Polynomial long division: Difference between revisions
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This program has some unnecessary features contributing to its length: |
This program has some unnecessary features contributing to its length: |
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* It creates polynomial objects rather than performing its operations directly on arrays. |
* It creates polynomial objects rather than performing its operations directly on arrays. |
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Revision as of 23:54, 25 September 2011
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 zeros 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
if degree(N) ≥ degree(D) then
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
else
q ← 0
r ← N
endif
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
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>
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;
Tuple!(double[],double[]) polyDiv(double[] N, in double[] D) {
static nothrow pure int degree(T)(auto ref T[] poly) {
while (poly.length && poly[$-1] == 0)
poly = poly[0 .. $-1];
return cast(int)poly.length - 1;
}
const dD = degree(D);
auto dN = degree(N);
double[] q, r;
if (dD < 0)
throw new Exception("ZeroDivisionError");
if (dN >= dD) {
q = array(take(repeat(0.0), dN));
while (dN >= dD) {
auto d = array(take(repeat(0.0), dN - dD)) ~ D;
auto mult = q[dN - dD] = N[$-1] / d[$-1];
d[] *= mult;
N[] -= d[];
dN = degree(N);
}
r = N;
} else {
q = [0.0];
r = N;
}
return tuple(q, r);
}
void main() {
auto N = [-42.0, 0.0, -12.0, 1.0];
auto D = [-3.0, 1.0, 0.0, 0.0];
const qr = polyDiv(N, D);
writefln("%s / %s = %s remainder %s", N, D, qr[0], qr[1]);
}</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)
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 .
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>
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- ( ( @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 - $^k ) } }
my @polys = [ [ 1, -12, 0, -42 ], [ 1, -3 ] ],
[ [ 1, -12, 0, -42 ], [ 1, 1, -3 ] ],
[ [ 1, 3, 2 ], [ 1, 1 ] ],
[ [ 1, -4, 6, 5, 3 ], [ 1, 2, 1 ] ];
say 'Failed to parse (unknown function "\begin{array}"): {\displaystyle \begin{array}{rr}'; for @polys -> [ @a, @b ] { printf "%s , & %s \\\\\n", poly_long_div( @a, @b ).map: { poly_print($_) }; } 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>
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 {|x| x.nonzero?}
idx.nil? ? -1 : idx
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) p [f, g, q, r]
- => [[-42, 0, -12, 1], [-3, 1, 0, 0], [-27, -9, 1, 0, 0], [-123, 0, 0, 0, 0]]
g = [-3, 1, 1, 0] q, r = polynomial_long_division(f, g) p [f, g, q, r]
- => [[-42, 0, -12, 1], [-3, 1, 0, 0], [-13, 1, 0, 0, 0], [-81, 16, 0, 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) p [f, g, q, r]
- => [[1, -12, 0, -42], [1, -3], [1, -9, -27], [-123]]
g = [1, 1, -3] q, r = polynomial_division(f, g) p [f, g, q, r]
- => [[1, -12, 0, -42], [1, 1, -3], [1, -13], [16, -81]]</lang>
Best of both worlds:
<lang ruby>def tcl_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, d = f.dup, g
while n.length >= d.length
q << Float(n[0]) / d[0]
n[0, d.length].zip(d).each_with_index do |pair, i|
n[i] = Float(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) p [f, g, q, r]
- => [[1, -12, 0, -42], [1, -3], [1, -9, -27], [-123]]
g = [1, 1, -3] q, r = polynomial_division(f, g) p [f, g, q, r]
- => [[1, -12, 0, -42], [1, 1, -3], [1, -13], [16, -81]]</lang>
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>
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>)