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* @license Apache-2.0
*
* Copyright (c) 2018 The Stdlib Authors.
*
* 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.
*
*
* ## Notice
*
* The original C++ code and copyright notice are from the [Boost library]{@link http://www.boost.org/doc/libs/1_92_0/boost/math/tools/roots.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* Copyright John Maddock 2006.
* Copyright Matt Borland 2024.
*
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE or copy at http://www.boost.org/LICENSE_1_0.txt)
* ```
*/
/* eslint-disable max-lines-per-function, max-statements, max-lines */
'use strict';
// MODULES //
var ulpDifference = require( '@stdlib/number/float64/base/ulp-difference' );
var frexp = require( '@stdlib/math/base/special/frexp' );
var ldexp = require( '@stdlib/math/base/special/ldexp' );
var floor = require( '@stdlib/math/base/special/floor' );
var sign = require( '@stdlib/math/base/special/signum' );
var abs = require( '@stdlib/math/base/special/abs' );
var max = require( '@stdlib/math/base/special/max' );
var MAX_VALUE = require( '@stdlib/constants/float64/max' );
var floatPrior = require( './float_prior.js' );
var floatNext = require( './float_next.js' );
// VARIABLES //
// Shared bracket state (`[ min, max, count ]`) used by the bracketing helpers:
var STATE = [ 0.0, 0.0, 0 ]; // WARNING: not thread safe
// FUNCTIONS //
/**
* Moves the current best guess toward the lower bracket until the root is bracketed, updating the brackets along the way.
*
* @private
* @param {Function} fun - function which returns the function value and its first two derivatives
* @param {number} x - current best guess
* @param {number} fx - function value at the position from which bracketing begins
* @param {NumericArray} state - bracket state (`[ min, max, count ]`), updated in place
* @returns {number} step which should be subtracted from the current best guess
*/
function bracketRootTowardsMin( fun, x, fx, state ) {
var multiplier;
var guess0;
var xc;
var fc;
var fe;
var e;
if ( state[ 2 ] < 2 ) {
// Not enough iterations left to do anything!
return x - ( ( state[ 1 ] + state[ 0 ] ) / 2.0 );
}
// Move `x` towards the lower bracket until we bracket the root, updating the brackets as we go:
fe = frexp( x / state[ 0 ] );
e = abs( fe[ 1 ] );
guess0 = x;
xc = x;
multiplier = ( e < 64 ) ? 2.0 : ldexp( 1.0, floor( e / 32 ) );
fc = fx;
state[ 2 ] -= 1;
if ( abs( state[ 0 ] ) < abs( state[ 1 ] ) ) {
while ( state[ 2 ] !== 0 && ( ( fc < 0.0 ) === ( fx < 0.0 ) ) ) {
state[ 1 ] = xc;
xc /= multiplier;
if ( xc < state[ 0 ] ) {
xc = state[ 0 ];
fc = -fc; // there must be a change of sign!
break;
}
multiplier *= ( e > 1024 ) ? 8.0 : 2.0;
fc = fun( xc )[ 0 ];
state[ 2 ] -= 1;
}
} else {
// If the brackets are negative we have to multiply to head towards the lower bracket:
while ( state[ 2 ] !== 0 && ( ( fc < 0.0 ) === ( fx < 0.0 ) ) ) {
state[ 1 ] = xc;
xc *= multiplier;
if ( xc < state[ 0 ] ) {
xc = state[ 0 ];
fc = -fc; // there must be a change of sign!
break;
}
multiplier *= ( e > 1024 ) ? 8.0 : 2.0;
fc = fun( xc )[ 0 ];
state[ 2 ] -= 1;
}
}
if ( state[ 2 ] !== 0 ) {
state[ 0 ] = xc;
if ( multiplier > 16.0 ) {
return ( guess0-xc ) + bracketRootTowardsMax( fun, xc, fc, state );
}
}
return guess0 - ( ( state[ 1 ] + state[ 0 ] ) / 2.0 );
}
/**
* Moves the current best guess toward the upper bracket until the root is bracketed, updating the brackets along the way.
*
* @private
* @param {Function} fun - function which returns the function value and its first two derivatives
* @param {number} x - current best guess
* @param {number} fx - function value at the position from which bracketing begins
* @param {NumericArray} state - bracket state (`[ min, max, count ]`), updated in place
* @returns {number} step which should be subtracted from the current best guess
*/
function bracketRootTowardsMax( fun, x, fx, state ) {
var multiplier;
var guess0;
var xc;
var fc;
var fe;
var e;
if ( state[ 2 ] < 2 ) {
// Not enough iterations left to do anything!
return x - ( ( state[ 1 ] + state[ 0 ] ) / 2.0 );
}
// Move `x` towards the upper bracket until we bracket the root, updating the brackets as we go:
fe = frexp( state[ 1 ] / x );
e = abs( fe[ 1 ] );
guess0 = x;
xc = x;
multiplier = ( e < 64 ) ? 2.0 : ldexp( 1.0, floor( e / 32 ) );
fc = fx;
state[ 2 ] -= 1;
if ( abs( state[ 0 ] ) < abs( state[ 1 ] ) ) {
while ( state[ 2 ] !== 0 && ( ( fc < 0.0 ) === ( fx < 0.0 ) ) ) {
state[ 0 ] = xc;
xc *= multiplier;
if ( xc > state[ 1 ] ) {
xc = state[ 1 ];
fc = -fc; // there must be a change of sign!
break;
}
multiplier *= ( e > 1024 ) ? 8.0 : 2.0;
fc = fun( xc )[ 0 ];
state[ 2 ] -= 1;
}
} else {
// If the brackets are negative we have to divide to head towards the upper bracket:
while ( state[ 2 ] !== 0 && ( ( fc < 0.0 ) === ( fx < 0.0 ) ) ) {
state[ 0 ] = xc;
xc /= multiplier;
if ( xc > state[ 1 ] ) {
xc = state[ 1 ];
fc = -fc; // there must be a change of sign!
break;
}
multiplier *= ( e > 1024 ) ? 8.0 : 2.0;
fc = fun( xc )[ 0 ];
state[ 2 ] -= 1;
}
}
if ( state[ 2 ] !== 0 ) {
state[ 1 ] = xc;
if ( multiplier > 16.0 ) {
return ( guess0-xc ) + bracketRootTowardsMin( fun, xc, fc, state );
}
}
return guess0 - ( ( state[ 1 ] + state[ 0 ] ) / 2.0 );
}
// MAIN //
/**
* Performs root finding via third order Halley iteration.
*
* @private
* @param {Function} fun - function which returns the function value and its first two derivatives
* @param {number} guess - initial starting value
* @param {number} minimum - minimum possible value for the result, used as initial lower bracket
* @param {number} maximum - maximum possible value for the result, used as initial upper bracket
* @param {PositiveInteger} digits - desired number of binary digits
* @param {PositiveInteger} maxIter - maximum number of iterations
* @returns {number} function value
*/
function halleyIterate( fun, guess, minimum, maximum, digits, maxIter ) {
var convergence;
var outOfBounds;
var maxRangeF;
var minRangeF;
var bigRange;
var delta1;
var delta2;
var factor;
var result;
var f0Last;
var count;
var delta;
var denom;
var diff;
var num;
var res;
var f0;
var f1;
var f2;
f0 = 0.0;
outOfBounds = false;
result = guess;
factor = ldexp( 1.0, 1.0-digits );
delta = max( 10000000*guess, 10000000 ); // Arbitrarily large delta...
f0Last = 0;
delta1 = delta;
delta2 = delta;
// We use these to sanity check that we do actually bracket a root; if we update both ends of the brackets and the function value has the same sign at both ends, then there is no root to be found:
maxRangeF = 0.0;
minRangeF = 0.0;
count = maxIter;
do {
f0Last = f0;
delta2 = delta1;
delta1 = delta;
res = fun( result );
f0 = res[ 0 ];
f1 = res[ 1 ];
f2 = res[ 2 ];
count -= 1;
if ( f0 === 0.0 ) {
break;
}
if ( f1 === 0.0 ) {
// Oops zero derivative!!!
if ( f0Last === 0.0 ) {
// Must be the first iteration, pretend that we had a previous one at either min or max:
if ( result === minimum ) {
guess = maximum;
} else {
guess = minimum;
}
f0Last = fun( guess )[ 0 ];
delta = guess - result;
}
if ( sign( f0Last ) * sign( f0 ) < 0 ) {
// We've crossed over so move in opposite direction to last step:
if ( delta < 0 ) {
delta = ( result-minimum ) / 2.0;
} else {
delta = ( result-maximum ) / 2.0;
}
// Move in same direction as last step:
} else if ( delta < 0 ) {
delta = (result-maximum) / 2.0;
} else {
delta = (result-minimum) / 2.0;
}
} else if ( f2 === 0.0 ) {
delta = f0 / f1;
} else {
denom = 2.0 * f0;
num = ( 2.0 * f1 ) - ( f0 * ( f2 / f1 ) );
if ( abs( num ) < 1.0 && ( abs( denom ) >= abs( num ) * MAX_VALUE ) ) {
// Possible overflow, use Newton step:
delta = f0 / f1;
} else {
delta = denom / num;
}
if ( delta * f1 / f0 < 0.0 ) {
// Probably cancellation error, try a Newton step instead:
delta = f0 / f1;
if ( abs( delta ) > 2.0 * abs( result ) ) {
delta = ( ( delta < 0.0 ) ? -2.0 : 2.0 ) * abs( result );
}
}
}
// We need to avoid delta/delta2 overflowing here:
if ( abs( delta2 ) > 1.0 || abs( MAX_VALUE*delta2 ) > abs( delta ) ) {
convergence = abs( delta / delta2 );
} else {
convergence = MAX_VALUE;
}
if ( convergence > 0.8 && convergence < 2.0 ) {
// Last two steps haven't converged...
if ( abs( minimum ) < 1.0 ) {
bigRange = ( abs( 1000.0*minimum ) < abs( maximum ) );
} else {
bigRange = ( abs( maximum/minimum ) > 1000.0 );
}
if ( bigRange ) {
STATE[ 0 ] = minimum;
STATE[ 1 ] = maximum;
STATE[ 2 ] = count;
if ( delta > 0.0 ) {
delta = bracketRootTowardsMin( fun, result, f0, STATE );
} else {
delta = bracketRootTowardsMax( fun, result, f0, STATE );
}
minimum = STATE[ 0 ];
maximum = STATE[ 1 ];
count = STATE[ 2 ];
} else {
delta = ( delta > 0.0 ) ? ( result-minimum )/2.0 : ( result-maximum )/2.0; // eslint-disable-line max-len
if ( result !== 0.0 && abs( delta ) > result ) {
delta = sign( delta ) * abs( result ) * 0.9; // Protect against huge jumps!
}
}
// Reset delta1/delta2 so that this branch will *not* be taken on the next iteration:
delta2 = delta * 3.0;
delta1 = delta * 3.0;
}
guess = result;
result -= delta;
// Check for out of bounds step:
if ( result < minimum ) {
if (
abs( minimum ) < 1.0 &&
abs( result ) > 1.0 &&
( MAX_VALUE / abs( result ) < abs( minimum ) )
) {
diff = 1000.0;
} else if (
abs( minimum ) < 1.0 &&
( abs( MAX_VALUE*minimum ) < abs( result ) )
) {
diff = ( ( minimum < 0.0 ) === ( result < 0.0 ) ) ? MAX_VALUE : -MAX_VALUE; // eslint-disable-line max-len
} else {
diff = result / minimum;
}
if ( abs( diff ) < 1.0 ) {
diff = 1.0 / diff;
}
if ( !outOfBounds && diff > 0.0 && diff < 3.0 ) {
// Only a small out of bounds step, let's assume that the result is probably approximately at minimum:
delta = 0.99 * (guess - minimum);
result = guess - delta;
outOfBounds = true; // Only take this branch once!
} else {
if ( ulpDifference( minimum, maximum ) < 2 ) {
result = ( minimum + maximum ) / 2.0;
guess = result;
break;
}
STATE[ 0 ] = minimum;
STATE[ 1 ] = maximum;
STATE[ 2 ] = count;
delta = bracketRootTowardsMin( fun, guess, f0, STATE );
minimum = STATE[ 0 ];
maximum = STATE[ 1 ];
count = STATE[ 2 ];
result = guess - delta;
if ( result <= minimum ) {
result = floatNext( minimum );
}
if ( result >= maximum ) {
result = floatPrior( maximum );
}
guess = minimum;
continue;
}
} else if ( result > maximum ) {
if (
abs( maximum ) < 1.0 &&
abs( result ) > 1.0 &&
MAX_VALUE / abs( result ) < abs( maximum )
) {
diff = 1000.0;
} else {
diff = result / maximum;
}
if ( abs( diff ) < 1.0 ) {
diff = 1.0 / diff;
}
if ( !outOfBounds && diff > 0.0 && diff < 3.0 ) {
// Only a small out of bounds step, let's assume that the result is probably approximately at minimum:
delta = 0.99 * (guess - maximum);
result = guess - delta;
outOfBounds = true; // Only take this branch once!
} else {
if ( ulpDifference( minimum, maximum ) < 2.0 ) {
result = ( minimum + maximum ) / 2.0;
guess = result;
break;
}
STATE[ 0 ] = minimum;
STATE[ 1 ] = maximum;
STATE[ 2 ] = count;
delta = bracketRootTowardsMax( fun, guess, f0, STATE );
minimum = STATE[ 0 ];
maximum = STATE[ 1 ];
count = STATE[ 2 ];
result = guess - delta;
if ( result >= maximum ) {
result = floatPrior( maximum );
}
if ( result <= minimum ) {
result = floatNext( minimum );
}
guess = minimum;
continue;
}
}
// Update brackets:
if ( delta > 0.0 ) {
maximum = guess;
maxRangeF = f0;
} else {
minimum = guess;
minRangeF = f0;
}
// Sanity check that we bracket the root:
if ( maxRangeF * minRangeF > 0.0 ) {
// There appears to be no root to be found (the original C++ implementation raises an evaluation error; we instead return the current best guess):
return guess;
}
} while ( count && ( abs( result * factor ) < abs( delta ) ) );
return result;
}
// EXPORTS //
module.exports = halleyIterate;
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