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quickjs/qjscalc.js

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/*
* QuickJS Javascript Calculator
*
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* Copyright (c) 2017-2020 Fabrice Bellard
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*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
* THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
"use strict";
"use math";
var Integer, Float, Fraction, Complex, Mod, Polynomial, PolyMod, RationalFunction, Series, Matrix;
(function(global) {
global.Integer = global.BigInt;
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global.Float = global.BigFloat;
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global.algebraicMode = true;
/* add non enumerable properties */
function add_props(obj, props) {
var i, val, prop, tab, desc;
tab = Reflect.ownKeys(props);
for(i = 0; i < tab.length; i++) {
prop = tab[i];
desc = Object.getOwnPropertyDescriptor(props, prop);
desc.enumerable = false;
if ("value" in desc) {
if (typeof desc.value !== "function") {
desc.writable = false;
desc.configurable = false;
}
} else {
/* getter/setter */
desc.configurable = false;
}
Object.defineProperty(obj, prop, desc);
}
}
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/* same as proto[Symbol.operatorSet] = Operators.create(..op_list)
but allow shortcuts: left: [], right: [] or both
*/
function operators_set(proto, ...op_list)
{
var new_op_list, i, a, j, b, k, obj, tab;
var fields = [ "left", "right" ];
new_op_list = [];
for(i = 0; i < op_list.length; i++) {
a = op_list[i];
if (a.left || a.right) {
tab = [ a.left, a.right ];
delete a.left;
delete a.right;
for(k = 0; k < 2; k++) {
obj = tab[k];
if (obj) {
if (!Array.isArray(obj)) {
obj = [ obj ];
}
for(j = 0; j < obj.length; j++) {
b = {};
Object.assign(b, a);
b[fields[k]] = obj[j];
new_op_list.push(b);
}
}
}
} else {
new_op_list.push(a);
}
}
proto[Symbol.operatorSet] =
Operators.create.call(null, ...new_op_list);
}
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/* Integer */
function generic_pow(a, b) {
var r, is_neg, i;
if (!Integer.isInteger(b)) {
return exp(log(a) * b);
}
if (Array.isArray(a) && !(a instanceof Polynomial ||
a instanceof Series)) {
r = idn(Matrix.check_square(a));
} else {
r = 1;
}
if (b == 0)
return r;
is_neg = false;
if (b < 0) {
is_neg = true;
b = -b;
}
r = a;
for(i = Integer.floorLog2(b) - 1; i >= 0; i--) {
r *= r;
if ((b >> i) & 1)
r *= a;
}
if (is_neg) {
if (typeof r.inverse != "function")
throw "negative powers are not supported for this type";
r = r.inverse();
}
return r;
}
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 ];
function miller_rabin_test(n, t) {
var d, r, s, i, j, a;
d = n - 1;
s = 0;
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while ((d & 1) == 0) {
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d >>= 1;
s++;
}
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if (small_primes.length < t)
t = small_primes.length;
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loop: for(j = 0; j < t; j++) {
a = small_primes[j];
r = Integer.pmod(a, d, n);
if (r == 1 || r == (n - 1))
continue;
for(i = 1; i < s; i++) {
r = (r * r) % n;
if (r == 1)
return false;
if (r == (n - 1))
continue loop;
}
return false; /* n is composite */
}
return true; /* n is probably prime with probability (1-0.5^t) */
}
function fact_rec(a, b) { /* assumes a <= b */
var i, r;
if ((b - a) <= 5) {
r = a;
for(i = a + 1; i <= b; i++)
r *= i;
return r;
} else {
/* to avoid a quadratic running time it is better to
multiply numbers of similar size */
i = (a + b) >> 1;
return fact_rec(a, i) * fact_rec(i + 1, b);
}
}
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/* math mode specific quirk to overload the integer division and power */
Operators.updateBigIntOperators(
{
"/"(a, b) {
if (algebraicMode) {
return Fraction.toFraction(a, b);
} else {
return Float(a) / Float(b);
}
},
"**"(a, b) {
if (algebraicMode) {
return generic_pow(a, b);
} else {
return Float(a) ** Float(b);
}
}
});
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add_props(Integer, {
isInteger(a) {
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/* integers are represented either as bigint or as number */
return typeof a === "bigint" ||
(typeof a === "number" && Number.isSafeInteger(a));
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},
gcd(a, b) {
var r;
while (b != 0) {
r = a % b;
a = b;
b = r;
}
return a;
},
fact(n) {
return n <= 0 ? 1 : fact_rec(1, n);
},
/* binomial coefficient */
comb(n, k) {
if (k < 0 || k > n)
return 0;
if (k > n - k)
k = n - k;
if (k == 0)
return 1;
return Integer.tdiv(fact_rec(n - k + 1, n), fact_rec(1, k));
},
/* inverse of x modulo y */
invmod(x, y) {
var q, u, v, a, c, t;
u = x;
v = y;
c = 1;
a = 0;
while (u != 0) {
t = Integer.fdivrem(v, u);
q = t[0];
v = u;
u = t[1];
t = c;
c = a - q * c;
a = t;
}
/* v = gcd(x, y) */
if (v != 1)
throw RangeError("not invertible");
return a % y;
},
/* return a ^ b modulo m */
pmod(a, b, m) {
var r;
if (b == 0)
return 1;
if (b < 0) {
a = Integer.invmod(a, m);
b = -b;
}
r = 1;
for(;;) {
if (b & 1) {
r = (r * a) % m;
}
b >>= 1;
if (b == 0)
break;
a = (a * a) % m;
}
return r;
},
/* return true if n is prime (or probably prime with
probability 1-0.5^t) */
isPrime(n, t) {
var i, d, n1;
if (!Integer.isInteger(n))
throw TypeError("invalid type");
if (n <= 1)
return false;
n1 = small_primes.length;
/* XXX: need Integer.sqrt() */
for(i = 0; i < n1; i++) {
d = small_primes[i];
if (d == n)
return true;
if (d > n)
return false;
if ((n % d) == 0)
return false;
}
if (n < d * d)
return true;
if (typeof t == "undefined")
t = 64;
return miller_rabin_test(n, t);
},
nextPrime(n) {
if (!Integer.isInteger(n))
throw TypeError("invalid type");
if (n < 1)
n = 1;
for(;;) {
n++;
if (Integer.isPrime(n))
return n;
}
},
factor(n) {
var r, d;
if (!Integer.isInteger(n))
throw TypeError("invalid type");
r = [];
if (abs(n) <= 1) {
r.push(n);
return r;
}
if (n < 0) {
r.push(-1);
n = -n;
}
while ((n % 2) == 0) {
n >>= 1;
r.push(2);
}
d = 3;
while (n != 1) {
if (Integer.isPrime(n)) {
r.push(n);
break;
}
/* we are sure there is at least one divisor, so one test */
for(;;) {
if ((n % d) == 0)
break;
d += 2;
}
for(;;) {
r.push(d);
n = Integer.tdiv(n, d);
if ((n % d) != 0)
break;
}
}
return r;
},
});
add_props(Integer.prototype, {
inverse() {
return 1 / this;
},
norm2() {
return this * this;
},
abs() {
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var v = this;
if (v < 0)
v = -v;
return v;
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},
conj() {
return this;
},
arg() {
if (this >= 0)
return 0;
else
return Float.PI;
},
exp() {
if (this == 0)
return 1;
else
return Float.exp(this);
},
log() {
if (this == 1)
return 0;
else
return Float(this).log();
},
});
/* Fraction */
Fraction = function Fraction(a, b)
{
var d, r, obj;
if (new.target)
throw TypeError("not a constructor");
if (a instanceof Fraction)
return a;
if (!Integer.isInteger(a))
throw TypeError("integer expected");
if (typeof b === "undefined") {
b = 1;
} else {
if (!Integer.isInteger(b))
throw TypeError("integer expected");
if (b == 0)
throw RangeError("division by zero");
d = Integer.gcd(a, b);
if (d != 1) {
a = Integer.tdiv(a, d);
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;
}
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function fraction_add(a, b) {
a = Fraction(a);
b = Fraction(b);
return Fraction.toFraction(a.num * b.den + a.den * b.num, a.den * b.den);
}
function fraction_sub(a, b) {
a = Fraction(a);
b = Fraction(b);
return Fraction.toFraction(a.num * b.den - a.den * b.num, a.den * b.den);
}
function fraction_mul(a, b) {
a = Fraction(a);
b = Fraction(b);
return Fraction.toFraction(a.num * b.num, a.den * b.den);
}
function fraction_div(a, b) {
a = Fraction(a);
b = Fraction(b);
return Fraction.toFraction(a.num * b.den, a.den * b.num);
}
function fraction_mod(a, b) {
var a1 = Fraction(a);
var b1 = Fraction(b);
return a - Integer.ediv(a1.num * b1.den, a1.den * b1.num) * b;
}
function fraction_eq(a, b) {
a = Fraction(a);
b = Fraction(b);
/* we assume the fractions are normalized */
return (a.num == b.num && a.den == b.den);
}
function fraction_lt(a, b) {
a = Fraction(a);
b = Fraction(b);
return (a.num * b.den < b.num * a.den);
}
/* operators are needed for fractions */
function float_add(a, b) {
return Float(a) + Float(b);
}
function float_sub(a, b) {
return Float(a) - Float(b);
}
function float_mul(a, b) {
return Float(a) * Float(b);
}
function float_div(a, b) {
return Float(a) / Float(b);
}
function float_mod(a, b) {
return Float(a) % Float(b);
}
function float_pow(a, b) {
return Float(a) ** Float(b);
}
function float_eq(a, b) {
/* XXX: may be better to use infinite precision for the comparison */
return Float(a) === Float(b);
}
function float_lt(a, b) {
a = Float(a);
b = Float(b);
/* XXX: may be better to use infinite precision for the comparison */
if (Float.isNaN(a) || Float.isNaN(b))
return undefined;
else
return a < b;
}
operators_set(Fraction.prototype,
{
"+": fraction_add,
"-": fraction_sub,
"*": fraction_mul,
"/": fraction_div,
"%": fraction_mod,
"**": generic_pow,
"==": fraction_eq,
"<": fraction_lt,
"pos"(a) {
return a;
},
"neg"(a) {
return Fraction(-a.num, a.den);
},
},
{
left: [Number, BigInt],
right: [Number, BigInt],
"+": fraction_add,
"-": fraction_sub,
"*": fraction_mul,
"/": fraction_div,
"%": fraction_mod,
"**": generic_pow,
"==": fraction_eq,
"<": fraction_lt,
},
{
left: Float,
right: Float,
"+": float_add,
"-": float_sub,
"*": float_mul,
"/": float_div,
"%": float_mod,
"**": float_pow,
"==": float_eq,
"<": float_lt,
});
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add_props(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;
},
});
add_props(Fraction.prototype, {
[Symbol.toPrimitive](hint) {
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if (hint === "string") {
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return this.toString();
} else {
return Float(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)
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return -this;
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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.prototype, {
inverse() {
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return 1 / this;
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},
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 */
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.prototype, {
inverse() {
return 1.0 / this;
},
norm2() {
return this * this;
},
abs() {
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return Float.abs(this);
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},
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;
}
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function complex_add(a, b) {
a = Complex(a);
b = Complex(b);
return Complex.toComplex(a.re + b.re, a.im + b.im);
}
function complex_sub(a, b) {
a = Complex(a);
b = Complex(b);
return Complex.toComplex(a.re - b.re, a.im - b.im);
}
function complex_mul(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);
}
function complex_div(a, b) {
a = Complex(a);
b = Complex(b);
return a * b.inverse();
}
function complex_eq(a, b) {
a = Complex(a);
b = Complex(b);
return a.re == b.re && a.im == b.im;
}
operators_set(Complex.prototype,
{
"+": complex_add,
"-": complex_sub,
"*": complex_mul,
"/": complex_div,
"**": generic_pow,
"==": complex_eq,
"pos"(a) {
return a;
},
"neg"(a) {
return Complex(-a.re, -a.im);
}
},
{
left: [Number, BigInt, Float, Fraction],
right: [Number, BigInt, Float, Fraction],
"+": complex_add,
"-": complex_sub,
"*": complex_mul,
"/": complex_div,
"**": generic_pow,
"==": complex_eq,
});
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add_props(Complex, {
/* simplify to real number when possible */
toComplex(re, im) {
if (algebraicMode && im == 0)
return re;
else
return Complex(re, im);
},
});
add_props(Complex.prototype, {
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;
};
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function mod_add(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);
}
}
function mod_sub(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);
}
}
function mod_mul(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);
}
}
function mod_div(a, b) {
if (!(b instanceof Mod))
b = Mod(b, a.mod);
return mod_mul(a, b.inverse());
}
function mod_eq(a, b) {
return (a.mod == b.mod && a.res == b.res);
}
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operators_set(Mod.prototype,
{
"+": mod_add,
"-": mod_sub,
"*": mod_mul,
"/": mod_div,
"**": generic_pow,
"==": mod_eq,
"pos"(a) {
return a;
},
"neg"(a) {
return Mod(-a.res, a.mod);
}
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},
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{
left: [Number, BigInt, Float, Fraction],
right: [Number, BigInt, Float, Fraction],
"+": mod_add,
"-": mod_sub,
"*": mod_mul,
"/": mod_div,
"**": generic_pow,
});
add_props(Mod.prototype, {
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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 + ")";
},
});
/* Polynomial */
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function polynomial_is_scalar(a)
{
if (typeof a === "number" ||
typeof a === "bigint" ||
typeof a === "bigfloat")
return true;
if (a instanceof Fraction ||
a instanceof Complex ||
a instanceof Mod)
return true;
return false;
}
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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();
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} else if (polynomial_is_scalar(a)) {
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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;
},
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;
},
});
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function polynomial_add(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);
}
function polynomial_sub(a, b) {
return polynomial_add(a, -b);
}
function polynomial_mul(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];
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}
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}
return Polynomial(r);
}
function polynomial_div_scalar(a, b) {
return a * (1 / b);
}
function polynomial_div(a, b)
{
return RationalFunction(Polynomial(a),
Polynomial(b));
}
function polynomial_mod(a, b) {
return Polynomial.divrem(a, b)[1];
}
function polynomial_eq(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])
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return false;
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}
return true;
}
operators_set(Polynomial.prototype,
{
"+": polynomial_add,
"-": polynomial_sub,
"*": polynomial_mul,
"/": polynomial_div,
"**": generic_pow,
"==": polynomial_eq,
"pos"(a) {
return a;
},
"neg"(a) {
var r, i, n, a;
n = a.length;
r = [];
for(i = 0; i < n; i++)
r[i] = -a[i];
return Polynomial(r);
},
},
{
left: [Number, BigInt, Float, Fraction, Complex, Mod],
"+": polynomial_add,
"-": polynomial_sub,
"*": polynomial_mul,
"/": polynomial_div,
"**": generic_pow, /* XXX: only for integer */
},
{
right: [Number, BigInt, Float, Fraction, Complex, Mod],
"+": polynomial_add,
"-": polynomial_sub,
"*": polynomial_mul,
"/": polynomial_div_scalar,
"**": generic_pow, /* XXX: only for integer */
});
add_props(Polynomial, {
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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;
};
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function polymod_add(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);
}
}
function polymod_sub(a, b) {
return polymod_add(a, -b);
}
function polymod_mul(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);
}
}
function polymod_div(a, b) {
if (!(b instanceof PolyMod))
b = PolyMod(b, a.mod);
return polymod_mul(a, b.inverse());
}
function polymod_eq(a, b) {
return (a.mod == b.mod && a.res == b.res);
}
operators_set(PolyMod.prototype,
{
"+": polymod_add,
"-": polymod_sub,
"*": polymod_mul,
"/": polymod_div,
"**": generic_pow,
"==": polymod_eq,
"pos"(a) {
return a;
},
"neg"(a) {
return PolyMod(-a.res, a.mod);
},
},
{
left: [Number, BigInt, Float, Fraction, Complex, Mod, Polynomial],
right: [Number, BigInt, Float, Fraction, Complex, Mod, Polynomial],
"+": polymod_add,
"-": polymod_sub,
"*": polymod_mul,
"/": polymod_div,
"**": generic_pow, /* XXX: only for integer */
});
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add_props(PolyMod.prototype, {
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 + ")";
},
});
/* 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, {
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);
},
});
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function ratfunc_add(a, b) {
a = RationalFunction.toRationalFunction(a);
b = RationalFunction.toRationalFunction(b);
return RationalFunction(a.num * b.den + a.den * b.num, a.den * b.den);
}
function ratfunc_sub(a, b) {
a = RationalFunction.toRationalFunction(a);
b = RationalFunction.toRationalFunction(b);
return RationalFunction(a.num * b.den - a.den * b.num, a.den * b.den);
}
function ratfunc_mul(a, b) {
a = RationalFunction.toRationalFunction(a);
b = RationalFunction.toRationalFunction(b);
return RationalFunction(a.num * b.num, a.den * b.den);
}
function ratfunc_div(a, b) {
a = RationalFunction.toRationalFunction(a);
b = RationalFunction.toRationalFunction(b);
return RationalFunction(a.num * b.den, a.den * b.num);
}
function ratfunc_eq(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);
}
operators_set(RationalFunction.prototype,
{
"+": ratfunc_add,
"-": ratfunc_sub,
"*": ratfunc_mul,
"/": ratfunc_div,
"**": generic_pow,
"==": ratfunc_eq,
"pos"(a) {
return a;
},
"neg"(a) {
return RationalFunction(-this.num, this.den);
},
},
{
left: [Number, BigInt, Float, Fraction, Complex, Mod, Polynomial],
right: [Number, BigInt, Float, Fraction, Complex, Mod, Polynomial],
"+": ratfunc_add,
"-": ratfunc_sub,
"*": ratfunc_mul,
"/": ratfunc_div,
"**": generic_pow, /* should only be used with integers */
});
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add_props(RationalFunction, {
/* 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;
}
},
});
/* 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;
};
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function series_is_scalar_or_polynomial(a)
{
return polynomial_is_scalar(a) ||
(a instanceof Polynomial);
}
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/* 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;
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} else if (series_is_scalar_or_polynomial(a)) {
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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");
}
};
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function series_add(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 (series_is_scalar_or_polynomial(v2)) {
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();
}
function series_sub(a, b) {
return series_add(a, -b);
}
function series_mul(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];
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}
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}
return r.trim();
}
function series_div(v1, v2) {
if (!(v2 instanceof Series))
v2 = Series(v2, v1.length);
return series_mul(v1, v2.inverse());
}
function series_pow(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);
}
}
function series_eq(a, b) {
var n, i;
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;
}
operators_set(Series.prototype,
{
"+": series_add,
"-": series_sub,
"*": series_mul,
"/": series_div,
"**": series_pow,
"==": series_eq,
"pos"(a) {
return a;
},
"neg"(a) {
var obj, n, i;
n = a.length;
obj = Series.zero(a.length, a.emin);
for(i = 0; i < n; i++) {
obj[i] = -a[i];
}
return obj;
},
},
{
left: [Number, BigInt, Float, Fraction, Complex, Mod, Polynomial],
right: [Number, BigInt, Float, Fraction, Complex, Mod, Polynomial],
"+": series_add,
"-": series_sub,
"*": series_mul,
"/": series_div,
"**": series_pow,
});
add_props(Series.prototype, {
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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)
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throw RangeError("cannot represent integ(1/X)");
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} else {
r[i] = a[i] / (j + 1);
}
}
return r.trim();
},
exp() {
var c, i, r, n, a = this;
if (a.emin < 0)
throw RangeError("negative exponent in exp");
n = a.emin + a.length;
if (a.emin > 0 || a[0] == 0) {
c = 1;
} else {
c = global.exp(a[0]);
a -= a[0];
}
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)
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throw RangeError("log argument must have a non zero constant term");
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r = integ(deriv(a) / a);
/* add the constant term */
r += global.log(a[0]);
return r;
},
});
add_props(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;
},
O(a) {
function ErrorO() {
return TypeError("invalid O() argument");
}
var n;
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if (series_is_scalar_or_polynomial(a)) {
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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;
},
});
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function array_add(a, b) {
var r, i, n;
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;
}
function array_sub(a, b) {
var r, i, n;
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;
}
function array_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;
}
function array_mul(a, b) {
var h, w, l, i, j, k, r, rl, sum, a_mat, b_mat;
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++) {
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sum = 0;
for(k = 0; k < l; k++) {
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sum += a[i][k] * b[k][j];
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}
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rl[j] = sum;
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}
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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];
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}
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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];
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}
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r[i] = rl;
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}
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} else {
for(i = 0; i < h; i++) {
r[i] = a[i] * b[0];
}
}
return r;
}
function array_div(a, b) {
return array_mul(a, b.inverse());
}
function array_scalar_div(a, b) {
return a * b.inverse();
}
function array_eq(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])
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return false;
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}
return true;
}
operators_set(Array.prototype,
{
"+": array_add,
"-": array_sub,
"*": array_mul,
"/": array_div,
"==": array_eq,
"pos"(a) {
return a;
},
"neg"(a) {
var i, n, r;
n = a.length;
r = [];
for(i = 0; i < n; i++)
r[i] = -a[i];
return r;
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}
},
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{
right: [Number, BigInt, Float, Fraction, Complex, Mod,
Polynomial, PolyMod, RationalFunction, Series],
"*": array_scalar_mul,
"/": array_scalar_div,
"**": generic_pow, /* XXX: only for integer */
},
{
left: [Number, BigInt, Float, Fraction, Complex, Mod,
Polynomial, PolyMod, RationalFunction, Series],
"*"(a, b) { return array_scalar_mul(b, a); },
"/"(a, b) { return array_scalar_div(b, a); },
});
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add_props(Array.prototype, {
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;
}
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function sinh(a)
{
var e = Float.exp(Float(a));
return (e - 1/e) * 0.5;
}
function cosh(a)
{
var e = Float.exp(Float(a));
return (e + 1/e) * 0.5;
}
function tanh(a)
{
var e = Float.exp(Float(a) * 2);
return (e - 1) / (e + 1);
}
function asinh(a)
{
var x = Float(a);
return log(sqrt(x * x + 1) + x);
}
function acosh(a)
{
var x = Float(a);
return log(sqrt(x * x - 1) + x);
}
function atanh(a)
{
var x = Float(a);
return 0.5 * log((1 + x) / (1 - x));
}
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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;
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