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A 100% Java sparse and dense matrix library.
/*
* Copyright 2011-2013, by Vladimir Kostyukov and Contributors.
*
* This file is part of la4j project (http://la4j.org)
*
* 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.
*
* Contributor(s): Maxim Samoylov
*
*/
package org.la4j.decomposition;
import org.la4j.factory.Factory;
import org.la4j.matrix.Matrices;
import org.la4j.matrix.Matrix;
import org.la4j.vector.Vector;
import org.la4j.vector.Vectors;
/**
* This class represents Eigen decomposition of matrices. More details
*
* here.
*
*/
public class EigenDecompositor extends AbstractDecompositor implements MatrixDecompositor {
public EigenDecompositor(Matrix matrix) {
super(matrix);
}
/**
* Returns the result of Eigen (EVD) decomposition of given matrix
*
* See
* http://mathworld.wolfram.com/EigenDecomposition.html for more
* details.
*
*
* @param factory
* @return { V, D }
*/
@Override
public Matrix[] decompose(Factory factory) {
if (matrix.is(Matrices.SYMMETRIC_MATRIX)) {
return decomposeSymmetricMatrix(matrix, factory);
} else if (matrix.rows() == matrix.columns()) {
return decomposeNonSymmetricMatrix(matrix, factory);
} else {
throw new IllegalArgumentException("Can't decompose rectangle matrix");
}
}
@Override
public boolean applicableTo(Matrix matrix) {
return matrix.rows() == matrix.columns();
}
/**
* Returns the result of Eigen decomposition for symmetric
* matrix
*
* See
* http://mathworld.wolfram.com/EigenDecomposition.html for more
* details.
*
*
* @param matrix
* @param factory
* @return { V, D }
*/
private Matrix[] decomposeSymmetricMatrix(Matrix matrix, Factory factory) {
Matrix d = matrix.copy();
Matrix v = factory.createIdentityMatrix(matrix.rows());
Vector r = generateR(d, factory);
Matrix u = factory.createIdentityMatrix(matrix.rows());
double n = Matrices.EPS;
double nn = r.norm();
int kk = 0, ll = 0;
while (Math.abs(n - nn) > Matrices.EPS) {
int k = findMax(r);
int l = findMax(d, k);
regenerateU(u, d, k, l, kk, ll);
kk = k;
ll = l;
v = v.multiply(u);
d = u.transpose().multiply(d.multiply(u));
r.set(k, generateRi(d, k));
r.set(l, generateRi(d, l));
n = nn;
nn = r.norm();
}
return new Matrix[] { v, d };
}
private int findMax(Vector vector) {
double value = vector.get(0);
int result = 0;
for (int i = 1; i < vector.length(); i++) {
double v = vector.get(i);
if (Math.abs(value) < Math.abs(v)) {
result = i;
value = v;
}
}
return result;
}
private int findMax(Matrix matrix, int i) {
double value = i > 0 ? matrix.get(i, 0) : matrix.get(i, 1);
int result = i > 0 ? 0 : 1;
for (int j = 0; j < matrix.columns(); j++) {
if (i != j) {
double v = matrix.get(i, j);
if (Math.abs(value) < Math.abs(v)) {
result = j;
value = v;
}
}
}
return result;
}
private Vector generateR(Matrix matrix, Factory factory) {
Vector result = factory.createVector(matrix.rows());
for (int i = 0; i < matrix.rows(); i++) {
result.set(i, generateRi(matrix, i));
}
return result;
}
private double generateRi(Matrix matrix, int i) {
double acc = 0;
for (int j = 0; j < matrix.columns(); j++) {
if (j != i) {
double value = matrix.get(i, j);
acc += value * value;
}
}
return acc;
}
private void regenerateU(Matrix u, Matrix matrix, int k, int l, int kk, int ll) {
u.set(kk, kk, 1.0);
u.set(ll, ll, 1.0);
u.set(kk, ll, 0.0);
u.set(ll, kk, 0.0);
double alpha = 0.0, beta = 0.0;
if (Math.abs(matrix.get(k, k) - matrix.get(l, l)) < Matrices.EPS) {
alpha = beta = Math.sqrt(0.5);
} else {
double mu = 2 * matrix.get(k, l) / (matrix.get(k, k) - matrix.get(l, l));
mu = 1.0 / Math.sqrt(1.0 + mu * mu);
alpha = Math.sqrt(0.5 * (1.0 + mu));
beta = Math.signum(mu) * Math.sqrt(0.5 * (1.0 - mu));
}
u.set(k, k, alpha);
u.set(l, l, alpha);
u.set(k, l, -beta);
u.set(l, k, beta);
}
/**
* Returns the result of Eigen decomposition for non-symmetric
* matrix
*
* See
* http://mathworld.wolfram.com/EigenDecomposition.html for more
* details.
*
*
* @param matrix
* @param factory
* @return { P, D }
*/
private Matrix[] decomposeNonSymmetricMatrix(Matrix matrix, Factory factory) {
Matrix A = matrix.copy();
int n = matrix.columns();
Matrix v = factory.createIdentityMatrix(n);
Vector d = factory.createVector(n);
Vector e = factory.createVector(n);
Matrix h = A.copy();
Vector ort = factory.createVector(n);
// Reduce to Hessenberg form.
orthes(h, v, ort);
// Reduce Hessenberg to real Schur form.
hqr2(h, v, d, e);
Matrix dd = factory.createMatrix(n, n);
for (int i = 0; i < n; i++) {
dd.set(i, i, d.get(i));
if (e.get(i) > 0) {
dd.set(i, i + 1, e.get(i));
} else if (e.get(i) < 0) {
dd.set(i, i - 1, e.get(i));
}
}
return new Matrix[] { v, dd };
}
// Nonsymmetric reduction to Hessenberg form.
private void orthes(Matrix h, Matrix v, Vector ort) {
// 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 n = ort.length();
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.get(i, m - 1));
}
if (scale != 0.0) {
// Compute Householder transformation.
double hh = 0.0;
for (int i = high; i >= m; i--) {
ort.set(i, h.get(i, m - 1) / scale);
hh += ort.get(i) * ort.get(i);
}
double g = Math.sqrt(hh);
if (ort.get(m) > Matrices.EPS) {
g = -g;
}
hh = hh - ort.get(m) * g;
ort.update(m, Vectors.asMinusFunction(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.get(i) * h.get(i, j);
}
f = f / hh;
for (int i = m; i <= high; i++) {
h.update(i, j, Matrices.asMinusFunction(f * ort.get(i)));
}
}
for (int i = 0; i <= high; i++) {
double f = 0.0;
for (int j = high; j >= m; j--) {
f += ort.get(j) * h.get(i, j);
}
f = f / hh;
for (int j = m; j <= high; j++) {
h.update(i, j, Matrices.asMinusFunction(f * ort.get(j)));
}
}
ort.set(m, scale * ort.get(m));
h.set(m, m - 1, scale * g);
}
}
// Accumulate transformations (Algol's ortran).
for (int m = high - 1; m >= low + 1; m--) {
if (Math.abs(h.get(m, m - 1)) > Matrices.EPS) {
for (int i = m + 1; i <= high; i++) {
ort.set(i, h.get(i, m - 1));
}
for (int j = m; j <= high; j++) {
double g = 0.0;
for (int i = m; i <= high; i++) {
g += ort.get(i) * v.get(i, j);
}
// Double division avoids possible underflow
g = (g / ort.get(m)) / h.get(m, m - 1);
for (int i = m; i <= high; i++) {
v.update(i, j, Matrices.asPlusFunction(g * ort.get(i)));
}
}
}
}
}
// Nonsymmetric reduction from Hessenberg to real Schur form.
private void hqr2(Matrix H, Matrix V, Vector d, Vector e) {
// 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 = e.length();
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.set(i, H.get(i, i));
e.set(i, 0.0);
}
for (int j = Math.max(i - 1, 0); j < nn; j++) {
norm = norm + Math.abs(H.get(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.get(l - 1, l - 1))
+ Math.abs(H.get(l, l));
if (s == 0.0) {
s = norm;
}
if (Math.abs(H.get(l, l - 1)) < eps * s) {
break;
}
l--;
}
// Check for convergence
// One root found
if (l == n) {
H.update(n, n, Matrices.asPlusFunction(exshift));
d.set(n, H.get(n, n));
e.set(n, 0.0);
n--;
iter = 0;
// Two roots found
} else if (l == n - 1) {
w = H.get(n, n - 1) * H.get(n - 1, n);
p = (H.get(n - 1, n - 1) - H.get(n, n)) / 2.0;
q = p * p + w;
z = Math.sqrt(Math.abs(q));
H.update(n, n, Matrices.asPlusFunction(exshift));
H.update(n - 1, n - 1, Matrices.asPlusFunction(exshift));
x = H.get(n, n);
// Real pair
if (q >= 0) {
if (p >= 0) {
z = p + z;
} else {
z = p - z;
}
d.set(n - 1, x + z);
d.set(n, d.get(n - 1));
if (z != 0.0) {
d.set(n, x - w / z);
}
e.set(n - 1, 0.0);
e.set(n, 0.0);
x = H.get(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.get(n - 1, j);
H.set(n - 1, j, q * z + p * H.get(n, j));
H.set(n, j, q * H.get(n, j) - p * z);
}
// Column modification
for (int i = 0; i <= n; i++) {
z = H.get(i, n - 1);
H.set(i, n - 1, q * z + p * H.get(i, n));
H.set(i, n, q * H.get(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.set(n - 1, x + p);
d.set(n, x + p);
e.set(n - 1, z);
e.set(n, -z);
}
n = n - 2;
iter = 0;
// No convergence yet
} else {
// Form shift
x = H.get(n, n);
y = 0.0;
w = 0.0;
if (l < n) {
y = H.get(n - 1, n - 1);
w = H.get(n, n - 1) * H.get(n - 1, n);
}
// Wilkinson's original ad hoc shift
if (iter == 10) {
exshift += x;
for (int i = low; i <= n; i++) {
H.update(i, i, Matrices.asMinusFunction(x));
}
s = Math.abs(H.get(n, n - 1))
+ Math.abs(H.get(n - 1, n - 2));
x = y = 0.75 * s; // WTF ???
w = -0.4375 * s * s; // Are you kidding me???
}
// 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.update(i, i, Matrices.asMinusFunction(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.get(m, m);
r = x - z;
s = y - z;
p = (r * s - w) / H.get(m + 1, m)
+ H.get(m, m + 1);
q = H.get(m + 1, m + 1) - z - r - s;
r = H.get(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.get(m, m - 1)) * (Math.abs(q) + Math.abs(r)) < eps
* (Math.abs(p) * (Math.abs(H.get(m - 1, m - 1))
+ Math.abs(z) + Math.abs(H.get(m + 1, m + 1))))) {
break;
}
m--;
}
for (int i = m + 2; i <= n; i++) {
H.set(i, i - 2, 0.0);
if (i > m + 2) {
H.set(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.get(k, k - 1);
q = H.get(k + 1, k - 1);
r = (notlast ? H.get(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.set(k, k - 1, -s * x);
} else if (l != m) {
H.update(k, k - 1, Matrices.INV_FUNCTION);
}
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.get(k, j) + q * H.get(k + 1, j);
if (notlast) {
p = p + r * H.get(k + 2, j);
H.update(k + 2, j,
Matrices.asMinusFunction(p * z));
}
H.update(k, j, Matrices.asMinusFunction(p * x));
H.update(k + 1, j, Matrices.asMinusFunction(p * y));
}
// Column modification
for (int i = 0; i <= Math.min(n, k + 3); i++) {
p = x * H.get(i, k) + y
* H.get(i, k + 1);
if (notlast) {
p = p + z * H.get(i, k + 2);
H.update(i, k + 2,
Matrices.asMinusFunction(p * r));
}
H.update(i, k, Matrices.asMinusFunction(p));
H.update(i, k + 1, Matrices.asMinusFunction(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.update(i, k + 2,
Matrices.asMinusFunction(p * r));
}
V.update(i, k, Matrices.asMinusFunction(p));
V.update(i, k + 1, Matrices.asMinusFunction(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.get(n);
q = e.get(n);
// Real vector
if (q == 0) {
int l = n;
H.set(n, n, 1.0);
for (int i = n - 1; i >= 0; i--) {
w = H.get(i, i) - p;
r = 0.0;
for (int j = l; j <= n; j++) {
r = r + H.get(i, j) * H.get(j, n);
}
if (e.get(i) < 0.0) {
z = w;
s = r;
} else {
l = i;
if (e.get(i) == 0.0) {
if (w != 0.0) {
H.set(i, n, -r / w);
} else {
H.set(i, n, -r / (eps * norm));
}
// Solve real equations
} else {
x = H.get(i, i + 1);
y = H.get(i + 1, i);
q = (d.get(i) - p) * (d.get(i) - p)
+ e.get(i) * e.get(i);
t = (x * s - z * r) / q;
H.set(i, n, t);
if (Math.abs(x) > Math.abs(z)) {
H.set(i + 1, n, (-r - w * t) / x);
} else {
H.set(i + 1, n, (-s - y * t) / z);
}
}
// Overflow control
t = Math.abs(H.get(i, n));
if ((eps * t) * t > 1) {
for (int j = i; j <= n; j++) {
H.update(j, n, Matrices.asDivFunction(t));
}
}
}
}
// Complex vector
} else if (q < 0) {
int l = n - 1;
// Last vector component imaginary so matrix is triangular
if (Math.abs(H.get(n, n - 1))
> Math.abs(H.get(n - 1, n))) {
H.set(n - 1, n - 1, q / H.get(n, n - 1));
H.set(n - 1, n, -(H.get(n, n) - p)
/ H.get(n, n - 1));
} else {
double cdiv[] = cdiv(0.0, -H.get(n - 1, n),
H.get(n - 1, n - 1) - p, q);
H.set(n - 1, n - 1, cdiv[0]);
H.set(n - 1, n, cdiv[1]);
}
H.set(n, n - 1, 0.0);
H.set(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.get(i, j) * H.get(j, n - 1);
sa = sa + H.get(i, j) * H.get(j, n);
}
w = H.get(i, i) - p;
if (e.get(i) < 0.0) {
z = w;
r = ra;
s = sa;
} else {
l = i;
if (e.get(i) == 0) {
double cdiv[] = cdiv(-ra, -sa, w, q);
H.set(i, n - 1, cdiv[0]);
H.set(i, n, cdiv[1]);
} else {
// Solve complex equations
x = H.get(i, i + 1);
y = H.get(i + 1, i);
vr = (d.get(i) - p) * (d.get(i) - p)
+ e.get(i) * e.get(i) - q * q;
vi = (d.get(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));
}
double cdiv[] = cdiv(x * r - z * ra + q * sa,
x * s - z * sa - q * ra, vr, vi);
H.set(i, n - 1, cdiv[0]);
H.set(i, n, cdiv[1]);
if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {
H.set(i + 1, n - 1, (-ra - w
* H.get(i, n - 1) + q
* H.get(i, n)) / x);
H.set(i + 1, n, (-sa - w
* H.get(i, n) - q
* H.get(i, n - 1)) / x);
} else {
cdiv = cdiv(-r - y
* H.get(i, n - 1), -s - y
* H.get(i, n), z, q);
H.set(i + 1, n - 1, cdiv[0]);
H.set(i + 1, n, cdiv[1]);
}
}
// Overflow control
t = Math.max(Math.abs(H.get(i, n - 1)),
Math.abs(H.get(i, n)));
if ((eps * t) * t > 1) {
for (int j = i; j <= n; j++) {
H.update(j, n - 1, Matrices.asDivFunction(t));
H.update(j, n, Matrices.asDivFunction(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.get(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.get(k, j);
}
V.set(i, j, z);
}
}
}
private double[] cdiv(double xr, double xi, double yr, double yi) {
double cdivr, cdivi;
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;
}
return new double[] { cdivr, cdivi };
}
}