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/*
* Arithmetic on polynomials with (integer) coefficients
*
* Copyright © 2002 Bart Massey.
* All Rights Reserved. See the file COPYING in this directory
* for licensing information.
*/
namespace Polynomial {
/*
* Change the typedef to rational for rational
* coefficients, then walk carefully through code below.
*/
typedef int coefficient;
/*
* Polynomials are stored as an array of
* coefficients indexed by degree. There
* is always a constant term, and the leading
* term of a polynomial is always nonzero (except
* for the zero polynomial).
*/
public typedef coefficient[*] polynomial;
/*
* Bug: There's currently no Nickle way to normalize the
* polynomial by shortening the array. This should be
* fixed. In the meantime, we will just make a copy.
*/
polynomial normalize(polynomial p) {
int n = dim(p) - 1;
if (p[n] != 0)
return p;
while (n > 0 && p[n] == 0)
--n;
coefficient[n + 1] q = { [i] = p[i] };
return q;
}
/*
* Addition is performed using an array comprehension
* block. I'm not sure whether this is better or worse
* than just doing it with a loop.
*/
public polynomial add(polynomial a, polynomial b) {
coefficient na = dim(a);
coefficient nb = dim(b);
coefficient nc = max(na, nb);
coefficient[nc] c = { [i] {
coefficient get_coeff(coefficient[*] a) {
if (i < dim(a))
return a[i];
return 0;
}
coefficient ai = get_coeff(a);
coefficient bi = get_coeff(b);
return ai + bi;
} };
return normalize(c);
}
/*
* Multiplication efficiency might be minutely improved by
* avoiding the all-zeros initialization. Or it might
* not...
*/
public polynomial mult(polynomial a, polynomial b) {
coefficient na = dim(a);
coefficient nb = dim(b);
coefficient nc = na + nb - 1;
coefficient[nc] c = {0 ...};
for (coefficient i = 0; i < na; i++)
for (coefficient j = 0; j < nb; j++)
c[i + j] += a[i] * b[j];
return c;
}
/*
* The quotient and remainder for polynomial division are
* computed simultaneously, and thus should be returned
* together.
*/
public typedef struct {
polynomial quot;
polynomial rem;
} quot_rem;
/*
* Return quotient and remainder of a div b. This code
* assumes that with integer coefficients, leading
* coefficient of b must be 1.
*/
public quot_rem div_mod(polynomial a, polynomial b) {
coefficient na = dim(a);
coefficient nb = dim(b);
if (b[nb - 1] != 1)
abort("leading integer coefficient in divisor should be 1");
coefficient nq = na - nb + 1;
if (nq <= 0)
return (quot_rem) { .quot = (polynomial){0}, .rem = b };
coefficient[nq] q = {0 ...};
while (na >= nb) {
coefficient xn = na - nb;
coefficient cx = a[na - 1] / b[nb - 1];
for (int i = 0; i < nb; i++)
a[i + xn] -= cx * b[i];
q[xn] += cx;
if (a[na - 1] != 0)
abort("leading coefficient error");
--na;
while (na >= 1 && a[na - 1] == 0)
--na;
}
return (quot_rem) { .quot = q, .rem = normalize(a) };
}
/*
* Returns b ** e computed using the usual fast algorithm
* (O(log e) multiplies).
*/
public polynomial
power(polynomial b, int e) {
if (e < 0)
abort("negative exponent error");
if (e == 0)
return (polynomial) {1};
if (e == 1)
return b;
polynomial tmp = power(b, e // 2);
polynomial tmp2 = mult(tmp, tmp);
if (e % 2 == 0)
return tmp2;
return mult(tmp2, b);
}
/*
* Returns b ** e (mod m) computed using the usual
* fast algorithm (O(log e) multiplies, each requiring
* O(|m|*|b**2|) primitive steps).
*/
public polynomial
power_mod(polynomial b, int e, polynomial m) {
if (e < 0)
abort("negative exponent error");
if (e == 0)
return (polynomial) {1};
if (e == 1)
return div_mod(b, m).rem;
polynomial tmp = power_mod(b, e // 2, m);
polynomial tmp2 = mult(tmp, tmp);
if (e % 2 == 0)
return div_mod(tmp2, m).rem;
return div_mod(mult(tmp2, b), m).rem;
}
/*
* Returns a string representation of p suitable for Maple
* input. x is the name of the polynomial basis variable
* (usually just "x"). There's a lot of boundary cases
* here.
*/
public string maple(polynomial p, string x) {
string s = "";
bool started = false;
for (int i = 0; i < dim(p); i++) {
/* Don't print absent terms. */
if (p[i] == 0)
continue;
/* If a term has already been printed, this one
must be added or subtracted */
if (started) {
if (p[i] > 0) {
s += " + ";
} else {
s += " - ";
p[i] = -p[i];
}
}
/* Don't print 1*x or 1*x^n. */
string star = "*";
if (p[i] == 1 && i > 0)
star = "";
else
s += sprintf("%d", p[i]);
/* Don't print x^0 or x^1. If there's no leading
coefficient due to previous logic, don't print
the multiply symbol. */
if (i == 1)
s += sprintf("%s%s", star, x);
else if (i > 1)
s += sprintf("%s%s^%d", star, x, i);
/* A term has now been printed */
started = true;
}
return s;
}
}
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