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/*
* Copyright (c) 2012-2013, Peter Abeles. All Rights Reserved.
*
* This file is part of DDogleg (http://ddogleg.org).
*
* Licensed 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.ddogleg.optimization.impl;
/**
*
* Contains interpolation functions for use in line searches. These interpolation
* algorithms are designed to meet the condition below, without being too small,
*
*
*
* Sufficient decrease equation:
* f(α) ≤ f(0) + c1αg(0)
* where f is the function, g is its derivative, and α is the step length.
*
*
*
* See Chapter 3 in "Numerical Optimization 2nd Ed." by Jorge Nocedal and Stephen J. Wright, 2006
*
*
* @author Peter Abeles
*/
public class SearchInterpolate {
/**
*
* Quadratic interpolation using two function values and one derivative.
*
*
* @param f0 Value of f(x0)
* @param g0 Derivative f'(x0)
* @param x0 First sample point
* @param f1 Value of f(x1)
* @param x1 Second sample point
* @return Interpolated point
*/
public static double quadratic( double f0 , double g0 , double x0 , double f1 , double x1 ) {
return x0 + ((g0/((f0-f1)/(x1-x0)+g0))/2.0)*(x1-x0);
}
/**
*
* Quadratic interpolation using two derivatives.
*
*
* @param g0 Derivative f'(x0)
* @param x0 First sample point
* @param g1 Derivative f'(x1)
* @param x1 Second sample point
* @return Interpolated point
*/
public static double quadratic2( double g0 , double x0 , double g1 , double x1 ) {
return x0 + (g0/(g0-g1))*(x1-x0);
}
/**
* Interpolates the next step using a cubic model. Interpolation works by solving for 'a' and 'b' in
* the equation below. Designed to minimize the number of times the derivative
* needs to be computed. Care has been taken reduce overflow/underflow by normalizing.
*
* φ(α) = a*α3 + b*α2 + α3 + α*φ'(0) + &phi(0)
*
* @param f0 Function value at f(0)
* @param g0 Derivative value at g(0)
* @param f1 Function value at f(a1)
* @param alpha1 value of a1
* @param f2 Function value at f(a2)
* @param alpha2 value of a2
*
* @return Interpolated step length
*/
public static double cubic( double f0 , double g0 ,
double f1 , double alpha1 ,
double f2 , double alpha2 ) {
// Several different formulation were considered for solving this equation.
// See ExamineCubicInterpolateStability in benchmark directory
// Turns out that the straight forward implementation is just about the best.
double denominator = alpha1*alpha1*alpha2*alpha2*(alpha2-alpha1);
double a11 = alpha1*alpha1/denominator;
double a12 = -alpha2*alpha2/denominator;
double a21 = -alpha1*a11;
double a22 = -alpha2*a12;
double y1 = f2 - f0 - g0*alpha2;
double y2 = f1 - f0 - g0*alpha1;
double a = a11*y1 + a12*y2;
double b = a21*y1 + a22*y2;
return (-b+Math.sqrt(b*b-3*a*g0))/(3.0*a);
}
/**
*
* Cubic interpolation using the function and derivative computed at two different points. This particular
* implementation taken from [1] and appears to be designed to maximize stability.
*
*
* [1] MINPACK-2 source code http://ftp.mcs.anl.gov/pub/MINPACK-2/dcstep.f
*
*
* @param f0 Value of f(x0)
* @param g0 Derivative f'(x0)
* @param x0 First sample point
* @param f1 Value of f(x1)
* @param g1 Derivative g'(x1)
* @param x1 Second sample point
* @return Interpolated point
*/
public static double cubic2( double f0 , double g0 , double x0 ,
double f1 , double g1 , double x1 )
{
double theta = 3.0*(f0-f1)/(x1-x0) + g0 + g1;
double s = Math.max(Math.abs(theta),Math.abs(g0));
s= Math.max(s,Math.abs(g1));
double gamma = s*Math.sqrt((theta/s)*(theta/s) - (g0/s)*(g1/s));
if( x0 > x1 )
gamma = -gamma;
double p = (gamma-g0) + theta;
double q = ((gamma-g0)+gamma) + g1;
return x0 + (p/q)*(x1-x0);
}
/**
*
* Use cubic interpolation only if the cubic tends to infinity in the direction of the step or if the minim of the
* cubic is beyond x1. Otherwise the the step will be max if x0 > x1 else it will be min.
*
*
* [1] MINPACK-2 source code http://ftp.mcs.anl.gov/pub/MINPACK-2/dcstep.f
*
*
* @param f0 Value of f(x0)
* @param g0 Derivative f'(x0)
* @param x0 First sample point
* @param f1 Value of f(x1)
* @param g1 Derivative g'(x1)
* @param x1 Second sample point
* @return Interpolated point
*/
public static double cubicSafe( double f0 , double g0 , double x0 ,
double f1 , double g1 , double x1 ,
double min , double max ) {
double theta = 3.0*(f0-f1)/(x1-x0) + g0 + g1;
double s = Math.max(Math.abs(theta),Math.abs(g0));
s= Math.max(s,Math.abs(g1));
double gamma = s*Math.sqrt((theta/s)*(theta/s) - (g0/s)*(g1/s));
if( x1 < x0 )
gamma = -gamma;
double p = (gamma-g0) + theta;
double q = (gamma+(g1-g0))+gamma;
double r = p/q;
// gamma == 0 only rises if the cubic does not tend to infinity in the direction of the step
if( r < 0 && gamma != 0 ) {
return x0 + r*(x1-x0);
} else if( x0 > x1 ) {
return max;
} else {
return min;
}
}
}