 1, 0); j < nn; j++) {
norm = norm + Math.abs(H[i][j]);
}
}
while (n >= low) {
l = n;
while (l > low) {
s = Math.abs(H[l - 1][l - 1]) + Math.abs(H[l][l]);
if (s === 0) {
s = norm;
}
if (Math.abs(H[l][l - 1]) < eps * s) {
break;
}
l--;
}
if (l === n) {
H[n][n] = H[n][n] + exshift;
d[n] = H[n][n];
e[n] = 0;
n--;
iter = 0;
} else if (l === n - 1) {
w = H[n][n - 1] * H[n - 1][n];
p = (H[n - 1][n - 1] - H[n][n]) / 2;
q = p * p + w;
z = Math.sqrt(Math.abs(q));
H[n][n] = H[n][n] + exshift;
H[n - 1][n - 1] = H[n - 1][n - 1] + exshift;
x = H[n][n];
if (q >= 0) {
z = (p >= 0) ? (p + z) : (p - z);
d[n - 1] = x + z;
d[n] = d[n - 1];
if (z !== 0) {
d[n] = x - w / z;
}
e[n - 1] = 0;
e[n] = 0;
x = H[n][n - 1];
s = Math.abs(x) + Math.abs(z);
p = x / s;
q = z / s;
r = Math.sqrt(p * p + q * q);
p = p / r;
q = q / r;
for (j = n - 1; j < nn; j++) {
z = H[n - 1][j];
H[n - 1][j] = q * z + p * H[n][j];
H[n][j] = q * H[n][j] - p * z;
}
for (i = 0; i <= n; i++) {
z = H[i][n - 1];
H[i][n - 1] = q * z + p * H[i][n];
H[i][n] = q * H[i][n] - p * z;
}
for (i = low; i <= high; i++) {
z = V[i][n - 1];
V[i][n - 1] = q * z + p * V[i][n];
V[i][n] = q * V[i][n] - p * z;
}
} else {
d[n - 1] = x + p;
d[n] = x + p;
e[n - 1] = z;
e[n] = -z;
}
n = n - 2;
iter = 0;
} else {
x = H[n][n];
y = 0;
w = 0;
if (l < n) {
y = H[n - 1][n - 1];
w = H[n][n - 1] * H[n - 1][n];
}
if (iter === 10) {
exshift += x;
for (i = low; i <= n; i++) {
H[i][i] -= x;
}
s = Math.abs(H[n][n - 1]) + Math.abs(H[n - 1][n - 2]);
x = y = 0.75 * s;
w = -0.4375 * s * s;
}
if (iter === 30) {
s = (y - x) / 2;
s = s * s + w;
if (s > 0) {
s = Math.sqrt(s);
if (y < x) {
s = -s;
}
s = x - w / ((y - x) / 2 + s);
for (i = low; i <= n; i++) {
H[i][i] -= s;
}
exshift += s;
x = y = w = 0.964;
}
}
iter = iter + 1;
m = n - 2;
while (m >= l) {
z = H[m][m];
r = x - z;
s = y - z;
p = (r * s - w) / H[m + 1][m] + H[m][m + 1];
q = H[m + 1][m + 1] - z - r - s;
r = H[m + 2][m + 1];
s = Math.abs(p) + Math.abs(q) + Math.abs(r);
p = p / s;
q = q / s;
r = r / s;
if (m === l) {
break;
}
if (Math.abs(H[m][m - 1]) * (Math.abs(q) + Math.abs(r)) < eps * (Math.abs(p) * (Math.abs(H[m - 1][m - 1]) + Math.abs(z) + Math.abs(H[m + 1][m + 1])))) {
break;
}
m--;
}
for (i = m + 2; i <= n; i++) {
H[i][i - 2] = 0;
if (i > m + 2) {
H[i][i - 3] = 0;
}
}
for (k = m; k <= n - 1; k++) {
notlast = (k !== n - 1);
if (k !== m) {
p = H[k][k - 1];
q = H[k + 1][k - 1];
r = (notlast ? H[k + 2][k - 1] : 0);
x = Math.abs(p) + Math.abs(q) + Math.abs(r);
if (x !== 0) {
p = p / x;
q = q / x;
r = r / x;
}
}
if (x === 0) {
break;
}
s = Math.sqrt(p * p + q * q + r * r);
if (p < 0) {
s = -s;
}
if (s !== 0) {
if (k !== m) {
H[k][k - 1] = -s * x;
} else if (l !== m) {
H[k][k - 1] = -H[k][k - 1];
}
p = p + s;
x = p / s;
y = q / s;
z = r / s;
q = q / p;
r = r / p;
for (j = k; j < nn; j++) {
p = H[k][j] + q * H[k + 1][j];
if (notlast) {
p = p + r * H[k + 2][j];
H[k + 2][j] = H[k + 2][j] - p * z;
}
H[k][j] = H[k][j] - p * x;
H[k + 1][j] = H[k + 1][j] - p * y;
}
for (i = 0; i <= Math.min(n, k + 3); i++) {
p = x * H[i][k] + y * H[i][k + 1];
if (notlast) {
p = p + z * H[i][k + 2];
H[i][k + 2] = H[i][k + 2] - p * r;
}
H[i][k] = H[i][k] - p;
H[i][k + 1] = H[i][k + 1] - p * q;
}
for (i = low; i <= high; i++) {
p = x * V[i][k] + y * V[i][k + 1];
if (notlast) {
p = p + z * V[i][k + 2];
V[i][k + 2] = V[i][k + 2] - p * r;
}
V[i][k] = V[i][k] - p;
V[i][k + 1] = V[i][k + 1] - p * q;
}
}
}
}
}
if (norm === 0) {
return;
}
for (n = nn - 1; n >= 0; n--) {
p = d[n];
q = e[n];
if (q === 0) {
l = n;
H[n][n] = 1;
for (i = n - 1; i >= 0; i--) {
w = H[i][i] - p;
r = 0;
for (j = l; j <= n; j++) {
r = r + H[i][j] * H[j][n];
}
if (e[i] < 0) {
z = w;
s = r;
} else {
l = i;
if (e[i] === 0) {
H[i][n] = (w !== 0) ? (-r / w) : (-r / (eps * norm));
} else {
x = H[i][i + 1];
y = H[i + 1][i];
q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
t = (x * s - z * r) / q;
H[i][n] = t;
H[i + 1][n] = (Math.abs(x) > Math.abs(z)) ? ((-r - w * t) / x) : ((-s - y * t) / z);
}
t = Math.abs(H[i][n]);
if ((eps * t) * t > 1) {
for (j = i; j <= n; j++) {
H[j][n] = H[j][n] / t;
}
}
}
}
} else if (q < 0) {
l = n - 1;
if (Math.abs(H[n][n - 1]) > Math.abs(H[n - 1][n])) {
H[n - 1][n - 1] = q / H[n][n - 1];
H[n - 1][n] = -(H[n][n] - p) / H[n][n - 1];
} else {
cdivres = cdiv(0, -H[n - 1][n], H[n - 1][n - 1] - p, q);
H[n - 1][n - 1] = cdivres[0];
H[n - 1][n] = cdivres[1];
}
H[n][n - 1] = 0;
H[n][n] = 1;
for (i = n - 2; i >= 0; i--) {
ra = 0;
sa = 0;
for (j = l; j <= n; j++) {
ra = ra + H[i][j] * H[j][n - 1];
sa = sa + H[i][j] * H[j][n];
}
w = H[i][i] - p;
if (e[i] < 0) {
z = w;
r = ra;
s = sa;
} else {
l = i;
if (e[i] === 0) {
cdivres = cdiv(-ra, -sa, w, q);
H[i][n - 1] = cdivres[0];
H[i][n] = cdivres[1];
} else {
x = H[i][i + 1];
y = H[i + 1][i];
vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
vi = (d[i] - p) * 2 * q;
if (vr === 0 && vi === 0) {
vr = eps * norm * (Math.abs(w) + Math.abs(q) + Math.abs(x) + Math.abs(y) + Math.abs(z));
}
cdivres = cdiv(x * r - z * ra + q * sa, x * s - z * sa - q * ra, vr, vi);
H[i][n - 1] = cdivres[0];
H[i][n] = cdivres[1];
if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {
H[i + 1][n - 1] = (-ra - w * H[i][n - 1] + q * H[i][n]) / x;
H[i + 1][n] = (-sa - w * H[i][n] - q * H[i][n - 1]) / x;
} else {
cdivres = cdiv(-r - y * H[i][n - 1], -s - y * H[i][n], z, q);
H[i + 1][n - 1] = cdivres[0];
H[i + 1][n] = cdivres[1];
}
}
t = Math.max(Math.abs(H[i][n - 1]), Math.abs(H[i][n]));
if ((eps * t) * t > 1) {
for (j = i; j <= n; j++) {
H[j][n - 1] = H[j][n - 1] / t;
H[j][n] = H[j][n] / t;
}
}
}
}
}
}
for (i = 0; i < nn; i++) {
if (i < low || i > high) {
for (j = i; j < nn; j++) {
V[i][j] = H[i][j];
}
}
}
for (j = nn - 1; j >= low; j--) {
for (i = low; i <= high; i++) {
z = 0;
for (k = low; k <= Math.min(j, high); k++) {
z = z + V[i][k] * H[k][j];
}
V[i][j] = z;
}
}
}
function cdiv(xr, xi, yr, yi) {
var r, d;
if (Math.abs(yr) > Math.abs(yi)) {
r = yi / yr;
d = yr + r * yi;
return [(xr + r * xi) / d, (xi - r * xr) / d];
}
else {
r = yr / yi;
d = yi + r * yr;
return [(r * xr + xi) / d, (r * xi - xr) / d];
}
}
module.exports = EigenvalueDecomposition;