Examples of org.apache.commons.math3.analysis.differentiation.DerivativeStructure

Package org.apache.commons.math3.analysis.differentiation

Examples of org.apache.commons.math3.analysis.differentiation.DerivativeStructure

org.apache.commons.math3.analysis.differentiation.DerivativeStructure
american.edu/cas/mathstat/People/kalman/pdffiles/mmgautodiff.pdf">Doubly Recursive Multivariate Automatic Differentiation, Mathematics Magazine, vol. 75, no. 3, June 2002.
. Rall's numbers are an extension to the real numbers used throughout mathematical expressions; they hold the derivative together with the value of a function. Dan Kalman's derivative structures hold all partial derivatives up to any specified order, with respect to any number of free parameters. Rall's numbers therefore can be seen as derivative structures for order one derivative and one free parameter, and real numbers can be seen as derivative structures with zero order derivative and no free parameters.

{@link DerivativeStructure} instances can be used directly thanks tothe arithmetic operators to the mathematical functions provided as methods by this class (+, -, *, /, %, sin, cos ...).

Implementing complex expressions by hand using these classes is a tedious and error-prone task but has the advantage of having no limitation on the derivation order despite no requiring users to compute the derivatives by themselves. Implementing complex expression can also be done by developing computation code using standard primitive double values and to use {@link UnivariateFunctionDifferentiator differentiators} to create the {@link DerivativeStructure}-based instances. This method is simpler but may be limited in the accuracy and derivation orders and may be computationally intensive (this is typically the case for {@link FiniteDifferencesDifferentiator finite differencesdifferentiator}.

Instances of this class are guaranteed to be immutable.
@see DSCompiler @since 3.1

                        final int n = x.length;


                        // delegate computation to underlying function
                        final DerivativeStructure[] dsX = new DerivativeStructure[n];
                        for (int i = 0; i < n; ++i) {
                            dsX[i] = new DerivativeStructure(n, 1, i, x[i]);
                        }
                        final DerivativeStructure[] y = f.value(dsX);


                        // extract Jacobian
                        final double[][] jacobian = new double[y.length][n];

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                        for (int j = 0; j < n; ++j) {
                            derivatives[i + 1] += jacobian[k][j] * t[j].getPartialDerivative(orders);
                        }
                        orders[i] = 0;
                    }
                    merged[k] = new DerivativeStructure(parameters, order, derivatives);
                }


                return merged;


            }

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        try {
            dataset = new StatisticalReferenceDataset(in) {


                @Override
                public DerivativeStructure getModelValue(final double x, final DerivativeStructure[] a) {
                    final DerivativeStructure p = a[0].add(a[1].add(a[2].multiply(x)).multiply(x));
                    final DerivativeStructure q = a[3].add(a[4].multiply(x)).multiply(x).add(1.0);
                    return p.divide(q);
                }


            };
        } finally {

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        try {
            dataset = new StatisticalReferenceDataset(in) {


                @Override
                public DerivativeStructure getModelValue(final double x, final DerivativeStructure[] a) {
                    final DerivativeStructure p = a[0].add(a[1].add(a[2].add(a[3].multiply(x)).multiply(x)).multiply(x));
                    final DerivativeStructure q = a[4].add(a[5].add(a[6].multiply(x)).multiply(x)).multiply(x).add(1.0);
                    return p.divide(q);
                }


            };
        } finally {

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                for (int j = 0; j < factors.getColumnDimension(); ++j) {
                    y[i] = y[i].add(variables[j].multiply(factors.getEntry(i, j)));
                }
            }


            DerivativeStructure sum = variables[0].getField().getZero();
            for (int i = 0; i < y.length; ++i) {
                DerivativeStructure ri = y[i].subtract(target[i]);
                sum = sum.add(ri.multiply(ri));
            }
            return sum;
        }

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    }


    @Test
    public void testComposeDifferentiable() {
        UnivariateDifferentiableFunction id = new Identity();
        Assert.assertEquals(1, FunctionUtils.compose(id, id, id).value(new DerivativeStructure(1, 1, 0, 3)).getPartialDerivative(1), EPS);


        UnivariateDifferentiableFunction c = new Constant(4);
        Assert.assertEquals(0, FunctionUtils.compose(id, c).value(new DerivativeStructure(1, 1, 0, 3)).getPartialDerivative(1), EPS);
        Assert.assertEquals(0, FunctionUtils.compose(c, id).value(new DerivativeStructure(1, 1, 0, 3)).getPartialDerivative(1), EPS);


        UnivariateDifferentiableFunction m = new Minus();
        Assert.assertEquals(-1, FunctionUtils.compose(m).value(new DerivativeStructure(1, 1, 0, 3)).getPartialDerivative(1), EPS);
        Assert.assertEquals(1, FunctionUtils.compose(m, m).value(new DerivativeStructure(1, 1, 0, 3)).getPartialDerivative(1), EPS);


        UnivariateDifferentiableFunction inv = new Inverse();
        Assert.assertEquals(0.25, FunctionUtils.compose(inv, m, id).value(new DerivativeStructure(1, 1, 0, 2)).getPartialDerivative(1), EPS);


        UnivariateDifferentiableFunction pow = new Power(2);
        Assert.assertEquals(108, FunctionUtils.compose(pow, pow).value(new DerivativeStructure(1, 1, 0, 3)).getPartialDerivative(1), EPS);


        UnivariateDifferentiableFunction log = new Log();
        double a = 9876.54321;
        Assert.assertEquals(pow.value(new DerivativeStructure(1, 1, 0, a)).getPartialDerivative(1) / pow.value(a),
                            FunctionUtils.compose(log, pow).value(new DerivativeStructure(1, 1, 0, a)).getPartialDerivative(1), EPS);
    }

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        UnivariateDifferentiableFunction m = new Minus();
        UnivariateDifferentiableFunction inv = new Inverse();


        final double a = 123.456;
        Assert.assertEquals(- 1 / (a * a) -1 + FastMath.cos(a),
                            FunctionUtils.add(inv, m, c, sin).value(new DerivativeStructure(1, 1, 0, a)).getPartialDerivative(1),
                            EPS);
    }

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    @Test
    public void testMultiplyDifferentiable() {
        UnivariateDifferentiableFunction c = new Constant(4);
        UnivariateDifferentiableFunction id = new Identity();
        final double a = 1.2345678;
        Assert.assertEquals(8 * a, FunctionUtils.multiply(c, id, id).value(new DerivativeStructure(1, 1, 0, a)).getPartialDerivative(1), EPS);


        UnivariateDifferentiableFunction inv = new Inverse();
        UnivariateDifferentiableFunction pow = new Power(2.5);
        UnivariateDifferentiableFunction cos = new Cos();
        Assert.assertEquals(1.5 * FastMath.sqrt(a) * FastMath.cos(a) - FastMath.pow(a, 1.5) * FastMath.sin(a),
                            FunctionUtils.multiply(inv, pow, cos).value(new DerivativeStructure(1, 1, 0, a)).getPartialDerivative(1), EPS);


        UnivariateDifferentiableFunction cosh = new Cosh();
        Assert.assertEquals(1.5 * FastMath.sqrt(a) * FastMath.cosh(a) + FastMath.pow(a, 1.5) * FastMath.sinh(a),
                            FunctionUtils.multiply(inv, pow, cosh).value(new DerivativeStructure(1, 1, 0, a)).getPartialDerivative(1), 8 * EPS);
    }

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        // Sin implements both UnivariateDifferentiableFunction and DifferentiableUnivariateFunction
        Sin sin = new Sin();
        UnivariateDifferentiableFunction converted = FunctionUtils.toUnivariateDifferential(sin);
        for (double x = 0.1; x < 0.5; x += 0.01) {
            DerivativeStructure t = new DerivativeStructure(2, 1, x, 1.0, 2.0);
            Assert.assertEquals(sin.value(t).getValue(), converted.value(t).getValue(), 1.0e-10);
            Assert.assertEquals(sin.value(t).getPartialDerivative(1, 0),
                                converted.value(t).getPartialDerivative(1, 0),
                                1.0e-10);
            Assert.assertEquals(sin.value(t).getPartialDerivative(0, 1),

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        MultivariateDifferentiableFunction converted = FunctionUtils.toMultivariateDifferentiableFunction(hypot);
        for (double x = 0.1; x < 0.5; x += 0.01) {
            for (double y = 0.1; y < 0.5; y += 0.01) {
                DerivativeStructure[] t = new DerivativeStructure[] {
                    new DerivativeStructure(3, 1, x, 1.0, 2.0, 3.0 ),
                    new DerivativeStructure(3, 1, y, 4.0, 5.0, 6.0 )
                };
                DerivativeStructure h = DerivativeStructure.hypot(t[0], t[1]);
                Assert.assertEquals(h.getValue(), converted.value(t).getValue(), 1.0e-10);
                Assert.assertEquals(h.getPartialDerivative(1, 0, 0),
                                    converted.value(t).getPartialDerivative(1, 0, 0),
                                    1.0e-10);
                Assert.assertEquals(h.getPartialDerivative(0, 1, 0),
                                    converted.value(t).getPartialDerivative(0, 1, 0),
                                    1.0e-10);
                Assert.assertEquals(h.getPartialDerivative(0, 0, 1),
                                    converted.value(t).getPartialDerivative(0, 0, 1),
                                    1.0e-10);
            }
        }
    }

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