/*
* Apache License
* Version 2.0, January 2004
* http://www.apache.org/licenses/
*
* Copyright 2013 Aurelian Tutuianu
* Copyright 2014 Aurelian Tutuianu
* Copyright 2015 Aurelian Tutuianu
* Copyright 2016 Aurelian Tutuianu
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
*/
package rapaio.math.linear.dense;
import rapaio.math.linear.RM;
import java.io.Serializable;
/**
* Eigenvalues and eigen vectors of a real matrix.
* <p>
* If A is symmetric, then A = RV*D*RV' where the eigenvalue matrix D is diagonal
* and the eigenvector matrix RV is orthogonal. I.e. A = RV.prod(D.prod(RV.t())) and RV.prod(RV.prod()) equals the
* identity matrix.
* <p>
* If A is not symmetric, then the eigenvalue matrix D is block diagonal with
* the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda +
* i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of RV represent
* the eigenvectors in the sense that A*RV = RV*D, i.e. A.times(RV) equals
* RV.times(D). The matrix RV may be badly conditioned, or even singular, so the
* validity of the equation A = RV*D*inverse(RV) depends upon RV.cond().
*/
public class EigenvalueDecomposition implements Serializable {
private static final long serialVersionUID = 5064091847331016868L;
/**
* Row and column dimension (square matrix).
*
* @serial matrix dimension.
*/
private int n;
/**
* Symmetry flag.
*
* @serial internal symmetry flag.
*/
private boolean issymmetric;
/**
* Arrays for internal storage of eigenvalues.
*
* @serial internal storage of eigenvalues.
*/
private double[] d, e;
/**
* Array for internal storage of eigenvectors.
*
* @serial internal storage of eigenvectors.
*/
private RM V;
/**
* Array for internal storage of nonsymmetric Hessenberg form.
*
* @serial internal storage of nonsymmetric Hessenberg form.
*/
private double[][] H;
/**
* Working storage for nonsymmetric algorithm.
*
* @serial working storage for nonsymmetric algorithm.
*/
private double[] ort;
/**
* Check for symmetry, then construct the eigenvalue decomposition
* Structure to access D and V.
*
* @param Arg Square matrix
*/
public EigenvalueDecomposition(RM Arg) {
RM A = Arg.solidCopy();
n = Arg.colCount();
V = SolidRM.empty(n, n);
d = new double[n];
e = new double[n];
issymmetric = true;
for (int j = 0; (j < n) & issymmetric; j++) {
for (int i = 0; (i < n) & issymmetric; i++) {
if (!(A.get(i, j) == A.get(j, i))) {
issymmetric = false;
break;
}
}
}
if (issymmetric) {
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
V.set(i, j, A.get(i, j));
}
}
// Tridiagonalize.
tred2();
// Diagonalize.
tql2();
} else {
H = new double[n][n];
ort = new double[n];
for (int j = 0; j < n; j++) {
for (int i = 0; i < n; i++) {
H[i][j] = A.get(i, j);
}
}
// Reduce to Hessenberg form.
orthes();
// Reduce Hessenberg to real Schur form.
hqr2();
}
}
/* ------------------------
Private Methods
* ------------------------ */
// Symmetric Householder reduction to tridiagonal form.
private void tred2() {
// This is derived from the Algol procedures tred2 by
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
for (int j = 0; j < n; j++) {
d[j] = V.get(n - 1, j);
}
// Householder reduction to tridiagonal form.
for (int i = n - 1; i > 0; i--) {
// Scale to avoid under/overflow.
double scale = 0.0;
double h = 0.0;
for (int k = 0; k < i; k++) {
scale = scale + Math.abs(d[k]);
}
if (scale == 0.0) {
e[i] = d[i - 1];
for (int j = 0; j < i; j++) {
d[j] = V.get(i - 1, j);
V.set(i, j, 0.0);
V.set(j, i, 0.0);
}
} else {
// Generate Householder vector.
for (int k = 0; k < i; k++) {
d[k] /= scale;
h += d[k] * d[k];
}
double f = d[i - 1];
double g = Math.sqrt(h);
if (f > 0) {
g = -g;
}
e[i] = scale * g;
h = h - f * g;
d[i - 1] = f - g;
for (int j = 0; j < i; j++) {
e[j] = 0.0;
}
// Apply similarity transformation to remaining columns.
for (int j = 0; j < i; j++) {
f = d[j];
V.set(j, i, f);
g = e[j] + V.get(j, j) * f;
for (int k = j + 1; k <= i - 1; k++) {
g += V.get(k, j) * d[k];
e[k] += V.get(k, j) * f;
}
e[j] = g;
}
f = 0.0;
for (int j = 0; j < i; j++) {
e[j] /= h;
f += e[j] * d[j];
}
double hh = f / (h + h);
for (int j = 0; j < i; j++) {
e[j] -= hh * d[j];
}
for (int j = 0; j < i; j++) {
f = d[j];
g = e[j];
for (int k = j; k <= i - 1; k++) {
V.increment(k, j, -(f * e[k] + g * d[k]));
}
d[j] = V.get(i - 1, j);
V.set(i, j, 0.0);
}
}
d[i] = h;
}
// Accumulate transformations.
for (int i = 0; i < n - 1; i++) {
V.set(n - 1, i, V.get(i, i));
V.set(i, i, 1.0);
double h = d[i + 1];
if (h != 0.0) {
for (int k = 0; k <= i; k++) {
d[k] = V.get(k, i + 1) / h;
}
for (int j = 0; j <= i; j++) {
double g = 0.0;
for (int k = 0; k <= i; k++) {
g += V.get(k, i + 1) * V.get(k, j);
}
for (int k = 0; k <= i; k++) {
V.increment(k, j, -g * d[k]);
}
}
}
for (int k = 0; k <= i; k++) {
V.set(k, i + 1, 0.0);
}
}
for (int j = 0; j < n; j++) {
d[j] = V.get(n - 1, j);
V.set(n - 1, j, 0.0);
}
V.set(n - 1, n - 1, 1.0);
e[0] = 0.0;
}
// Symmetric tridiagonal QL algorithm.
private void tql2() {
// This is derived from the Algol procedures tql2, by
// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
for (int i = 1; i < n; i++) {
e[i - 1] = e[i];
}
e[n - 1] = 0.0;
double f = 0.0;
double tst1 = 0.0;
double eps = Math.pow(2.0, -52.0);
for (int l = 0; l < n; l++) {
// Find small subdiagonal element
tst1 = Math.max(tst1, Math.abs(d[l]) + Math.abs(e[l]));
int m = l;
while (m < n) {
if (Math.abs(e[m]) <= eps * tst1) {
break;
}
m++;
}
// If m == l, d[l] is an eigenvalue,
// otherwise, iterate.
if (m > l) {
int iter = 0;
do {
iter = iter + 1; // (Could check iteration count here.)
// Compute implicit shift
double g = d[l];
double p = (d[l + 1] - g) / (2.0 * e[l]);
double r = Math.hypot(p, 1.0);
if (p < 0) {
r = -r;
}
d[l] = e[l] / (p + r);
d[l + 1] = e[l] * (p + r);
double dl1 = d[l + 1];
double h = g - d[l];
for (int i = l + 2; i < n; i++) {
d[i] -= h;
}
f = f + h;
// Implicit QL transformation.
p = d[m];
double c = 1.0;
double c2 = c;
double c3 = c;
double el1 = e[l + 1];
double s = 0.0;
double s2 = 0.0;
for (int i = m - 1; i >= l; i--) {
c3 = c2;
c2 = c;
s2 = s;
g = c * e[i];
h = c * p;
r = Math.hypot(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]);
// Accumulate transformation.
for (int k = 0; k < n; k++) {
h = V.get(k, i + 1);
V.set(k, i + 1, s * V.get(k, i) + c * h);
V.set(k, i, c * V.get(k, i) - s * h);
}
}
p = -s * s2 * c3 * el1 * e[l] / dl1;
e[l] = s * p;
d[l] = c * p;
// Check for convergence.
} while (Math.abs(e[l]) > eps * tst1);
}
d[l] = d[l] + f;
e[l] = 0.0;
}
// Sort eigenvalues and corresponding vectors.
for (int i = 0; i < n - 1; i++) {
int k = i;
double p = d[i];
for (int 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 (int j = 0; j < n; j++) {
p = V.get(j, i);
V.set(j, i, V.get(j, k));
V.set(j, k, p);
}
}
}
}
// Nonsymmetric reduction to Hessenberg form.
private void orthes() {
// This is derived from the Algol procedures orthes and ortran,
// by Martin and Wilkinson, Handbook for Auto. Comp.,
// Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutines in EISPACK.
int low = 0;
int high = n - 1;
for (int m = low + 1; m <= high - 1; m++) {
// Scale column.
double scale = 0.0;
for (int i = m; i <= high; i++) {
scale = scale + Math.abs(H[i][m - 1]);
}
if (scale != 0.0) {
// Compute Householder transformation.
double h = 0.0;
for (int i = high; i >= m; i--) {
ort[i] = H[i][m - 1] / scale;
h += ort[i] * ort[i];
}
double g = Math.sqrt(h);
if (ort[m] > 0) {
g = -g;
}
h = h - ort[m] * g;
ort[m] = ort[m] - g;
// Apply Householder similarity transformation
// H = (I-u*u'/h)*H*(I-u*u')/h)
for (int j = m; j < n; j++) {
double f = 0.0;
for (int i = high; i >= m; i--) {
f += ort[i] * H[i][j];
}
f = f / h;
for (int i = m; i <= high; i++) {
H[i][j] -= f * ort[i];
}
}
for (int i = 0; i <= high; i++) {
double f = 0.0;
for (int j = high; j >= m; j--) {
f += ort[j] * H[i][j];
}
f = f / h;
for (int j = m; j <= high; j++) {
H[i][j] -= f * ort[j];
}
}
ort[m] = scale * ort[m];
H[m][m - 1] = scale * g;
}
}
// Accumulate transformations (Algol's ortran).
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
V.set(i, j, (i == j ? 1.0 : 0.0));
}
}
for (int m = high - 1; m >= low + 1; m--) {
if (H[m][m - 1] != 0.0) {
for (int i = m + 1; i <= high; i++) {
ort[i] = H[i][m - 1];
}
for (int j = m; j <= high; j++) {
double g = 0.0;
for (int i = m; i <= high; i++) {
g += ort[i] * V.get(i, j);
}
// Double division avoids possible underflow
g = (g / ort[m]) / H[m][m - 1];
for (int i = m; i <= high; i++) {
V.increment(i, j, g * ort[i]);
}
}
}
}
}
// Complex scalar division.
private transient double cdivr, cdivi;
private void cdiv(double xr, double xi, double yr, double yi) {
double r, d;
if (Math.abs(yr) > Math.abs(yi)) {
r = yi / yr;
d = yr + r * yi;
cdivr = (xr + r * xi) / d;
cdivi = (xi - r * xr) / d;
} else {
r = yr / yi;
d = yi + r * yr;
cdivr = (r * xr + xi) / d;
cdivi = (r * xi - xr) / d;
}
}
// Nonsymmetric reduction from Hessenberg to real Schur form.
private void hqr2() {
// This is derived from the Algol procedure hqr2,
// by Martin and Wilkinson, Handbook for Auto. Comp.,
// Vol.ii-Linear Algebra, and the corresponding
// Fortran subroutine in EISPACK.
// Initialize
int nn = this.n;
int n = nn - 1;
int low = 0;
int high = nn - 1;
double eps = Math.pow(2.0, -52.0);
double exshift = 0.0;
double p = 0, q = 0, r = 0, s = 0, z = 0, t, w, x, y;
// Store roots isolated by balanc and compute matrix norm
double norm = 0.0;
for (int i = 0; i < nn; i++) {
if (i < low | i > high) {
d[i] = H[i][i];
e[i] = 0.0;
}
for (int j = Math.max(i - 1, 0); j < nn; j++) {
norm = norm + Math.abs(H[i][j]);
}
}
// Outer loop over eigenvalue index
int iter = 0;
while (n >= low) {
// Look for single small sub-diagonal element
int l = n;
while (l > low) {
s = Math.abs(H[l - 1][l - 1]) + Math.abs(H[l][l]);
if (s == 0.0) {
s = norm;
}
if (Math.abs(H[l][l - 1]) < eps * s) {
break;
}
l--;
}
// Check for convergence
// One root found
if (l == n) {
H[n][n] = H[n][n] + exshift;
d[n] = H[n][n];
e[n] = 0.0;
n--;
iter = 0;
// Two roots found
} else if (l == n - 1) {
w = H[n][n - 1] * H[n - 1][n];
p = (H[n - 1][n - 1] - H[n][n]) / 2.0;
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];
// Real pair
if (q >= 0) {
if (p >= 0) {
z = p + z;
} else {
z = p - z;
}
d[n - 1] = x + z;
d[n] = d[n - 1];
if (z != 0.0) {
d[n] = x - w / z;
}
e[n - 1] = 0.0;
e[n] = 0.0;
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;
// Row modification
for (int 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;
}
// Column modification
for (int 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;
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
z = V.get(i, n - 1);
V.set(i, n - 1, q * z + p * V.get(i, n));
V.set(i, n, q * V.get(i, n) - p * z);
}
// Complex pair
} else {
d[n - 1] = x + p;
d[n] = x + p;
e[n - 1] = z;
e[n] = -z;
}
n = n - 2;
iter = 0;
// No convergence yet
} else {
// Form shift
x = H[n][n];
y = 0.0;
w = 0.0;
if (l < n) {
y = H[n - 1][n - 1];
w = H[n][n - 1] * H[n - 1][n];
}
// Wilkinson's original ad hoc shift
if (iter == 10) {
exshift += x;
for (int 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;
}
// MATLAB's new ad hoc shift
if (iter == 30) {
s = (y - x) / 2.0;
s = s * s + w;
if (s > 0) {
s = Math.sqrt(s);
if (y < x) {
s = -s;
}
s = x - w / ((y - x) / 2.0 + s);
for (int i = low; i <= n; i++) {
H[i][i] -= s;
}
exshift += s;
x = y = w = 0.964;
}
}
iter = iter + 1; // (Could check iteration count here.)
// Look for two consecutive small sub-diagonal elements
int 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 (int i = m + 2; i <= n; i++) {
H[i][i - 2] = 0.0;
if (i > m + 2) {
H[i][i - 3] = 0.0;
}
}
// Double QR step involving rows l:n and columns m:n
for (int k = m; k <= n - 1; k++) {
boolean 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] : 0.0);
x = Math.abs(p) + Math.abs(q) + Math.abs(r);
if (x == 0.0) {
continue;
}
p = p / x;
q = q / x;
r = r / x;
}
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;
// Row modification
for (int 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;
}
// Column modification
for (int 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;
}
// Accumulate transformations
for (int i = low; i <= high; i++) {
p = x * V.get(i, k) + y * V.get(i, k + 1);
if (notlast) {
p = p + z * V.get(i, k + 2);
V.set(i, k + 2, V.get(i, k + 2) - p * r);
}
V.set(i, k, V.get(i, k) - p);
V.set(i, k + 1, V.get(i, k + 1) - p * q);
}
} // (s != 0)
} // k loop
} // check convergence
} // while (n >= low)
// Backsubstitute to find vectors of upper triangular form
if (norm == 0.0) {
return;
}
for (n = nn - 1; n >= 0; n--) {
p = d[n];
q = e[n];
// Real vector
if (q == 0) {
int l = n;
H[n][n] = 1.0;
for (int i = n - 1; i >= 0; i--) {
w = H[i][i] - p;
r = 0.0;
for (int j = l; j <= n; j++) {
r = r + H[i][j] * H[j][n];
}
if (e[i] < 0.0) {
z = w;
s = r;
} else {
l = i;
if (e[i] == 0.0) {
if (w != 0.0) {
H[i][n] = -r / w;
} else {
H[i][n] = -r / (eps * norm);
}
// Solve real equations
} 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;
if (Math.abs(x) > Math.abs(z)) {
H[i + 1][n] = (-r - w * t) / x;
} else {
H[i + 1][n] = (-s - y * t) / z;
}
}
// Overflow control
t = Math.abs(H[i][n]);
if ((eps * t) * t > 1) {
for (int j = i; j <= n; j++) {
H[j][n] = H[j][n] / t;
}
}
}
}
// Complex vector
} else if (q < 0) {
int l = n - 1;
// Last vector component imaginary so matrix is triangular
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 {
cdiv(0.0, -H[n - 1][n], H[n - 1][n - 1] - p, q);
H[n - 1][n - 1] = cdivr;
H[n - 1][n] = cdivi;
}
H[n][n - 1] = 0.0;
H[n][n] = 1.0;
for (int i = n - 2; i >= 0; i--) {
double ra, sa, vr, vi;
ra = 0.0;
sa = 0.0;
for (int 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] < 0.0) {
z = w;
r = ra;
s = sa;
} else {
l = i;
if (e[i] == 0) {
cdiv(-ra, -sa, w, q);
H[i][n - 1] = cdivr;
H[i][n] = cdivi;
} else {
// Solve complex equations
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) * 2.0 * q;
if (vr == 0.0 & vi == 0.0) {
vr = eps * norm * (Math.abs(w) + Math.abs(q) +
Math.abs(x) + Math.abs(y) + Math.abs(z));
}
cdiv(x * r - z * ra + q * sa, x * s - z * sa - q * ra, vr, vi);
H[i][n - 1] = cdivr;
H[i][n] = cdivi;
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 {
cdiv(-r - y * H[i][n - 1], -s - y * H[i][n], z, q);
H[i + 1][n - 1] = cdivr;
H[i + 1][n] = cdivi;
}
}
// Overflow control
t = Math.max(Math.abs(H[i][n - 1]), Math.abs(H[i][n]));
if ((eps * t) * t > 1) {
for (int j = i; j <= n; j++) {
H[j][n - 1] = H[j][n - 1] / t;
H[j][n] = H[j][n] / t;
}
}
}
}
}
}
// Vectors of isolated roots
for (int i = 0; i < nn; i++) {
if (i < low | i > high) {
for (int j = i; j < nn; j++) {
V.set(i, j, H[i][j]);
}
}
}
// Back transformation to get eigenvectors of original matrix
for (int j = nn - 1; j >= low; j--) {
for (int i = low; i <= high; i++) {
z = 0.0;
for (int k = low; k <= Math.min(j, high); k++) {
z = z + V.get(i, k) * H[k][j];
}
V.set(i, j, z);
}
}
}
/**
* Return the eigen vector matrix
*
* @return V
*/
public RM getV() {
return V;
}
/**
* Return the real parts of the eigenvalues
*
* @return real(diag(D))
*/
public double[] getRealEigenvalues() {
return d;
}
/**
* Return the imaginary parts of the eigenvalues
*
* @return imag(diag(D))
*/
public double[] getImagEigenvalues() {
return e;
}
/**
* Return the block diagonal eigenvalue matrix
*
* @return D
*/
public RM getD() {
RM X = SolidRM.empty(n, n);
for (int i = 0; i < n; i++) {
X.set(i, i, d[i]);
if (e[i] > 0) {
X.set(i, i + 1, e[i]);
} else if (e[i] < 0) {
X.set(i, i - 1, e[i]);
}
}
return X;
}
}