Polynomial long division
You are encouraged to solve this task according to the task description, using any language you may know.
This page uses content from Wikipedia. The original article was at Polynomial long division. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
- In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree.
Let us suppose a polynomial is represented by a vector, (i.e., an ordered collection of coefficients) so that the th element keeps the coefficient of , and the multiplication by a monomial is a shift of the vector's elements "towards right" (injecting ones from left) followed by a multiplication of each element by the coefficient of the monomial.
Then a pseudocode for the polynomial long division using the conventions described above could be:
degree(P): return the index of the last non-zero element of P; if all elements are 0, return -∞ polynomial_long_division(N, D) returns (q, r): // N, D, q, r are vectors if degree(D) < 0 then error q ← 0 while degree(N) ≥ degree(D) d ← D shifted right by (degree(N) - degree(D)) q(degree(N) - degree(D)) ← N(degree(N)) / d(degree(d)) // by construction, degree(d) = degree(N) of course d ← d * q(degree(N) - degree(D)) N ← N - d endwhile r ← N return (q, r)
Note: vector * scalar
multiplies each element of the vector by the scalar; vectorA - vectorB
subtracts each element of the vectorB from the element of the vectorA with "the same index". The vectors in the pseudocode are zero-based.
- Error handling (for allocations or for wrong inputs) is not mandatory.
- Conventions can be different; in particular, note that if the first coefficient in the vector is the highest power of x for the polynomial represented by the vector, then the algorithm becomes simpler.
Example for clarification
This example is from Wikipedia, but changed to show how the given pseudocode works.
0 1 2 3 ---------------------- N: -42 0 -12 1 degree = 3 D: -3 1 0 0 degree = 1 d(N) - d(D) = 2, so let's shift D towards right by 2: N: -42 0 -12 1 d: 0 0 -3 1 N(3)/d(3) = 1, so d is unchanged. Now remember that "shifting by 2" is like multiplying by x2, and the final multiplication (here by 1) is the coefficient of this monomial. Let's store this into q: 0 1 2 --------------- q: 0 0 1 now compute N - d, and let it be the "new" N, and let's loop N: -42 0 -9 0 degree = 2 D: -3 1 0 0 degree = 1 d(N) - d(D) = 1, right shift D by 1 and let it be d N: -42 0 -9 0 d: 0 -3 1 0 * -9/1 = -9 q: 0 -9 1 d: 0 27 -9 0 N ← N - d N: -42 -27 0 0 degree = 1 D: -3 1 0 0 degree = 1 looping again... d(N)-d(D)=0, so no shift is needed; we multiply D by -27 (= -27/1) storing the result in d, then q: -27 -9 1 and N: -42 -27 0 0 - d: 81 -27 0 0 = N: -123 0 0 0 (last N) d(N) < d(D), so now r ← N, and the result is: 0 1 2 ------------- q: -27 -9 1 → x2 - 9x - 27 r: -123 0 0 → -123
Ada
long_division.adb: <lang Ada>with Ada.Text_IO; use Ada.Text_IO;
procedure Long_Division is
package Int_IO is new Ada.Text_IO.Integer_IO (Integer); use Int_IO;
type Degrees is range -1 .. Integer'Last; subtype Valid_Degrees is Degrees range 0 .. Degrees'Last; type Polynom is array (Valid_Degrees range <>) of Integer;
function Degree (P : Polynom) return Degrees is begin for I in reverse P'Range loop if P (I) /= 0 then return I; end if; end loop; return -1; end Degree;
function Shift_Right (P : Polynom; D : Valid_Degrees) return Polynom is Result : Polynom (0 .. P'Last + D) := (others => 0); begin Result (Result'Last - P'Length + 1 .. Result'Last) := P; return Result; end Shift_Right;
function "*" (Left : Polynom; Right : Integer) return Polynom is Result : Polynom (Left'Range); begin for I in Result'Range loop Result (I) := Left (I) * Right; end loop; return Result; end "*";
function "-" (Left, Right : Polynom) return Polynom is Result : Polynom (Left'Range); begin for I in Result'Range loop if I in Right'Range then Result (I) := Left (I) - Right (I); else Result (I) := Left (I); end if; end loop; return Result; end "-";
procedure Poly_Long_Division (Num, Denom : Polynom; Q, R : out Polynom) is N : Polynom := Num; D : Polynom := Denom; begin if Degree (D) < 0 then raise Constraint_Error; end if; Q := (others => 0); while Degree (N) >= Degree (D) loop declare T : Polynom := Shift_Right (D, Degree (N) - Degree (D)); begin Q (Degree (N) - Degree (D)) := N (Degree (N)) / T (Degree (T)); T := T * Q (Degree (N) - Degree (D)); N := N - T; end; end loop; R := N; end Poly_Long_Division;
procedure Output (P : Polynom) is First : Boolean := True; begin for I in reverse P'Range loop if P (I) /= 0 then if First then First := False; else Put (" + "); end if; if I > 0 then if P (I) /= 1 then Put (P (I), 0); Put ("*"); end if; Put ("x"); if I > 1 then Put ("^"); Put (Integer (I), 0); end if; elsif P (I) /= 0 then Put (P (I), 0); end if; end if; end loop; New_Line; end Output;
Test_N : constant Polynom := (0 => -42, 1 => 0, 2 => -12, 3 => 1); Test_D : constant Polynom := (0 => -3, 1 => 1); Test_Q : Polynom (Test_N'Range); Test_R : Polynom (Test_N'Range);
begin
Poly_Long_Division (Test_N, Test_D, Test_Q, Test_R); Put_Line ("Dividing Polynoms:"); Put ("N: "); Output (Test_N); Put ("D: "); Output (Test_D); Put_Line ("-------------------------"); Put ("Q: "); Output (Test_Q); Put ("R: "); Output (Test_R);
end Long_Division;</lang>
output:
Dividing Polynoms: N: x^3 + -12*x^2 + -42 D: x + -3 ------------------------- Q: x^2 + -9*x + -27 R: -123
APL
<lang APL>div←{
{ q r d←⍵ (≢d) > n←≢r : q r c ← (⊃⌽r) ÷ ⊃⌽d ∇ (c,q) ((¯1↓r) - c × ¯1↓(-n)↑d) d } ⍬ ⍺ ⍵
} </lang>
- Output:
N←¯42 0 ¯12 1 D←¯3 1 ⍪N div D ¯27 ¯9 1 ¯123
BBC BASIC
<lang bbcbasic> DIM N%(3) : N%() = -42, 0, -12, 1
DIM D%(3) : D%() = -3, 1, 0, 0 DIM q%(3), r%(3) PROC_poly_long_div(N%(), D%(), q%(), r%()) PRINT "Quotient = "; FNcoeff(q%(2)) "x^2" FNcoeff(q%(1)) "x" FNcoeff(q%(0)) PRINT "Remainder = " ; r%(0) END DEF PROC_poly_long_div(N%(), D%(), q%(), r%()) LOCAL d%(), i%, s% DIM d%(DIM(N%(),1)) s% = FNdegree(N%()) - FNdegree(D%()) IF s% >= 0 THEN q%() = 0 WHILE s% >= 0 FOR i% = 0 TO DIM(d%(),1) - s% d%(i%+s%) = D%(i%) NEXT q%(s%) = N%(FNdegree(N%())) DIV d%(FNdegree(d%())) d%() = d%() * q%(s%) N%() -= d%() s% = FNdegree(N%()) - FNdegree(D%()) ENDWHILE r%() = N%() ELSE q%() = 0 r%() = N%() ENDIF ENDPROC DEF FNdegree(a%()) LOCAL i% i% = DIM(a%(),1) WHILE a%(i%)=0 i% -= 1 IF i%<0 EXIT WHILE ENDWHILE = i% DEF FNcoeff(n%) IF n%=0 THEN = "" IF n%<0 THEN = " - " + STR$(-n%) IF n%=1 THEN = " + " = " + " + STR$(n%)</lang>
Output:
Quotient = + x^2 - 9x - 27 Remainder = -123
C
<lang c>#include <stdio.h>
- include <stdlib.h>
- include <stdarg.h>
- include <assert.h>
- include <gsl/gsl_vector.h>
- define MAX(A,B) (((A)>(B))?(A):(B))
void reoshift(gsl_vector *v, int h) {
if ( h > 0 ) { gsl_vector *temp = gsl_vector_alloc(v->size); gsl_vector_view p = gsl_vector_subvector(v, 0, v->size - h); gsl_vector_view p1 = gsl_vector_subvector(temp, h, v->size - h); gsl_vector_memcpy(&p1.vector, &p.vector); p = gsl_vector_subvector(temp, 0, h); gsl_vector_set_zero(&p.vector); gsl_vector_memcpy(v, temp); gsl_vector_free(temp); }
}
gsl_vector *poly_long_div(gsl_vector *n, gsl_vector *d, gsl_vector **r) {
gsl_vector *nt = NULL, *dt = NULL, *rt = NULL, *d2 = NULL, *q = NULL; int gn, gt, gd;
if ( (n->size >= d->size) && (d->size > 0) && (n->size > 0) ) { nt = gsl_vector_alloc(n->size); assert(nt != NULL); dt = gsl_vector_alloc(n->size); assert(dt != NULL); rt = gsl_vector_alloc(n->size); assert(rt != NULL); d2 = gsl_vector_alloc(n->size); assert(d2 != NULL); gsl_vector_memcpy(nt, n); gsl_vector_set_zero(dt); gsl_vector_set_zero(rt); gsl_vector_view p = gsl_vector_subvector(dt, 0, d->size); gsl_vector_memcpy(&p.vector, d); gsl_vector_memcpy(d2, dt); gn = n->size - 1; gd = d->size - 1; gt = 0;
while( gsl_vector_get(d, gd) == 0 ) gd--; while ( gn >= gd ) { reoshift(dt, gn-gd); double v = gsl_vector_get(nt, gn)/gsl_vector_get(dt, gn); gsl_vector_set(rt, gn-gd, v); gsl_vector_scale(dt, v); gsl_vector_sub(nt, dt); gt = MAX(gt, gn-gd); while( (gn>=0) && (gsl_vector_get(nt, gn) == 0.0) ) gn--; gsl_vector_memcpy(dt, d2); }
q = gsl_vector_alloc(gt+1); assert(q != NULL); p = gsl_vector_subvector(rt, 0, gt+1); gsl_vector_memcpy(q, &p.vector); if ( r != NULL ) { if ( (gn+1) > 0 ) {
*r = gsl_vector_alloc(gn+1); assert( *r != NULL ); p = gsl_vector_subvector(nt, 0, gn+1); gsl_vector_memcpy(*r, &p.vector);
} else {
*r = gsl_vector_alloc(1); assert( *r != NULL ); gsl_vector_set_zero(*r);
} } gsl_vector_free(nt); gsl_vector_free(dt); gsl_vector_free(rt); gsl_vector_free(d2); return q; } else { q = gsl_vector_alloc(1); assert( q != NULL ); gsl_vector_set_zero(q); if ( r != NULL ) { *r = gsl_vector_alloc(n->size); assert( *r != NULL ); gsl_vector_memcpy(*r, n); } return q; }
}
void poly_print(gsl_vector *p) {
int i; for(i=p->size-1; i >= 0; i--) { if ( i > 0 ) printf("%lfx^%d + ",
gsl_vector_get(p, i), i);
else printf("%lf\n", gsl_vector_get(p, i)); }
}
gsl_vector *create_poly(int d, ...) {
va_list al; int i; gsl_vector *r = NULL;
va_start(al, d); r = gsl_vector_alloc(d); assert( r != NULL ); for(i=0; i < d; i++) gsl_vector_set(r, i, va_arg(al, double));
return r;
}</lang>
<lang c>int main() {
int i; gsl_vector *q, *r; gsl_vector *nv, *dv; //nv = create_poly(4, -42., 0., -12., 1.); //dv = create_poly(2, -3., 1.); //nv = create_poly(3, 2., 3., 1.); //dv = create_poly(2, 1., 1.); nv = create_poly(4, -42., 0., -12., 1.); dv = create_poly(3, -3., 1., 1.);
q = poly_long_div(nv, dv, &r);
poly_print(q); poly_print(r);
gsl_vector_free(q); gsl_vector_free(r);
return 0;
}</lang>
Another version
Without outside libs, for clarity. Note that polys are stored and show with zero-degree term first:<lang C>#include <stdio.h>
- include <stdlib.h>
- include <stdarg.h>
- include <string.h>
typedef struct {
int power; double * coef;
} poly_t, *poly;
- define E(x, i) (x)->coef[i]
/* passing in negative power to have a zeroed poly */ poly p_new(int power, ...) {
int i, zeroed = 0; va_list ap;
if (power < 0) { power = -power; zeroed = 1; }
poly p = malloc(sizeof(poly_t)); p->power = power; p->coef = malloc(sizeof(double) * ++power);
if (zeroed) for (i = 0; i < power; i++) p->coef[i] = 0; else { va_start(ap, power); for (i = 0; i < power; i++) E(p, i) = va_arg(ap, double); va_end(ap); }
return p;
}
void p_del(poly p) {
free(p->coef); free(p);
}
void p_print(poly p) {
int i; for (i = 0; i <= p->power; i++) printf("%g ", E(p, i)); printf("\n");
}
poly p_copy(poly p) {
poly q = p_new(-p->power); memcpy(q->coef, p->coef, sizeof(double) * (1 + p->power)); return q;
}
/* p: poly; d: divisor; r: remainder; returns quotient */ poly p_div(poly p, poly d, poly* r) {
poly q; int i, j; int power = p->power - d->power; double ratio;
if (power < 0) return 0;
q = p_new(-power); *r= p_copy(p);
for (i = p->power; i >= d->power; i--) { E(q, i - d->power) = ratio = E(*r, i) / E(d, d->power); E(*r ,i) = 0;
for (j = 0; j < d->power; j++) E(*r, i - d->power + j) -= E(d, j) * ratio; } while (! E(*r, --(*r)->power));
return q;
}
int main() {
poly p = p_new(3, 1., 2., 3., 4.); poly d = p_new(2, 1., 2., 1.); poly r; poly q = p_div(p, d, &r);
printf("poly: "); p_print(p); printf("div: "); p_print(d); printf("quot: "); p_print(q); printf("rem: "); p_print(r);
p_del(p); p_del(q); p_del(r); p_del(d);
return 0;
}</lang>
C++
<lang c++>
- include <iostream>
- include <math.h>
using namespace std;
// does: prints all members of vector // input: c - ASCII char with the name of the vector // d - degree of vector // A - pointer to vector void Print(char c, int d, double* A) { int i;
for (i=0; i < d+1; i++) cout << c << "[" << i << "]= " << A[i] << endl; cout << "Degree of " << c << ": " << d << endl << endl; }</lang>
<lang c++>int main() { double *N,*D,*d,*q,*r; // vectors - N / D = q N % D = r int dN, dD, dd, dq, dr; // degrees of vectors int i; // iterators
// setting the degrees of vectors cout << "Enter the degree of N:"; cin >> dN; cout << "Enter the degree of D:"; cin >> dD; dq = dN-dD; dr = dN-dD;
// allocation and initialization of vectors
N=new double [dN+1];
cout << "Enter the coefficients of N:"<<endl;
for ( i = 0; i < dN+1; i++ ) {
cout << "N[" << i << "]= " << endl;
cin >> N[i];
}
D=new double [dN+1]; cout << "Enter the coefficients of D:"<<endl; for ( i = 0; i < dD+1; i++ ) { cout << "D[" << i << "]= " << endl; cin >> D[i]; }
d=new double [dN+1]; for( i = dD+1 ; i < dN+1; i++ ) { D[i] = 0; }
q=new double [dq+1]; for( i = 0 ; i < dq + 1 ; i++ ) { q[i] = 0; }
r=new double [dr+1]; for( i = 0 ; i < dr + 1 ; i++ ) { r[i] = 0; }
if( dD < 0) { cout << "Degree of D is less than zero. Error!"; }
cout << "-- Procedure --" << endl << endl; if( dN >= dD ) { while(dN >= dD) { // d equals D shifted right for( i = 0 ; i < dN + 1 ; i++ ) { d[i] = 0; } for( i = 0 ; i < dD + 1 ; i++ ) { d[i+dN-dD] = D[i]; } dd = dN;
Print( 'd', dd, d );
// calculating one element of q q[dN-dD] = N[dN]/d[dd];
Print( 'q', dq, q );
// d equals d * q[dN-dD] for( i = 0 ; i < dq + 1 ; i++ ) { d[i] = d[i] * q[dN-dD]; }
Print( 'd', dd, d );
// N equals N - d for( i = 0 ; i < dN + 1 ; i++ ) { N[i] = N[i] - d[i]; } dN--;
Print( 'N', dN, N ); cout << "-----------------------" << endl << endl;
}
}
// r equals N for( i = 0 ; i < dN + 1 ; i++ ) { r[i] = N[i]; } dr = dN;
cout << "=========================" << endl << endl; cout << "-- Result --" << endl << endl;
Print( 'q', dq, q ); Print( 'r', dr, r );
// dealocation delete [] N; delete [] D; delete [] d; delete [] q; delete [] r; }</lang>
Clojure
This example performs multivariate polynomial division using Buchberger's algorithm to decompose a polynomial into its Gröbner bases. Polynomials are represented as hash-maps of monomials with tuples of exponents as keys and their corresponding coefficients as values: e.g. 2xy + 3x + 5y + 7 is represented as {[1 1] 2, [1 0] 3, [0 1] 5, [0 0] 7}.
Since this algorithm is much more efficient when the input is in graded reverse lexicographic (grevlex) order a comparator is included to be used with Clojure's sorted-map—(into (sorted-map-by grevlex) ...)
—as well as necessary functions to compute polynomial multiplication, monomial complements, and S-polynomials.
<lang clojure>(defn grevlex [term1 term2]
(let [grade1 (reduce +' term1) grade2 (reduce +' term2) comp (- grade2 grade1)] ;; total degree (if (not= 0 comp) comp (loop [term1 term1 term2 term2] (if (empty? term1) 0 (let [grade1 (last term1) grade2 (last term2) comp (- grade1 grade2)] ;; differs from grlex because terms are flipped from above (if (not= 0 comp) comp (recur (pop term1) (pop term2)))))))))
(defn mul
;; transducer ([poly1] ;; completion (fn ([] poly1) ([poly2] (mul poly1 poly2)) ([poly2 & more] (mul poly1 poly2 more)))) ([poly1 poly2] (let [product (atom (transient (sorted-map-by grevlex)))] (doall ;; `for` is lazy so must to be forced for side-effects (for [term1 poly1 term2 poly2 :let [vars (mapv +' (key term1) (key term2)) coeff (* (val term1) (val term2))]] (if (contains? @product vars) (swap! product assoc! vars (+ (get @product vars) coeff)) (swap! product assoc! vars coeff)))) (->> product (deref) (persistent!) (denull)))) ([poly1 poly2 & more] (reduce mul (mul poly1 poly2) more)))
(defn compl [term1 term2]
(map (fn [x y] (cond (and (zero? x) (not= 0 y)) nil (< x y) nil (>= x y) (- x y))) term1 term2))
(defn s-poly [f g]
(let [f-vars (first f) g-vars (first g) lcm (compl f-vars g-vars)] (if (not-any? nil? lcm) {(vec lcm) (/ (second f) (second g))})))
(defn divide [f g]
(loop [f f g g result (transient {}) remainder {}] (if (empty? f) (list (persistent! result) (->> remainder (filter #(not (nil? %))) (into (sorted-map-by grevlex)))) (let [term1 (first f) term2 (first g) s-term (s-poly term1 term2)] (if (nil? s-term) (recur (dissoc f (first term1)) (dissoc g (first term2)) result (conj remainder term1)) (recur (sub f (mul g s-term)) g (conj! result s-term) remainder))))))
(deftest divide-tests
(is (= (divide {[1 1] 2, [1 0] 3, [0 1] 5, [0 0] 7} {[1 1] 2, [1 0] 3, [0 1] 5, [0 0] 7}) '({[0 0] 1} {}))) (is (= (divide {[1 1] 2, [1 0] 3, [0 1] 5, [0 0] 7} {[0 0] 1}) '({[1 1] 2, [1 0] 3, [0 1] 5, [0 0] 7} {}))) (is (= (divide {[1 1] 2, [1 0] 10, [0 1] 3, [0 0] 15} {[0 1] 1, [0 0] 5}) '({[1 0] 2, [0 0] 3} {}))) (is (= (divide {[1 1] 2, [1 0] 10, [0 1] 3, [0 0] 15} {[1 0] 2, [0 0] 3}) '({[0 1] 1, [0 0] 5} {}))))</lang>
Common Lisp
Polynomials are represented as lists of degree/coefficient pairs ordered by degree (highest degree first), and pairs with zero coefficients can be omitted. Multiply
and divide
perform long multiplication and long division, respectively. multiply
returns one value, the product, and divide
returns two, the quotient and the remainder.
<lang lisp>(defun add (p1 p2)
(do ((sum '())) ((and (endp p1) (endp p2)) (nreverse sum)) (let ((pd1 (if (endp p1) -1 (caar p1))) (pd2 (if (endp p2) -1 (caar p2)))) (multiple-value-bind (c1 c2) (cond ((> pd1 pd2) (values (cdr (pop p1)) 0)) ((< pd1 pd2) (values 0 (cdr (pop p2)))) (t (values (cdr (pop p1)) (cdr (pop p2))))) (let ((csum (+ c1 c2))) (unless (zerop csum) (setf sum (acons (max pd1 pd2) csum sum))))))))
(defun multiply (p1 p2)
(flet ((*p2 (p) (destructuring-bind (d . c) p (loop for (pd . pc) in p2 collecting (cons (+ d pd) (* c pc)))))) (reduce 'add (mapcar #'*p2 p1) :initial-value '())))
(defun subtract (p1 p2)
(add p1 (multiply '((0 . -1)) p2)))
(defun divide (dividend divisor &aux (sum '()))
(assert (not (endp divisor)) (divisor) 'division-by-zero :operation 'divide :operands (list dividend divisor)) (flet ((floor1 (dividend divisor) (if (endp dividend) (values '() ()) (destructuring-bind (d1 . c1) (first dividend) (destructuring-bind (d2 . c2) (first divisor) (if (> d2 d1) (values '() dividend) (let* ((quot (list (cons (- d1 d2) (/ c1 c2)))) (rem (subtract dividend (multiply divisor quot)))) (values quot rem)))))))) (loop (multiple-value-bind (quotient remainder) (floor1 dividend divisor) (if (endp quotient) (return (values sum remainder)) (setf dividend remainder sum (add quotient sum)))))))</lang>
The wikipedia example:
<lang lisp>> (divide '((3 . 1) (2 . -12) (0 . -42)) ; x^3 - 12x^2 - 42
'((1 . 1) (0 . -3))) ; x - 3
((2 . 1) (1 . -9) (0 . -27)) ; x^2 - 9x - 27 ((0 . -123)) ; -123</lang>
D
<lang d>import std.stdio, std.range, std.algorithm, std.typecons, std.conv;
Tuple!(double[], double[]) polyDiv(in double[] inN, in double[] inD) nothrow pure @safe {
// Code smell: a function that does two things. static int trimAndDegree(T)(ref T[] poly) nothrow pure @safe @nogc { poly = poly.retro.find!q{ a != b }(0.0).retro; return poly.length.signed - 1; }
auto N = inN.dup; const(double)[] D = inD; const dD = trimAndDegree(D); auto dN = trimAndDegree(N); double[] q; if (dD < 0) throw new Error("ZeroDivisionError"); if (dN >= dD) { q = [0.0].replicate(dN); while (dN >= dD) { auto d = [0.0].replicate(dN - dD) ~ D; immutable mult = q[dN - dD] = N[$ - 1] / d[$ - 1]; d[] *= mult; N[] -= d[]; dN = trimAndDegree(N); } } else q = [0.0]; return tuple(q, N);
}
int trimAndDegree1(T)(ref T[] poly) nothrow pure @safe @nogc {
poly.length -= poly.retro.countUntil!q{ a != 0 }; return poly.length.signed - 1;
}
void main() {
immutable N = [-42.0, 0.0, -12.0, 1.0]; immutable D = [-3.0, 1.0, 0.0, 0.0]; writefln("%s / %s = %s remainder %s", N, D, polyDiv(N, D)[]);
}</lang>
- Output:
[-42, 0, -12, 1] / [-3, 1, 0, 0] = [-27, -9, 1] remainder [-123]
E
This program has some unnecessary features contributing to its length:
- It creates polynomial objects rather than performing its operations directly on arrays.
- It includes code for printing polynomials nicely.
- It prints the intermediate steps of the division.
pragma.syntax("0.9") pragma.enable("accumulator") def superscript(x, out) { if (x >= 10) { superscript(x // 10) } out.print("⁰¹²³⁴⁵⁶⁷⁸⁹"[x %% 10]) } def makePolynomial(initCoeffs :List) { def degree := { var i := initCoeffs.size() - 1 while (i >= 0 && initCoeffs[i] <=> 0) { i -= 1 } if (i < 0) { -Infinity } else { i } } def coeffs := initCoeffs(0, if (degree == -Infinity) { [] } else { degree + 1 }) def polynomial { /** Print the polynomial (not necessary for the task) */ to __printOn(out) { out.print("(λx.") var first := true for i in (0..!(coeffs.size())).descending() { def coeff := coeffs[i] if (coeff <=> 0) { continue } out.print(" ") if (coeff <=> 1 && !(i <=> 0)) { # no coefficient written if it's 1 and not the constant term } else if (first) { out.print(coeff) } else if (coeff > 0) { out.print("+ ", coeff) } else { out.print("- ", -coeff) } if (i <=> 0) { # no x if it's the constant term } else if (i <=> 1) { out.print("x") } else { out.print("x"); superscript(i, out) } first := false } out.print(")") } /** Evaluate the polynomial (not necessary for the task) */ to run(x) { return accum 0 for i => c in coeffs { _ + c * x**i } } to degree() { return degree } to coeffs() { return coeffs } to highestCoeff() { return coeffs[degree] } /** Could support another polynomial, but not part of this task. Computes this * x**power. */ to timesXToThe(power) { return makePolynomial([0] * power + coeffs) } /** Multiply (by a scalar only). */ to multiply(scalar) { return makePolynomial(accum [] for x in coeffs { _.with(x * scalar) }) } /** Subtract (by another polynomial only). */ to subtract(other) { def oc := other.coeffs() :List return makePolynomial(accum [] for i in 0..(coeffs.size().max(oc.size())) { _.with(coeffs.fetch(i, fn{0}) - oc.fetch(i, fn{0})) }) } /** Polynomial long division. */ to quotRem(denominator, trace) { var numerator := polynomial require(denominator.degree() >= 0) if (numerator.degree() < denominator.degree()) { return [makePolynomial([]), denominator] } else { var quotientCoeffs := [0] * (numerator.degree() - denominator.degree()) while (numerator.degree() >= denominator.degree()) { trace.print(" ", numerator, "\n") def qCoeff := numerator.highestCoeff() / denominator.highestCoeff() def qPower := numerator.degree() - denominator.degree() quotientCoeffs with= (qPower, qCoeff) def d := denominator.timesXToThe(qPower) * qCoeff trace.print("- ", d, " (= ", denominator, " * ", qCoeff, "x"); superscript(qPower, trace); trace.print(")\n") numerator -= d trace.print(" -------------------------- (Quotient so far: ", makePolynomial(quotientCoeffs), ")\n") } return [makePolynomial(quotientCoeffs), numerator] } } } return polynomial }
<lang e>def n := makePolynomial([-42, 0, -12, 1]) def d := makePolynomial([-3, 1]) println("Numerator: ", n) println("Denominator: ", d) def [q, r] := n.quotRem(d, stdout) println("Quotient: ", q) println("Remainder: ", r)</lang>
Output:
Numerator: (λx. x³ - 12x² - 42) Denominator: (λx. x - 3) (λx. x³ - 12x² - 42) - (λx. x³ - 3.0x²) (= (λx. x - 3) * 1.0x²) -------------------------- (Quotient so far: (λx. x²)) (λx. -9.0x² - 42.0) - (λx. -9.0x² + 27.0x) (= (λx. x - 3) * -9.0x¹) -------------------------- (Quotient so far: (λx. x² - 9.0x)) (λx. -27.0x - 42.0) - (λx. -27.0x + 81.0) (= (λx. x - 3) * -27.0x⁰) -------------------------- (Quotient so far: (λx. x² - 9.0x - 27.0)) Quotient: (λx. x² - 9.0x - 27.0) Remainder: (λx. -123.0)
Elixir
<lang elixir>defmodule Polynomial do
def division(_, []), do: raise ArgumentError, "denominator is zero" def division(_, [0]), do: raise ArgumentError, "denominator is zero" def division(f, g) when length(f) < length(g), do: {[0], f} def division(f, g) do {q, r} = division(g, [], f) if q==[], do: q = [0] if r==[], do: r = [0] {q, r} end defp division(g, q, r) when length(r) < length(g), do: {q, r} defp division(g, q, r) do p = hd(r) / hd(g) r2 = Enum.zip(r, g) |> Enum.with_index |> Enum.reduce(r, fn {{pn,pg},i},acc -> List.replace_at(acc, i, pn - p * pg) end) division(g, q++[p], tl(r2)) end
end
[ { [1, -12, 0, -42], [1, -3] },
{ [1, -12, 0, -42], [1, 1, -3] }, { [1, 3, 2], [1, 1] }, { [1, -4, 6, 5, 3], [1, 2, 1] } ]
|> Enum.each(fn {f,g} ->
{q, r} = Polynomial.division(f, g) IO.puts "#{inspect f} / #{inspect g} => #{inspect q} remainder #{inspect r}" end)</lang>
- Output:
[1, -12, 0, -42] / [1, -3] => [1.0, -9.0, -27.0] remainder [-123.0] [1, -12, 0, -42] / [1, 1, -3] => [1.0, -13.0] remainder [16.0, -81.0] [1, 3, 2] / [1, 1] => [1.0, 2.0] remainder [0.0] [1, -4, 6, 5, 3] / [1, 2, 1] => [1.0, -6.0, 17.0] remainder [-23.0, -14.0]
Fortran
<lang fortran>module Polynom
implicit none
contains
subroutine poly_long_div(n, d, q, r) real, dimension(:), intent(in) :: n, d real, dimension(:), intent(out), allocatable :: q real, dimension(:), intent(out), allocatable, optional :: r
real, dimension(:), allocatable :: nt, dt, rt integer :: gn, gt, gd
if ( (size(n) >= size(d)) .and. (size(d) > 0) .and. (size(n) > 0) ) then allocate(nt(size(n)), dt(size(n)), rt(size(n)))
nt = n dt = 0 dt(1:size(d)) = d rt = 0 gn = size(n)-1 gd = size(d)-1 gt = 0
do while ( d(gd+1) == 0 ) gd = gd - 1 end do
do while( gn >= gd ) dt = eoshift(dt, -(gn-gd)) rt(gn-gd+1) = nt(gn+1) / dt(gn+1) nt = nt - dt * rt(gn-gd+1) gt = max(gt, gn-gd) do gn = gn - 1 if ( nt(gn+1) /= 0 ) exit end do dt = 0 dt(1:size(d)) = d end do
allocate(q(gt+1)) q = rt(1:gt+1) if ( present(r) ) then if ( (gn+1) > 0 ) then allocate(r(gn+1)) r = nt(1:gn+1) else allocate(r(1)) r = 0.0 end if end if deallocate(nt, dt, rt) else allocate(q(1)) q = 0 if ( present(r) ) then allocate(r(size(n))) r = n end if end if
end subroutine poly_long_div
subroutine poly_print(p) real, dimension(:), intent(in) :: p
integer :: i
do i = size(p), 1, -1 if ( i > 1 ) then write(*, '(F0.2,"x^",I0," + ")', advance="no") p(i), i-1 else write(*, '(F0.2)') p(i) end if end do
end subroutine poly_print
end module Polynom</lang>
<lang fortran>program PolyDivTest
use Polynom implicit none
real, dimension(:), allocatable :: q real, dimension(:), allocatable :: r
!! three tests from Wikipedia, plus an extra !call poly_long_div( (/ -3., 1. /), (/ -42., 0.0, -12., 1. /), q, r) call poly_long_div( (/ -42., 0.0, -12., 1. /), (/ -3., 1. /), q, r) !call poly_long_div( (/ -42., 0.0, -12., 1. /), (/ -3., 1., 1. /), q, r) !call poly_long_div( (/ 2., 3., 1. /), (/ 1., 1. /), q, r)
call poly_print(q) call poly_print(r) deallocate(q, r)
end program PolyDivTest</lang>
Go
By the convention and pseudocode given in the task: <lang go>package main
import "fmt"
func main() {
n := []float64{-42, 0, -12, 1} d := []float64{-3, 1} fmt.Println("N:", n) fmt.Println("D:", d) q, r, ok := pld(n, d) if ok { fmt.Println("Q:", q) fmt.Println("R:", r) } else { fmt.Println("error") }
}
func degree(p []float64) int {
for d := len(p) - 1; d >= 0; d-- { if p[d] != 0 { return d } } return -1
}
func pld(nn, dd []float64) (q, r []float64, ok bool) {
if degree(dd) < 0 { return } nn = append(r, nn...) if degree(nn) >= degree(dd) { q = make([]float64, degree(nn)-degree(dd)+1) for degree(nn) >= degree(dd) { d := make([]float64, degree(nn)+1) copy(d[degree(nn)-degree(dd):], dd) q[degree(nn)-degree(dd)] = nn[degree(nn)] / d[degree(d)] for i := range d { d[i] *= q[degree(nn)-degree(dd)] nn[i] -= d[i] } } } return q, nn, true
}</lang> Output:
N: [-42 0 -12 1] D: [-3 1] Q: [-27 -9 1] R: [-123 0 0 0]
GAP
GAP has built-in functions for computations with polynomials. <lang gap>x := Indeterminate(Rationals, "x"); p := x^11 + 3*x^8 + 7*x^2 + 3; q := x^7 + 5*x^3 + 1; QuotientRemainder(p, q);
- [ x^4+3*x-5, -16*x^4+25*x^3+7*x^2-3*x+8 ]</lang>
Haskell
Translated from the OCaml code elsewhere on the page.
<lang haskell>import Data.List
shift n l = l ++ replicate n 0
pad n l = replicate n 0 ++ l
norm :: Fractional a => [a] -> [a] norm = dropWhile (== 0)
deg l = length (norm l) - 1
zipWith' op p q = zipWith op (pad (-d) p) (pad d q)
where d = (length p) - (length q)
polydiv f g = aux (norm f) (norm g) []
where aux f s q | ddif < 0 = (q, f) | otherwise = aux f' s q' where ddif = (deg f) - (deg s) k = (head f) / (head s) ks = map (* k) $ shift ddif s q' = zipWith' (+) q $ shift ddif [k] f' = norm $ tail $ zipWith' (-) f ks</lang>
And this is the also-translated pretty printing function.
<lang haskell>str_poly l = intercalate " + " $ terms l
where term v 0 = show v term 1 1 = "x" term v 1 = (show v) ++ "x" term 1 p = "x^" ++ (show p) term v p = (show v) ++ "x^" ++ (show p)
terms :: Fractional a => [a] -> [String] terms [] = [] terms (0:t) = terms t terms (h:t) = (term h (length t)) : (terms t)</lang>
J
From http://www.jsoftware.com/jwiki/Phrases/Polynomials
<lang J>divmod=:[: (}: ; {:) ([ (] -/@,:&}. (* {:)) ] , %&{.~)^:(>:@-~&#)&.|.~</lang>
Wikipedia example: <lang J>_42 0 _12 1 divmod _3 1</lang> This produces the result:
┌────────┬────┐ │_27 _9 1│_123│ └────────┴────┘
This means that divided by produces with a remainder of .
Java
<lang Java>import java.util.Arrays;
public class PolynomialLongDivision {
private static class Solution { double[] quotient, remainder;
Solution(double[] q, double[] r) { this.quotient = q; this.remainder = r; } }
private static int polyDegree(double[] p) { for (int i = p.length - 1; i >= 0; --i) { if (p[i] != 0.0) return i; } return Integer.MIN_VALUE; }
private static double[] polyShiftRight(double[] p, int places) { if (places <= 0) return p; int pd = polyDegree(p); if (pd + places >= p.length) { throw new IllegalArgumentException("The number of places to be shifted is too large"); } double[] d = Arrays.copyOf(p, p.length); for (int i = pd; i >= 0; --i) { d[i + places] = d[i]; d[i] = 0.0; } return d; }
private static void polyMultiply(double[] p, double m) { for (int i = 0; i < p.length; ++i) { p[i] *= m; } }
private static void polySubtract(double[] p, double[] s) { for (int i = 0; i < p.length; ++i) { p[i] -= s[i]; } }
private static Solution polyLongDiv(double[] n, double[] d) { if (n.length != d.length) { throw new IllegalArgumentException("Numerator and denominator vectors must have the same size"); } int nd = polyDegree(n); int dd = polyDegree(d); if (dd < 0) { throw new IllegalArgumentException("Divisor must have at least one one-zero coefficient"); } if (nd < dd) { throw new IllegalArgumentException("The degree of the divisor cannot exceed that of the numerator"); } double[] n2 = Arrays.copyOf(n, n.length); double[] q = new double[n.length]; while (nd >= dd) { double[] d2 = polyShiftRight(d, nd - dd); q[nd - dd] = n2[nd] / d2[nd]; polyMultiply(d2, q[nd - dd]); polySubtract(n2, d2); nd = polyDegree(n2); } return new Solution(q, n2); }
private static void polyShow(double[] p) { int pd = polyDegree(p); for (int i = pd; i >= 0; --i) { double coeff = p[i]; if (coeff == 0.0) continue; if (coeff == 1.0) { if (i < pd) { System.out.print(" + "); } } else if (coeff == -1.0) { if (i < pd) { System.out.print(" - "); } else { System.out.print("-"); } } else if (coeff < 0.0) { if (i < pd) { System.out.printf(" - %.1f", -coeff); } else { System.out.print(coeff); } } else { if (i < pd) { System.out.printf(" + %.1f", coeff); } else { System.out.print(coeff); } } if (i > 1) System.out.printf("x^%d", i); else if (i == 1) System.out.print("x"); } System.out.println(); }
public static void main(String[] args) { double[] n = new double[]{-42.0, 0.0, -12.0, 1.0}; double[] d = new double[]{-3.0, 1.0, 0.0, 0.0}; System.out.print("Numerator : "); polyShow(n); System.out.print("Denominator : "); polyShow(d); System.out.println("-------------------------------------"); Solution sol = polyLongDiv(n, d); System.out.print("Quotient : "); polyShow(sol.quotient); System.out.print("Remainder : "); polyShow(sol.remainder); }
}</lang>
- Output:
Numerator : x^3 - 12.0x^2 - 42.0 Denominator : x - 3.0 ------------------------------------- Quotient : x^2 - 9.0x - 27.0 Remainder : -123.0
Julia
This task is straightforward with the help of Julia's Polynomials package. <lang Julia> using Polynomials
p = Poly([-42,0,-12,1]) q = Poly([-3,1])
d, r = divrem(p,q)
println(p, " divided by ", q, " is ", d, " with remainder ", r, ".") </lang>
- Output:
-42 - 12x^2 + x^3 divided by -3 + x is -27.0 - 9.0x + x^2 with remainder -123.0.
Kotlin
<lang scala>// version 1.1.51
typealias IAE = IllegalArgumentException
data class Solution(val quotient: DoubleArray, val remainder: DoubleArray)
fun polyDegree(p: DoubleArray): Int {
for (i in p.size - 1 downTo 0) { if (p[i] != 0.0) return i } return Int.MIN_VALUE
}
fun polyShiftRight(p: DoubleArray, places: Int): DoubleArray {
if (places <= 0) return p val pd = polyDegree(p) if (pd + places >= p.size) { throw IAE("The number of places to be shifted is too large") } val d = p.copyOf() for (i in pd downTo 0) { d[i + places] = d[i] d[i] = 0.0 } return d
}
fun polyMultiply(p: DoubleArray, m: Double) {
for (i in 0 until p.size) p[i] *= m
}
fun polySubtract(p: DoubleArray, s: DoubleArray) {
for (i in 0 until p.size) p[i] -= s[i]
}
fun polyLongDiv(n: DoubleArray, d: DoubleArray): Solution {
if (n.size != d.size) { throw IAE("Numerator and denominator vectors must have the same size") } var nd = polyDegree(n) val dd = polyDegree(d) if (dd < 0) { throw IAE("Divisor must have at least one one-zero coefficient") } if (nd < dd) { throw IAE("The degree of the divisor cannot exceed that of the numerator") } val n2 = n.copyOf() val q = DoubleArray(n.size) // all elements zero by default while (nd >= dd) { val d2 = polyShiftRight(d, nd - dd) q[nd - dd] = n2[nd] / d2[nd] polyMultiply(d2, q[nd - dd]) polySubtract(n2, d2) nd = polyDegree(n2) } return Solution(q, n2)
}
fun polyShow(p: DoubleArray) {
val pd = polyDegree(p) for (i in pd downTo 0) { val coeff = p[i] if (coeff == 0.0) continue print (when { coeff == 1.0 -> if (i < pd) " + " else "" coeff == -1.0 -> if (i < pd) " - " else "-" coeff < 0.0 -> if (i < pd) " - ${-coeff}" else "$coeff" else -> if (i < pd) " + $coeff" else "$coeff" }) if (i > 1) print("x^$i") else if (i == 1) print("x") } println()
}
fun main(args: Array<String>) {
val n = doubleArrayOf(-42.0, 0.0, -12.0, 1.0) val d = doubleArrayOf( -3.0, 1.0, 0.0, 0.0) print("Numerator : ") polyShow(n) print("Denominator : ") polyShow(d) println("-------------------------------------") val (q, r) = polyLongDiv(n, d) print("Quotient : ") polyShow(q) print("Remainder : ") polyShow(r)
}</lang>
- Output:
Numerator : x^3 - 12.0x^2 - 42.0 Denominator : x - 3.0 ------------------------------------- Quotient : x^2 - 9.0x - 27.0 Remainder : -123.0
Maple
As Maple is a symbolic computation system, polynomial arithmetic is, of course, provided by the language runtime. The remainder (rem) and quotient (quo) operations each allow for the other to be computed simultaneously by passing an unassigned name as an optional fourth argument. Since rem and quo deal also with multivariate polynomials, the indeterminate is passed as the third argument. <lang Maple> > p := randpoly( x ); # pick a random polynomial in x
5 4 3 2 p := -56 - 7 x + 22 x - 55 x - 94 x + 87 x
> rem( p, x^2 + 2, x, 'q' ); # remainder
220 + 169 x
> q; # quotient
3 2 -7 x + 22 x - 41 x - 138
> quo( p, x^2 + 2, x, 'r' ); # quotient
3 2 -7 x + 22 x - 41 x - 138
> r; # remainder
220 + 169 x
> expand( (x^2+2)*q + r - p ); # check
0
</lang>
Mathematica
<lang Mathematica>PolynomialQuotientRemainder[x^3-12 x^2-42,x-3,x]</lang> output:
{-27 - 9 x + x^2, -123}
OCaml
First define some utility operations on polynomials as lists (with highest power coefficient first). <lang ocaml>let rec shift n l = if n <= 0 then l else shift (pred n) (l @ [0.0]) let rec pad n l = if n <= 0 then l else pad (pred n) (0.0 :: l) let rec norm = function | 0.0 :: tl -> norm tl | x -> x let deg l = List.length (norm l) - 1
let zip op p q =
let d = (List.length p) - (List.length q) in List.map2 op (pad (-d) p) (pad d q)</lang>
Then the main polynomial division function <lang ocaml>let polydiv f g =
let rec aux f s q = let ddif = (deg f) - (deg s) in if ddif < 0 then (q, f) else let k = (List.hd f) /. (List.hd s) in let ks = List.map (( *.) k) (shift ddif s) in let q' = zip (+.) q (shift ddif [k]) and f' = norm (List.tl (zip (-.) f ks)) in aux f' s q' in aux (norm f) (norm g) []</lang>
For output we need a pretty-printing function <lang ocaml>let str_poly l =
let term v p = match (v, p) with | ( _, 0) -> string_of_float v | (1.0, 1) -> "x" | ( _, 1) -> (string_of_float v) ^ "*x" | (1.0, _) -> "x^" ^ (string_of_int p) | _ -> (string_of_float v) ^ "*x^" ^ (string_of_int p) in let rec terms = function | [] -> [] | h :: t -> if h = 0.0 then (terms t) else (term h (List.length t)) :: (terms t) in String.concat " + " (terms l)</lang>
and then the example <lang ocaml>let _ =
let f = [1.0; -4.0; 6.0; 5.0; 3.0] and g = [1.0; 2.0; 1.0] in let q, r = polydiv f g in Printf.printf " (%s) div (%s)\ngives\nquotient:\t(%s)\nremainder:\t(%s)\n" (str_poly f) (str_poly g) (str_poly q) (str_poly r)</lang>
gives the output:
(x^4 + -4.*x^3 + 6.*x^2 + 5.*x + 3.) div (x^2 + 2.*x + 1.) gives quotient: (x^2 + -6.*x + 17.) remainder: (-23.*x + -14.)
Octave
Octave has already facilities to divide two polynomials (deconv(n,d)
); and the reason to adopt the convention of keeping the highest power coefficient first, is to make the code compatible with builtin functions: we can use polyout to output the result.
<lang octave>function [q, r] = poly_long_div(n, d)
gd = length(d); pv = zeros(1, length(n)); pv(1:gd) = d; if ( length(n) >= gd ) q = []; while ( length(n) >= gd ) q = [q, n(1)/pv(1)]; n = n - pv .* (n(1)/pv(1)); n = shift(n, -1); % tn = n(1:length(n)-1); % eat the higher power term n = tn; % tp = pv(1:length(pv)-1); pv = tp; % make pv the same length of n endwhile r = n; else q = [0]; r = n; endif
endfunction
[q, r] = poly_long_div([1,-12,0,-42], [1,-3]); polyout(q, 'x'); polyout(r, 'x'); disp(""); [q, r] = poly_long_div([1,-12,0,-42], [1,1,-3]); polyout(q, 'x'); polyout(r, 'x'); disp(""); [q, r] = poly_long_div([1,3,2], [1,1]); polyout(q, 'x'); polyout(r, 'x'); disp(""); [q, r] = poly_long_div([1,3], [1,-12,0,-42]); polyout(q, 'x'); polyout(r, 'x');</lang>
PARI/GP
This uses the built-in PARI polynomials. <lang parigp>poldiv(a,b)={
my(rem=a%b); [(a - rem)/b, rem]
};
poldiv(x^9+1, x^3+x-3)</lang>
Alternately, use the built-in function divrem
:
<lang parigp>divrem(x^9+1, x^3+x-3)~</lang>
Perl
This solution keeps the highest power coefficient first, like OCaml solution and Octave solution.
<lang perl>use strict; use List::Util qw(min);
sub poly_long_div {
my ($rn, $rd) = @_; my @n = @$rn; my $gd = scalar(@$rd); if ( scalar(@n) >= $gd ) {
my @q = (); while ( scalar(@n) >= $gd ) { my $piv = $n[0]/$rd->[0]; push @q, $piv; $n[$_] -= $rd->[$_] * $piv foreach ( 0 .. min(scalar(@n), $gd)-1 ); shift @n; } return ( \@q, \@n );
} else {
return ( [0], $rn );
}
}</lang>
<lang perl>sub poly_print {
my @c = @_; my $l = scalar(@c); for(my $i=0; $i < $l; $i++) {
print $c[$i]; print "x^" . ($l-$i-1) . " + " if ($i < ($l-1));
} print "\n";
}</lang>
<lang perl>my ($q, $r);
($q, $r) = poly_long_div([1, -12, 0, -42], [1, -3]); poly_print(@$q); poly_print(@$r); print "\n"; ($q, $r) = poly_long_div([1,-12,0,-42], [1,1,-3]); poly_print(@$q); poly_print(@$r); print "\n"; ($q, $r) = poly_long_div([1,3,2], [1,1]); poly_print(@$q); poly_print(@$r); print "\n";
- the example from the OCaml solution
($q, $r) = poly_long_div([1,-4,6,5,3], [1,2,1]); poly_print(@$q); poly_print(@$r);</lang>
Perl 6
for the core algorithm; original code for LaTeX pretty-printing.
<lang perl6>sub poly_long_div ( @n is copy, @d ) {
return [0], |@n if +@n < +@d;
my @q = gather while +@n >= +@d { @n = @n Z- flat ( ( @d X* take ( @n[0] / @d[0] ) ), 0 xx * ); @n.shift; }
return $(@q), $(@n);
}
sub xP ( $power ) { $power>1 ?? "x^$power" !! $power==1 ?? 'x' !! } sub poly_print ( @c ) { join ' + ', @c.kv.map: { $^v ~ xP( @c.end - $^k ) } }
my @polys = [ [ 1, -12, 0, -42 ], [ 1, -3 ] ],
[ [ 1, -12, 0, -42 ], [ 1, 1, -3 ] ], [ [ 1, 3, 2 ], [ 1, 1 ] ], [ [ 1, -4, 6, 5, 3 ], [ 1, 2, 1 ] ];
say 'Failed to parse (unknown function "\begin{array}"): {\displaystyle \begin{array}{rr}'; for @polys -> [ @a, @b ] { printf Q"%s , & %s \\\\\n", poly_long_div( @a, @b ).map: { poly_print($_) }; } say '\end{array}} ';</lang>
Output:
PicoLisp
<lang PicoLisp>(de degree (P)
(let I NIL (for (N . C) P (or (=0 C) (setq I N)) ) (dec I) ) )
(de divPoly (N D)
(if (lt0 (degree D)) (quit "Div/0" D) (let (Q NIL Diff) (while (ge0 (setq Diff (- (degree N) (degree D)))) (setq Q (need (- -1 Diff) Q 0)) (let E D (do Diff (push 'E 0)) (let F (/ (get N (inc (degree N))) (get E (inc (degree E)))) (set (nth Q (inc Diff)) F) (setq N (mapcar '((N E) (- N (* E F))) N E)) ) ) ) (list Q N) ) ) )</lang>
Output:
: (divPoly (-42 0 -12 1) (-3 1 0 0)) -> ((-27 -9 1) (-123 0 0 0))
Python
<lang python># -*- coding: utf-8 -*-
from itertools import izip from math import fabs
def degree(poly):
while poly and poly[-1] == 0: poly.pop() # normalize return len(poly)-1
def poly_div(N, D):
dD = degree(D) dN = degree(N) if dD < 0: raise ZeroDivisionError if dN >= dD: q = [0] * dN while dN >= dD: d = [0]*(dN - dD) + D mult = q[dN - dD] = N[-1] / float(d[-1]) d = [coeff*mult for coeff in d] N = [fabs ( coeffN - coeffd ) for coeffN, coeffd in izip(N, d)] dN = degree(N) r = N else: q = [0] r = N return q, r
if __name__ == '__main__':
print "POLYNOMIAL LONG DIVISION" N = [-42, 0, -12, 1] D = [-3, 1, 0, 0] print " %s / %s =" % (N,D), print " %s remainder %s" % poly_div(N, D)</lang>
Sample output:
POLYNOMIAL LONG DIVISION [-42, 0, -12, 1] / [-3, 1, 0, 0] = [-27.0, -9.0, 1.0] remainder [-123.0]
R
<lang R>polylongdiv <- function(n,d) {
gd <- length(d) pv <- vector("numeric", length(n)) pv[1:gd] <- d if ( length(n) >= gd ) { q <- c() while ( length(n) >= gd ) { q <- c(q, n[1]/pv[1]) n <- n - pv * (n[1]/pv[1]) n <- n[2:length(n)] pv <- pv[1:(length(pv)-1)] } list(q=q, r=n) } else { list(q=c(0), r=n) }
}
- an utility function to print polynomial
print.polynomial <- function(p) {
i <- length(p)-1 for(a in p) { if ( i == 0 ) { cat(a, "\n") } else { cat(a, "x^", i, " + ", sep="") } i <- i - 1 }
}
r <- polylongdiv(c(1,-12,0,-42), c(1,-3)) print.polynomial(r$q) print.polynomial(r$r)</lang>
Racket
<lang racket>
- lang racket
(define (deg p)
(for/fold ([d -inf.0]) ([(pi i) (in-indexed p)]) (if (zero? pi) d i)))
(define (lead p) (vector-ref p (deg p))) (define (mono c d) (build-vector (+ d 1) (λ(i) (if (= i d) c 0)))) (define (poly*cx^n c n p) (vector-append (make-vector n 0) (for/vector ([pi p]) (* c pi)))) (define (poly+ p q) (poly/lin 1 p 1 q)) (define (poly- p q) (poly/lin 1 p -1 q)) (define (poly/lin a p b q)
(cond [(< (deg p) 0) q] [(< (deg q) 0) p] [(< (deg p) (deg q)) (poly/lin b q a p)] [else (define ap+bq (for/vector #:length (+ (deg p) 1) #:fill 0 ([pi p] [qi q]) (+ (* a pi) (* b qi)))) (for ([i (in-range (+ (deg q) 1) (+ (deg p) 1))]) (vector-set! ap+bq i (* a (vector-ref p i)))) ap+bq]))
(define (poly/ n d)
(define N (deg n)) (define D (deg d)) (cond [(< N 0) (error 'poly/ "can't divide by zero")] [(< N D) (values 0 n)] [else (define c (/ (lead n) (lead d))) (define q (mono c (- N D))) (define r (poly- n (poly*cx^n c (- N D) d))) (define-values (q1 r1) (poly/ r d)) (values (poly+ q q1) r1)]))
- Example
(poly/ #(-42 0 -12 1) #(-3 1))
- Output
'#(-27 -9 1) '#(-123 0) </lang>
REXX
<lang rexx>/* REXX needed by some... */ z='1 -12 0 -42' /* Numerator */ n='1 -3' /* Denominator */ zx=z nx=n copies('0 ',words(z)-words(n)) qx= /* Quotient */ Do Until words(zx)<words(n)
Parse Value div(zx,nx) With q zx qx=qx q nx=subword(nx,1,words(nx)-1) End
Say '('show(z)')/('show(n)')=('show(qx)')' Say 'Remainder:' show(zx) Exit div: Procedure Parse Arg z,n q=word(z,1)/word(n,1) zz= Do i=1 To words(z)
zz=zz word(z,i)-q*word(n,i) End
Return q subword(zz,2)
show: Procedure Parse Arg poly d=words(poly)-1 res= Do i=1 To words(poly)
Select When d>1 Then fact='*x**'d When d=1 Then fact='*x' Otherwise fact= End Select When word(poly,i)=0 Then p= When word(poly,i)=1 Then p='+'substr(fact,2) When word(poly,i)=-1 Then p='-'substr(fact,2) When word(poly,i)<0 Then p=word(poly,i)||fact Otherwise p='+'word(poly,i)||fact End res=res p d=d-1 End
Return strip(space(res,0),'L','+')</lang>
- Output:
(x**3-12*x**2-42)/(x-3)=(x**2-9*x-27) Remainder: -123
Ruby
Implementing the algorithm given in the task description: <lang ruby>def polynomial_long_division(numerator, denominator)
dd = degree(denominator) raise ArgumentError, "denominator is zero" if dd < 0 if dd == 0 return [multiply(numerator, 1.0/denominator[0]), [0]*numerator.length] end q = [0] * numerator.length while (dn = degree(numerator)) >= dd d = shift_right(denominator, dn - dd) q[dn-dd] = numerator[dn] / d[degree(d)] d = multiply(d, q[dn-dd]) numerator = subtract(numerator, d) end [q, numerator]
end
def degree(ary)
idx = ary.rindex(&:nonzero?) idx ? idx : -1
end
def shift_right(ary, n)
[0]*n + ary[0, ary.length - n]
end
def subtract(a1, a2)
a1.zip(a2).collect {|v1,v2| v1 - v2}
end
def multiply(ary, num)
ary.collect {|x| x * num}
end
f = [-42, 0, -12, 1] g = [-3, 1, 0, 0] q, r = polynomial_long_division(f, g) puts "#{f} / #{g} => #{q} remainder #{r}"
- => [-42, 0, -12, 1] / [-3, 1, 0, 0] => [-27, -9, 1, 0] remainder [-123, 0, 0, 0]
g = [-3, 1, 1, 0] q, r = polynomial_long_division(f, g) puts "#{f} / #{g} => #{q} remainder #{r}"
- => [-42, 0, -12, 1] / [-3, 1, 1, 0] => [-13, 1, 0, 0] remainder [-81, 16, 0, 0]</lang>
Implementing the algorithms on the wikipedia page -- uglier code but nicer user interface <lang ruby>def polynomial_division(f, g)
if g.length == 0 or (g.length == 1 and g[0] == 0) raise ArgumentError, "denominator is zero" elsif g.length == 1 [f.collect {|x| Float(x)/g[0]}, [0]] elsif g.length == 2 synthetic_division(f, g) else higher_degree_synthetic_division(f, g) end
end
def synthetic_division(f, g)
board = [f] << Array.new(f.length) << Array.new(f.length) board[2][0] = board[0][0] 1.upto(f.length - 1).each do |i| board[1][i] = board[2][i-1] * -g[1] board[2][i] = board[0][i] + board[1][i] end [board[2][0..-2], [board[2][-1]]]
end
- an ugly mess of array index arithmetic
- http://en.wikipedia.org/wiki/Polynomial_long_division#Higher_degree_synthetic_division
def higher_degree_synthetic_division(f, g)
# [use] the negative coefficients of the denominator following the leading term lhs = g[1..-1].collect {|x| -x} board = [f] q = [] 1.upto(f.length - lhs.length).each do |i| n = 2*i - 1 # underline the leading coefficient of the right-hand side, multiply it by # the left-hand coefficients and write the products beneath the next columns # on the right. q << board[n-1][i-1] board << Array.new(f.length).fill(0, i) # row n (lhs.length).times do |j| board[n][i+j] = q[-1]*lhs[j] end # perform an addition board << Array.new(f.length).fill(0, i) # row n+1 (lhs.length + 1).times do |j| board[n+1][i+j] = board[n-1][i+j] + board[n][i+j] if i+j < f.length end end # the remaining numbers in the bottom row correspond to the coefficients of the remainder r = board[-1].compact q = [0] if q.empty? [q, r]
end
f = [1, -12, 0, -42] g = [1, -3] q, r = polynomial_division(f, g) puts "#{f} / #{g} => #{q} remainder #{r}"
- => [1, -12, 0, -42] / [1, -3] => [1, -9, -27] remainder [-123]
g = [1, 1, -3] q, r = polynomial_division(f, g) puts "#{f} / #{g} => #{q} remainder #{r}"
- => [1, -12, 0, -42] / [1, 1, -3] => [1, -13] remainder [16, -81]</lang>
Best of both worlds:
<lang ruby>def polynomial_division(f, g)
if g.length == 0 or (g.length == 1 and g[0] == 0) raise ArgumentError, "denominator is zero" end return [[0], f] if f.length < g.length q, n = [], f.dup while n.length >= g.length q << Float(n[0]) / g[0] n[0, g.length].zip(g).each_with_index do |pair, i| n[i] = pair[0] - q[-1] * pair[1] end n.shift end q = [0] if q.empty? n = [0] if n.empty? [q, n]
end
f = [1, -12, 0, -42] g = [1, -3] q, r = polynomial_division(f, g) puts "#{f} / #{g} => #{q} remainder #{r}"
- => [1, -12, 0, -42] / [1, -3] => [1.0, -9.0, -27.0] remainder [-123.0]
g = [1, 1, -3] q, r = polynomial_division(f, g) puts "#{f} / #{g} => #{q} remainder #{r}"
- => [1, -12, 0, -42] / [1, 1, -3] => [1.0, -13.0] remainder [16.0, -81.0]</lang>
Sidef
<lang ruby>func poly_long_div(rn, rd) {
var n = rn.map{_} var gd = rd.len
if (n.len >= gd) { return(gather { while (n.len >= gd) { var piv = n[0]/rd[0] take(piv) { |i| n[i] -= (rd[i] * piv) } << ^(n.len `min` gd) n.shift } }, n) }
return([0], rn)
}</lang>
Example:
<lang ruby>func poly_print(c) {
var l = c.len c.each_kv {|i, n| print n print("x^", (l - i - 1), " + ") if (i < l-1) } print "\n";
}
var poly = [
Pair([1,-12,0,-42], [1, -3]), Pair([1,-12,0,-42], [1,1,-3]), Pair( [1,3,2], [1,1]), Pair( [1,-4,6,5,3], [1,2,1]),
]
poly.each { |pair|
var (q, r) = poly_long_div(pair.first, pair.second) poly_print(q) poly_print(r) print "\n"
}</lang>
- Output:
1x^2 + -9x^1 + -27 -123 1x^1 + -13 16x^1 + -81 1x^1 + 2 0 1x^2 + -6x^1 + 17 -23x^1 + -14
Slate
<lang Slate>define: #Polynomial &parents: {Comparable} &slots: {#coefficients -> ExtensibleArray new}.
p@(Polynomial traits) new &capacity: n [
p cloneSettingSlots: #(coefficients) to: {p coefficients new &capacity: n}
].
p@(Polynomial traits) newFrom: seq@(Sequence traits) [
p clone `>> [coefficients: (seq as: p coefficients). normalize. ]
].
p@(Polynomial traits) copy [
p cloneSettingSlots: #(coefficients) to: {p coefficients copy}
].
p1@(Polynomial traits) >= p2@(Polynomial traits) [p1 degree >= p2 degree].
p@(Polynomial traits) degree [p coefficients indexOfLastSatisfying: [| :n | n isZero not]].
p@(Polynomial traits) normalize [
[p degree isPositive /\ [p coefficients last isZero]] whileTrue: [p coefficients removeLast]
].
p@(Polynomial traits) * n@(Number traits) [
p newFrom: (p coefficients collect: [| :x | x * n])
].
p@(Polynomial traits) / n@(Number traits) [
p newFrom: (p coefficients collect: [| :x | x / n])
].
p1@(Polynomial traits) minusCoefficients: p2@(Polynomial traits) [
p1 newFrom: (p1 coefficients with: p2 coefficients collect: #- `er)
].
p@(Polynomial traits) / denom@(Polynomial traits) [
p >= denom ifTrue: [| n q | n: p copy. q: p new. [n >= denom] whileTrue: [| piv |
piv: p coefficients last / denom coefficients last. q coefficients add: piv. n: (n minusCoefficients: denom * piv). n normalize].
n coefficients isEmpty ifTrue: [n coefficients add: 0]. {q. n}] ifFalse: [{p newFrom: #(0). p copy}]
].</lang>
Smalltalk
<lang smalltalk>Object subclass: Polynomial [
|coeffs| Polynomial class >> new [ ^ super basicNew init ] init [ coeffs := OrderedCollection new. ^ self ] Polynomial class >> newWithCoefficients: coefficients [ |r| r := super basicNew. ^ r initWithCoefficients: coefficients ] initWithCoefficients: coefficients [ coeffs := coefficients asOrderedCollection. ^ self ] / denominator [ |n q| n := self deepCopy. self >= denominator ifTrue: [ q := Polynomial new. [ n >= denominator ] whileTrue: [ |piv| piv := (n coeff: 0) / (denominator coeff: 0).
q addCoefficient: piv. n := n - (denominator * piv). n clean
]. ^ { q . (n degree) > 0 ifTrue: [ n ] ifFalse: [ n addCoefficient: 0. n ] } ] ifFalse: [ ^ { Polynomial newWithCoefficients: #( 0 ) . self deepCopy } ] ] * constant [ |r| r := self deepCopy. 1 to: (coeffs size) do: [ :i | r at: i put: ((r at: i) * constant) ]. ^ r ] at: index [ ^ coeffs at: index ] at: index put: obj [ ^ coeffs at: index put: obj ] >= anotherPoly [ ^ (self degree) >= (anotherPoly degree) ] degree [ ^ coeffs size ] - anotherPoly [ "This is not a real subtraction between Polynomial: it is an internal method ..." |a| a := self deepCopy. 1 to: ( (coeffs size) min: (anotherPoly degree) ) do: [ :i | a at: i put: ( (a at: i) - (anotherPoly at: i) ) ]. ^ a ] coeff: index [ ^ coeffs at: (index + 1) ] addCoefficient: coeff [ coeffs add: coeff ] clean [ [ (coeffs size) > 0 ifTrue: [ (coeffs at: 1) = 0 ] ifFalse: [ false ] ] whileTrue: [ coeffs removeFirst ]. ] display [ 1 to: (coeffs size) do: [ :i | (coeffs at: i) display. i < (coeffs size) ifTrue: [ ('x^%1 + ' % {(coeffs size) - i} ) display ] ] ] displayNl [ self display. Character nl display ]
].</lang>
<lang smalltalk>|res| res := OrderedCollection new.
res add: ((Polynomial newWithCoefficients: #( 1 -12 0 -42) ) /
(Polynomial newWithCoefficients: #( 1 -3 ) )) ; add: ((Polynomial newWithCoefficients: #( 1 -12 0 -42) ) / (Polynomial newWithCoefficients: #( 1 1 -3 ) )).
res do: [ :o |
(o at: 1) display. ' with rest: ' display. (o at: 2) displayNl
]</lang>
SPAD
<lang SPAD>(1) -> monicDivide(x^3-12*x^2-42,x-3,'x)
2 (1) [quotient = x - 9x - 27,remainder = - 123]
Type: Record(quotient: Polynomial(Integer),remainder: Polynomial(Integer))</lang>
Domain:[1]
Tcl
<lang tcl># poldiv - Divide two polynomials n and d.
- Result is a list of two polynomials, q and r, where n = qd + r
- and the degree of r is less than the degree of b.
- Polynomials are represented as lists, where element 0 is the
- x**0 coefficient, element 1 is the x**1 coefficient, and so on.
proc poldiv {a b} {
# Toss out leading zero coefficients efficiently while {[lindex $a end] == 0} {set a [lrange $a[set a {}] 0 end-1]} while {[lindex $b end] == 0} {set b [lrange $b[set b {}] 0 end-1]} if {[llength $a] < [llength $b]} { return [list 0 $a] }
# Rearrange the terms to put highest powers first set n [lreverse $a] set d [lreverse $b]
# Carry out classical long division, accumulating quotient coefficients # in q, and replacing n with the remainder. set q {} while {[llength $n] >= [llength $d]} { set qd [expr {[lindex $n 0] / [lindex $d 0]}] set i 0 foreach nd [lrange $n 0 [expr {[llength $d] - 1}]] dd $d { lset n $i [expr {$nd - $qd * $dd}] incr i } lappend q $qd set n [lrange $n 1 end] }
# Return quotient and remainder, constant term first return [list [lreverse $q] [lreverse $n]]
}
- Demonstration
lassign [poldiv {-42. 0. -12. 1.} {-3. 1. 0. 0.}] Q R puts [list Q = $Q] puts [list R = $R]</lang>
Ursala
The input is a pair of lists of coefficients in order of increasing degree. Trailing zeros can be omitted. The output is a pair of lists (q,r), the quotient and remainder polynomial coefficients. This is a straightforward implementation of the algorithm in terms of list operations (fold, zip, map, distribute, etc.) instead of array indexing, hence not unnecessarily verbose. <lang ursala>#import std
- import flo
polydiv =
zeroid~-l~~; leql?rlX\~&NlX ^H\(@rNrNSPXlHDlS |\ :/0.) @NlX //=> ?(
@lrrPX ==!| zipp0.; @x not zeroid+ ==@h->hr ~&t, (^lryPX/~&lrrl2C minus^*p/~&rrr times*lrlPD)^/div@bzPrrPlXO ~&, @r ^|\~& ~&i&& :/0.)</lang>
test program: <lang Ursala>#cast %eLW
example = polydiv(<-42.,0.,-12.,1.>,<-3.,1.,0.,0.>)</lang> output:
( <-2.700000e+01,-9.000000e+00,1.000000e+00>, <-1.230000e+02>)
zkl
<lang zkl>fcn polyLongDivision(a,b){ // (a0 + a1x + a2x^2 + a3x^3 ...)
_assert_(degree(b)>=0,"degree(%s) < 0".fmt(b)); q:=List.createLong(a.len(),0.0); while((ad:=degree(a)) >= (bd:=degree(b))){ z,d,m := ad-bd, List.createLong(z,0.0).extend(b), a[ad]/b[bd];; q[z]=m; d,a = d.apply('*(m)), a.zipWith('-,d); } return(q,a); // may have trailing zero elements
} fcn degree(v){ // -1,0,..len(v)-1, -1 if v==0
v.len() - v.copy().reverse().filter1n('!=(0)) - 1;
} fcn polyString(terms){ // (a0,a1,a2...)-->"a0 + a1x + a2x^2 ..."
str:=[0..].zipWith('wrap(n,a){ if(a) "+ %sx^%s ".fmt(a,n) else "" },terms) .pump(String) .replace("x^0 "," ").replace(" 1x"," x").replace("x^1 ","x ") .replace("+ -", "- "); if(not str) return(" "); // all zeros if(str[0]=="+") str[1,*]; // leave leading space else String("-",str[2,*]);
}</lang> <lang zkl>q,r:=polyLongDivision(T(-42.0, 0.0, -12.0, 1.0),T(-3.0, 1.0)); println("Quotient = ",polyString(q)); println("Remainder = ",polyString(r));</lang>
- Output:
Quotient = -27 - 9x + x^2 Remainder = -123
- Programming Tasks
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