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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)
* ```
*/
'use strict';
// MODULES //
var sign = require( '@stdlib/math/base/special/signum' );
var abs = require( '@stdlib/math/base/special/abs' );
var ldexp = require( '@stdlib/math/base/special/ldexp' );
var MAX_VALUE = require( '@stdlib/constants/float64/max' );
// MAIN //
/**
* Performs root finding via second order Newton-Raphson iteration.
*
* @private
* @param {Function} fun - function which returns the function value and its first derivative
* @param {number} guess - initial starting value
* @param {number} min - minimum possible value for the result, used as initial lower bracket
* @param {number} max - 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 newtonRaphsonIterate( fun, guess, min, max, digits, maxIter ) {
var maxRangeF;
var minRangeF;
var f0last;
var delta1;
var delta2;
var factor;
var result;
var count;
var delta;
var shift;
var res;
var f0;
var f1;
f0 = 0.0;
f0last = 0.0;
result = guess;
factor = ldexp( 1.0, 1.0 - digits );
delta = MAX_VALUE;
delta1 = MAX_VALUE;
delta2 = MAX_VALUE;
// 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 ];
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 === min ) {
guess = max;
} else {
guess = min;
}
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 - min) / 2.0;
} else {
delta = (result - max) / 2.0;
}
} else if ( delta < 0 ) {
delta = (result - max) / 2.0;
} else {
delta = (result - min) / 2.0;
}
} else {
delta = f0 / f1;
}
if ( abs( delta * 2.0 ) > abs( delta2 ) ) {
// Last two steps haven't converged:
shift = ( delta > 0.0 ) ? (result-min) / 2.0 : (result-max) / 2.0;
if ( result !== 0.0 && abs( shift ) > abs( result ) ) {
delta = sign( delta ) * abs( result ); // Protect against huge jumps!
} else {
delta = shift;
}
// Reset delta1/delta2 so that this branch will *not* be taken on the next iteration:
delta1 = 3.0 * delta;
delta2 = 3.0 * delta;
}
guess = result;
result -= delta;
if ( result <= min ) {
delta = 0.5 * (guess - min);
result = guess - delta;
if ( result === min || result === max ) {
break;
}
} else if ( result >= max ) {
delta = 0.5 * (guess - max);
result = guess - delta;
if ( result === min || result === max ) {
break;
}
}
// Update brackets:
if ( delta > 0.0 ) {
max = guess;
maxRangeF = f0;
} else {
min = 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 = newtonRaphsonIterate;
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