MATH-621

Original contribution due to Dietmar Wolz: Fortran code translated in Java.
This commit is for reference only; work is under way to adapt the code into
a more maintainable version.


git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1154543 13f79535-47bb-0310-9956-ffa450edef68

src/main/java/org/apache/commons/math/exception/util/LocalizedFormats.java

@@ -162,6 +162,7 @@ public enum LocalizedFormats implements Localizable {
     NEGATIVE_NUMBER_OF_SUCCESSES("number of successes must be non-negative ({0})"),
     NUMBER_OF_SUCCESSES("number of successes ({0})"), /* keep */
     NEGATIVE_NUMBER_OF_TRIALS("number of trials must be non-negative ({0})"),
+    NUMBER_OF_INTERPOLATION_POINTS("number of interpolation points ({0})"), /* keep */
     NUMBER_OF_TRIALS("number of trials ({0})"),
     ROBUSTNESS_ITERATIONS("number of robustness iterations ({0})"),
     START_POSITION("start position ({0})"), /* keep */
@@ -261,6 +262,7 @@ public enum LocalizedFormats implements Localizable {
     N_POINTS_GAUSS_LEGENDRE_INTEGRATOR_NOT_SUPPORTED("{0} points Legendre-Gauss integrator not supported, number of points must be in the {1}-{2} range"),
     OBSERVED_COUNTS_ALL_ZERO("observed counts are all 0 in observed array {0}"),
     OBSERVED_COUNTS_BOTTH_ZERO_FOR_ENTRY("observed counts are both zero for entry {0}"),
+    BOBYQA_BOUND_DIFFERENCE_CONDITION("the difference between the upper and lower bound must be larger than twice the initial trust region radius ({0})"),
     OUT_OF_BOUNDS_QUANTILE_VALUE("out of bounds quantile value: {0}, must be in (0, 100]"),
     OUT_OF_BOUND_SIGNIFICANCE_LEVEL("out of bounds significance level {0}, must be between {1} and {2}"),
     SIGNIFICANCE_LEVEL("significance level ({0})"), /* keep */
@@ -295,11 +297,13 @@ public enum LocalizedFormats implements Localizable {
     SUBARRAY_ENDS_AFTER_ARRAY_END("subarray ends after array end"),
     TOO_LARGE_CUTOFF_SINGULAR_VALUE("cutoff singular value is {0}, should be at most {1}"),
     TOO_MANY_ELEMENTS_TO_DISCARD_FROM_ARRAY("cannot discard {0} elements from a {1} elements array"),
+    TOO_MUCH_CANCELLATION("too much cancellation in a denominator"),
     TOO_MANY_REGRESSORS("too many regressors ({0}) specified, only {1} in the model"),
     TOO_SMALL_COST_RELATIVE_TOLERANCE("cost relative tolerance is too small ({0}), no further reduction in the sum of squares is possible"),
     TOO_SMALL_INTEGRATION_INTERVAL("too small integration interval: length = {0}"),
     TOO_SMALL_ORTHOGONALITY_TOLERANCE("orthogonality tolerance is too small ({0}), solution is orthogonal to the jacobian"),
     TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE("parameters relative tolerance is too small ({0}), no further improvement in the approximate solution is possible"),
+    TRUST_REGION_STEP_FAILED("trust region step has failed to reduce Q"),
     TWO_OR_MORE_CATEGORIES_REQUIRED("two or more categories required, got {0}"),
     TWO_OR_MORE_VALUES_IN_CATEGORY_REQUIRED("two or more values required in each category, one has {0}"),
     UNABLE_TO_BRACKET_OPTIMUM_IN_LINE_SEARCH("unable to bracket optimum in line search"),

src/main/resources/META-INF/localization/LocalizedFormats_fr.properties

@@ -133,6 +133,7 @@ NEGATIVE_ELEMENT_AT_INDEX = l''\u00e9l\u00e9ment {0} est n\u00e9gatif : {1}
 NEGATIVE_NUMBER_OF_SUCCESSES = le nombre de succ\u00e8s ne doit pas \u00eatre n\u00e9gatif ({0})
 NUMBER_OF_SUCCESSES = nombre de succ\u00e8s ({0})
 NEGATIVE_NUMBER_OF_TRIALS = le nombre d''essais ne doit pas \u00eatre n\u00e9gatif ({0})
+NUMBER_OF_INTERPOLATION_POINTS = nombre de points d''interpolation ({0})
 NUMBER_OF_TRIALS = nombre d''essais ({0})
 NEGATIVE_ROBUSTNESS_ITERATIONS = le nombre d''it\u00e9rations robuste ne peut \u00eatre n\u00e9gatif, alors qu''il est de {0}
 START_POSITION = position de d\u00e9part
@@ -231,6 +232,7 @@ NUMERATOR_OVERFLOW_AFTER_MULTIPLY = d\u00e9passement de capacit\u00e9 pour le nu
 N_POINTS_GAUSS_LEGENDRE_INTEGRATOR_NOT_SUPPORTED = l''int\u00e9grateur de Legendre-Gauss en {0} points n''est pas disponible, le nombre de points doit \u00eatre entre {1} et {2}
 OBSERVED_COUNTS_ALL_ZERO = aucune occurrence dans le tableau des observations {0}
 OBSERVED_COUNTS_BOTTH_ZERO_FOR_ENTRY = les occurrences observ\u00e9es sont toutes deux nulles pour l''entr\u00e9e {0}
+BOBYQA_BOUND_DIFFERENCE_CONDITION = la diff\u00e9rence entre la contrainte sup\u00e9rieure et inf\u00e9rieure doit \u00eatre plus grande que deux fois le rayon de la r\u00e9gion de confiance initiale ({0})
 OUT_OF_BOUNDS_QUANTILE_VALUE = valeur de quantile {0} hors bornes, doit \u00eatre dans l''intervalle ]0, 100]
 OUT_OF_BOUND_SIGNIFICANCE_LEVEL = niveau de signification {0} hors domaine, doit \u00eatre entre {1} et {2}
 SIGNIFICANCE_LEVEL = niveau de signification ({0})

src/test/java/org/apache/commons/math/optimization/direct/BOBYQAOptimizerTest.java

@@ -0,0 +1,588 @@
+/*
+ * 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.math.optimization.direct;
+
+import static org.junit.Assert.fail;
+
+import java.util.Arrays;
+import java.util.Random;
+
+import org.apache.commons.math.analysis.MultivariateRealFunction;
+import org.apache.commons.math.exception.MultiDimensionMismatchException;
+import org.apache.commons.math.exception.NoDataException;
+import org.apache.commons.math.exception.OutOfRangeException;
+import org.apache.commons.math.exception.TooManyEvaluationsException;
+import org.apache.commons.math.optimization.GoalType;
+import org.apache.commons.math.optimization.MultivariateRealOptimizer;
+import org.apache.commons.math.optimization.RealPointValuePair;
+import org.junit.Assert;
+import org.junit.Test;
+
+/**
+ * Test for {@link BOBYQAOptimizer}.
+ */
+public class BOBYQAOptimizerTest {
+
+    static final int DIM = 13;
+ 
+    @Test(expected = OutOfRangeException.class)
+    public void testInitOutofbounds() {
+        double[] startPoint = point(DIM,3);
+        double[][] boundaries = boundaries(DIM,-1,2);
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,1.0),0.0);
+        doTest(new Rosen(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 1e-6, 2000, expected);
+    }
+    
+    @Test(expected = MultiDimensionMismatchException.class)
+    public void testBoundariesDimensionMismatch() {
+        double[] startPoint = point(DIM,0.5);
+        double[][] boundaries = boundaries(DIM+1,-1,2);
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,1.0),0.0);
+        doTest(new Rosen(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 1e-6, 2000, expected);
+    }
+
+    @Test(expected = NoDataException.class)
+    public void testBoundariesNoData() {
+        double[] startPoint = point(DIM,0.5);
+        double[][] boundaries = boundaries(DIM,-1,2);
+        boundaries[1] = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,1.0),0.0);
+        doTest(new Rosen(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 1e-6, 2000, expected);
+    }
+    
+    @Test
+    public void testRosen() {
+        double[] startPoint = point(DIM,0.1);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,1.0),0.0);
+        doTest(new Rosen(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 1e-6, 2000, expected);
+     }
+    
+    @Test
+    public void testRescue() {
+        double[] startPoint = point(13,1.0);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(13,0.0),0);
+        try {
+            doTest(new MinusElli(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 1e-6, 1000, expected);
+            fail("An TooManyEvaluationsException should have been thrown");
+        } catch(TooManyEvaluationsException e) {
+        }
+    }
+
+    @Test
+    public void testMaximize() {
+        double[] startPoint = point(DIM,1.0);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,0.0),1.0);
+        doTest(new MinusElli(), startPoint, boundaries,
+                GoalType.MAXIMIZE, 
+                2e-10, 5e-6, 1000, expected);
+        boundaries = boundaries(DIM,-0.3,0.3); 
+        startPoint = point(DIM,0.1);
+        doTest(new MinusElli(), startPoint, boundaries,
+                GoalType.MAXIMIZE, 
+                2e-10, 5e-6, 1000, expected);
+    }
+
+    @Test
+    public void testEllipse() {
+        double[] startPoint = point(DIM,1.0);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,0.0),0.0);
+        doTest(new Elli(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 1e-6, 1000, expected);
+     }
+
+    @Test
+    public void testElliRotated() {
+        double[] startPoint = point(DIM,1.0);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,0.0),0.0);
+        doTest(new ElliRotated(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-12, 1e-6, 10000, expected);
+    }
+
+    @Test
+    public void testCigar() {
+        double[] startPoint = point(DIM,1.0);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,0.0),0.0);
+        doTest(new Cigar(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 1e-6, 100, expected);
+    }
+
+    @Test
+    public void testTwoAxes() {
+        double[] startPoint = point(DIM,1.0);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,0.0),0.0);
+        doTest(new TwoAxes(), startPoint, boundaries,
+                GoalType.MINIMIZE, 2*
+                1e-13, 1e-6, 100, expected);
+     }
+
+    @Test
+    public void testCigTab() {
+        double[] startPoint = point(DIM,1.0);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,0.0),0.0);
+        doTest(new CigTab(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 5e-5, 100, expected);
+     }
+
+    @Test
+    public void testSphere() {
+        double[] startPoint = point(DIM,1.0);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,0.0),0.0);
+        doTest(new Sphere(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 1e-6, 100, expected);
+    }
+
+    @Test
+    public void testTablet() {
+        double[] startPoint = point(DIM,1.0); 
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,0.0),0.0);
+        doTest(new Tablet(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 1e-6, 100, expected);
+    }
+
+    @Test
+    public void testDiffPow() {
+        double[] startPoint = point(DIM/2,1.0);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM/2,0.0),0.0);
+        doTest(new DiffPow(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-8, 1e-1, 12000, expected);
+    }
+
+    @Test
+    public void testSsDiffPow() {
+        double[] startPoint = point(DIM/2,1.0);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM/2,0.0),0.0);
+        doTest(new SsDiffPow(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-2, 1.3e-1, 50000, expected);
+    }
+
+    @Test
+    public void testAckley() {
+        double[] startPoint = point(DIM,0.1);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,0.0),0.0);
+        doTest(new Ackley(), startPoint, boundaries,
+                GoalType.MINIMIZE,
+                1e-8, 1e-5, 1000, expected);
+    }
+
+    @Test
+    public void testRastrigin() {
+        double[] startPoint = point(DIM,1.0);
+
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,0.0),0.0);
+        doTest(new Rastrigin(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-13, 1e-6, 1000, expected);
+    }
+
+    @Test
+    public void testConstrainedRosen() {
+        double[] startPoint = point(DIM,0.1);
+
+        double[][] boundaries = boundaries(DIM,-1,2);
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,1.0),0.0);
+        doTest(new Rosen(), startPoint, boundaries,
+                GoalType.MINIMIZE,
+                1e-13, 1e-6, 2000, expected);
+    }
+
+    @Test
+    public void testDiagonalRosen() {
+        double[] startPoint = point(DIM,0.1);
+        double[][] boundaries = null;
+        RealPointValuePair expected =
+            new RealPointValuePair(point(DIM,1.0),0.0);
+        doTest(new Rosen(), startPoint, boundaries,
+                GoalType.MINIMIZE, 
+                1e-10, 1e-4, 2000, expected);
+     }
+
+    /**
+     * @param func Function to optimize.
+     * @param startPoint Starting point.
+     * @param boundaries Upper / lower point limit.
+     * @param goal Minimization or maximization.
+     * @param fTol Tolerance relative error on the objective function.
+     * @param pointTol Tolerance for checking that the optimum is correct.
+     * @param maxEvaluations Maximum number of evaluations.
+     * @param expected Expected point / value.
+     */
+    private void doTest(MultivariateRealFunction func,
+            double[] startPoint,
+            double[][] boundaries,
+            GoalType goal,
+            double fTol,
+            double pointTol,
+            int maxEvaluations,
+            RealPointValuePair expected) {
+        int dim = startPoint.length;
+//        MultivariateRealOptimizer optim =
+//            new PowellOptimizer(1e-13, Math.ulp(1d));
+//        RealPointValuePair result = optim.optimize(100000, func, goal, startPoint);
+        MultivariateRealOptimizer optim =
+            new BOBYQAOptimizer(boundaries);
+        RealPointValuePair result = optim.optimize(maxEvaluations, func, goal, startPoint);        
+//        System.out.println(func.getClass().getName() + " = " 
+//        		+ optim.getEvaluations() + " f(");
+//        for (double x: result.getPoint())  System.out.print(x + " ");
+//        System.out.println(") = " +  result.getValue());
+        Assert.assertEquals(expected.getValue(),
+                result.getValue(), fTol);
+        for (int i = 0; i < dim; i++) {
+            Assert.assertEquals(expected.getPoint()[i],
+                    result.getPoint()[i], pointTol);
+        }
+    }
+
+    private static double[] point(int n, double value) {
+        double[] ds = new double[n];
+        Arrays.fill(ds, value);
+        return ds;
+    }
+
+    private static double[][] boundaries(int dim,
+            double lower, double upper) {
+        double[][] boundaries = new double[2][dim];
+        for (int i = 0; i < dim; i++)
+            boundaries[0][i] = lower;
+        for (int i = 0; i < dim; i++)
+            boundaries[1][i] = upper;
+        return boundaries;
+    }
+
+    private static class Sphere implements MultivariateRealFunction {
+
+        public double value(double[] x) {
+            double f = 0;
+            for (int i = 0; i < x.length; ++i)
+                f += x[i] * x[i];
+            return f;
+        }
+    }
+
+    private static class Cigar implements MultivariateRealFunction {
+        private double factor;
+
+        Cigar() {
+            this(1e3);
+        }
+
+        Cigar(double axisratio) {
+            factor = axisratio * axisratio;
+        }
+
+        public double value(double[] x) {
+            double f = x[0] * x[0];
+            for (int i = 1; i < x.length; ++i)
+                f += factor * x[i] * x[i];
+            return f;
+        }
+    }
+
+    private static class Tablet implements MultivariateRealFunction {
+        private double factor;
+
+        Tablet() {
+            this(1e3);
+        }
+
+        Tablet(double axisratio) {
+            factor = axisratio * axisratio;
+        }
+
+        public double value(double[] x) {
+            double f = factor * x[0] * x[0];
+            for (int i = 1; i < x.length; ++i)
+                f += x[i] * x[i];
+            return f;
+        }
+    }
+
+    private static class CigTab implements MultivariateRealFunction {
+        private double factor;
+
+        CigTab() {
+            this(1e4);
+        }
+
+        CigTab(double axisratio) {
+            factor = axisratio;
+        }
+
+        public double value(double[] x) {
+            int end = x.length - 1;
+            double f = x[0] * x[0] / factor + factor * x[end] * x[end];
+            for (int i = 1; i < end; ++i)
+                f += x[i] * x[i];
+            return f;
+        }
+    }
+
+    private static class TwoAxes implements MultivariateRealFunction {
+
+        private double factor;
+
+        TwoAxes() {
+            this(1e6);
+        }
+
+        TwoAxes(double axisratio) {
+            factor = axisratio * axisratio;
+        }
+
+        public double value(double[] x) {
+            double f = 0;
+            for (int i = 0; i < x.length; ++i)
+                f += (i < x.length / 2 ? factor : 1) * x[i] * x[i];
+            return f;
+        }
+    }
+
+    private static class ElliRotated implements MultivariateRealFunction {
+        private Basis B = new Basis();
+        private double factor;
+
+        ElliRotated() {
+            this(1e3);
+        }
+
+        ElliRotated(double axisratio) {
+            factor = axisratio * axisratio;
+        }
+
+        public double value(double[] x) {
+            double f = 0;
+            x = B.Rotate(x);
+            for (int i = 0; i < x.length; ++i)
+                f += Math.pow(factor, i / (x.length - 1.)) * x[i] * x[i];
+            return f;
+        }
+    }
+
+    private static class Elli implements MultivariateRealFunction {
+
+        private double factor;
+
+        Elli() {
+            this(1e3);
+        }
+
+        Elli(double axisratio) {
+            factor = axisratio * axisratio;
+        }
+
+        public double value(double[] x) {
+            double f = 0;
+            for (int i = 0; i < x.length; ++i)
+                f += Math.pow(factor, i / (x.length - 1.)) * x[i] * x[i];
+            return f;
+        }
+    }
+
+    private static class MinusElli implements MultivariateRealFunction {
+        private int fcount = 0;
+        public double value(double[] x) {
+          double f = 1.0-(new Elli().value(x));
+//          System.out.print("" + (fcount++) + ") ");
+//          for (int i = 0; i < x.length; i++)
+//              System.out.print(x[i] +  " ");
+//          System.out.println(" = " + f);
+          return f;
+       }
+    }
+
+    private static class DiffPow implements MultivariateRealFunction {
+        private int fcount = 0;
+        public double value(double[] x) {
+            double f = 0;
+            for (int i = 0; i < x.length; ++i)
+                f += Math.pow(Math.abs(x[i]), 2. + 10 * (double) i
+                        / (x.length - 1.));
+//            System.out.print("" + (fcount++) + ") ");
+//            for (int i = 0; i < x.length; i++)
+//                System.out.print(x[i] +  " ");
+//            System.out.println(" = " + f);
+            return f;
+        }
+    }
+
+    private static class SsDiffPow implements MultivariateRealFunction {
+
+        public double value(double[] x) {
+            double f = Math.pow(new DiffPow().value(x), 0.25);
+            return f;
+        }
+    }
+
+    private static class Rosen implements MultivariateRealFunction {
+
+        public double value(double[] x) {
+            double f = 0;
+            for (int i = 0; i < x.length - 1; ++i)
+                f += 1e2 * (x[i] * x[i] - x[i + 1]) * (x[i] * x[i] - x[i + 1])
+                + (x[i] - 1.) * (x[i] - 1.);
+            return f;
+        }
+    }
+
+    private static class Ackley implements MultivariateRealFunction {
+        private double axisratio;
+
+        Ackley(double axra) {
+            axisratio = axra;
+        }
+
+        public Ackley() {
+            this(1);
+        }
+
+        public double value(double[] x) {
+            double f = 0;
+            double res2 = 0;
+            double fac = 0;
+            for (int i = 0; i < x.length; ++i) {
+                fac = Math.pow(axisratio, (i - 1.) / (x.length - 1.));
+                f += fac * fac * x[i] * x[i];
+                res2 += Math.cos(2. * Math.PI * fac * x[i]);
+            }
+            f = (20. - 20. * Math.exp(-0.2 * Math.sqrt(f / x.length))
+                    + Math.exp(1.) - Math.exp(res2 / x.length));
+            return f;
+        }
+    }
+
+    private static class Rastrigin implements MultivariateRealFunction {
+
+        private double axisratio;
+        private double amplitude;
+
+        Rastrigin() {
+            this(1, 10);
+        }
+
+        Rastrigin(double axisratio, double amplitude) {
+            this.axisratio = axisratio;
+            this.amplitude = amplitude;
+        }
+
+        public double value(double[] x) {
+            double f = 0;
+            double fac;
+            for (int i = 0; i < x.length; ++i) {
+                fac = Math.pow(axisratio, (i - 1.) / (x.length - 1.));
+                if (i == 0 && x[i] < 0)
+                    fac *= 1.;
+                f += fac * fac * x[i] * x[i] + amplitude
+                * (1. - Math.cos(2. * Math.PI * fac * x[i]));
+            }
+            return f;
+        }
+    }
+
+    private static class Basis {
+        double[][] basis;
+        Random rand = new Random(2); // use not always the same basis
+
+        double[] Rotate(double[] x) {
+            GenBasis(x.length);
+            double[] y = new double[x.length];
+            for (int i = 0; i < x.length; ++i) {
+                y[i] = 0;
+                for (int j = 0; j < x.length; ++j)
+                    y[i] += basis[i][j] * x[j];
+            }
+            return y;
+        }
+
+        void GenBasis(int DIM) {
+            if (basis != null ? basis.length == DIM : false)
+                return;
+
+            double sp;
+            int i, j, k;
+
+            /* generate orthogonal basis */
+            basis = new double[DIM][DIM];
+            for (i = 0; i < DIM; ++i) {
+                /* sample components gaussian */
+                for (j = 0; j < DIM; ++j)
+                    basis[i][j] = rand.nextGaussian();
+                /* substract projection of previous vectors */
+                for (j = i - 1; j >= 0; --j) {
+                    for (sp = 0., k = 0; k < DIM; ++k)
+                        sp += basis[i][k] * basis[j][k]; /* scalar product */
+                    for (k = 0; k < DIM; ++k)
+                        basis[i][k] -= sp * basis[j][k]; /* substract */
+                }
+                /* normalize */
+                for (sp = 0., k = 0; k < DIM; ++k)
+                    sp += basis[i][k] * basis[i][k]; /* squared norm */
+                for (k = 0; k < DIM; ++k)
+                    basis[i][k] /= Math.sqrt(sp);
+            }
+        }
+    }
+}