fibonacci.php

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<?php
//
// +----------------------------------------------------------------------+
// | PHP Version 4                                                        |
// +----------------------------------------------------------------------+
// | Copyright (c) 1997-2002 The PHP Group                                |
// +----------------------------------------------------------------------+
// | This source file is subject to version 2.0 of the PHP license,       |
// | that is bundled with this package in the file LICENSE, and is        |
// | available at through the world-wide-web at                           |
// | http://www.php.net/license/2_02.txt.                                 |
// | If you did not receive a copy of the PHP license and are unable to   |
// | obtain it through the world-wide-web, please send a note to          |
// | [e-mail] so we can mail you a copy immediately.               |
// +----------------------------------------------------------------------+
// | Authors: Jesus M. Castagnetto <[e-mail]>                |
// +----------------------------------------------------------------------+
//
// $Id: Fibonacci.php,v 1.1 2003/01/02 01:56:59 jmcastagnetto Exp $
//

include_once 'PEAR.php';
include_once 'Math/IntegerOp.php';
include_once 'Math/Fibonacci/_fibonacciTable.php';

// PHI and phi are the roots of: x^2 - x - 1 = 0
// PHI is called the golden ratio. Also phi = -1/PHI = 1 - PHI
/**
 * MATH_PHI: The golden ratio = (1 + sqrt(5))/2
 */
define ('MATH_PHI', (1 + sqrt(5))/2);
/**
 * MATH_PHI_x100000: (int) (MATH_PHI * 100000)
 */
define ('MATH_PHI_x100000', intval(MATH_PHI * 100000));

/**
 * MATH_phi: The reciprocal of PHI = (1 - sqrt(5))/2
 */
define ('MATH_phi', (1 - sqrt(5))/2);
/**
 * MATH_phi_x100000: (int) (MATH_phi * 100000)
 */
define ('MATH_phi_x100000', intval(MATH_phi * 100000));

/**
 * MATH_SQRT5_x100000: (int) (sqrt(5) * 100000)
 */
define ('MATH_SQRT5_x100000', intval(sqrt(5) * 100000));

/**
 * MATH_LNSQRT5: ln(sqrt(5))
 */
define ('MATH_LNSQRT5', log(sqrt(5)));

/**
 * MATH_LNPHI: ln(PHI)
 */
define ('MATH_LNPHI', log(MATH_PHI));

/**
 * Math_Fibonacci: class to calculate, validate, and decompose into Fibonacci numbers
 *
 * Examples:
 * <pre>
 * include_once 'Math/Fibonacci.php';
 * 
 * $idx = 20;
 * echo "Calculate F($idx), fast equation = ";
 * $fib =& Math_Fibonacci::term($idx);
 * echo $fib->toString()."\n";
 * // Calculate F(20), fast equation = 6765
 * 
 * $idx = 55;
 * echo "Calculate F($idx), lookup table = ";
 * $fib =& Math_Fibonacci::term($idx);
 * echo $fib->toString()."\n";
 * // Calculate F(55), lookup table = 139583862445
 * 
 * $idx = 502;
 * echo "Calculate F($idx), addition loop = ";
 * $fib = Math_Fibonacci::term($idx);
 * echo $fib->toString()."\n";
 * // Calculate F(502), addition loop = 365014740723634211012237077906479355996081581501455497852747829366800199361550174096573645929019489792751
 * 
 * echo "\nSeries from F(0) to F(10):\n";
 * $series = Math_Fibonacci::series(10);
 * foreach ($series as $n=>$fib) {
 *     echo "n = $n, F(n) = ".$fib->toString()."\n";
 * }
 * // Series from F(0) to F(10):
 * // n = 0, F(n) = 0
 * // n = 1, F(n) = 1
 * // n = 2, F(n) = 1
 * // n = 3, F(n) = 2
 * // n = 4, F(n) = 3
 * // n = 5, F(n) = 5
 * // n = 6, F(n) = 8
 * // n = 7, F(n) = 13
 * // n = 8, F(n) = 21
 * // n = 9, F(n) = 34
 * // n = 10, F(n) = 55
 * 
 * echo "\nand now from F(11) to F(19):\n";
 * $series = Math_Fibonacci::series(11, 19);
 * foreach ($series as $n=>$fib) {
 *     echo "n = $n, F(n) = ".$fib->toString()."\n";
 * }
 * // and now from F(11) to F(19):
 * // n = 11, F(n) = 89
 * // n = 12, F(n) = 144
 * // n = 13, F(n) = 233
 * // n = 14, F(n) = 377
 * // n = 15, F(n) = 610
 * // n = 16, F(n) = 987
 * // n = 17, F(n) = 1597
 * // n = 18, F(n) = 2584
 * // n = 19, F(n) = 4181
 *
 * echo "\nChecking if 26 and 4181 are Fibonacci numbers\n";
 * $verb = Math_Fibonacci::isFibonacci(new Math_Integer(26)) ? 'is' : 'is not';
 * echo "26 $verb a Fibonacci number\n";
 * // 26 is not a Fibonacci number
 * $verb = Math_Fibonacci::isFibonacci(new Math_Integer(4181)) ? 'is' : 'is not';
 * echo "4181 $verb a Fibonacci number\n";
 * // 4181 is a Fibonacci number
 * 
 * echo "\nDecompose 34512\n";
 * $decarr = Math_Fibonacci::decompose(new Math_Integer(34512));
 * foreach ($decarr as $fib) {
 *     $index = Math_Fibonacci::getIndexOf($fib);
 *     echo "F(".$index->toString().") = ".$fib->toString()."\n";
 * }
 * // Decompose 34512
 * // F(23) = 28657
 * // F(19) = 4181
 * // F(17) = 1597
 * // F(10) = 55
 * // F(8) = 21
 * // F(2) = 1
 * 
 * echo "\nF(n) closest to 314156 is: ";
 * $fib = Math_Fibonacci::closestTo(new Math_Integer(314156));
 * echo $fib->toString()."\n\n";
 * // F(n) closest to 314156 is: 317811
 * 
 * echo 'The index for 1597 is : ';
 * $idx = Math_Fibonacci::getIndexOf(new Math_Integer(1597));
 * echo $idx->toString()."\n\n";
 * // The index for 1597 is : 17
 * 
 * $bigint = '3141579834521345220291';
 * echo "Finding the Fibonacci numbers that add up to $bigint\n";
 * $series = Math_Fibonacci::decompose(new Math_Integer($bigint));
 * foreach ($series as $fib) {
 *     $index = Math_Fibonacci::getIndexOf($fib);
 *     echo "F(".$index->toString().") = ".$fib->toString()."\n";
 * }
 * // Finding the Fibonacci numbers that add up to 3141579834521345220291
 * // F(104) = 2427893228399975082453
 * // F(101) = 573147844013817084101
 * // F(98) = 135301852344706746049
 * // F(91) = 4660046610375530309
 * // F(86) = 420196140727489673
 * // F(83) = 99194853094755497
 * // F(81) = 37889062373143906
 * // F(79) = 14472334024676221
 * // F(76) = 3416454622906707
 * // F(74) = 1304969544928657
 * // F(71) = 308061521170129
 * // F(68) = 72723460248141
 * // F(63) = 6557470319842
 * // F(60) = 1548008755920
 * // F(57) = 365435296162
 * // F(53) = 53316291173
 * // F(51) = 20365011074
 * // F(49) = 7778742049
 * // F(44) = 701408733
 * // F(37) = 24157817
 * // F(31) = 1346269
 * // F(26) = 121393
 * // F(20) = 6765
 * // F(16) = 987
 * // F(13) = 233
 * // F(8) = 21
 * // F(6) = 8
 * // F(3) = 2
 * 
 * </pre>
 *
 * @author  Jesus M. Castagnetto <[e-mail]>
 * @version 0.8
 * @access  public
 * @package Math_Fibonacci
 */
class Math_Fibonacci {/*{{{*/

    /**
     * Calculates a Fibonacci number using the (exact) golden ratio equation:
     * F(n) = (PHI^n - phi^n)/sqrt(5) [Lucas formula]
     * for terms from [0,46]
     * from [47,500] it uses a lookup table
     * from then on uses the recursive addition
     *
     * @param mixed $n the index of the Fibonacci number to calculate, as an integer or a Math_Integer number
     * @return mixed numeric on success, PEAR_Error otherwise
     * @access public
     */
    function &term($n) {/*{{{*/
        if (Math_IntegerOp::isMath_Integer($n)) {
            $idx =& $n;
        } elseif (is_numeric($n) && $n >= 0) {
            $idx =& new Math_Integer($n);
        } else {
            return PEAR::raiseError("The parameter $n is not a Math_Integer object");
        }
        
        $table =& $GLOBALS['_fibonacciTable'];

        // shortcut and check if it is already a cached value
        if (isset($table[$idx->toString()])) {
            return new Math_Integer($table[$idx->toString()]);
        }
        
        // from index [0,46] use the fast algorithm
        $cmp_0 = Math_IntegerOp::compare($idx, new Math_Integer(0));
        $cmp_46 = Math_IntegerOp::compare($idx, new Math_Integer(46));
        if ( ($cmp_0 >= 0) && ($cmp_46 <= 0) ) {
            $val = intval($idx->toString());
            $fn = (pow(MATH_PHI, $val) - pow(MATH_phi, $val))/sqrt(5);
            // add to lookup table
            $table[$val] = $fn;
            return new Math_Integer(strval($fn));
        }
        // from [47,500] use the lookup table
        $cmp_500 = Math_IntegerOp::compare($idx, new Math_Integer(500));
        if ( ($cmp_46 > 0) && ($cmp_500 <= 0) ) {
            return new Math_Integer($table[$idx->toString()]);
        } else {
            // calculate and cache the values
            $a = new Math_Integer($table['499']);
            $b = new Math_Integer($table['500']);
            $pos = new Math_Integer('501');
            $one = new Math_Integer('1');
            while (Math_IntegerOp::compare($pos,$idx) <= 0) {