/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You 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 org.apache.commons.math4.linear;
import java.util.Arrays;
import java.util.Random;
import org.apache.commons.math4.distribution.RealDistribution;
import org.apache.commons.math4.distribution.NormalDistribution;
import org.apache.commons.math4.exception.MathUnsupportedOperationException;
import org.apache.commons.math4.linear.ArrayRealVector;
import org.apache.commons.math4.linear.EigenDecomposition;
import org.apache.commons.math4.linear.LUDecomposition;
import org.apache.commons.math4.linear.MatrixUtils;
import org.apache.commons.math4.linear.RealMatrix;
import org.apache.commons.math4.linear.RealVector;
import org.apache.commons.math4.linear.TriDiagonalTransformer;
import org.apache.commons.math4.util.FastMath;
import org.apache.commons.math4.util.MathArrays;
import org.apache.commons.numbers.core.Precision;
import org.apache.commons.rng.simple.RandomSource;
import org.junit.After;
import org.junit.Assert;
import org.junit.Before;
import org.junit.Ignore;
import org.junit.Test;
public class EigenDecompositionTest {
private double[] refValues;
private RealMatrix matrix;
@Test
public void testDimension1() {
RealMatrix matrix =
MatrixUtils.createRealMatrix(new double[][] { { 1.5 } });
EigenDecomposition ed;
ed = new EigenDecomposition(matrix);
Assert.assertEquals(1.5, ed.getRealEigenvalue(0), 1.0e-15);
}
@Test
public void testDimension2() {
RealMatrix matrix =
MatrixUtils.createRealMatrix(new double[][] {
{ 59.0, 12.0 },
{ 12.0, 66.0 }
});
EigenDecomposition ed;
ed = new EigenDecomposition(matrix);
Assert.assertEquals(75.0, ed.getRealEigenvalue(0), 1.0e-15);
Assert.assertEquals(50.0, ed.getRealEigenvalue(1), 1.0e-15);
}
@Test
public void testDimension3() {
RealMatrix matrix =
MatrixUtils.createRealMatrix(new double[][] {
{ 39632.0, -4824.0, -16560.0 },
{ -4824.0, 8693.0, 7920.0 },
{ -16560.0, 7920.0, 17300.0 }
});
EigenDecomposition ed;
ed = new EigenDecomposition(matrix);
Assert.assertEquals(50000.0, ed.getRealEigenvalue(0), 3.0e-11);
Assert.assertEquals(12500.0, ed.getRealEigenvalue(1), 3.0e-11);
Assert.assertEquals( 3125.0, ed.getRealEigenvalue(2), 3.0e-11);
}
@Test
public void testDimension3MultipleRoot() {
RealMatrix matrix =
MatrixUtils.createRealMatrix(new double[][] {
{ 5, 10, 15 },
{ 10, 20, 30 },
{ 15, 30, 45 }
});
EigenDecomposition ed;
ed = new EigenDecomposition(matrix);
Assert.assertEquals(70.0, ed.getRealEigenvalue(0), 3.0e-11);
Assert.assertEquals(0.0, ed.getRealEigenvalue(1), 3.0e-11);
Assert.assertEquals(0.0, ed.getRealEigenvalue(2), 3.0e-11);
}
@Test
public void testDimension4WithSplit() {
RealMatrix matrix =
MatrixUtils.createRealMatrix(new double[][] {
{ 0.784, -0.288, 0.000, 0.000 },
{ -0.288, 0.616, 0.000, 0.000 },
{ 0.000, 0.000, 0.164, -0.048 },
{ 0.000, 0.000, -0.048, 0.136 }
});
EigenDecomposition ed;
ed = new EigenDecomposition(matrix);
Assert.assertEquals(1.0, ed.getRealEigenvalue(0), 1.0e-15);
Assert.assertEquals(0.4, ed.getRealEigenvalue(1), 1.0e-15);
Assert.assertEquals(0.2, ed.getRealEigenvalue(2), 1.0e-15);
Assert.assertEquals(0.1, ed.getRealEigenvalue(3), 1.0e-15);
}
@Test
public void testDimension4WithoutSplit() {
RealMatrix matrix =
MatrixUtils.createRealMatrix(new double[][] {
{ 0.5608, -0.2016, 0.1152, -0.2976 },
{ -0.2016, 0.4432, -0.2304, 0.1152 },
{ 0.1152, -0.2304, 0.3088, -0.1344 },
{ -0.2976, 0.1152, -0.1344, 0.3872 }
});
EigenDecomposition ed;
ed = new EigenDecomposition(matrix);
Assert.assertEquals(1.0, ed.getRealEigenvalue(0), 1.0e-15);
Assert.assertEquals(0.4, ed.getRealEigenvalue(1), 1.0e-15);
Assert.assertEquals(0.2, ed.getRealEigenvalue(2), 1.0e-15);
Assert.assertEquals(0.1, ed.getRealEigenvalue(3), 1.0e-15);
}
// the following test triggered an ArrayIndexOutOfBoundsException in commons-math 2.0
@Test
public void testMath308() {
double[] mainTridiagonal = {
22.330154644539597, 46.65485522478641, 17.393672330044705, 54.46687435351116, 80.17800767709437
};
double[] secondaryTridiagonal = {
13.04450406501361, -5.977590941539671, 2.9040909856707517, 7.1570352792841225
};
// the reference values have been computed using routine DSTEMR
// from the fortran library LAPACK version 3.2.1
double[] refEigenValues = {
82.044413207204002, 53.456697699894512, 52.536278520113882, 18.847969733754262, 14.138204224043099
};
RealVector[] refEigenVectors = {
new ArrayRealVector(new double[] { -0.000462690386766, -0.002118073109055, 0.011530080757413, 0.252322434584915, 0.967572088232592 }),
new ArrayRealVector(new double[] { 0.314647769490148, 0.750806415553905, -0.167700312025760, -0.537092972407375, 0.143854968127780 }),
new ArrayRealVector(new double[] { 0.222368839324646, 0.514921891363332, -0.021377019336614, 0.801196801016305, -0.207446991247740 }),
new ArrayRealVector(new double[] { -0.713933751051495, 0.190582113553930, -0.671410443368332, 0.056056055955050, -0.006541576993581 }),
new ArrayRealVector(new double[] { -0.584677060845929, 0.367177264979103, 0.721453187784497, -0.052971054621812, 0.005740715188257 })
};
EigenDecomposition decomposition;
decomposition = new EigenDecomposition(mainTridiagonal, secondaryTridiagonal);
double[] eigenValues = decomposition.getRealEigenvalues();
for (int i = 0; i < refEigenValues.length; ++i) {
Assert.assertEquals(refEigenValues[i], eigenValues[i], 1.0e-5);
Assert.assertEquals(0, refEigenVectors[i].subtract(decomposition.getEigenvector(i)).getNorm(), 2.0e-7);
}
}
@Test
public void testMathpbx02() {
double[] mainTridiagonal = {
7484.860960227216, 18405.28129035345, 13855.225609560746,
10016.708722343366, 559.8117399576674, 6750.190788301587,
71.21428769782159
};
double[] secondaryTridiagonal = {
-4175.088570476366,1975.7955858241994,5193.178422374075,
1995.286659169179,75.34535882933804,-234.0808002076056
};
// the reference values have been computed using routine DSTEMR
// from the fortran library LAPACK version 3.2.1
double[] refEigenValues = {
20654.744890306974412,16828.208208485466457,
6893.155912634994820,6757.083016675340332,
5887.799885688558788,64.309089923240379,
57.992628792736340
};
RealVector[] refEigenVectors = {
new ArrayRealVector(new double[] {-0.270356342026904, 0.852811091326997, 0.399639490702077, 0.198794657813990, 0.019739323307666, 0.000106983022327, -0.000001216636321}),
new ArrayRealVector(new double[] {0.179995273578326,-0.402807848153042,0.701870993525734,0.555058211014888,0.068079148898236,0.000509139115227,-0.000007112235617}),
new ArrayRealVector(new double[] {-0.399582721284727,-0.056629954519333,-0.514406488522827,0.711168164518580,0.225548081276367,0.125943999652923,-0.004321507456014}),
new ArrayRealVector(new double[] {0.058515721572821,0.010200130057739,0.063516274916536,-0.090696087449378,-0.017148420432597,0.991318870265707,-0.034707338554096}),
new ArrayRealVector(new double[] {0.855205995537564,0.327134656629775,-0.265382397060548,0.282690729026706,0.105736068025572,-0.009138126622039,0.000367751821196}),
new ArrayRealVector(new double[] {-0.002913069901144,-0.005177515777101,0.041906334478672,-0.109315918416258,0.436192305456741,0.026307315639535,0.891797507436344}),
new ArrayRealVector(new double[] {-0.005738311176435,-0.010207611670378,0.082662420517928,-0.215733886094368,0.861606487840411,-0.025478530652759,-0.451080697503958})
};
// the following line triggers the exception
EigenDecomposition decomposition;
decomposition = new EigenDecomposition(mainTridiagonal, secondaryTridiagonal);
double[] eigenValues = decomposition.getRealEigenvalues();
for (int i = 0; i < refEigenValues.length; ++i) {
Assert.assertEquals(refEigenValues[i], eigenValues[i], 1.0e-3);
if (refEigenVectors[i].dotProduct(decomposition.getEigenvector(i)) < 0) {
Assert.assertEquals(0, refEigenVectors[i].add(decomposition.getEigenvector(i)).getNorm(), 1.0e-5);
} else {
Assert.assertEquals(0, refEigenVectors[i].subtract(decomposition.getEigenvector(i)).getNorm(), 1.0e-5);
}
}
}
@Test
public void testMathpbx03() {
double[] mainTridiagonal = {
1809.0978259647177,3395.4763425956166,1832.1894584712693,3804.364873592377,
806.0482458637571,2403.656427234185,28.48691431556015
};
double[] secondaryTridiagonal = {
-656.8932064545833,-469.30804108920734,-1021.7714889369421,
-1152.540497328983,-939.9765163817368,-12.885877015422391
};
// the reference values have been computed using routine DSTEMR
// from the fortran library LAPACK version 3.2.1
double[] refEigenValues = {
4603.121913685183245,3691.195818048970978,2743.442955402465032,1657.596442107321764,
1336.797819095331306,30.129865209677519,17.035352085224986
};
RealVector[] refEigenVectors = {
new ArrayRealVector(new double[] {-0.036249830202337,0.154184732411519,-0.346016328392363,0.867540105133093,-0.294483395433451,0.125854235969548,-0.000354507444044}),
new ArrayRealVector(new double[] {-0.318654191697157,0.912992309960507,-0.129270874079777,-0.184150038178035,0.096521712579439,-0.070468788536461,0.000247918177736}),
new ArrayRealVector(new double[] {-0.051394668681147,0.073102235876933,0.173502042943743,-0.188311980310942,-0.327158794289386,0.905206581432676,-0.004296342252659}),
new ArrayRealVector(new double[] {0.838150199198361,0.193305209055716,-0.457341242126146,-0.166933875895419,0.094512811358535,0.119062381338757,-0.000941755685226}),
new ArrayRealVector(new double[] {0.438071395458547,0.314969169786246,0.768480630802146,0.227919171600705,-0.193317045298647,-0.170305467485594,0.001677380536009}),
new ArrayRealVector(new double[] {-0.003726503878741,-0.010091946369146,-0.067152015137611,-0.113798146542187,-0.313123000097908,-0.118940107954918,0.932862311396062}),
new ArrayRealVector(new double[] {0.009373003194332,0.025570377559400,0.170955836081348,0.291954519805750,0.807824267665706,0.320108347088646,0.360202112392266}),
};
// the following line triggers the exception
EigenDecomposition decomposition;
decomposition = new EigenDecomposition(mainTridiagonal, secondaryTridiagonal);
double[] eigenValues = decomposition.getRealEigenvalues();
for (int i = 0; i < refEigenValues.length; ++i) {
Assert.assertEquals(refEigenValues[i], eigenValues[i], 1.0e-4);
if (refEigenVectors[i].dotProduct(decomposition.getEigenvector(i)) < 0) {
Assert.assertEquals(0, refEigenVectors[i].add(decomposition.getEigenvector(i)).getNorm(), 1.0e-5);
} else {
Assert.assertEquals(0, refEigenVectors[i].subtract(decomposition.getEigenvector(i)).getNorm(), 1.0e-5);
}
}
}
/** test a matrix already in tridiagonal form. */
@Test
public void testTridiagonal() {
Random r = new Random(4366663527842l);
double[] ref = new double[30];
for (int i = 0; i < ref.length; ++i) {
if (i < 5) {
ref[i] = 2 * r.nextDouble() - 1;
} else {
ref[i] = 0.0001 * r.nextDouble() + 6;
}
}
Arrays.sort(ref);
TriDiagonalTransformer t =
new TriDiagonalTransformer(createTestMatrix(r, ref));
EigenDecomposition ed;
ed = new EigenDecomposition(t.getMainDiagonalRef(), t.getSecondaryDiagonalRef());
double[] eigenValues = ed.getRealEigenvalues();
Assert.assertEquals(ref.length, eigenValues.length);
for (int i = 0; i < ref.length; ++i) {
Assert.assertEquals(ref[ref.length - i - 1], eigenValues[i], 2.0e-14);
}
}
/** test dimensions */
@Test
public void testDimensions() {
final int m = matrix.getRowDimension();
EigenDecomposition ed;
ed = new EigenDecomposition(matrix);
Assert.assertEquals(m, ed.getV().getRowDimension());
Assert.assertEquals(m, ed.getV().getColumnDimension());
Assert.assertEquals(m, ed.getD().getColumnDimension());
Assert.assertEquals(m, ed.getD().getColumnDimension());
Assert.assertEquals(m, ed.getVT().getRowDimension());
Assert.assertEquals(m, ed.getVT().getColumnDimension());
}
/** test eigenvalues */
@Test
public void testEigenvalues() {
EigenDecomposition ed;
ed = new EigenDecomposition(matrix);
double[] eigenValues = ed.getRealEigenvalues();
Assert.assertEquals(refValues.length, eigenValues.length);
for (int i = 0; i < refValues.length; ++i) {
Assert.assertEquals(refValues[i], eigenValues[i], 3.0e-15);
}
}
/** test eigenvalues for a big matrix. */
@Test
public void testBigMatrix() {
Random r = new Random(17748333525117l);
double[] bigValues = new double[200];
for (int i = 0; i < bigValues.length; ++i) {
bigValues[i] = 2 * r.nextDouble() - 1;
}
Arrays.sort(bigValues);
EigenDecomposition ed;
ed = new EigenDecomposition(createTestMatrix(r, bigValues));
double[] eigenValues = ed.getRealEigenvalues();
Assert.assertEquals(bigValues.length, eigenValues.length);
for (int i = 0; i < bigValues.length; ++i) {
Assert.assertEquals(bigValues[bigValues.length - i - 1], eigenValues[i], 2.0e-14);
}
}
@Test
public void testSymmetric() {
RealMatrix symmetric = MatrixUtils.createRealMatrix(new double[][] {
{4, 1, 1},
{1, 2, 3},
{1, 3, 6}
});
EigenDecomposition ed;
ed = new EigenDecomposition(symmetric);
RealMatrix d = ed.getD();
RealMatrix v = ed.getV();
RealMatrix vT = ed.getVT();
double norm = v.multiply(d).multiply(vT).subtract(symmetric).getNorm();
Assert.assertEquals(0, norm, 6.0e-13);
}
@Test
public void testSquareRoot() {
final double[][] data = {
{ 33, 24, 7 },
{ 24, 57, 11 },
{ 7, 11, 9 }
};
final EigenDecomposition dec = new EigenDecomposition(MatrixUtils.createRealMatrix(data));
final RealMatrix sqrtM = dec.getSquareRoot();
// Reconstruct initial matrix.
final RealMatrix m = sqrtM.multiply(sqrtM);
final int dim = data.length;
for (int r = 0; r < dim; r++) {
for (int c = 0; c < dim; c++) {
Assert.assertEquals("m[" + r + "][" + c + "]",
data[r][c], m.getEntry(r, c), 1e-13);
}
}
}
@Test(expected=MathUnsupportedOperationException.class)
public void testSquareRootNonSymmetric() {
final double[][] data = {
{ 1, 2, 4 },
{ 2, 3, 5 },
{ 11, 5, 9 }
};
final EigenDecomposition dec = new EigenDecomposition(MatrixUtils.createRealMatrix(data));
@SuppressWarnings("unused")
final RealMatrix sqrtM = dec.getSquareRoot();
}
@Test(expected=MathUnsupportedOperationException.class)
public void testSquareRootNonPositiveDefinite() {
final double[][] data = {
{ 1, 2, 4 },
{ 2, 3, 5 },
{ 4, 5, -9 }
};
final EigenDecomposition dec = new EigenDecomposition(MatrixUtils.createRealMatrix(data));
@SuppressWarnings("unused")
final RealMatrix sqrtM = dec.getSquareRoot();
}
@Test
public void testUnsymmetric() {
// Vandermonde matrix V(x;i,j) = x_i^{n - j} with x = (-1,-2,3,4)
double[][] vData = { { -1.0, 1.0, -1.0, 1.0 },
{ -8.0, 4.0, -2.0, 1.0 },
{ 27.0, 9.0, 3.0, 1.0 },
{ 64.0, 16.0, 4.0, 1.0 } };
checkUnsymmetricMatrix(MatrixUtils.createRealMatrix(vData));
RealMatrix randMatrix = MatrixUtils.createRealMatrix(new double[][] {
{0, 1, 0, 0},
{1, 0, 2.e-7, 0},
{0, -2.e-7, 0, 1},
{0, 0, 1, 0}
});
checkUnsymmetricMatrix(randMatrix);
// from http://eigen.tuxfamily.org/dox/classEigen_1_1RealSchur.html
double[][] randData2 = {
{ 0.680, -0.3300, -0.2700, -0.717, -0.687, 0.0259 },
{ -0.211, 0.5360, 0.0268, 0.214, -0.198, 0.6780 },
{ 0.566, -0.4440, 0.9040, -0.967, -0.740, 0.2250 },
{ 0.597, 0.1080, 0.8320, -0.514, -0.782, -0.4080 },
{ 0.823, -0.0452, 0.2710, -0.726, 0.998, 0.2750 },
{ -0.605, 0.2580, 0.4350, 0.608, -0.563, 0.0486 }
};
checkUnsymmetricMatrix(MatrixUtils.createRealMatrix(randData2));
}
@Test
@Ignore
public void testRandomUnsymmetricMatrix() {
for (int run = 0; run < 100; run++) {
Random r = new Random(System.currentTimeMillis());
// matrix size
int size = r.nextInt(20) + 4;
double[][] data = new double[size][size];
for (int i = 0; i < size; i++) {
for (int j = 0; j < size; j++) {
data[i][j] = r.nextInt(100);
}
}
RealMatrix m = MatrixUtils.createRealMatrix(data);
checkUnsymmetricMatrix(m);
}
}
/**
* Tests the porting of a bugfix in Jama-1.0.3 (from changelog):
*
* Patched hqr2 method in Jama.EigenvalueDecomposition to avoid infinite loop;
* Thanks Frederic Devernay <frederic.devernay@m4x.org>
*/
@Test
public void testMath1051() {
double[][] data = {
{0,0,0,0,0},
{0,0,0,0,1},
{0,0,0,1,0},
{1,1,0,0,1},
{1,0,1,0,1}
};
RealMatrix m = MatrixUtils.createRealMatrix(data);
checkUnsymmetricMatrix(m);
}
@Test
@Ignore
public void testNormalDistributionUnsymmetricMatrix() {
for (int run = 0; run < 100; run++) {
Random r = new Random(System.currentTimeMillis());
RealDistribution.Sampler dist
= new NormalDistribution(0.0, r.nextDouble() * 5).createSampler(RandomSource.create(RandomSource.WELL_512_A,
64925784252L));
// matrix size
int size = r.nextInt(20) + 4;
double[][] data = new double[size][size];
for (int i = 0; i < size; i++) {
for (int j = 0; j < size; j++) {
data[i][j] = dist.sample();
}
}
RealMatrix m = MatrixUtils.createRealMatrix(data);
checkUnsymmetricMatrix(m);
}
}
@Test
public void testMath848() {
double[][] data = {
{ 0.1849449280, -0.0646971046, 0.0774755812, -0.0969651755, -0.0692648806, 0.3282344352, -0.0177423074, 0.2063136340},
{-0.0742700134, -0.0289063030, -0.0017269460, -0.0375550146, -0.0487737922, -0.2616837868, -0.0821201295, -0.2530000167},
{ 0.2549910127, 0.0995733692, -0.0009718388, 0.0149282808, 0.1791878897, -0.0823182816, 0.0582629256, 0.3219545182},
{-0.0694747557, -0.1880649148, -0.2740630911, 0.0720096468, -0.1800836914, -0.3518996425, 0.2486747833, 0.6257938167},
{ 0.0536360918, -0.1339297778, 0.2241579764, -0.0195327484, -0.0054103808, 0.0347564518, 0.5120802482, -0.0329902864},
{-0.5933332356, -0.2488721082, 0.2357173629, 0.0177285473, 0.0856630593, -0.3567126300, -0.1600668126, -0.1010899621},
{-0.0514349819, -0.0854319435, 0.1125050061, 0.0063453560, -0.2250000688, -0.2209343090, 0.1964623477, -0.1512329924},
{ 0.0197395947, -0.1997170581, -0.1425959019, -0.2749477910, -0.0969467073, 0.0603688520, -0.2826905192, 0.1794315473}};
RealMatrix m = MatrixUtils.createRealMatrix(data);
checkUnsymmetricMatrix(m);
}
/**
* Checks that the eigen decomposition of a general (unsymmetric) matrix is valid by
* checking: A*V = V*D
*/
private void checkUnsymmetricMatrix(final RealMatrix m) {
try {
EigenDecomposition ed = new EigenDecomposition(m);
RealMatrix d = ed.getD();
RealMatrix v = ed.getV();
//RealMatrix vT = ed.getVT();
RealMatrix x = m.multiply(v);
RealMatrix y = v.multiply(d);
double diffNorm = x.subtract(y).getNorm();
Assert.assertTrue("The norm of (X-Y) is too large: " + diffNorm + ", matrix=" + m.toString(),
x.subtract(y).getNorm() < 1000 * Precision.EPSILON * FastMath.max(x.getNorm(), y.getNorm()));
RealMatrix invV = new LUDecomposition(v).getSolver().getInverse();
double norm = v.multiply(d).multiply(invV).subtract(m).getNorm();
Assert.assertEquals(0.0, norm, 1.0e-10);
} catch (Exception e) {
Assert.fail("Failed to create EigenDecomposition for matrix " + m.toString() + ", ex=" + e.toString());
}
}
/** test eigenvectors */
@Test
public void testEigenvectors() {
EigenDecomposition ed;
ed = new EigenDecomposition(matrix);
for (int i = 0; i < matrix.getRowDimension(); ++i) {
double lambda = ed.getRealEigenvalue(i);
RealVector v = ed.getEigenvector(i);
RealVector mV = matrix.operate(v);
Assert.assertEquals(0, mV.subtract(v.mapMultiplyToSelf(lambda)).getNorm(), 1.0e-13);
}
}
/** test A = VDVt */
@Test
public void testAEqualVDVt() {
EigenDecomposition ed;
ed = new EigenDecomposition(matrix);
RealMatrix v = ed.getV();
RealMatrix d = ed.getD();
RealMatrix vT = ed.getVT();
double norm = v.multiply(d).multiply(vT).subtract(matrix).getNorm();
Assert.assertEquals(0, norm, 6.0e-13);
}
/** test that V is orthogonal */
@Test
public void testVOrthogonal() {
RealMatrix v = new EigenDecomposition(matrix).getV();
RealMatrix vTv = v.transpose().multiply(v);
RealMatrix id = MatrixUtils.createRealIdentityMatrix(vTv.getRowDimension());
Assert.assertEquals(0, vTv.subtract(id).getNorm(), 2.0e-13);
}
/** test diagonal matrix */
@Test
public void testDiagonal() {
double[] diagonal = new double[] { -3.0, -2.0, 2.0, 5.0 };
RealMatrix m = MatrixUtils.createRealDiagonalMatrix(diagonal);
EigenDecomposition ed;
ed = new EigenDecomposition(m);
Assert.assertEquals(diagonal[0], ed.getRealEigenvalue(3), 2.0e-15);
Assert.assertEquals(diagonal[1], ed.getRealEigenvalue(2), 2.0e-15);
Assert.assertEquals(diagonal[2], ed.getRealEigenvalue(1), 2.0e-15);
Assert.assertEquals(diagonal[3], ed.getRealEigenvalue(0), 2.0e-15);
}
/**
* Matrix with eigenvalues {8, -1, -1}
*/
@Test
public void testRepeatedEigenvalue() {
RealMatrix repeated = MatrixUtils.createRealMatrix(new double[][] {
{3, 2, 4},
{2, 0, 2},
{4, 2, 3}
});
EigenDecomposition ed;
ed = new EigenDecomposition(repeated);
checkEigenValues((new double[] {8, -1, -1}), ed, 1E-12);
checkEigenVector((new double[] {2, 1, 2}), ed, 1E-12);
}
/**
* Matrix with eigenvalues {2, 0, 12}
*/
@Test
public void testDistinctEigenvalues() {
RealMatrix distinct = MatrixUtils.createRealMatrix(new double[][] {
{3, 1, -4},
{1, 3, -4},
{-4, -4, 8}
});
EigenDecomposition ed;
ed = new EigenDecomposition(distinct);
checkEigenValues((new double[] {2, 0, 12}), ed, 1E-12);
checkEigenVector((new double[] {1, -1, 0}), ed, 1E-12);
checkEigenVector((new double[] {1, 1, 1}), ed, 1E-12);
checkEigenVector((new double[] {-1, -1, 2}), ed, 1E-12);
}
/**
* Verifies operation on indefinite matrix
*/
@Test
public void testZeroDivide() {
RealMatrix indefinite = MatrixUtils.createRealMatrix(new double [][] {
{ 0.0, 1.0, -1.0 },
{ 1.0, 1.0, 0.0 },
{ -1.0,0.0, 1.0 }
});
EigenDecomposition ed;
ed = new EigenDecomposition(indefinite);
checkEigenValues((new double[] {2, 1, -1}), ed, 1E-12);
double isqrt3 = 1/FastMath.sqrt(3.0);
checkEigenVector((new double[] {isqrt3,isqrt3,-isqrt3}), ed, 1E-12);
double isqrt2 = 1/FastMath.sqrt(2.0);
checkEigenVector((new double[] {0.0,-isqrt2,-isqrt2}), ed, 1E-12);
double isqrt6 = 1/FastMath.sqrt(6.0);
checkEigenVector((new double[] {2*isqrt6,-isqrt6,isqrt6}), ed, 1E-12);
}
/**
* Verifies operation on very small values.
* Matrix with eigenvalues {2e-100, 0, 12e-100}
*/
@Test
public void testTinyValues() {
final double tiny = 1e-100;
RealMatrix distinct = MatrixUtils.createRealMatrix(new double[][] {
{3, 1, -4},
{1, 3, -4},
{-4, -4, 8}
});
distinct = distinct.scalarMultiply(tiny);
final EigenDecomposition ed = new EigenDecomposition(distinct);
checkEigenValues(MathArrays.scale(tiny, new double[] {2, 0, 12}), ed, 1e-12 * tiny);
checkEigenVector(new double[] {1, -1, 0}, ed, 1e-12);
checkEigenVector(new double[] {1, 1, 1}, ed, 1e-12);
checkEigenVector(new double[] {-1, -1, 2}, ed, 1e-12);
}
/**
* Verifies that the given EigenDecomposition has eigenvalues equivalent to
* the targetValues, ignoring the order of the values and allowing
* values to differ by tolerance.
*/
protected void checkEigenValues(double[] targetValues,
EigenDecomposition ed, double tolerance) {
double[] observed = ed.getRealEigenvalues();
for (int i = 0; i < observed.length; i++) {
Assert.assertTrue(isIncludedValue(observed[i], targetValues, tolerance));
Assert.assertTrue(isIncludedValue(targetValues[i], observed, tolerance));
}
}
/**
* Returns true iff there is an entry within tolerance of value in
* searchArray.
*/
private boolean isIncludedValue(double value, double[] searchArray,
double tolerance) {
boolean found = false;
int i = 0;
while (!found && i < searchArray.length) {
if (FastMath.abs(value - searchArray[i]) < tolerance) {
found = true;
}
i++;
}
return found;
}
/**
* Returns true iff eigenVector is a scalar multiple of one of the columns
* of ed.getV(). Does not try linear combinations - i.e., should only be
* used to find vectors in one-dimensional eigenspaces.
*/
protected void checkEigenVector(double[] eigenVector,
EigenDecomposition ed, double tolerance) {
Assert.assertTrue(isIncludedColumn(eigenVector, ed.getV(), tolerance));
}
/**
* Returns true iff there is a column that is a scalar multiple of column
* in searchMatrix (modulo tolerance)
*/
private boolean isIncludedColumn(double[] column, RealMatrix searchMatrix,
double tolerance) {
boolean found = false;
int i = 0;
while (!found && i < searchMatrix.getColumnDimension()) {
double multiplier = 1.0;
boolean matching = true;
int j = 0;
while (matching && j < searchMatrix.getRowDimension()) {
double colEntry = searchMatrix.getEntry(j, i);
// Use the first entry where both are non-zero as scalar
if (FastMath.abs(multiplier - 1.0) <= FastMath.ulp(1.0) && FastMath.abs(colEntry) > 1E-14
&& FastMath.abs(column[j]) > 1e-14) {
multiplier = colEntry / column[j];
}
if (FastMath.abs(column[j] * multiplier - colEntry) > tolerance) {
matching = false;
}
j++;
}
found = matching;
i++;
}
return found;
}
@Before
public void setUp() {
refValues = new double[] {
2.003, 2.002, 2.001, 1.001, 1.000, 0.001
};
matrix = createTestMatrix(new Random(35992629946426l), refValues);
}
@After
public void tearDown() {
refValues = null;
matrix = null;
}
static RealMatrix createTestMatrix(final Random r, final double[] eigenValues) {
final int n = eigenValues.length;
final RealMatrix v = createOrthogonalMatrix(r, n);
final RealMatrix d = MatrixUtils.createRealDiagonalMatrix(eigenValues);
return v.multiply(d).multiply(v.transpose());
}
public static RealMatrix createOrthogonalMatrix(final Random r, final int size) {
final double[][] data = new double[size][size];
for (int i = 0; i < size; ++i) {
final double[] dataI = data[i];
double norm2 = 0;
do {
// generate randomly row I
for (int j = 0; j < size; ++j) {
dataI[j] = 2 * r.nextDouble() - 1;
}
// project the row in the subspace orthogonal to previous rows
for (int k = 0; k < i; ++k) {
final double[] dataK = data[k];
double dotProduct = 0;
for (int j = 0; j < size; ++j) {
dotProduct += dataI[j] * dataK[j];
}
for (int j = 0; j < size; ++j) {
dataI[j] -= dotProduct * dataK[j];
}
}
// normalize the row
norm2 = 0;
for (final double dataIJ : dataI) {
norm2 += dataIJ * dataIJ;
}
final double inv = 1.0 / FastMath.sqrt(norm2);
for (int j = 0; j < size; ++j) {
dataI[j] *= inv;
}
} while (norm2 * size < 0.01);
}
return MatrixUtils.createRealMatrix(data);
}
}