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

               }
           };


       for (double x = 1e-30; x < 1e10; x *= 2) {
           final double fX = f.value(x);
           final double sX = sPrime.value(new DerivativeStructure(1, 1, 0, x)).getPartialDerivative(1);
           Assert.assertEquals("x=" + x, fX, sX, FastMath.ulp(fX));
       }
   }

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       }
   }


   @Test
   public void testDerivativesHighOrder() {
       DerivativeStructure s = new Sqrt().value(new DerivativeStructure(1, 5, 0, 1.2));
       Assert.assertEquals(1.0954451150103322269, s.getPartialDerivative(0), 1.0e-16);
       Assert.assertEquals(0.45643546458763842789, s.getPartialDerivative(1), 1.0e-16);
       Assert.assertEquals(-0.1901814435781826783,  s.getPartialDerivative(2), 1.0e-16);
       Assert.assertEquals(0.23772680447272834785,  s.getPartialDerivative(3), 1.0e-16);
       Assert.assertEquals(-0.49526417598485072465,   s.getPartialDerivative(4), 1.0e-16);
       Assert.assertEquals(1.4445205132891479465,  s.getPartialDerivative(5), 5.0e-16);
   }

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        // safety check
        checkInterpolation();


        final DerivativeStructure[] value = new DerivativeStructure[topDiagonal.get(0).length];
        Arrays.fill(value, x.getField().getZero());
        DerivativeStructure valueCoeff = x.getField().getOne();
        for (int i = 0; i < topDiagonal.size(); ++i) {
            double[] dividedDifference = topDiagonal.get(i);
            for (int k = 0; k < value.length; ++k) {
                value[k] = value[k].add(valueCoeff.multiply(dividedDifference[k]));
            }
            final DerivativeStructure deltaX = x.subtract(abscissae.get(i));
            valueCoeff = valueCoeff.multiply(deltaX);
        }


        return value;

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        ++jacobianEvaluations;


        final DerivativeStructure[] dsPoint = new DerivativeStructure[params.length];
        final int nC = params.length;
        for (int i = 0; i < nC; ++i) {
            dsPoint[i] = new DerivativeStructure(nC, 1, i, params[i]);
        }
        final DerivativeStructure[] dsValue = jF.value(dsPoint);
        final int nR = getTarget().length;
        if (dsValue.length != nR) {
            throw new DimensionMismatchException(dsValue.length, nR);

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            int i = 0;
            for (WeightedObservedPoint observed : observations) {


                // build the DerivativeStructure by adding first the value as a constant
                // and then adding derivatives
                DerivativeStructure vi = new DerivativeStructure(point.length, 1, f.value(observed.getX(), parameters));
                for (int k = 0; k < point.length; ++k) {
                    vi = vi.add(new DerivativeStructure(point.length, 1, k, 0.0));
                }


                values[i++] = vi;


            }

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     */
    public DerivativeStructure value(final DerivativeStructure t) {
        verifyInputArray(a, c);


        final int n = c.length;
        DerivativeStructure value = new DerivativeStructure(t.getFreeParameters(), t.getOrder(), a[n]);
        for (int i = n - 1; i >= 0; i--) {
            value = t.subtract(c[i]).multiply(value).add(a[i]);
        }


        return value;

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        for (int i = 0; i < CardanOrders.length; ++i) {
            for (double alpha1 = 0.1; alpha1 < 6.2; alpha1 += 0.3) {
                for (double alpha2 = -1.55; alpha2 < 1.55; alpha2 += 0.3) {
                    for (double alpha3 = 0.1; alpha3 < 6.2; alpha3 += 0.3) {
                        FieldRotation<DerivativeStructure> r = new FieldRotation<DerivativeStructure>(CardanOrders[i],
                                                      new DerivativeStructure(3, 1, 0, alpha1),
                                                      new DerivativeStructure(3, 1, 1, alpha2),
                                                      new DerivativeStructure(3, 1, 2, alpha3));
                        DerivativeStructure[] angles = r.getAngles(CardanOrders[i]);
                        checkAngle(angles[0], alpha1);
                        checkAngle(angles[1], alpha2);
                        checkAngle(angles[2], alpha3);
                    }
                }
            }
        }


        RotationOrder[] EulerOrders = {
            RotationOrder.XYX, RotationOrder.XZX, RotationOrder.YXY,
            RotationOrder.YZY, RotationOrder.ZXZ, RotationOrder.ZYZ
        };


        for (int i = 0; i < EulerOrders.length; ++i) {
            for (double alpha1 = 0.1; alpha1 < 6.2; alpha1 += 0.3) {
                for (double alpha2 = 0.05; alpha2 < 3.1; alpha2 += 0.3) {
                    for (double alpha3 = 0.1; alpha3 < 6.2; alpha3 += 0.3) {
                        FieldRotation<DerivativeStructure> r = new FieldRotation<DerivativeStructure>(EulerOrders[i],
                                                      new DerivativeStructure(3, 1, 0, alpha1),
                                                      new DerivativeStructure(3, 1, 1, alpha2),
                                                      new DerivativeStructure(3, 1, 2, alpha3));
                        DerivativeStructure[] angles = r.getAngles(EulerOrders[i]);
                        checkAngle(angles[0], alpha1);
                        checkAngle(angles[1], alpha2);
                        checkAngle(angles[2], alpha3);
                    }

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        double[] singularCardanAngle = { FastMath.PI / 2, -FastMath.PI / 2 };
        for (int i = 0; i < CardanOrders.length; ++i) {
            for (int j = 0; j < singularCardanAngle.length; ++j) {
                FieldRotation<DerivativeStructure> r = new FieldRotation<DerivativeStructure>(CardanOrders[i],
                                              new DerivativeStructure(3, 1, 0, 0.1),
                                              new DerivativeStructure(3, 1, 1, singularCardanAngle[j]),
                                              new DerivativeStructure(3, 1, 2, 0.3));
                try {
                    r.getAngles(CardanOrders[i]);
                    Assert.fail("an exception should have been caught");
                } catch (CardanEulerSingularityException cese) {
                    // expected behavior
                }
            }
        }


        RotationOrder[] EulerOrders = {
            RotationOrder.XYX, RotationOrder.XZX, RotationOrder.YXY,
            RotationOrder.YZY, RotationOrder.ZXZ, RotationOrder.ZYZ
        };


        double[] singularEulerAngle = { 0, FastMath.PI };
        for (int i = 0; i < EulerOrders.length; ++i) {
            for (int j = 0; j < singularEulerAngle.length; ++j) {
                FieldRotation<DerivativeStructure> r = new FieldRotation<DerivativeStructure>(EulerOrders[i],
                                              new DerivativeStructure(3, 1, 0, 0.1),
                                              new DerivativeStructure(3, 1, 1, singularEulerAngle[j]),
                                              new DerivativeStructure(3, 1, 2, 0.3));
                try {
                    r.getAngles(EulerOrders[i]);
                    Assert.fail("an exception should have been caught");
                } catch (CardanEulerSingularityException cese) {
                    // expected behavior

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        UnitSphereRandomVectorGenerator g = new UnitSphereRandomVectorGenerator(3, random);
        for (int i = 0; i < 10; ++i) {
            double[] unit1 = g.nextVector();
            Rotation r1 = new Rotation(new Vector3D(unit1[0], unit1[1], unit1[2]),
                                      random.nextDouble());
            FieldRotation<DerivativeStructure> r1Prime = new FieldRotation<DerivativeStructure>(new DerivativeStructure(4, 1, 0, r1.getQ0()),
                                                new DerivativeStructure(4, 1, 1, r1.getQ1()),
                                                new DerivativeStructure(4, 1, 2, r1.getQ2()),
                                                new DerivativeStructure(4, 1, 3, r1.getQ3()),
                                                false);
            double[] unit2 = g.nextVector();
            FieldRotation<DerivativeStructure> r2 = new FieldRotation<DerivativeStructure>(createVector(unit2[0], unit2[1], unit2[2]),
                                           createAngle(random.nextDouble()));

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        Assert.assertEquals(0, FieldRotation.distance(r, rPrime).getReal(), 1.0e-12);
    }


    private FieldRotation<DerivativeStructure> createRotation(double q0, double q1, double q2, double q3,
                                      boolean needsNormalization) {
        return new FieldRotation<DerivativeStructure>(new DerivativeStructure(4, 1, 0, q0),
                              new DerivativeStructure(4, 1, 1, q1),
                              new DerivativeStructure(4, 1, 2, q2),
                              new DerivativeStructure(4, 1, 3, q3),
                              needsNormalization);
    }

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org.apache.commons.math3.analysis.differentiation.MultivariateDifferentiableVectorFunction

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org.apache.commons.math3.analysis.function.LogitTest

org.apache.commons.math3.analysis.function.SigmoidTest

org.apache.commons.math3.analysis.function.Sin

org.apache.commons.math3.analysis.function.Sinc

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org.apache.commons.math3.analysis.function.SqrtTest

org.apache.commons.math3.analysis.FunctionUtils

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