2455 lines
69 KiB
JavaScript
2455 lines
69 KiB
JavaScript
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/*
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* QuickJS Javascript Calculator
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*
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* Copyright (c) 2017-2018 Fabrice Bellard
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*
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* Permission is hereby granted, free of charge, to any person obtaining a copy
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* of this software and associated documentation files (the "Software"), to deal
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* in the Software without restriction, including without limitation the rights
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* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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* copies of the Software, and to permit persons to whom the Software is
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* furnished to do so, subject to the following conditions:
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*
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* The above copyright notice and this permission notice shall be included in
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* all copies or substantial portions of the Software.
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*
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* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
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* THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
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* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
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* THE SOFTWARE.
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*/
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"use strict";
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"use math";
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var Integer, Float, Fraction, Complex, Mod, Polynomial, PolyMod, RationalFunction, Series, Matrix;
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(function(global) {
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/* the types index are used to dispatch the operator functions */
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var OT_INT = 0;
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var OT_FRACTION = 10;
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var OT_FLOAT64 = 19;
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var OT_FLOAT = 20;
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var OT_COMPLEX = 30;
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var OT_MOD = 40;
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var OT_POLY = 50;
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var OT_POLYMOD = 55;
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var OT_RFUNC = 60;
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var OT_SERIES = 70;
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var OT_ARRAY = 80;
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global.Integer = global.BigInt;
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global.algebraicMode = true;
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/* add non enumerable properties */
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function add_props(obj, props) {
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var i, val, prop, tab, desc, prop1;
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tab = Reflect.ownKeys(props);
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for(i = 0; i < tab.length; i++) {
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prop = tab[i];
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desc = Object.getOwnPropertyDescriptor(props, prop);
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desc.enumerable = false;
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if ("value" in desc) {
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if (typeof desc.value !== "function") {
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desc.writable = false;
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desc.configurable = false;
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}
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} else {
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/* getter/setter */
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desc.configurable = false;
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}
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Object.defineProperty(obj, prop, desc);
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if (obj !== Integer) {
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/* by default also add the non math operators */
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switch(prop) {
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case Symbol.operatorMathDiv:
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prop1 = Symbol.operatorDiv;
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break;
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case Symbol.operatorMathMod:
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prop1 = Symbol.operatorMod;
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break;
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case Symbol.operatorMathPow:
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prop1 = Symbol.operatorPow;
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break;
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default:
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prop1 = null;
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break;
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}
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if (prop1)
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Object.defineProperty(obj, prop1, desc);
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}
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}
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}
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/* Integer */
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function generic_pow(a, b) {
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var r, is_neg, i;
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if (!Integer.isInteger(b)) {
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return exp(log(a) * b);
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}
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if (Array.isArray(a) && !(a instanceof Polynomial ||
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a instanceof Series)) {
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r = idn(Matrix.check_square(a));
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} else {
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r = 1;
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}
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if (b == 0)
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return r;
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is_neg = false;
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if (b < 0) {
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is_neg = true;
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b = -b;
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}
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r = a;
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for(i = Integer.floorLog2(b) - 1; i >= 0; i--) {
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r *= r;
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if ((b >> i) & 1)
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r *= a;
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}
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if (is_neg) {
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if (typeof r.inverse != "function")
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throw "negative powers are not supported for this type";
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r = r.inverse();
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}
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return r;
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}
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var small_primes = [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499 ];
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function miller_rabin_test(n, t) {
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var d, r, s, i, j, a;
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d = n - 1;
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s = 0;
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while (d & 1) {
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d >>= 1;
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s++;
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}
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t = Math.min(t, small_primes.length);
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loop: for(j = 0; j < t; j++) {
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a = small_primes[j];
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r = Integer.pmod(a, d, n);
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if (r == 1 || r == (n - 1))
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continue;
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for(i = 1; i < s; i++) {
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r = (r * r) % n;
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if (r == 1)
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return false;
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if (r == (n - 1))
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continue loop;
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}
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return false; /* n is composite */
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}
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return true; /* n is probably prime with probability (1-0.5^t) */
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}
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function fact_rec(a, b) { /* assumes a <= b */
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var i, r;
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if ((b - a) <= 5) {
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r = a;
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for(i = a + 1; i <= b; i++)
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r *= i;
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return r;
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} else {
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/* to avoid a quadratic running time it is better to
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multiply numbers of similar size */
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i = (a + b) >> 1;
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return fact_rec(a, i) * fact_rec(i + 1, b);
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}
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}
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add_props(Integer, {
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isInteger(a) {
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return typeof a === "bigint";
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},
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[Symbol.operatorOrder]: OT_INT,
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[Symbol.operatorMathDiv](a, b) {
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if (algebraicMode) {
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return Fraction.toFraction(a, b);
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} else {
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return Float(a) / Float(b);
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}
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},
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[Symbol.operatorMathPow](a, b) {
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if (algebraicMode) {
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return generic_pow(a, b);
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} else {
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return Float(a) ** Float(b);
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}
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},
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gcd(a, b) {
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var r;
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while (b != 0) {
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r = a % b;
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a = b;
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b = r;
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}
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return a;
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},
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fact(n) {
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return n <= 0 ? 1 : fact_rec(1, n);
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},
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/* binomial coefficient */
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comb(n, k) {
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if (k < 0 || k > n)
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return 0;
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if (k > n - k)
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k = n - k;
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if (k == 0)
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return 1;
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return Integer.tdiv(fact_rec(n - k + 1, n), fact_rec(1, k));
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},
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/* inverse of x modulo y */
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invmod(x, y) {
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var q, u, v, a, c, t;
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u = x;
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v = y;
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c = 1;
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a = 0;
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while (u != 0) {
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t = Integer.fdivrem(v, u);
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q = t[0];
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v = u;
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u = t[1];
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t = c;
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c = a - q * c;
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a = t;
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}
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/* v = gcd(x, y) */
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if (v != 1)
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throw RangeError("not invertible");
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return a % y;
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},
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/* return a ^ b modulo m */
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pmod(a, b, m) {
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var r;
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if (b == 0)
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return 1;
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if (b < 0) {
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a = Integer.invmod(a, m);
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b = -b;
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}
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r = 1;
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for(;;) {
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if (b & 1) {
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r = (r * a) % m;
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}
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b >>= 1;
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if (b == 0)
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break;
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a = (a * a) % m;
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}
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return r;
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},
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/* return true if n is prime (or probably prime with
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probability 1-0.5^t) */
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isPrime(n, t) {
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var i, d, n1;
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if (!Integer.isInteger(n))
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throw TypeError("invalid type");
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if (n <= 1)
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return false;
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n1 = small_primes.length;
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/* XXX: need Integer.sqrt() */
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for(i = 0; i < n1; i++) {
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d = small_primes[i];
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if (d == n)
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return true;
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if (d > n)
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return false;
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if ((n % d) == 0)
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return false;
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}
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if (n < d * d)
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return true;
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if (typeof t == "undefined")
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t = 64;
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return miller_rabin_test(n, t);
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},
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nextPrime(n) {
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if (!Integer.isInteger(n))
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throw TypeError("invalid type");
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if (n < 1)
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n = 1;
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for(;;) {
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n++;
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if (Integer.isPrime(n))
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return n;
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}
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},
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factor(n) {
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var r, d;
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if (!Integer.isInteger(n))
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throw TypeError("invalid type");
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r = [];
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if (abs(n) <= 1) {
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r.push(n);
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return r;
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}
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if (n < 0) {
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r.push(-1);
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n = -n;
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}
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while ((n % 2) == 0) {
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n >>= 1;
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r.push(2);
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}
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d = 3;
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while (n != 1) {
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if (Integer.isPrime(n)) {
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r.push(n);
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break;
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}
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/* we are sure there is at least one divisor, so one test */
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for(;;) {
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if ((n % d) == 0)
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break;
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d += 2;
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}
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for(;;) {
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r.push(d);
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n = Integer.tdiv(n, d);
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if ((n % d) != 0)
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break;
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}
|
||
|
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}
|
||
|
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return r;
|
||
|
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},
|
||
|
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});
|
||
|
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|
||
|
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add_props(Integer.prototype, {
|
||
|
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inverse() {
|
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|
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return 1 / this;
|
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|
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},
|
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norm2() {
|
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return this * this;
|
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},
|
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abs() {
|
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|
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return Math.abs(this);
|
||
|
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},
|
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|
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conj() {
|
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return this;
|
||
|
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},
|
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|
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arg() {
|
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|
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if (this >= 0)
|
||
|
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return 0;
|
||
|
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else
|
||
|
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return Float.PI;
|
||
|
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},
|
||
|
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exp() {
|
||
|
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if (this == 0)
|
||
|
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return 1;
|
||
|
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else
|
||
|
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return Float.exp(this);
|
||
|
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},
|
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|
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log() {
|
||
|
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if (this == 1)
|
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|
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return 0;
|
||
|
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else
|
||
|
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return Float(this).log();
|
||
|
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},
|
||
|
|
});
|
||
|
|
|
||
|
|
/* Fraction */
|
||
|
|
|
||
|
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Fraction = function Fraction(a, b)
|
||
|
|
{
|
||
|
|
var d, r, obj;
|
||
|
|
|
||
|
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if (new.target)
|
||
|
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throw TypeError("not a constructor");
|
||
|
|
if (a instanceof Fraction)
|
||
|
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return a;
|
||
|
|
if (!Integer.isInteger(a))
|
||
|
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throw TypeError("integer expected");
|
||
|
|
if (typeof b === "undefined") {
|
||
|
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b = 1;
|
||
|
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} else {
|
||
|
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if (!Integer.isInteger(b))
|
||
|
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throw TypeError("integer expected");
|
||
|
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if (b == 0)
|
||
|
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throw RangeError("division by zero");
|
||
|
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d = Integer.gcd(a, b);
|
||
|
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if (d != 1) {
|
||
|
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a = Integer.tdiv(a, d);
|
||
|
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b = Integer.tdiv(b, d);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* the fractions are normalized with den > 0 */
|
||
|
|
if (b < 0) {
|
||
|
|
a = -a;
|
||
|
|
b = -b;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
obj = Object.create(Fraction.prototype);
|
||
|
|
obj.num = a;
|
||
|
|
obj.den = b;
|
||
|
|
return obj;
|
||
|
|
}
|
||
|
|
|
||
|
|
add_props(Fraction, {
|
||
|
|
[Symbol.operatorOrder]: OT_FRACTION,
|
||
|
|
/* (internal use) simplify 'a' to an integer when possible */
|
||
|
|
toFraction(a, b) {
|
||
|
|
var r = Fraction(a, b);
|
||
|
|
if (algebraicMode && r.den == 1)
|
||
|
|
return r.num;
|
||
|
|
else
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
|
||
|
|
[Symbol.operatorAdd](a, b) {
|
||
|
|
a = Fraction(a);
|
||
|
|
b = Fraction(b);
|
||
|
|
return Fraction.toFraction(a.num * b.den + a.den * b.num, a.den * b.den);
|
||
|
|
},
|
||
|
|
[Symbol.operatorSub](a, b) {
|
||
|
|
a = Fraction(a);
|
||
|
|
b = Fraction(b);
|
||
|
|
return Fraction.toFraction(a.num * b.den - a.den * b.num, a.den * b.den);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMul](a, b) {
|
||
|
|
a = Fraction(a);
|
||
|
|
b = Fraction(b);
|
||
|
|
return Fraction.toFraction(a.num * b.num, a.den * b.den);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathDiv](a, b) {
|
||
|
|
a = Fraction(a);
|
||
|
|
b = Fraction(b);
|
||
|
|
return Fraction.toFraction(a.num * b.den, a.den * b.num);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathMod](a, b) {
|
||
|
|
var a1 = Fraction(a);
|
||
|
|
var b1 = Fraction(b);
|
||
|
|
return a - Integer.ediv(a1.num * b1.den, a1.den * b1.num) * b;
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathPow]: generic_pow,
|
||
|
|
[Symbol.operatorCmpEQ](a, b) {
|
||
|
|
a = Fraction(a);
|
||
|
|
b = Fraction(b);
|
||
|
|
/* we assume the fractions are normalized */
|
||
|
|
return (a.num == b.num && a.den == b.den);
|
||
|
|
},
|
||
|
|
[Symbol.operatorCmpLT](a, b) {
|
||
|
|
a = Fraction(a);
|
||
|
|
b = Fraction(b);
|
||
|
|
return (a.num * b.den < b.num * a.den);
|
||
|
|
},
|
||
|
|
[Symbol.operatorCmpLE](a, b) {
|
||
|
|
a = Fraction(a);
|
||
|
|
b = Fraction(b);
|
||
|
|
return (a.num * b.den <= b.num * a.den);
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(Fraction.prototype, {
|
||
|
|
[Symbol.toPrimitive](hint) {
|
||
|
|
if (hint === "integer") {
|
||
|
|
return Integer.tdiv(this.num, this.den);
|
||
|
|
} else if (hint === "string") {
|
||
|
|
return this.toString();
|
||
|
|
} else {
|
||
|
|
return Float(this.num) / this.den;
|
||
|
|
}
|
||
|
|
},
|
||
|
|
[Symbol.operatorPlus]() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
[Symbol.operatorNeg]() {
|
||
|
|
return Fraction(-this.num, this.den);
|
||
|
|
},
|
||
|
|
inverse() {
|
||
|
|
return Fraction(this.den, this.num);
|
||
|
|
},
|
||
|
|
toString() {
|
||
|
|
return this.num + "/" + this.den;
|
||
|
|
},
|
||
|
|
norm2() {
|
||
|
|
return this * this;
|
||
|
|
},
|
||
|
|
abs() {
|
||
|
|
if (this.num < 0)
|
||
|
|
return this[Symbol.operatorNeg]();
|
||
|
|
else
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
conj() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
arg() {
|
||
|
|
if (this.num >= 0)
|
||
|
|
return 0;
|
||
|
|
else
|
||
|
|
return Float.PI;
|
||
|
|
},
|
||
|
|
exp() {
|
||
|
|
return Float.exp(Float(this));
|
||
|
|
},
|
||
|
|
log() {
|
||
|
|
return Float(this).log();
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
/* Number (Float64) */
|
||
|
|
|
||
|
|
add_props(Number, {
|
||
|
|
[Symbol.operatorOrder]: OT_FLOAT64,
|
||
|
|
/* operators are needed for fractions */
|
||
|
|
[Symbol.operatorAdd](a, b) {
|
||
|
|
return Number(a) + Number(b);
|
||
|
|
},
|
||
|
|
[Symbol.operatorSub](a, b) {
|
||
|
|
return Number(a) - Number(b);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMul](a, b) {
|
||
|
|
return Number(a) * Number(b);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathDiv](a, b) {
|
||
|
|
return Number(a) / Number(b);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathPow](a, b) {
|
||
|
|
return Number(a) ** Number(b);
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(Number.prototype, {
|
||
|
|
inverse() {
|
||
|
|
return 1.0 / this;
|
||
|
|
},
|
||
|
|
norm2() {
|
||
|
|
return this * this;
|
||
|
|
},
|
||
|
|
abs() {
|
||
|
|
return Math.abs(this);
|
||
|
|
},
|
||
|
|
conj() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
arg() {
|
||
|
|
if (this >= 0)
|
||
|
|
return 0;
|
||
|
|
else
|
||
|
|
return Float.PI;
|
||
|
|
},
|
||
|
|
exp() {
|
||
|
|
return Float.exp(this);
|
||
|
|
},
|
||
|
|
log() {
|
||
|
|
if (this < 0) {
|
||
|
|
return Complex(this).log();
|
||
|
|
} else {
|
||
|
|
return Float.log(this);
|
||
|
|
}
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
/* Float */
|
||
|
|
|
||
|
|
global.Float = global.BigFloat;
|
||
|
|
|
||
|
|
var const_tab = [];
|
||
|
|
|
||
|
|
/* we cache the constants for small precisions */
|
||
|
|
function get_const(n) {
|
||
|
|
var t, c, p;
|
||
|
|
t = const_tab[n];
|
||
|
|
p = BigFloatEnv.prec;
|
||
|
|
if (t && t.prec == p) {
|
||
|
|
return t.val;
|
||
|
|
} else {
|
||
|
|
switch(n) {
|
||
|
|
case 0: c = Float.exp(1); break;
|
||
|
|
case 1: c = Float.log(10); break;
|
||
|
|
// case 2: c = Float.log(2); break;
|
||
|
|
case 3: c = 1/Float.log(2); break;
|
||
|
|
case 4: c = 1/Float.log(10); break;
|
||
|
|
// case 5: c = Float.atan(1) * 4; break;
|
||
|
|
case 6: c = Float.sqrt(0.5); break;
|
||
|
|
case 7: c = Float.sqrt(2); break;
|
||
|
|
}
|
||
|
|
if (p <= 1024) {
|
||
|
|
const_tab[n] = { prec: p, val: c };
|
||
|
|
}
|
||
|
|
return c;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
add_props(Float, {
|
||
|
|
isFloat(a) {
|
||
|
|
return typeof a === "number" || typeof a === "bigfloat";
|
||
|
|
},
|
||
|
|
bestappr(u, b) {
|
||
|
|
var num1, num0, den1, den0, u, num, den, n;
|
||
|
|
|
||
|
|
if (typeof b === "undefined")
|
||
|
|
throw TypeError("second argument expected");
|
||
|
|
num1 = 1;
|
||
|
|
num0 = 0;
|
||
|
|
den1 = 0;
|
||
|
|
den0 = 1;
|
||
|
|
for(;;) {
|
||
|
|
n = Integer(Float.floor(u));
|
||
|
|
num = n * num1 + num0;
|
||
|
|
den = n * den1 + den0;
|
||
|
|
if (den > b)
|
||
|
|
break;
|
||
|
|
u = 1.0 / (u - n);
|
||
|
|
num0 = num1;
|
||
|
|
num1 = num;
|
||
|
|
den0 = den1;
|
||
|
|
den1 = den;
|
||
|
|
}
|
||
|
|
return Fraction(num1, den1);
|
||
|
|
},
|
||
|
|
/* similar constants as Math.x */
|
||
|
|
get E() { return get_const(0); },
|
||
|
|
get LN10() { return get_const(1); },
|
||
|
|
// get LN2() { return get_const(2); },
|
||
|
|
get LOG2E() { return get_const(3); },
|
||
|
|
get LOG10E() { return get_const(4); },
|
||
|
|
// get PI() { return get_const(5); },
|
||
|
|
get SQRT1_2() { return get_const(6); },
|
||
|
|
get SQRT2() { return get_const(7); },
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(Float, {
|
||
|
|
[Symbol.operatorOrder]: OT_FLOAT,
|
||
|
|
/* operators are needed for fractions */
|
||
|
|
[Symbol.operatorAdd](a, b) {
|
||
|
|
return Float(a) + Float(b);
|
||
|
|
},
|
||
|
|
[Symbol.operatorSub](a, b) {
|
||
|
|
return Float(a) - Float(b);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMul](a, b) {
|
||
|
|
return Float(a) * Float(b);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathDiv](a, b) {
|
||
|
|
return Float(a) / Float(b);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathPow](a, b) {
|
||
|
|
return Float(a) ** Float(b);
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(Float.prototype, {
|
||
|
|
inverse() {
|
||
|
|
return 1.0 / this;
|
||
|
|
},
|
||
|
|
norm2() {
|
||
|
|
return this * this;
|
||
|
|
},
|
||
|
|
abs() {
|
||
|
|
return Math.abs(this);
|
||
|
|
},
|
||
|
|
conj() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
arg() {
|
||
|
|
if (this >= 0)
|
||
|
|
return 0;
|
||
|
|
else
|
||
|
|
return Float.PI;
|
||
|
|
},
|
||
|
|
exp() {
|
||
|
|
return Float.exp(this);
|
||
|
|
},
|
||
|
|
log() {
|
||
|
|
if (this < 0) {
|
||
|
|
return Complex(this).log();
|
||
|
|
} else {
|
||
|
|
return Float.log(this);
|
||
|
|
}
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
/* Complex */
|
||
|
|
|
||
|
|
Complex = function Complex(re, im)
|
||
|
|
{
|
||
|
|
var obj;
|
||
|
|
if (new.target)
|
||
|
|
throw TypeError("not a constructor");
|
||
|
|
if (re instanceof Complex)
|
||
|
|
return re;
|
||
|
|
if (typeof im === "undefined") {
|
||
|
|
im = 0;
|
||
|
|
}
|
||
|
|
obj = Object.create(Complex.prototype);
|
||
|
|
obj.re = re;
|
||
|
|
obj.im = im;
|
||
|
|
return obj;
|
||
|
|
}
|
||
|
|
|
||
|
|
add_props(Complex, {
|
||
|
|
[Symbol.operatorOrder]: OT_COMPLEX,
|
||
|
|
/* simplify to real number when possible */
|
||
|
|
toComplex(re, im) {
|
||
|
|
if (algebraicMode && im == 0)
|
||
|
|
return re;
|
||
|
|
else
|
||
|
|
return Complex(re, im);
|
||
|
|
},
|
||
|
|
|
||
|
|
[Symbol.operatorAdd](a, b) {
|
||
|
|
a = Complex(a);
|
||
|
|
b = Complex(b);
|
||
|
|
return Complex.toComplex(a.re + b.re, a.im + b.im);
|
||
|
|
},
|
||
|
|
[Symbol.operatorSub](a, b) {
|
||
|
|
a = Complex(a);
|
||
|
|
b = Complex(b);
|
||
|
|
return Complex.toComplex(a.re - b.re, a.im - b.im);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMul](a, b) {
|
||
|
|
a = Complex(a);
|
||
|
|
b = Complex(b);
|
||
|
|
return Complex.toComplex(a.re * b.re - a.im * b.im,
|
||
|
|
a.re * b.im + a.im * b.re);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathDiv](a, b) {
|
||
|
|
a = Complex(a);
|
||
|
|
b = Complex(b);
|
||
|
|
return a * b.inverse();
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathPow]: generic_pow,
|
||
|
|
[Symbol.operatorCmpEQ](a, b) {
|
||
|
|
a = Complex(a);
|
||
|
|
b = Complex(b);
|
||
|
|
return a.re == b.re && a.im == b.im;
|
||
|
|
}
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(Complex.prototype, {
|
||
|
|
[Symbol.operatorPlus]() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
[Symbol.operatorNeg]() {
|
||
|
|
return Complex(-this.re, -this.im);
|
||
|
|
},
|
||
|
|
inverse() {
|
||
|
|
var c = this.norm2();
|
||
|
|
return Complex(this.re / c, -this.im / c);
|
||
|
|
},
|
||
|
|
toString() {
|
||
|
|
var v, s = "", a = this;
|
||
|
|
if (a.re != 0)
|
||
|
|
s += a.re.toString();
|
||
|
|
if (a.im == 1) {
|
||
|
|
if (s != "")
|
||
|
|
s += "+";
|
||
|
|
s += "I";
|
||
|
|
} else if (a.im == -1) {
|
||
|
|
s += "-I";
|
||
|
|
} else {
|
||
|
|
v = a.im.toString();
|
||
|
|
if (v[0] != "-" && s != "")
|
||
|
|
s += "+";
|
||
|
|
s += v + "*I";
|
||
|
|
}
|
||
|
|
return s;
|
||
|
|
},
|
||
|
|
norm2() {
|
||
|
|
return this.re * this.re + this.im * this.im;
|
||
|
|
},
|
||
|
|
abs() {
|
||
|
|
return Float.sqrt(norm2(this));
|
||
|
|
},
|
||
|
|
conj() {
|
||
|
|
return Complex(this.re, -this.im);
|
||
|
|
},
|
||
|
|
arg() {
|
||
|
|
return Float.atan2(this.im, this.re);
|
||
|
|
},
|
||
|
|
exp() {
|
||
|
|
var arg = this.im, r = this.re.exp();
|
||
|
|
return Complex(r * cos(arg), r * sin(arg));
|
||
|
|
},
|
||
|
|
log() {
|
||
|
|
return Complex(abs(this).log(), atan2(this.im, this.re));
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
/* Mod */
|
||
|
|
|
||
|
|
Mod = function Mod(a, m) {
|
||
|
|
var obj, t;
|
||
|
|
if (new.target)
|
||
|
|
throw TypeError("not a constructor");
|
||
|
|
obj = Object.create(Mod.prototype);
|
||
|
|
if (Integer.isInteger(m)) {
|
||
|
|
if (m <= 0)
|
||
|
|
throw RangeError("the modulo cannot be <= 0");
|
||
|
|
if (Integer.isInteger(a)) {
|
||
|
|
a %= m;
|
||
|
|
} else if (a instanceof Fraction) {
|
||
|
|
return Mod(a.num, m) / a.den;
|
||
|
|
} else {
|
||
|
|
throw TypeError("invalid types");
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
throw TypeError("invalid types");
|
||
|
|
}
|
||
|
|
obj.res = a;
|
||
|
|
obj.mod = m;
|
||
|
|
return obj;
|
||
|
|
};
|
||
|
|
|
||
|
|
add_props(Mod.prototype, {
|
||
|
|
[Symbol.operatorPlus]() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
[Symbol.operatorNeg]() {
|
||
|
|
return Mod(-this.res, this.mod);
|
||
|
|
},
|
||
|
|
inverse() {
|
||
|
|
var a = this, m = a.mod;
|
||
|
|
if (Integer.isInteger(m)) {
|
||
|
|
return Mod(Integer.invmod(a.res, m), m);
|
||
|
|
} else {
|
||
|
|
throw TypeError("unsupported type");
|
||
|
|
}
|
||
|
|
},
|
||
|
|
toString() {
|
||
|
|
return "Mod(" + this.res + "," + this.mod + ")";
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(Mod, {
|
||
|
|
[Symbol.operatorOrder]: OT_MOD,
|
||
|
|
[Symbol.operatorAdd](a, b) {
|
||
|
|
if (!(a instanceof Mod)) {
|
||
|
|
return Mod(a + b.res, b.mod);
|
||
|
|
} else if (!(b instanceof Mod)) {
|
||
|
|
return Mod(a.res + b, a.mod);
|
||
|
|
} else {
|
||
|
|
if (a.mod != b.mod)
|
||
|
|
throw TypeError("different modulo for binary operator");
|
||
|
|
return Mod(a.res + b.res, a.mod);
|
||
|
|
}
|
||
|
|
},
|
||
|
|
[Symbol.operatorSub](a, b) {
|
||
|
|
if (!(a instanceof Mod)) {
|
||
|
|
return Mod(a - b.res, b.mod);
|
||
|
|
} else if (!(b instanceof Mod)) {
|
||
|
|
return Mod(a.res - b, a.mod);
|
||
|
|
} else {
|
||
|
|
if (a.mod != b.mod)
|
||
|
|
throw TypeError("different modulo for binary operator");
|
||
|
|
return Mod(a.res - b.res, a.mod);
|
||
|
|
}
|
||
|
|
},
|
||
|
|
[Symbol.operatorMul](a, b) {
|
||
|
|
if (!(a instanceof Mod)) {
|
||
|
|
return Mod(a * b.res, b.mod);
|
||
|
|
} else if (!(b instanceof Mod)) {
|
||
|
|
return Mod(a.res * b, a.mod);
|
||
|
|
} else {
|
||
|
|
if (a.mod != b.mod)
|
||
|
|
throw TypeError("different modulo for binary operator");
|
||
|
|
return Mod(a.res * b.res, a.mod);
|
||
|
|
}
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathDiv](a, b) {
|
||
|
|
if (!(b instanceof Mod))
|
||
|
|
b = Mod(b, a.mod);
|
||
|
|
return Mod[Symbol.operatorMul](a, b.inverse());
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathPow]: generic_pow,
|
||
|
|
[Symbol.operatorCmpEQ](a, b) {
|
||
|
|
if (!(a instanceof Mod) ||
|
||
|
|
!(b instanceof Mod))
|
||
|
|
return false;
|
||
|
|
return (a.mod == b.mod && a.res == b.res);
|
||
|
|
}
|
||
|
|
});
|
||
|
|
|
||
|
|
/* Polynomial */
|
||
|
|
|
||
|
|
Polynomial = function Polynomial(a)
|
||
|
|
{
|
||
|
|
if (new.target)
|
||
|
|
throw TypeError("not a constructor");
|
||
|
|
if (a instanceof Polynomial) {
|
||
|
|
return a;
|
||
|
|
} else if (Array.isArray(a)) {
|
||
|
|
if (a.length == 0)
|
||
|
|
a = [ 0 ];
|
||
|
|
Object.setPrototypeOf(a, Polynomial.prototype);
|
||
|
|
return a.trim();
|
||
|
|
} else if (a.constructor[Symbol.operatorOrder] <= OT_MOD) {
|
||
|
|
a = [a];
|
||
|
|
Object.setPrototypeOf(a, Polynomial.prototype);
|
||
|
|
return a;
|
||
|
|
} else {
|
||
|
|
throw TypeError("invalid type");
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
function number_need_paren(c)
|
||
|
|
{
|
||
|
|
return !(Integer.isInteger(c) ||
|
||
|
|
Float.isFloat(c) ||
|
||
|
|
c instanceof Fraction ||
|
||
|
|
(c instanceof Complex && c.re == 0));
|
||
|
|
}
|
||
|
|
|
||
|
|
/* string for c*X^i */
|
||
|
|
function monomial_toString(c, i)
|
||
|
|
{
|
||
|
|
var str1;
|
||
|
|
if (i == 0) {
|
||
|
|
str1 = c.toString();
|
||
|
|
} else {
|
||
|
|
if (c == 1) {
|
||
|
|
str1 = "";
|
||
|
|
} else if (c == -1) {
|
||
|
|
str1 = "-";
|
||
|
|
} else {
|
||
|
|
if (number_need_paren(c)) {
|
||
|
|
str1 = "(" + c + ")";
|
||
|
|
} else {
|
||
|
|
str1 = String(c);
|
||
|
|
}
|
||
|
|
str1 += "*";
|
||
|
|
}
|
||
|
|
str1 += "X";
|
||
|
|
if (i != 1) {
|
||
|
|
str1 += "^" + i;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return str1;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* find one complex root of 'p' starting from z at precision eps using
|
||
|
|
at most max_it iterations. Return null if could not find root. */
|
||
|
|
function poly_root_laguerre1(p, z, max_it)
|
||
|
|
{
|
||
|
|
var p1, p2, i, z0, z1, z2, d, t0, t1, d1, d2, e, el, zl;
|
||
|
|
|
||
|
|
d = p.deg();
|
||
|
|
if (d == 1) {
|
||
|
|
/* monomial case */
|
||
|
|
return -p[0] / p[1];
|
||
|
|
}
|
||
|
|
/* trivial zero */
|
||
|
|
if (p[0] == 0)
|
||
|
|
return 0.0;
|
||
|
|
|
||
|
|
p1 = p.deriv();
|
||
|
|
p2 = p1.deriv();
|
||
|
|
el = 0.0;
|
||
|
|
zl = 0.0;
|
||
|
|
for(i = 0; i < max_it; i++) {
|
||
|
|
z0 = p.apply(z);
|
||
|
|
if (z0 == 0)
|
||
|
|
return z; /* simple exit case */
|
||
|
|
|
||
|
|
/* Ward stopping criteria */
|
||
|
|
e = abs(z - zl);
|
||
|
|
// print("e", i, e);
|
||
|
|
if (i >= 2 && e >= el) {
|
||
|
|
if (abs(zl) < 1e-4) {
|
||
|
|
if (e < 1e-7)
|
||
|
|
return zl;
|
||
|
|
} else {
|
||
|
|
if (e < abs(zl) * 1e-3)
|
||
|
|
return zl;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
el = e;
|
||
|
|
zl = z;
|
||
|
|
|
||
|
|
z1 = p1.apply(z);
|
||
|
|
z2 = p2.apply(z);
|
||
|
|
t0 = (d - 1) * z1;
|
||
|
|
t0 = t0 * t0;
|
||
|
|
t1 = d * (d - 1) * z0 * z2;
|
||
|
|
t0 = sqrt(t0 - t1);
|
||
|
|
d1 = z1 + t0;
|
||
|
|
d2 = z1 - t0;
|
||
|
|
if (norm2(d2) > norm2(d1))
|
||
|
|
d1 = d2;
|
||
|
|
if (d1 == 0)
|
||
|
|
return null;
|
||
|
|
z = z - d * z0 / d1;
|
||
|
|
}
|
||
|
|
return null;
|
||
|
|
}
|
||
|
|
|
||
|
|
function poly_roots(p)
|
||
|
|
{
|
||
|
|
var d, i, roots, j, z, eps;
|
||
|
|
var start_points = [ 0.1, -1.4, 1.7 ];
|
||
|
|
|
||
|
|
if (!(p instanceof Polynomial))
|
||
|
|
throw TypeError("polynomial expected");
|
||
|
|
d = p.deg();
|
||
|
|
if (d <= 0)
|
||
|
|
return [];
|
||
|
|
eps = 2.0 ^ (-BigFloatEnv.prec);
|
||
|
|
roots = [];
|
||
|
|
for(i = 0; i < d; i++) {
|
||
|
|
/* XXX: should select another start point if error */
|
||
|
|
for(j = 0; j < 3; j++) {
|
||
|
|
z = poly_root_laguerre1(p, start_points[j], 100);
|
||
|
|
if (z !== null)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
if (j == 3)
|
||
|
|
throw RangeError("error in root finding algorithm");
|
||
|
|
roots[i] = z;
|
||
|
|
p = Polynomial.divrem(p, X - z)[0];
|
||
|
|
}
|
||
|
|
return roots;
|
||
|
|
}
|
||
|
|
|
||
|
|
add_props(Polynomial.prototype, {
|
||
|
|
trim() {
|
||
|
|
var a = this, i;
|
||
|
|
i = a.length;
|
||
|
|
while (i > 1 && a[i - 1] == 0)
|
||
|
|
i--;
|
||
|
|
a.length = i;
|
||
|
|
return a;
|
||
|
|
},
|
||
|
|
[Symbol.operatorPlus]() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
[Symbol.operatorNeg]() {
|
||
|
|
var r, i, n, a;
|
||
|
|
a = this;
|
||
|
|
n = a.length;
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i] = -a[i];
|
||
|
|
return Polynomial(r);
|
||
|
|
},
|
||
|
|
conj() {
|
||
|
|
var r, i, n, a;
|
||
|
|
a = this;
|
||
|
|
n = a.length;
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i] = a[i].conj();
|
||
|
|
return Polynomial(r);
|
||
|
|
},
|
||
|
|
inverse() {
|
||
|
|
return RationalFunction(Polynomial([1]), this);
|
||
|
|
},
|
||
|
|
toString() {
|
||
|
|
var i, str, str1, c, a = this;
|
||
|
|
if (a.length == 1) {
|
||
|
|
return a[0].toString();
|
||
|
|
}
|
||
|
|
str="";
|
||
|
|
for(i = a.length - 1; i >= 0; i--) {
|
||
|
|
c = a[i];
|
||
|
|
if (c == 0 ||
|
||
|
|
(c instanceof Mod) && c.res == 0)
|
||
|
|
continue;
|
||
|
|
str1 = monomial_toString(c, i);
|
||
|
|
if (str1[0] != "-") {
|
||
|
|
if (str != "")
|
||
|
|
str += "+";
|
||
|
|
}
|
||
|
|
str += str1;
|
||
|
|
}
|
||
|
|
return str;
|
||
|
|
},
|
||
|
|
deg() {
|
||
|
|
if (this.length == 1 && this[0] == 0)
|
||
|
|
return -Infinity;
|
||
|
|
else
|
||
|
|
return this.length - 1;
|
||
|
|
},
|
||
|
|
apply(b) {
|
||
|
|
var i, n, r, a = this;
|
||
|
|
n = a.length - 1;
|
||
|
|
r = a[n];
|
||
|
|
while (n > 0) {
|
||
|
|
n--;
|
||
|
|
r = r * b + a[n];
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
deriv() {
|
||
|
|
var a = this, n, r, i;
|
||
|
|
n = a.length;
|
||
|
|
if (n == 1) {
|
||
|
|
return Polynomial(0);
|
||
|
|
} else {
|
||
|
|
r = [];
|
||
|
|
for(i = 1; i < n; i++) {
|
||
|
|
r[i - 1] = i * a[i];
|
||
|
|
}
|
||
|
|
return Polynomial(r);
|
||
|
|
}
|
||
|
|
},
|
||
|
|
integ() {
|
||
|
|
var a = this, n, r, i;
|
||
|
|
n = a.length;
|
||
|
|
r = [0];
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
r[i + 1] = a[i] / (i + 1);
|
||
|
|
}
|
||
|
|
return Polynomial(r);
|
||
|
|
},
|
||
|
|
norm2() {
|
||
|
|
var a = this, n, r, i;
|
||
|
|
n = a.length;
|
||
|
|
r = 0;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
r += a[i].norm2();
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(Polynomial, {
|
||
|
|
[Symbol.operatorOrder]: OT_POLY,
|
||
|
|
[Symbol.operatorAdd](a, b) {
|
||
|
|
var tmp, r, i, n1, n2;
|
||
|
|
a = Polynomial(a);
|
||
|
|
b = Polynomial(b);
|
||
|
|
if (a.length < b.length) {
|
||
|
|
tmp = a;
|
||
|
|
a = b;
|
||
|
|
b = tmp;
|
||
|
|
}
|
||
|
|
n1 = b.length;
|
||
|
|
n2 = a.length;
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < n1; i++)
|
||
|
|
r[i] = a[i] + b[i];
|
||
|
|
for(i = n1; i < n2; i++)
|
||
|
|
r[i] = a[i];
|
||
|
|
return Polynomial(r);
|
||
|
|
},
|
||
|
|
[Symbol.operatorSub](a, b) {
|
||
|
|
return Polynomial[Symbol.operatorAdd](a, -b);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMul](a, b) {
|
||
|
|
var i, j, n1, n2, n, r;
|
||
|
|
a = Polynomial(a);
|
||
|
|
b = Polynomial(b);
|
||
|
|
n1 = a.length;
|
||
|
|
n2 = b.length;
|
||
|
|
n = n1 + n2 - 1;
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i] = 0;
|
||
|
|
for(i = 0; i < n1; i++) {
|
||
|
|
for(j = 0; j < n2; j++) {
|
||
|
|
r[i + j] += a[i] * b[j];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return Polynomial(r);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathDiv](a, b) {
|
||
|
|
if (b.constructor[Symbol.operatorOrder] <= OT_COMPLEX)
|
||
|
|
return a * (1 / b);
|
||
|
|
else
|
||
|
|
return RationalFunction(Polynomial(a),
|
||
|
|
Polynomial(b));
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathPow]: generic_pow,
|
||
|
|
[Symbol.operatorMathMod](a, b) {
|
||
|
|
return Polynomial.divrem(a, b)[1];
|
||
|
|
},
|
||
|
|
[Symbol.operatorCmpEQ](a, b) {
|
||
|
|
var n, i;
|
||
|
|
if (!(a instanceof Polynomial) ||
|
||
|
|
!(b instanceof Polynomial))
|
||
|
|
return false;
|
||
|
|
n = a.length;
|
||
|
|
if (n != b.length)
|
||
|
|
return false;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
if (a[i] != b[i])
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
return true;
|
||
|
|
},
|
||
|
|
divrem(a, b) {
|
||
|
|
var n1, n2, i, j, q, r, n, c;
|
||
|
|
if (b.deg() < 0)
|
||
|
|
throw RangeError("division by zero");
|
||
|
|
n1 = a.length;
|
||
|
|
n2 = b.length;
|
||
|
|
if (n1 < n2)
|
||
|
|
return [Polynomial([0]), a];
|
||
|
|
r = Array.prototype.dup.call(a);
|
||
|
|
q = [];
|
||
|
|
n2--;
|
||
|
|
n = n1 - n2;
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
q[i] = 0;
|
||
|
|
for(i = n - 1; i >= 0; i--) {
|
||
|
|
c = r[i + n2];
|
||
|
|
if (c != 0) {
|
||
|
|
c = c / b[n2];
|
||
|
|
r[i + n2] = 0;
|
||
|
|
for(j = 0; j < n2; j++) {
|
||
|
|
r[i + j] -= b[j] * c;
|
||
|
|
}
|
||
|
|
q[i] = c;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return [Polynomial(q), Polynomial(r)];
|
||
|
|
},
|
||
|
|
gcd(a, b) {
|
||
|
|
var t;
|
||
|
|
while (b.deg() >= 0) {
|
||
|
|
t = Polynomial.divrem(a, b);
|
||
|
|
a = b;
|
||
|
|
b = t[1];
|
||
|
|
}
|
||
|
|
/* convert to monic form */
|
||
|
|
return a / a[a.length - 1];
|
||
|
|
},
|
||
|
|
invmod(x, y) {
|
||
|
|
var q, u, v, a, c, t;
|
||
|
|
u = x;
|
||
|
|
v = y;
|
||
|
|
c = Polynomial([1]);
|
||
|
|
a = Polynomial([0]);
|
||
|
|
while (u.deg() >= 0) {
|
||
|
|
t = Polynomial.divrem(v, u);
|
||
|
|
q = t[0];
|
||
|
|
v = u;
|
||
|
|
u = t[1];
|
||
|
|
t = c;
|
||
|
|
c = a - q * c;
|
||
|
|
a = t;
|
||
|
|
}
|
||
|
|
/* v = gcd(x, y) */
|
||
|
|
if (v.deg() > 0)
|
||
|
|
throw RangeError("not invertible");
|
||
|
|
return Polynomial.divrem(a, y)[1];
|
||
|
|
},
|
||
|
|
roots(p) {
|
||
|
|
return poly_roots(p);
|
||
|
|
}
|
||
|
|
});
|
||
|
|
|
||
|
|
/* Polynomial Modulo Q */
|
||
|
|
|
||
|
|
PolyMod = function PolyMod(a, m) {
|
||
|
|
var obj, t;
|
||
|
|
if (new.target)
|
||
|
|
throw TypeError("not a constructor");
|
||
|
|
obj = Object.create(PolyMod.prototype);
|
||
|
|
if (m instanceof Polynomial) {
|
||
|
|
if (m.deg() <= 0)
|
||
|
|
throw RangeError("the modulo cannot have a degree <= 0");
|
||
|
|
if (a instanceof RationalFunction) {
|
||
|
|
return PolyMod(a.num, m) / a.den;
|
||
|
|
} else {
|
||
|
|
a = Polynomial(a);
|
||
|
|
t = Polynomial.divrem(a, m);
|
||
|
|
a = t[1];
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
throw TypeError("invalid types");
|
||
|
|
}
|
||
|
|
obj.res = a;
|
||
|
|
obj.mod = m;
|
||
|
|
return obj;
|
||
|
|
};
|
||
|
|
|
||
|
|
add_props(PolyMod.prototype, {
|
||
|
|
[Symbol.operatorPlus]() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
[Symbol.operatorNeg]() {
|
||
|
|
return PolyMod(-this.res, this.mod);
|
||
|
|
},
|
||
|
|
inverse() {
|
||
|
|
var a = this, m = a.mod;
|
||
|
|
if (m instanceof Polynomial) {
|
||
|
|
return PolyMod(Polynomial.invmod(a.res, m), m);
|
||
|
|
} else {
|
||
|
|
throw TypeError("unsupported type");
|
||
|
|
}
|
||
|
|
},
|
||
|
|
toString() {
|
||
|
|
return "PolyMod(" + this.res + "," + this.mod + ")";
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(PolyMod, {
|
||
|
|
[Symbol.operatorOrder]: OT_POLYMOD,
|
||
|
|
[Symbol.operatorAdd](a, b) {
|
||
|
|
if (!(a instanceof PolyMod)) {
|
||
|
|
return PolyMod(a + b.res, b.mod);
|
||
|
|
} else if (!(b instanceof PolyMod)) {
|
||
|
|
return PolyMod(a.res + b, a.mod);
|
||
|
|
} else {
|
||
|
|
if (a.mod != b.mod)
|
||
|
|
throw TypeError("different modulo for binary operator");
|
||
|
|
return PolyMod(a.res + b.res, a.mod);
|
||
|
|
}
|
||
|
|
},
|
||
|
|
[Symbol.operatorSub](a, b) {
|
||
|
|
if (!(a instanceof PolyMod)) {
|
||
|
|
return PolyMod(a - b.res, b.mod);
|
||
|
|
} else if (!(b instanceof PolyMod)) {
|
||
|
|
return PolyMod(a.res - b, a.mod);
|
||
|
|
} else {
|
||
|
|
if (a.mod != b.mod)
|
||
|
|
throw TypeError("different modulo for binary operator");
|
||
|
|
return PolyMod(a.res - b.res, a.mod);
|
||
|
|
}
|
||
|
|
},
|
||
|
|
[Symbol.operatorMul](a, b) {
|
||
|
|
if (!(a instanceof PolyMod)) {
|
||
|
|
return PolyMod(a * b.res, b.mod);
|
||
|
|
} else if (!(b instanceof PolyMod)) {
|
||
|
|
return PolyMod(a.res * b, a.mod);
|
||
|
|
} else {
|
||
|
|
if (a.mod != b.mod)
|
||
|
|
throw TypeError("different modulo for binary operator");
|
||
|
|
return PolyMod(a.res * b.res, a.mod);
|
||
|
|
}
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathDiv](a, b) {
|
||
|
|
if (!(b instanceof PolyMod))
|
||
|
|
b = PolyMod(b, a.mod);
|
||
|
|
return PolyMod[Symbol.operatorMul](a, b.inverse());
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathPow]: generic_pow,
|
||
|
|
[Symbol.operatorCmpEQ](a, b) {
|
||
|
|
if (!(a instanceof PolyMod) ||
|
||
|
|
!(b instanceof PolyMod))
|
||
|
|
return false;
|
||
|
|
return (a.mod == b.mod && a.res == b.res);
|
||
|
|
}
|
||
|
|
});
|
||
|
|
|
||
|
|
/* Rational function */
|
||
|
|
|
||
|
|
RationalFunction = function RationalFunction(a, b)
|
||
|
|
{
|
||
|
|
var t, r, d, obj;
|
||
|
|
if (new.target)
|
||
|
|
throw TypeError("not a constructor");
|
||
|
|
if (!(a instanceof Polynomial) ||
|
||
|
|
!(b instanceof Polynomial))
|
||
|
|
throw TypeError("polynomial expected");
|
||
|
|
t = Polynomial.divrem(a, b);
|
||
|
|
r = t[1];
|
||
|
|
if (r.deg() < 0)
|
||
|
|
return t[0]; /* no need for a fraction */
|
||
|
|
d = Polynomial.gcd(b, r);
|
||
|
|
if (d.deg() > 0) {
|
||
|
|
a = Polynomial.divrem(a, d)[0];
|
||
|
|
b = Polynomial.divrem(b, d)[0];
|
||
|
|
}
|
||
|
|
obj = Object.create(RationalFunction.prototype);
|
||
|
|
obj.num = a;
|
||
|
|
obj.den = b;
|
||
|
|
return obj;
|
||
|
|
}
|
||
|
|
|
||
|
|
add_props(RationalFunction.prototype, {
|
||
|
|
[Symbol.operatorPlus]() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
[Symbol.operatorNeg]() {
|
||
|
|
return RationalFunction(-this.num, this.den);
|
||
|
|
},
|
||
|
|
inverse() {
|
||
|
|
return RationalFunction(this.den, this.num);
|
||
|
|
},
|
||
|
|
conj() {
|
||
|
|
return RationalFunction(this.num.conj(), this.den.conj());
|
||
|
|
},
|
||
|
|
toString() {
|
||
|
|
var str;
|
||
|
|
if (this.num.deg() <= 0 &&
|
||
|
|
!number_need_paren(this.num[0]))
|
||
|
|
str = this.num.toString();
|
||
|
|
else
|
||
|
|
str = "(" + this.num.toString() + ")";
|
||
|
|
str += "/(" + this.den.toString() + ")"
|
||
|
|
return str;
|
||
|
|
},
|
||
|
|
apply(b) {
|
||
|
|
return this.num.apply(b) / this.den.apply(b);
|
||
|
|
},
|
||
|
|
deriv() {
|
||
|
|
var n = this.num, d = this.den;
|
||
|
|
return RationalFunction(n.deriv() * d - n * d.deriv(), d * d);
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(RationalFunction, {
|
||
|
|
[Symbol.operatorOrder]: OT_RFUNC,
|
||
|
|
/* This function always return a RationalFunction object even
|
||
|
|
if it could simplified to a polynomial, so it is not
|
||
|
|
equivalent to RationalFunction(a) */
|
||
|
|
toRationalFunction(a) {
|
||
|
|
var obj;
|
||
|
|
if (a instanceof RationalFunction) {
|
||
|
|
return a;
|
||
|
|
} else {
|
||
|
|
obj = Object.create(RationalFunction.prototype);
|
||
|
|
obj.num = Polynomial(a);
|
||
|
|
obj.den = Polynomial(1);
|
||
|
|
return obj;
|
||
|
|
}
|
||
|
|
},
|
||
|
|
[Symbol.operatorAdd](a, b) {
|
||
|
|
a = RationalFunction.toRationalFunction(a);
|
||
|
|
b = RationalFunction.toRationalFunction(b);
|
||
|
|
return RationalFunction(a.num * b.den + a.den * b.num, a.den * b.den);
|
||
|
|
},
|
||
|
|
[Symbol.operatorSub](a, b) {
|
||
|
|
a = RationalFunction.toRationalFunction(a);
|
||
|
|
b = RationalFunction.toRationalFunction(b);
|
||
|
|
return RationalFunction(a.num * b.den - a.den * b.num, a.den * b.den);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMul](a, b) {
|
||
|
|
a = RationalFunction.toRationalFunction(a);
|
||
|
|
b = RationalFunction.toRationalFunction(b);
|
||
|
|
return RationalFunction(a.num * b.num, a.den * b.den);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathDiv](a, b) {
|
||
|
|
a = RationalFunction.toRationalFunction(a);
|
||
|
|
b = RationalFunction.toRationalFunction(b);
|
||
|
|
return RationalFunction(a.num * b.den, a.den * b.num);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathPow]: generic_pow,
|
||
|
|
[Symbol.operatorCmpEQ](a, b) {
|
||
|
|
a = RationalFunction.toRationalFunction(a);
|
||
|
|
b = RationalFunction.toRationalFunction(b);
|
||
|
|
/* we assume the fractions are normalized */
|
||
|
|
return (a.num == b.num && a.den == b.den);
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
/* Power series */
|
||
|
|
|
||
|
|
/* 'a' is an array */
|
||
|
|
function get_emin(a) {
|
||
|
|
var i, n;
|
||
|
|
n = a.length;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
if (a[i] != 0)
|
||
|
|
return i;
|
||
|
|
}
|
||
|
|
return n;
|
||
|
|
};
|
||
|
|
|
||
|
|
/* n is the maximum number of terms if 'a' is not a serie */
|
||
|
|
Series = function Series(a, n) {
|
||
|
|
var emin, r, i;
|
||
|
|
|
||
|
|
if (a instanceof Series) {
|
||
|
|
return a;
|
||
|
|
} else if (a.constructor[Symbol.operatorOrder] <= OT_POLY) {
|
||
|
|
if (n <= 0) {
|
||
|
|
/* XXX: should still use the polynomial degree */
|
||
|
|
return Series.zero(0, 0);
|
||
|
|
} else {
|
||
|
|
a = Polynomial(a);
|
||
|
|
emin = get_emin(a);
|
||
|
|
r = Series.zero(n, emin);
|
||
|
|
n = Math.min(a.length - emin, n);
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i] = a[i + emin];
|
||
|
|
return r;
|
||
|
|
}
|
||
|
|
} else if (a instanceof RationalFunction) {
|
||
|
|
return Series(a.num, n) / a.den;
|
||
|
|
} else {
|
||
|
|
throw TypeError("invalid type");
|
||
|
|
}
|
||
|
|
};
|
||
|
|
|
||
|
|
add_props(Series.prototype, {
|
||
|
|
[Symbol.operatorPlus]() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
[Symbol.operatorNeg]() {
|
||
|
|
var obj, n, i;
|
||
|
|
n = this.length;
|
||
|
|
obj = Series.zero(this.length, this.emin);
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
obj[i] = -this[i];
|
||
|
|
}
|
||
|
|
return obj;
|
||
|
|
},
|
||
|
|
conj() {
|
||
|
|
var obj, n, i;
|
||
|
|
n = this.length;
|
||
|
|
obj = Series.zero(this.length, this.emin);
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
obj[i] = this[i].conj();
|
||
|
|
}
|
||
|
|
return obj;
|
||
|
|
},
|
||
|
|
inverse() {
|
||
|
|
var r, n, i, j, sum, v1 = this;
|
||
|
|
n = v1.length;
|
||
|
|
if (n == 0)
|
||
|
|
throw RangeError("division by zero");
|
||
|
|
r = Series.zero(n, -v1.emin);
|
||
|
|
r[0] = 1 / v1[0];
|
||
|
|
for(i = 1; i < n; i++) {
|
||
|
|
sum = 0;
|
||
|
|
for(j = 1; j <= i; j++) {
|
||
|
|
sum += v1[j] * r[i - j];
|
||
|
|
}
|
||
|
|
r[i] = -sum * r[0];
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
/* remove leading zero terms */
|
||
|
|
trim() {
|
||
|
|
var i, j, n, r, v1 = this;
|
||
|
|
n = v1.length;
|
||
|
|
i = 0;
|
||
|
|
while (i < n && v1[i] == 0)
|
||
|
|
i++;
|
||
|
|
if (i == 0)
|
||
|
|
return v1;
|
||
|
|
for(j = i; j < n; j++)
|
||
|
|
v1[j - i] = v1[j];
|
||
|
|
v1.length = n - i;
|
||
|
|
v1.__proto__.emin += i;
|
||
|
|
return v1;
|
||
|
|
},
|
||
|
|
toString() {
|
||
|
|
var i, j, str, str1, c, a = this, emin, n;
|
||
|
|
str="";
|
||
|
|
emin = this.emin;
|
||
|
|
n = this.length;
|
||
|
|
for(j = 0; j < n; j++) {
|
||
|
|
i = j + emin;
|
||
|
|
c = a[j];
|
||
|
|
if (c != 0) {
|
||
|
|
str1 = monomial_toString(c, i);
|
||
|
|
if (str1[0] != "-") {
|
||
|
|
if (str != "")
|
||
|
|
str += "+";
|
||
|
|
}
|
||
|
|
str += str1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
if (str != "")
|
||
|
|
str += "+";
|
||
|
|
str += "O(" + monomial_toString(1, n + emin) + ")";
|
||
|
|
return str;
|
||
|
|
},
|
||
|
|
apply(b) {
|
||
|
|
var i, n, r, a = this;
|
||
|
|
n = a.length;
|
||
|
|
if (n == 0)
|
||
|
|
return 0;
|
||
|
|
r = a[--n];
|
||
|
|
while (n > 0) {
|
||
|
|
n--;
|
||
|
|
r = r * b + a[n];
|
||
|
|
}
|
||
|
|
if (a.emin != 0)
|
||
|
|
r *= b ^ a.emin;
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
deriv() {
|
||
|
|
var a = this, n = a.length, emin = a.emin, r, i, j;
|
||
|
|
if (n == 0 && emin == 0) {
|
||
|
|
return Series.zero(0, 0);
|
||
|
|
} else {
|
||
|
|
r = Series.zero(n, emin - 1);
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
j = emin + i;
|
||
|
|
if (j == 0)
|
||
|
|
r[i] = 0;
|
||
|
|
else
|
||
|
|
r[i] = j * a[i];
|
||
|
|
}
|
||
|
|
return r.trim();
|
||
|
|
}
|
||
|
|
},
|
||
|
|
integ() {
|
||
|
|
var a = this, n = a.length, emin = a.emin, i, j, r;
|
||
|
|
r = Series.zero(n, emin + 1);
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
j = emin + i;
|
||
|
|
if (j == -1) {
|
||
|
|
if (a[i] != 0)
|
||
|
|
throw RangError("cannot represent integ(1/X)");
|
||
|
|
} else {
|
||
|
|
r[i] = a[i] / (j + 1);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return r.trim();
|
||
|
|
},
|
||
|
|
exp() {
|
||
|
|
var c, i, r, n, a = this;
|
||
|
|
n = a.emin;
|
||
|
|
if (n < 0)
|
||
|
|
throw RangeError("negative exponent in exp");
|
||
|
|
if (n > 0 || a[0] == 0) {
|
||
|
|
c = 1;
|
||
|
|
} else {
|
||
|
|
c = global.exp(a[0]);
|
||
|
|
a -= a[0];
|
||
|
|
}
|
||
|
|
n += a.length;
|
||
|
|
r = Series.zero(n, 0);
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
r[i] = c / fact(i);
|
||
|
|
}
|
||
|
|
return r.apply(a);
|
||
|
|
},
|
||
|
|
log() {
|
||
|
|
var a = this, r;
|
||
|
|
if (a.emin != 0)
|
||
|
|
throw Range("log argument must have a non zero constant term");
|
||
|
|
r = integ(deriv(a) / a);
|
||
|
|
/* add the constant term */
|
||
|
|
r += global.log(a[0]);
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(Series, {
|
||
|
|
[Symbol.operatorOrder]: OT_SERIES,
|
||
|
|
/* new series of length n and first exponent emin */
|
||
|
|
zero(n, emin) {
|
||
|
|
var r, i, obj;
|
||
|
|
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i] = 0;
|
||
|
|
/* we return an array and store emin in its prototype */
|
||
|
|
obj = Object.create(Series.prototype);
|
||
|
|
obj.emin = emin;
|
||
|
|
Object.setPrototypeOf(r, obj);
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
[Symbol.operatorAdd](v1, v2) {
|
||
|
|
var tmp, d, emin, n, r, i, j, v2_emin, c1, c2;
|
||
|
|
if (!(v1 instanceof Series)) {
|
||
|
|
tmp = v1;
|
||
|
|
v1 = v2;
|
||
|
|
v2 = tmp;
|
||
|
|
}
|
||
|
|
d = v1.emin + v1.length;
|
||
|
|
if (v2.constructor[Symbol.operatorOrder] <= OT_POLY) {
|
||
|
|
v2 = Polynomial(v2);
|
||
|
|
if (d <= 0)
|
||
|
|
return v1;
|
||
|
|
v2_emin = 0;
|
||
|
|
} else if (v2 instanceof RationalFunction) {
|
||
|
|
/* compute the emin of the rational fonction */
|
||
|
|
i = get_emin(v2.num) - get_emin(v2.den);
|
||
|
|
if (d <= i)
|
||
|
|
return v1;
|
||
|
|
/* compute the serie with the required terms */
|
||
|
|
v2 = Series(v2, d - i);
|
||
|
|
v2_emin = v2.emin;
|
||
|
|
} else {
|
||
|
|
v2_emin = v2.emin;
|
||
|
|
d = Math.min(d, v2_emin + v2.length);
|
||
|
|
}
|
||
|
|
emin = Math.min(v1.emin, v2_emin);
|
||
|
|
n = d - emin;
|
||
|
|
r = Series.zero(n, emin);
|
||
|
|
/* XXX: slow */
|
||
|
|
for(i = emin; i < d; i++) {
|
||
|
|
j = i - v1.emin;
|
||
|
|
if (j >= 0 && j < v1.length)
|
||
|
|
c1 = v1[j];
|
||
|
|
else
|
||
|
|
c1 = 0;
|
||
|
|
j = i - v2_emin;
|
||
|
|
if (j >= 0 && j < v2.length)
|
||
|
|
c2 = v2[j];
|
||
|
|
else
|
||
|
|
c2 = 0;
|
||
|
|
r[i - emin] = c1 + c2;
|
||
|
|
}
|
||
|
|
return r.trim();
|
||
|
|
},
|
||
|
|
[Symbol.operatorSub](a, b) {
|
||
|
|
return Series[Symbol.operatorAdd](a, -b);
|
||
|
|
},
|
||
|
|
[Symbol.operatorMul](v1, v2) {
|
||
|
|
var n, i, j, r, n, emin, n1, n2, k;
|
||
|
|
if (!(v1 instanceof Series))
|
||
|
|
v1 = Series(v1, v2.length);
|
||
|
|
else if (!(v2 instanceof Series))
|
||
|
|
v2 = Series(v2, v1.length);
|
||
|
|
emin = v1.emin + v2.emin;
|
||
|
|
n = Math.min(v1.length, v2.length);
|
||
|
|
n1 = v1.length;
|
||
|
|
n2 = v2.length;
|
||
|
|
r = Series.zero(n, emin);
|
||
|
|
for(i = 0; i < n1; i++) {
|
||
|
|
k = Math.min(n2, n - i);
|
||
|
|
for(j = 0; j < k; j++) {
|
||
|
|
r[i + j] += v1[i] * v2[j];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return r.trim();
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathDiv](v1, v2) {
|
||
|
|
if (!(v2 instanceof Series))
|
||
|
|
v2 = Series(v2, v1.length);
|
||
|
|
return Series[Symbol.operatorMul](v1, v2.inverse());
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathPow](a, b) {
|
||
|
|
if (Integer.isInteger(b)) {
|
||
|
|
return generic_pow(a, b);
|
||
|
|
} else {
|
||
|
|
if (!(a instanceof Series))
|
||
|
|
a = Series(a, b.length);
|
||
|
|
return exp(log(a) * b);
|
||
|
|
}
|
||
|
|
},
|
||
|
|
[Symbol.operatorCmpEQ](a, b) {
|
||
|
|
var n, i;
|
||
|
|
if (!(a instanceof Series) ||
|
||
|
|
!(b instanceof Series))
|
||
|
|
return false;
|
||
|
|
if (a.emin != b.emin)
|
||
|
|
return false;
|
||
|
|
n = a.length;
|
||
|
|
if (n != b.length)
|
||
|
|
return false;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
if (a[i] != b[i])
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
return true;
|
||
|
|
},
|
||
|
|
O(a) {
|
||
|
|
function ErrorO() {
|
||
|
|
return TypeError("invalid O() argument");
|
||
|
|
}
|
||
|
|
var n;
|
||
|
|
if (a.constructor[Symbol.operatorOrder] <= OT_POLY) {
|
||
|
|
a = Polynomial(a);
|
||
|
|
n = a.deg();
|
||
|
|
if (n < 0)
|
||
|
|
throw ErrorO();
|
||
|
|
} else if (a instanceof RationalFunction) {
|
||
|
|
if (a.num.deg() != 0)
|
||
|
|
throw ErrorO();
|
||
|
|
n = a.den.deg();
|
||
|
|
if (n < 0)
|
||
|
|
throw ErrorO();
|
||
|
|
n = -n;
|
||
|
|
} else
|
||
|
|
throw ErrorO();
|
||
|
|
return Series.zero(0, n);
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
/* Array (Matrix) */
|
||
|
|
|
||
|
|
Matrix = function Matrix(h, w) {
|
||
|
|
var i, j, r, rl;
|
||
|
|
if (typeof w === "undefined")
|
||
|
|
w = h;
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < h; i++) {
|
||
|
|
rl = [];
|
||
|
|
for(j = 0; j < w; j++)
|
||
|
|
rl[j] = 0;
|
||
|
|
r[i] = rl;
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
};
|
||
|
|
|
||
|
|
add_props(Matrix, {
|
||
|
|
idn(n) {
|
||
|
|
var r, i;
|
||
|
|
r = Matrix(n, n);
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i][i] = 1;
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
diag(a) {
|
||
|
|
var r, i, n;
|
||
|
|
n = a.length;
|
||
|
|
r = Matrix(n, n);
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i][i] = a[i];
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
hilbert(n) {
|
||
|
|
var i, j, r;
|
||
|
|
r = Matrix(n);
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
for(j = 0; j < n; j++) {
|
||
|
|
r[i][j] = 1 / (1 + i + j);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
trans(a) {
|
||
|
|
var h, w, r, i, j;
|
||
|
|
if (!Array.isArray(a))
|
||
|
|
throw TypeError("matrix expected");
|
||
|
|
h = a.length;
|
||
|
|
if (!Array.isArray(a[0])) {
|
||
|
|
w = 1;
|
||
|
|
r = Matrix(w, h);
|
||
|
|
for(i = 0; i < h; i++) {
|
||
|
|
r[0][i] = a[i];
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
w = a[0].length;
|
||
|
|
r = Matrix(w, h);
|
||
|
|
for(i = 0; i < h; i++) {
|
||
|
|
for(j = 0; j < w; j++) {
|
||
|
|
r[j][i] = a[i][j];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
check_square(a) {
|
||
|
|
var a, n;
|
||
|
|
if (!Array.isArray(a))
|
||
|
|
throw TypeError("array expected");
|
||
|
|
n = a.length;
|
||
|
|
if (!Array.isArray(a[0]) || n != a[0].length)
|
||
|
|
throw TypeError("square matrix expected");
|
||
|
|
return n;
|
||
|
|
},
|
||
|
|
trace(a) {
|
||
|
|
var n, r, i;
|
||
|
|
n = Matrix.check_square(a);
|
||
|
|
r = a[0][0];
|
||
|
|
for(i = 1; i < n; i++) {
|
||
|
|
r += a[i][i];
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
charpoly(a) {
|
||
|
|
var n, p, c, i, j, coef;
|
||
|
|
n = Matrix.check_square(a);
|
||
|
|
p = [];
|
||
|
|
for(i = 0; i < n + 1; i++)
|
||
|
|
p[i] = 0;
|
||
|
|
p[n] = 1;
|
||
|
|
c = Matrix.idn(n);
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
c = c * a;
|
||
|
|
coef = -trace(c) / (i + 1);
|
||
|
|
p[n - i - 1] = coef;
|
||
|
|
for(j = 0; j < n; j++)
|
||
|
|
c[j][j] += coef;
|
||
|
|
}
|
||
|
|
return Polynomial(p);
|
||
|
|
},
|
||
|
|
eigenvals(a) {
|
||
|
|
return Polynomial.roots(Matrix.charpoly(a));
|
||
|
|
},
|
||
|
|
det(a) {
|
||
|
|
var n, i, j, k, s, src, v, c;
|
||
|
|
|
||
|
|
n = Matrix.check_square(a);
|
||
|
|
s = 1;
|
||
|
|
src = a.dup();
|
||
|
|
for(i=0;i<n;i++) {
|
||
|
|
for(j = i; j < n; j++) {
|
||
|
|
if (src[j][i] != 0)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
if (j == n)
|
||
|
|
return 0;
|
||
|
|
if (j != i) {
|
||
|
|
for(k = 0;k < n; k++) {
|
||
|
|
v = src[j][k];
|
||
|
|
src[j][k] = src[i][k];
|
||
|
|
src[i][k] = v;
|
||
|
|
}
|
||
|
|
s = -s;
|
||
|
|
}
|
||
|
|
c = src[i][i].inverse();
|
||
|
|
for(j = i + 1; j < n; j++) {
|
||
|
|
v = c * src[j][i];
|
||
|
|
for(k = 0;k < n; k++) {
|
||
|
|
src[j][k] -= src[i][k] * v;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
c = s;
|
||
|
|
for(i=0;i<n;i++)
|
||
|
|
c *= src[i][i];
|
||
|
|
return c;
|
||
|
|
},
|
||
|
|
inverse(a) {
|
||
|
|
var n, dst, src, i, j, k, n2, r, c, v;
|
||
|
|
n = Matrix.check_square(a);
|
||
|
|
src = a.dup();
|
||
|
|
dst = Matrix.idn(n);
|
||
|
|
for(i=0;i<n;i++) {
|
||
|
|
for(j = i; j < n; j++) {
|
||
|
|
if (src[j][i] != 0)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
if (j == n)
|
||
|
|
throw RangeError("matrix is not invertible");
|
||
|
|
if (j != i) {
|
||
|
|
/* swap lines in src and dst */
|
||
|
|
v = src[j];
|
||
|
|
src[j] = src[i];
|
||
|
|
src[i] = v;
|
||
|
|
v = dst[j];
|
||
|
|
dst[j] = dst[i];
|
||
|
|
dst[i] = v;
|
||
|
|
}
|
||
|
|
|
||
|
|
c = src[i][i].inverse();
|
||
|
|
for(k = 0; k < n; k++) {
|
||
|
|
src[i][k] *= c;
|
||
|
|
dst[i][k] *= c;
|
||
|
|
}
|
||
|
|
|
||
|
|
for(j = 0; j < n; j++) {
|
||
|
|
if (j != i) {
|
||
|
|
c = src[j][i];
|
||
|
|
for(k = i; k < n; k++) {
|
||
|
|
src[j][k] -= src[i][k] * c;
|
||
|
|
}
|
||
|
|
for(k = 0; k < n; k++) {
|
||
|
|
dst[j][k] -= dst[i][k] * c;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return dst;
|
||
|
|
},
|
||
|
|
rank(a) {
|
||
|
|
var src, i, j, k, w, h, l, c;
|
||
|
|
|
||
|
|
if (!Array.isArray(a) ||
|
||
|
|
!Array.isArray(a[0]))
|
||
|
|
throw TypeError("matrix expected");
|
||
|
|
h = a.length;
|
||
|
|
w = a[0].length;
|
||
|
|
src = a.dup();
|
||
|
|
l = 0;
|
||
|
|
for(i=0;i<w;i++) {
|
||
|
|
for(j = l; j < h; j++) {
|
||
|
|
if (src[j][i] != 0)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
if (j == h)
|
||
|
|
continue;
|
||
|
|
if (j != l) {
|
||
|
|
/* swap lines */
|
||
|
|
for(k = 0; k < w; k++) {
|
||
|
|
v = src[j][k];
|
||
|
|
src[j][k] = src[l][k];
|
||
|
|
src[l][k] = v;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
c = src[l][i].inverse();
|
||
|
|
for(k = 0; k < w; k++) {
|
||
|
|
src[l][k] *= c;
|
||
|
|
}
|
||
|
|
|
||
|
|
for(j = l + 1; j < h; j++) {
|
||
|
|
c = src[j][i];
|
||
|
|
for(k = i; k < w; k++) {
|
||
|
|
src[j][k] -= src[l][k] * c;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
l++;
|
||
|
|
}
|
||
|
|
return l;
|
||
|
|
},
|
||
|
|
ker(a) {
|
||
|
|
var src, i, j, k, w, h, l, m, r, im_cols, ker_dim, c;
|
||
|
|
|
||
|
|
if (!Array.isArray(a) ||
|
||
|
|
!Array.isArray(a[0]))
|
||
|
|
throw TypeError("matrix expected");
|
||
|
|
h = a.length;
|
||
|
|
w = a[0].length;
|
||
|
|
src = a.dup();
|
||
|
|
im_cols = [];
|
||
|
|
l = 0;
|
||
|
|
for(i=0;i<w;i++) {
|
||
|
|
im_cols[i] = false;
|
||
|
|
for(j = l; j < h; j++) {
|
||
|
|
if (src[j][i] != 0)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
if (j == h)
|
||
|
|
continue;
|
||
|
|
im_cols[i] = true;
|
||
|
|
if (j != l) {
|
||
|
|
/* swap lines */
|
||
|
|
for(k = 0; k < w; k++) {
|
||
|
|
v = src[j][k];
|
||
|
|
src[j][k] = src[l][k];
|
||
|
|
src[l][k] = v;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
c = src[l][i].inverse();
|
||
|
|
for(k = 0; k < w; k++) {
|
||
|
|
src[l][k] *= c;
|
||
|
|
}
|
||
|
|
|
||
|
|
for(j = 0; j < h; j++) {
|
||
|
|
if (j != l) {
|
||
|
|
c = src[j][i];
|
||
|
|
for(k = i; k < w; k++) {
|
||
|
|
src[j][k] -= src[l][k] * c;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
l++;
|
||
|
|
// log_str("m=" + cval_toString(v1) + "\n");
|
||
|
|
}
|
||
|
|
// log_str("im cols="+im_cols+"\n");
|
||
|
|
|
||
|
|
/* build the kernel vectors */
|
||
|
|
ker_dim = w - l;
|
||
|
|
r = Matrix(w, ker_dim);
|
||
|
|
k = 0;
|
||
|
|
for(i = 0; i < w; i++) {
|
||
|
|
if (!im_cols[i]) {
|
||
|
|
/* select this column from the matrix */
|
||
|
|
l = 0;
|
||
|
|
m = 0;
|
||
|
|
for(j = 0; j < w; j++) {
|
||
|
|
if (im_cols[j]) {
|
||
|
|
r[j][k] = -src[m][i];
|
||
|
|
m++;
|
||
|
|
} else {
|
||
|
|
if (l == k) {
|
||
|
|
r[j][k] = 1;
|
||
|
|
} else {
|
||
|
|
r[j][k] = 0;
|
||
|
|
}
|
||
|
|
l++;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
k++;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
dp(a, b) {
|
||
|
|
var i, n, r;
|
||
|
|
n = a.length;
|
||
|
|
if (n != b.length)
|
||
|
|
throw TypeError("incompatible array length");
|
||
|
|
/* XXX: could do complex product */
|
||
|
|
r = 0;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
r += a[i] * b[i];
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
/* cross product */
|
||
|
|
cp(v1, v2) {
|
||
|
|
var r;
|
||
|
|
if (v1.length != 3 || v2.length != 3)
|
||
|
|
throw TypeError("vectors must have 3 elements");
|
||
|
|
r = [];
|
||
|
|
r[0] = v1[1] * v2[2] - v1[2] * v2[1];
|
||
|
|
r[1] = v1[2] * v2[0] - v1[0] * v2[2];
|
||
|
|
r[2] = v1[0] * v2[1] - v1[1] * v2[0];
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(Array, {
|
||
|
|
[Symbol.operatorOrder]: OT_ARRAY,
|
||
|
|
[Symbol.operatorAdd](a, b) {
|
||
|
|
var r, i, n;
|
||
|
|
if (!Array.isArray(a) || !Array.isArray(b))
|
||
|
|
throw TypeError("array expected");
|
||
|
|
n = a.length;
|
||
|
|
if (n != b.length)
|
||
|
|
throw TypeError("incompatible array size");
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i] = a[i] + b[i];
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
[Symbol.operatorSub](a, b) {
|
||
|
|
var r, i, n;
|
||
|
|
n = a.length;
|
||
|
|
if (!Array.isArray(a) || !Array.isArray(b))
|
||
|
|
throw TypeError("array expected");
|
||
|
|
if (n != b.length)
|
||
|
|
throw TypeError("incompatible array size");
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i] = a[i] - b[i];
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
scalar_mul(a, b) {
|
||
|
|
var r, i, n;
|
||
|
|
n = a.length;
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i] = a[i] * b;
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
[Symbol.operatorMul](a, b) {
|
||
|
|
var h, w, l, i, j, k, r, rl, sum, a_mat, b_mat, a_is_array, b_is_array;
|
||
|
|
a_is_array = Array.isArray(a);
|
||
|
|
b_is_array = Array.isArray(b);
|
||
|
|
if (!a_is_array && !b_is_array) {
|
||
|
|
throw TypeError("array expected");
|
||
|
|
} else if (!a_is_array && b_is_array) {
|
||
|
|
return Array.scalar_mul(b, a);
|
||
|
|
} else if (a_is_array && !b_is_array) {
|
||
|
|
return Array.scalar_mul(a, b);
|
||
|
|
}
|
||
|
|
h = a.length;
|
||
|
|
a_mat = Array.isArray(a[0]);
|
||
|
|
if (a_mat) {
|
||
|
|
l = a[0].length;
|
||
|
|
} else {
|
||
|
|
l = 1;
|
||
|
|
}
|
||
|
|
if (l != b.length)
|
||
|
|
throw RangeError("incompatible matrix size");
|
||
|
|
b_mat = Array.isArray(b[0]);
|
||
|
|
if (b_mat)
|
||
|
|
w = b[0].length;
|
||
|
|
else
|
||
|
|
w = 1;
|
||
|
|
r = [];
|
||
|
|
if (a_mat && b_mat) {
|
||
|
|
for(i = 0; i < h; i++) {
|
||
|
|
rl = [];
|
||
|
|
for(j = 0; j < w; j++) {
|
||
|
|
sum = 0;
|
||
|
|
for(k = 0; k < l; k++) {
|
||
|
|
sum += a[i][k] * b[k][j];
|
||
|
|
}
|
||
|
|
rl[j] = sum;
|
||
|
|
}
|
||
|
|
r[i] = rl;
|
||
|
|
}
|
||
|
|
} else if (a_mat && !b_mat) {
|
||
|
|
for(i = 0; i < h; i++) {
|
||
|
|
sum = 0;
|
||
|
|
for(k = 0; k < l; k++) {
|
||
|
|
sum += a[i][k] * b[k];
|
||
|
|
}
|
||
|
|
r[i] = sum;
|
||
|
|
}
|
||
|
|
} else if (!a_mat && b_mat) {
|
||
|
|
for(i = 0; i < h; i++) {
|
||
|
|
rl = [];
|
||
|
|
for(j = 0; j < w; j++) {
|
||
|
|
rl[j] = a[i] * b[0][j];
|
||
|
|
}
|
||
|
|
r[i] = rl;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
for(i = 0; i < h; i++) {
|
||
|
|
r[i] = a[i] * b[0];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathDiv](a, b) {
|
||
|
|
return Array[Symbol.operatorMul](a, b.inverse());
|
||
|
|
},
|
||
|
|
[Symbol.operatorMathPow]: generic_pow,
|
||
|
|
[Symbol.operatorCmpEQ](a, b) {
|
||
|
|
var n, i;
|
||
|
|
n = a.length;
|
||
|
|
if (n != b.length)
|
||
|
|
return false;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
if (a[i] != b[i])
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
return true;
|
||
|
|
},
|
||
|
|
});
|
||
|
|
|
||
|
|
add_props(Array.prototype, {
|
||
|
|
[Symbol.operatorPlus]() {
|
||
|
|
return this;
|
||
|
|
},
|
||
|
|
[Symbol.operatorNeg]() {
|
||
|
|
var i, n, r;
|
||
|
|
n = this.length;
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i] = -this[i];
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
conj() {
|
||
|
|
var i, n, r;
|
||
|
|
n = this.length;
|
||
|
|
r = [];
|
||
|
|
for(i = 0; i < n; i++)
|
||
|
|
r[i] = this[i].conj();
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
dup() {
|
||
|
|
var r, i, n, el, a = this;
|
||
|
|
r = [];
|
||
|
|
n = a.length;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
el = a[i];
|
||
|
|
if (Array.isArray(el))
|
||
|
|
el = el.dup();
|
||
|
|
r[i] = el;
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
},
|
||
|
|
inverse() {
|
||
|
|
return Matrix.inverse(this);
|
||
|
|
},
|
||
|
|
norm2: Polynomial.prototype.norm2,
|
||
|
|
});
|
||
|
|
|
||
|
|
})(this);
|
||
|
|
|
||
|
|
/* global definitions */
|
||
|
|
var I = Complex(0, 1);
|
||
|
|
var X = Polynomial([0, 1]);
|
||
|
|
var O = Series.O;
|
||
|
|
|
||
|
|
Object.defineProperty(this, "PI", { get: function () { return Float.PI } });
|
||
|
|
|
||
|
|
/* put frequently used functions in the global context */
|
||
|
|
var gcd = Integer.gcd;
|
||
|
|
var fact = Integer.fact;
|
||
|
|
var comb = Integer.comb;
|
||
|
|
var pmod = Integer.pmod;
|
||
|
|
var invmod = Integer.invmod;
|
||
|
|
var factor = Integer.factor;
|
||
|
|
var isprime = Integer.isPrime;
|
||
|
|
var nextprime = Integer.nextPrime;
|
||
|
|
|
||
|
|
function deriv(a)
|
||
|
|
{
|
||
|
|
return a.deriv();
|
||
|
|
}
|
||
|
|
|
||
|
|
function integ(a)
|
||
|
|
{
|
||
|
|
return a.integ();
|
||
|
|
}
|
||
|
|
|
||
|
|
function norm2(a)
|
||
|
|
{
|
||
|
|
return a.norm2();
|
||
|
|
}
|
||
|
|
|
||
|
|
function abs(a)
|
||
|
|
{
|
||
|
|
return a.abs();
|
||
|
|
}
|
||
|
|
|
||
|
|
function conj(a)
|
||
|
|
{
|
||
|
|
return a.conj();
|
||
|
|
}
|
||
|
|
|
||
|
|
function arg(a)
|
||
|
|
{
|
||
|
|
return a.arg();
|
||
|
|
}
|
||
|
|
|
||
|
|
function inverse(a)
|
||
|
|
{
|
||
|
|
return a.inverse();
|
||
|
|
}
|
||
|
|
|
||
|
|
function trunc(a)
|
||
|
|
{
|
||
|
|
if (Integer.isInteger(a)) {
|
||
|
|
return a;
|
||
|
|
} else if (a instanceof Fraction) {
|
||
|
|
return Integer.tdiv(a.num, a.den);
|
||
|
|
} else if (a instanceof Polynomial) {
|
||
|
|
return a;
|
||
|
|
} else if (a instanceof RationalFunction) {
|
||
|
|
return Polynomial.divrem(a.num, a.den)[0];
|
||
|
|
} else {
|
||
|
|
return Float.ceil(a);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
function floor(a)
|
||
|
|
{
|
||
|
|
if (Integer.isInteger(a)) {
|
||
|
|
return a;
|
||
|
|
} else if (a instanceof Fraction) {
|
||
|
|
return Integer.fdiv(a.num, a.den);
|
||
|
|
} else {
|
||
|
|
return Float.floor(a);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
function ceil(a)
|
||
|
|
{
|
||
|
|
if (Integer.isInteger(a)) {
|
||
|
|
return a;
|
||
|
|
} else if (a instanceof Fraction) {
|
||
|
|
return Integer.cdiv(a.num, a.den);
|
||
|
|
} else {
|
||
|
|
return Float.ceil(a);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
function sqrt(a)
|
||
|
|
{
|
||
|
|
var t, u, re, im;
|
||
|
|
if (a instanceof Series) {
|
||
|
|
return a ^ (1/2);
|
||
|
|
} else if (a instanceof Complex) {
|
||
|
|
t = abs(a);
|
||
|
|
u = a.re;
|
||
|
|
re = sqrt((t + u) / 2);
|
||
|
|
im = sqrt((t - u) / 2);
|
||
|
|
if (a.im < 0)
|
||
|
|
im = -im;
|
||
|
|
return Complex.toComplex(re, im);
|
||
|
|
} else {
|
||
|
|
a = Float(a);
|
||
|
|
if (a < 0) {
|
||
|
|
return Complex(0, Float.sqrt(-a));
|
||
|
|
} else {
|
||
|
|
return Float.sqrt(a);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
function exp(a)
|
||
|
|
{
|
||
|
|
return a.exp();
|
||
|
|
}
|
||
|
|
|
||
|
|
function log(a)
|
||
|
|
{
|
||
|
|
return a.log();
|
||
|
|
}
|
||
|
|
|
||
|
|
function log2(a)
|
||
|
|
{
|
||
|
|
return log(a) * Float.LOG2E;
|
||
|
|
}
|
||
|
|
|
||
|
|
function log10(a)
|
||
|
|
{
|
||
|
|
return log(a) * Float.LOG10E;
|
||
|
|
}
|
||
|
|
|
||
|
|
function todb(a)
|
||
|
|
{
|
||
|
|
return log10(a) * 10;
|
||
|
|
}
|
||
|
|
|
||
|
|
function fromdb(a)
|
||
|
|
{
|
||
|
|
return 10 ^ (a / 10);
|
||
|
|
}
|
||
|
|
|
||
|
|
function sin(a)
|
||
|
|
{
|
||
|
|
var t;
|
||
|
|
if (a instanceof Complex || a instanceof Series) {
|
||
|
|
t = exp(a * I);
|
||
|
|
return (t - 1/t) / (2 * I);
|
||
|
|
} else {
|
||
|
|
return Float.sin(Float(a));
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
function cos(a)
|
||
|
|
{
|
||
|
|
var t;
|
||
|
|
if (a instanceof Complex || a instanceof Series) {
|
||
|
|
t = exp(a * I);
|
||
|
|
return (t + 1/t) / 2;
|
||
|
|
} else {
|
||
|
|
return Float.cos(Float(a));
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
function tan(a)
|
||
|
|
{
|
||
|
|
if (a instanceof Complex || a instanceof Series) {
|
||
|
|
return sin(a) / cos(a);
|
||
|
|
} else {
|
||
|
|
return Float.tan(Float(a));
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
function asin(a)
|
||
|
|
{
|
||
|
|
return Float.asin(Float(a));
|
||
|
|
}
|
||
|
|
|
||
|
|
function acos(a)
|
||
|
|
{
|
||
|
|
return Float.acos(Float(a));
|
||
|
|
}
|
||
|
|
|
||
|
|
function atan(a)
|
||
|
|
{
|
||
|
|
return Float.atan(Float(a));
|
||
|
|
}
|
||
|
|
|
||
|
|
function atan2(a, b)
|
||
|
|
{
|
||
|
|
return Float.atan2(Float(a), Float(b));
|
||
|
|
}
|
||
|
|
|
||
|
|
function sinc(a)
|
||
|
|
{
|
||
|
|
if (a == 0) {
|
||
|
|
return 1;
|
||
|
|
} else {
|
||
|
|
a *= Float.PI;
|
||
|
|
return sin(a) / a;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
function todeg(a)
|
||
|
|
{
|
||
|
|
return a * 180 / Float.PI;
|
||
|
|
}
|
||
|
|
|
||
|
|
function fromdeg(a)
|
||
|
|
{
|
||
|
|
return a * Float.PI / 180;
|
||
|
|
}
|
||
|
|
|
||
|
|
var idn = Matrix.idn;
|
||
|
|
var diag = Matrix.diag;
|
||
|
|
var trans = Matrix.trans;
|
||
|
|
var trace = Matrix.trace;
|
||
|
|
var charpoly = Matrix.charpoly;
|
||
|
|
var eigenvals = Matrix.eigenvals;
|
||
|
|
var det = Matrix.det;
|
||
|
|
var rank = Matrix.rank;
|
||
|
|
var ker = Matrix.ker;
|
||
|
|
var cp = Matrix.cp;
|
||
|
|
var dp = Matrix.dp;
|
||
|
|
|
||
|
|
var polroots = Polynomial.roots;
|
||
|
|
var bestappr = Float.bestappr;
|