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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
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/*
* 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.math3.analysis.differentiation;
import java.io.Serializable;
import java.util.Collections;
import java.util.HashMap;
import java.util.Map;
import org.apache.commons.math3.Field;
import org.apache.commons.math3.FieldElement;
import org.apache.commons.math3.RealFieldElement;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathArrays;
import org.apache.commons.math3.util.MathUtils;
import org.apache.commons.math3.util.Precision;
/**
* First derivative computation with large number of variables.
*
* This class plays a similar role to {@link DerivativeStructure}, with
* a focus on efficiency when dealing with large number of independent variables
* and most computation depend only on a few of them, and when only first derivative
* is desired. When these conditions are met, this class should be much faster than
* {@link DerivativeStructure} and use less memory.
*
*
* @since 3.3
*/
public class SparseGradient implements RealFieldElement, Serializable {
/** Serializable UID. */
private static final long serialVersionUID = 20131025L;
/** Value of the calculation. */
private double value;
/** Stored derivative, each key representing a different independent variable. */
private final Map derivatives;
/** Internal constructor.
* @param value value of the function
* @param derivatives derivatives map, a deep copy will be performed,
* so the map given here will remain safe from changes in the new instance,
* may be null to create an empty derivatives map, i.e. a constant value
*/
private SparseGradient(final double value, final Map derivatives) {
this.value = value;
this.derivatives = new HashMap();
if (derivatives != null) {
this.derivatives.putAll(derivatives);
}
}
/** Internal constructor.
* @param value value of the function
* @param scale scaling factor to apply to all derivatives
* @param derivatives derivatives map, a deep copy will be performed,
* so the map given here will remain safe from changes in the new instance,
* may be null to create an empty derivatives map, i.e. a constant value
*/
private SparseGradient(final double value, final double scale,
final Map derivatives) {
this.value = value;
this.derivatives = new HashMap();
if (derivatives != null) {
for (final Map.Entry entry : derivatives.entrySet()) {
this.derivatives.put(entry.getKey(), scale * entry.getValue());
}
}
}
/** Factory method creating a constant.
* @param value value of the constant
* @return a new instance
*/
public static SparseGradient createConstant(final double value) {
return new SparseGradient(value, Collections. emptyMap());
}
/** Factory method creating an independent variable.
* @param idx index of the variable
* @param value value of the variable
* @return a new instance
*/
public static SparseGradient createVariable(final int idx, final double value) {
return new SparseGradient(value, Collections.singletonMap(idx, 1.0));
}
/**
* Find the number of variables.
* @return number of variables
*/
public int numVars() {
return derivatives.size();
}
/**
* Get the derivative with respect to a particular index variable.
*
* @param index index to differentiate with.
* @return derivative with respect to a particular index variable
*/
public double getDerivative(final int index) {
final Double out = derivatives.get(index);
return (out == null) ? 0.0 : out;
}
/**
* Get the value of the function.
* @return value of the function.
*/
public double getValue() {
return value;
}
/** {@inheritDoc} */
public double getReal() {
return value;
}
/** {@inheritDoc} */
public SparseGradient add(final SparseGradient a) {
final SparseGradient out = new SparseGradient(value + a.value, derivatives);
for (Map.Entry entry : a.derivatives.entrySet()) {
final int id = entry.getKey();
final Double old = out.derivatives.get(id);
if (old == null) {
out.derivatives.put(id, entry.getValue());
} else {
out.derivatives.put(id, old + entry.getValue());
}
}
return out;
}
/**
* Add in place.
*
* This method is designed to be faster when used multiple times in a loop.
*
*
* The instance is changed here, in order to not change the
* instance the {@link #add(SparseGradient)} method should
* be used.
*
* @param a instance to add
*/
public void addInPlace(final SparseGradient a) {
value += a.value;
for (final Map.Entry entry : a.derivatives.entrySet()) {
final int id = entry.getKey();
final Double old = derivatives.get(id);
if (old == null) {
derivatives.put(id, entry.getValue());
} else {
derivatives.put(id, old + entry.getValue());
}
}
}
/** {@inheritDoc} */
public SparseGradient add(final double c) {
final SparseGradient out = new SparseGradient(value + c, derivatives);
return out;
}
/** {@inheritDoc} */
public SparseGradient subtract(final SparseGradient a) {
final SparseGradient out = new SparseGradient(value - a.value, derivatives);
for (Map.Entry entry : a.derivatives.entrySet()) {
final int id = entry.getKey();
final Double old = out.derivatives.get(id);
if (old == null) {
out.derivatives.put(id, -entry.getValue());
} else {
out.derivatives.put(id, old - entry.getValue());
}
}
return out;
}
/** {@inheritDoc} */
public SparseGradient subtract(double c) {
return new SparseGradient(value - c, derivatives);
}
/** {@inheritDoc} */
public SparseGradient multiply(final SparseGradient a) {
final SparseGradient out =
new SparseGradient(value * a.value, Collections. emptyMap());
// Derivatives.
for (Map.Entry entry : derivatives.entrySet()) {
out.derivatives.put(entry.getKey(), a.value * entry.getValue());
}
for (Map.Entry entry : a.derivatives.entrySet()) {
final int id = entry.getKey();
final Double old = out.derivatives.get(id);
if (old == null) {
out.derivatives.put(id, value * entry.getValue());
} else {
out.derivatives.put(id, old + value * entry.getValue());
}
}
return out;
}
/**
* Multiply in place.
*
* This method is designed to be faster when used multiple times in a loop.
*
*
* The instance is changed here, in order to not change the
* instance the {@link #add(SparseGradient)} method should
* be used.
*
* @param a instance to multiply
*/
public void multiplyInPlace(final SparseGradient a) {
// Derivatives.
for (Map.Entry entry : derivatives.entrySet()) {
derivatives.put(entry.getKey(), a.value * entry.getValue());
}
for (Map.Entry entry : a.derivatives.entrySet()) {
final int id = entry.getKey();
final Double old = derivatives.get(id);
if (old == null) {
derivatives.put(id, value * entry.getValue());
} else {
derivatives.put(id, old + value * entry.getValue());
}
}
value *= a.value;
}
/** {@inheritDoc} */
public SparseGradient multiply(final double c) {
return new SparseGradient(value * c, c, derivatives);
}
/** {@inheritDoc} */
public SparseGradient multiply(final int n) {
return new SparseGradient(value * n, n, derivatives);
}
/** {@inheritDoc} */
public SparseGradient divide(final SparseGradient a) {
final SparseGradient out = new SparseGradient(value / a.value, Collections. emptyMap());
// Derivatives.
for (Map.Entry entry : derivatives.entrySet()) {
out.derivatives.put(entry.getKey(), entry.getValue() / a.value);
}
for (Map.Entry entry : a.derivatives.entrySet()) {
final int id = entry.getKey();
final Double old = out.derivatives.get(id);
if (old == null) {
out.derivatives.put(id, -out.value / a.value * entry.getValue());
} else {
out.derivatives.put(id, old - out.value / a.value * entry.getValue());
}
}
return out;
}
/** {@inheritDoc} */
public SparseGradient divide(final double c) {
return new SparseGradient(value / c, 1.0 / c, derivatives);
}
/** {@inheritDoc} */
public SparseGradient negate() {
return new SparseGradient(-value, -1.0, derivatives);
}
/** {@inheritDoc} */
public Field getField() {
return new Field() {
/** {@inheritDoc} */
public SparseGradient getZero() {
return createConstant(0);
}
/** {@inheritDoc} */
public SparseGradient getOne() {
return createConstant(1);
}
/** {@inheritDoc} */
public Class> getRuntimeClass() {
return SparseGradient.class;
}
};
}
/** {@inheritDoc} */
public SparseGradient remainder(final double a) {
return new SparseGradient(FastMath.IEEEremainder(value, a), derivatives);
}
/** {@inheritDoc} */
public SparseGradient remainder(final SparseGradient a) {
// compute k such that lhs % rhs = lhs - k rhs
final double rem = FastMath.IEEEremainder(value, a.value);
final double k = FastMath.rint((value - rem) / a.value);
return subtract(a.multiply(k));
}
/** {@inheritDoc} */
public SparseGradient abs() {
if (Double.doubleToLongBits(value) < 0) {
// we use the bits representation to also handle -0.0
return negate();
} else {
return this;
}
}
/** {@inheritDoc} */
public SparseGradient ceil() {
return createConstant(FastMath.ceil(value));
}
/** {@inheritDoc} */
public SparseGradient floor() {
return createConstant(FastMath.floor(value));
}
/** {@inheritDoc} */
public SparseGradient rint() {
return createConstant(FastMath.rint(value));
}
/** {@inheritDoc} */
public long round() {
return FastMath.round(value);
}
/** {@inheritDoc} */
public SparseGradient signum() {
return createConstant(FastMath.signum(value));
}
/** {@inheritDoc} */
public SparseGradient copySign(final SparseGradient sign) {
final long m = Double.doubleToLongBits(value);
final long s = Double.doubleToLongBits(sign.value);
if ((m >= 0 && s >= 0) || (m < 0 && s < 0)) { // Sign is currently OK
return this;
}
return negate(); // flip sign
}
/** {@inheritDoc} */
public SparseGradient copySign(final double sign) {
final long m = Double.doubleToLongBits(value);
final long s = Double.doubleToLongBits(sign);
if ((m >= 0 && s >= 0) || (m < 0 && s < 0)) { // Sign is currently OK
return this;
}
return negate(); // flip sign
}
/** {@inheritDoc} */
public SparseGradient scalb(final int n) {
final SparseGradient out = new SparseGradient(FastMath.scalb(value, n), Collections. emptyMap());
for (Map.Entry entry : derivatives.entrySet()) {
out.derivatives.put(entry.getKey(), FastMath.scalb(entry.getValue(), n));
}
return out;
}
/** {@inheritDoc} */
public SparseGradient hypot(final SparseGradient y) {
if (Double.isInfinite(value) || Double.isInfinite(y.value)) {
return createConstant(Double.POSITIVE_INFINITY);
} else if (Double.isNaN(value) || Double.isNaN(y.value)) {
return createConstant(Double.NaN);
} else {
final int expX = FastMath.getExponent(value);
final int expY = FastMath.getExponent(y.value);
if (expX > expY + 27) {
// y is negligible with respect to x
return abs();
} else if (expY > expX + 27) {
// x is negligible with respect to y
return y.abs();
} else {
// find an intermediate scale to avoid both overflow and underflow
final int middleExp = (expX + expY) / 2;
// scale parameters without losing precision
final SparseGradient scaledX = scalb(-middleExp);
final SparseGradient scaledY = y.scalb(-middleExp);
// compute scaled hypotenuse
final SparseGradient scaledH =
scaledX.multiply(scaledX).add(scaledY.multiply(scaledY)).sqrt();
// remove scaling
return scaledH.scalb(middleExp);
}
}
}
/**
* Returns the hypotenuse of a triangle with sides {@code x} and {@code y}
* - sqrt(x2 +y2)
* avoiding intermediate overflow or underflow.
*
*
* - If either argument is infinite, then the result is positive infinity.
* - else, if either argument is NaN then the result is NaN.
*
*
* @param x a value
* @param y a value
* @return sqrt(x2 +y2)
*/
public static SparseGradient hypot(final SparseGradient x, final SparseGradient y) {
return x.hypot(y);
}
/** {@inheritDoc} */
public SparseGradient reciprocal() {
return new SparseGradient(1.0 / value, -1.0 / (value * value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient sqrt() {
final double sqrt = FastMath.sqrt(value);
return new SparseGradient(sqrt, 0.5 / sqrt, derivatives);
}
/** {@inheritDoc} */
public SparseGradient cbrt() {
final double cbrt = FastMath.cbrt(value);
return new SparseGradient(cbrt, 1.0 / (3 * cbrt * cbrt), derivatives);
}
/** {@inheritDoc} */
public SparseGradient rootN(final int n) {
if (n == 2) {
return sqrt();
} else if (n == 3) {
return cbrt();
} else {
final double root = FastMath.pow(value, 1.0 / n);
return new SparseGradient(root, 1.0 / (n * FastMath.pow(root, n - 1)), derivatives);
}
}
/** {@inheritDoc} */
public SparseGradient pow(final double p) {
return new SparseGradient(FastMath.pow(value, p), p * FastMath.pow(value, p - 1), derivatives);
}
/** {@inheritDoc} */
public SparseGradient pow(final int n) {
if (n == 0) {
return getField().getOne();
} else {
final double valueNm1 = FastMath.pow(value, n - 1);
return new SparseGradient(value * valueNm1, n * valueNm1, derivatives);
}
}
/** {@inheritDoc} */
public SparseGradient pow(final SparseGradient e) {
return log().multiply(e).exp();
}
/** Compute ax where a is a double and x a {@link SparseGradient}
* @param a number to exponentiate
* @param x power to apply
* @return ax
*/
public static SparseGradient pow(final double a, final SparseGradient x) {
if (a == 0) {
if (x.value == 0) {
return x.compose(1.0, Double.NEGATIVE_INFINITY);
} else if (x.value < 0) {
return x.compose(Double.NaN, Double.NaN);
} else {
return x.getField().getZero();
}
} else {
final double ax = FastMath.pow(a, x.value);
return new SparseGradient(ax, ax * FastMath.log(a), x.derivatives);
}
}
/** {@inheritDoc} */
public SparseGradient exp() {
final double e = FastMath.exp(value);
return new SparseGradient(e, e, derivatives);
}
/** {@inheritDoc} */
public SparseGradient expm1() {
return new SparseGradient(FastMath.expm1(value), FastMath.exp(value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient log() {
return new SparseGradient(FastMath.log(value), 1.0 / value, derivatives);
}
/** Base 10 logarithm.
* @return base 10 logarithm of the instance
*/
public SparseGradient log10() {
return new SparseGradient(FastMath.log10(value), 1.0 / (FastMath.log(10.0) * value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient log1p() {
return new SparseGradient(FastMath.log1p(value), 1.0 / (1.0 + value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient cos() {
return new SparseGradient(FastMath.cos(value), -FastMath.sin(value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient sin() {
return new SparseGradient(FastMath.sin(value), FastMath.cos(value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient tan() {
final double t = FastMath.tan(value);
return new SparseGradient(t, 1 + t * t, derivatives);
}
/** {@inheritDoc} */
public SparseGradient acos() {
return new SparseGradient(FastMath.acos(value), -1.0 / FastMath.sqrt(1 - value * value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient asin() {
return new SparseGradient(FastMath.asin(value), 1.0 / FastMath.sqrt(1 - value * value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient atan() {
return new SparseGradient(FastMath.atan(value), 1.0 / (1 + value * value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient atan2(final SparseGradient x) {
// compute r = sqrt(x^2+y^2)
final SparseGradient r = multiply(this).add(x.multiply(x)).sqrt();
final SparseGradient a;
if (x.value >= 0) {
// compute atan2(y, x) = 2 atan(y / (r + x))
a = divide(r.add(x)).atan().multiply(2);
} else {
// compute atan2(y, x) = +/- pi - 2 atan(y / (r - x))
final SparseGradient tmp = divide(r.subtract(x)).atan().multiply(-2);
a = tmp.add(tmp.value <= 0 ? -FastMath.PI : FastMath.PI);
}
// fix value to take special cases (+0/+0, +0/-0, -0/+0, -0/-0, +/-infinity) correctly
a.value = FastMath.atan2(value, x.value);
return a;
}
/** Two arguments arc tangent operation.
* @param y first argument of the arc tangent
* @param x second argument of the arc tangent
* @return atan2(y, x)
*/
public static SparseGradient atan2(final SparseGradient y, final SparseGradient x) {
return y.atan2(x);
}
/** {@inheritDoc} */
public SparseGradient cosh() {
return new SparseGradient(FastMath.cosh(value), FastMath.sinh(value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient sinh() {
return new SparseGradient(FastMath.sinh(value), FastMath.cosh(value), derivatives);
}
/** {@inheritDoc} */
public SparseGradient tanh() {
final double t = FastMath.tanh(value);
return new SparseGradient(t, 1 - t * t, derivatives);
}
/** {@inheritDoc} */
public SparseGradient acosh() {
return new SparseGradient(FastMath.acosh(value), 1.0 / FastMath.sqrt(value * value - 1.0), derivatives);
}
/** {@inheritDoc} */
public SparseGradient asinh() {
return new SparseGradient(FastMath.asinh(value), 1.0 / FastMath.sqrt(value * value + 1.0), derivatives);
}
/** {@inheritDoc} */
public SparseGradient atanh() {
return new SparseGradient(FastMath.atanh(value), 1.0 / (1.0 - value * value), derivatives);
}
/** Convert radians to degrees, with error of less than 0.5 ULP
* @return instance converted into degrees
*/
public SparseGradient toDegrees() {
return new SparseGradient(FastMath.toDegrees(value), FastMath.toDegrees(1.0), derivatives);
}
/** Convert degrees to radians, with error of less than 0.5 ULP
* @return instance converted into radians
*/
public SparseGradient toRadians() {
return new SparseGradient(FastMath.toRadians(value), FastMath.toRadians(1.0), derivatives);
}
/** Evaluate Taylor expansion of a sparse gradient.
* @param delta parameters offsets (Δx, Δy, ...)
* @return value of the Taylor expansion at x + Δx, y + Δy, ...
*/
public double taylor(final double ... delta) {
double y = value;
for (int i = 0; i < delta.length; ++i) {
y += delta[i] * getDerivative(i);
}
return y;
}
/** Compute composition of the instance by a univariate function.
* @param f0 value of the function at (i.e. f({@link #getValue()}))
* @param f1 first derivative of the function at
* the current point (i.e. f'({@link #getValue()}))
* @return f(this)
*/
public SparseGradient compose(final double f0, final double f1) {
return new SparseGradient(f0, f1, derivatives);
}
/** {@inheritDoc} */
public SparseGradient linearCombination(final SparseGradient[] a,
final SparseGradient[] b)
throws DimensionMismatchException {
// compute a simple value, with all partial derivatives
SparseGradient out = a[0].getField().getZero();
for (int i = 0; i < a.length; ++i) {
out = out.add(a[i].multiply(b[i]));
}
// recompute an accurate value, taking care of cancellations
final double[] aDouble = new double[a.length];
for (int i = 0; i < a.length; ++i) {
aDouble[i] = a[i].getValue();
}
final double[] bDouble = new double[b.length];
for (int i = 0; i < b.length; ++i) {
bDouble[i] = b[i].getValue();
}
out.value = MathArrays.linearCombination(aDouble, bDouble);
return out;
}
/** {@inheritDoc} */
public SparseGradient linearCombination(final double[] a, final SparseGradient[] b) {
// compute a simple value, with all partial derivatives
SparseGradient out = b[0].getField().getZero();
for (int i = 0; i < a.length; ++i) {
out = out.add(b[i].multiply(a[i]));
}
// recompute an accurate value, taking care of cancellations
final double[] bDouble = new double[b.length];
for (int i = 0; i < b.length; ++i) {
bDouble[i] = b[i].getValue();
}
out.value = MathArrays.linearCombination(a, bDouble);
return out;
}
/** {@inheritDoc} */
public SparseGradient linearCombination(final SparseGradient a1, final SparseGradient b1,
final SparseGradient a2, final SparseGradient b2) {
// compute a simple value, with all partial derivatives
SparseGradient out = a1.multiply(b1).add(a2.multiply(b2));
// recompute an accurate value, taking care of cancellations
out.value = MathArrays.linearCombination(a1.value, b1.value, a2.value, b2.value);
return out;
}
/** {@inheritDoc} */
public SparseGradient linearCombination(final double a1, final SparseGradient b1,
final double a2, final SparseGradient b2) {
// compute a simple value, with all partial derivatives
SparseGradient out = b1.multiply(a1).add(b2.multiply(a2));
// recompute an accurate value, taking care of cancellations
out.value = MathArrays.linearCombination(a1, b1.value, a2, b2.value);
return out;
}
/** {@inheritDoc} */
public SparseGradient linearCombination(final SparseGradient a1, final SparseGradient b1,
final SparseGradient a2, final SparseGradient b2,
final SparseGradient a3, final SparseGradient b3) {
// compute a simple value, with all partial derivatives
SparseGradient out = a1.multiply(b1).add(a2.multiply(b2)).add(a3.multiply(b3));
// recompute an accurate value, taking care of cancellations
out.value = MathArrays.linearCombination(a1.value, b1.value,
a2.value, b2.value,
a3.value, b3.value);
return out;
}
/** {@inheritDoc} */
public SparseGradient linearCombination(final double a1, final SparseGradient b1,
final double a2, final SparseGradient b2,
final double a3, final SparseGradient b3) {
// compute a simple value, with all partial derivatives
SparseGradient out = b1.multiply(a1).add(b2.multiply(a2)).add(b3.multiply(a3));
// recompute an accurate value, taking care of cancellations
out.value = MathArrays.linearCombination(a1, b1.value,
a2, b2.value,
a3, b3.value);
return out;
}
/** {@inheritDoc} */
public SparseGradient linearCombination(final SparseGradient a1, final SparseGradient b1,
final SparseGradient a2, final SparseGradient b2,
final SparseGradient a3, final SparseGradient b3,
final SparseGradient a4, final SparseGradient b4) {
// compute a simple value, with all partial derivatives
SparseGradient out = a1.multiply(b1).add(a2.multiply(b2)).add(a3.multiply(b3)).add(a4.multiply(b4));
// recompute an accurate value, taking care of cancellations
out.value = MathArrays.linearCombination(a1.value, b1.value,
a2.value, b2.value,
a3.value, b3.value,
a4.value, b4.value);
return out;
}
/** {@inheritDoc} */
public SparseGradient linearCombination(final double a1, final SparseGradient b1,
final double a2, final SparseGradient b2,
final double a3, final SparseGradient b3,
final double a4, final SparseGradient b4) {
// compute a simple value, with all partial derivatives
SparseGradient out = b1.multiply(a1).add(b2.multiply(a2)).add(b3.multiply(a3)).add(b4.multiply(a4));
// recompute an accurate value, taking care of cancellations
out.value = MathArrays.linearCombination(a1, b1.value,
a2, b2.value,
a3, b3.value,
a4, b4.value);
return out;
}
/**
* Test for the equality of two sparse gradients.
*
* Sparse gradients are considered equal if they have the same value
* and the same derivatives.
*
* @param other Object to test for equality to this
* @return true if two sparse gradients are equal
*/
@Override
public boolean equals(Object other) {
if (this == other) {
return true;
}
if (other instanceof SparseGradient) {
final SparseGradient rhs = (SparseGradient)other;
if (!Precision.equals(value, rhs.value, 1)) {
return false;
}
if (derivatives.size() != rhs.derivatives.size()) {
return false;
}
for (final Map.Entry entry : derivatives.entrySet()) {
if (!rhs.derivatives.containsKey(entry.getKey())) {
return false;
}
if (!Precision.equals(entry.getValue(), rhs.derivatives.get(entry.getKey()), 1)) {
return false;
}
}
return true;
}
return false;
}
/**
* Get a hashCode for the derivative structure.
* @return a hash code value for this object
* @since 3.2
*/
@Override
public int hashCode() {
return 743 + 809 * MathUtils.hash(value) + 167 * derivatives.hashCode();
}
}