Polynomial synthetic division
Polynomial synthetic division is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
| This page uses content from Wikipedia. The original article was at Synthetic 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 synthetic division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree in an efficient way using a trick involving clever manipulations of coefficients, which results in a lower time complexity than polynomial long division.
11l
F extended_synthetic_division(dividend, divisor)
‘Fast polynomial division by using Extended Synthetic Division. Also works with non-monic polynomials.’
V out = copy(dividend)
V normalizer = divisor[0]
L(i) 0 .< dividend.len - (divisor.len - 1)
out[i] /= normalizer
V coef = out[i]
I coef != 0
L(j) 1 .< divisor.len
out[i + j] += -divisor[j] * coef
V separator = divisor.len - 1
R (out[0 .< (len)-separator], out[(len)-separator..])
print(‘POLYNOMIAL SYNTHETIC DIVISION’)
V n = [1, -12, 0, -42]
V D = [1, -3]
print(‘ #. / #. =’.format(n, D), end' ‘ ’)
V (a, b) = extended_synthetic_division(n, D)
print(‘#. remainder #.’.format(a, b))- Output:
POLYNOMIAL SYNTHETIC DIVISION [1, -12, 0, -42] / [1, -3] = [1, -9, -27] remainder [-123]
C++
/*
* C++ Polynomial Sythetic Division
* GNU Compile example for filename <synthdiv.cpp>
* g++ -std=c++11 -o synthdiv synthdiv.cpp
*/
#include <iostream>
#include <vector>
#include <string>
#include <cmath>
/*
* frmtPolynomial method
* Returns string for formatted
* polynomial from int vector of coefs.
* String looks like ax^2 + bx + c,
* a, b, and c being the integer
* coefs in the vector.
*/
std::string frmtPolynomial(std::vector<int> polynomial, bool remainder = false)
{
std::string r = "";
if (remainder)
{
r = " r: " + std::to_string(polynomial.back());
polynomial.pop_back();
}
std::string formatted = "";
int degree = polynomial.size() - 1;
int d = degree;
for (int i : polynomial)
{
if (d < degree)
{
if (i >= 0)
{
formatted += " + ";
}
else
{
formatted += " - ";
}
}
formatted += std::to_string(abs(i));
if (d > 1)
{
formatted += "x^" + std::to_string(d);
}
else if (d == 1)
{
formatted += "x";
}
d--;
}
return formatted;
}
/*
* syntheticDiv Method
* Performs Integer Polynomial Sythetic Division
* on polynomials expressed as vectors of coefs.
* Takes int vector param for dividend and
* divisor, and returns int vector quotient.
*/
std::vector<int> syntheticDiv(std::vector<int> dividend, std::vector<int> divisor)
{
std::vector<int> quotient;
quotient = dividend;
int normalizer = divisor[0];
for (int i = 0; i < dividend.size() - (divisor.size() - 1); i++)
{
quotient[i] /= normalizer;
int coef = quotient[i];
if (coef != 0)
{
for (int j = 1; j < divisor.size(); j++)
{
quotient[i + j] += -divisor[j] * coef;
}
}
}
return quotient;
}
/*
* Example of using the syntheticDiv method
* and the frmtPolynomial method.
* Assigns dividend and divisor polynomials:
* dividend: 1x^3 - 12x^2 + 0x - 42
* divisor: 1x - 3
* Outputs both to cout using frmtPolynomial.
* Printed polynomials look like above format.
* Processes dividend and divisor in the
* syntheticDiv method, returns quotient.
* Outputs quotient to cout using frmtPolynomial again.
* quotient: 1x^2 - 9x - 27 r: -123
*/
int main(int argc, char **argv)
{
std::vector<int> dividend{ 1, -12, 0, -42};
std::vector<int> divisor{ 1, -3};
std::cout << frmtPolynomial(dividend) << "\n";
std::cout << frmtPolynomial(divisor) << "\n";
std::vector<int> quotient = syntheticDiv(dividend, divisor);
std::cout << frmtPolynomial(quotient, true) << "\n";
}
C#
using System;
using System.Collections.Generic;
using System.Linq;
namespace SyntheticDivision
{
class Program
{
static (List<int>,List<int>) extendedSyntheticDivision(List<int> dividend, List<int> divisor)
{
List<int> output = dividend.ToList();
int normalizer = divisor[0];
for (int i = 0; i < dividend.Count() - (divisor.Count() - 1); i++)
{
output[i] /= normalizer;
int coef = output[i];
if (coef != 0)
{
for (int j = 1; j < divisor.Count(); j++)
output[i + j] += -divisor[j] * coef;
}
}
int separator = output.Count() - (divisor.Count() - 1);
return (
output.GetRange(0, separator),
output.GetRange(separator, output.Count() - separator)
);
}
static void Main(string[] args)
{
List<int> N = new List<int>{ 1, -12, 0, -42 };
List<int> D = new List<int> { 1, -3 };
var (quotient, remainder) = extendedSyntheticDivision(N, D);
Console.WriteLine("[ {0} ] / [ {1} ] = [ {2} ], remainder [ {3} ]" ,
string.Join(",", N),
string.Join(",", D),
string.Join(",", quotient),
string.Join(",", remainder)
);
}
}
}
Delphi
Thanks Rudy Velthuis for the Velthuis.BigRationals library.
program Polynomial_synthetic_division;
{$APPTYPE CONSOLE}
uses
System.SysUtils,
Velthuis.BigRationals;
type
TPollynomy = record
public
Terms: TArray<BigRational>;
class operator Divide(a, b: TPollynomy): TArray<TPollynomy>;
constructor Create(Terms: TArray<BigRational>);
function ToString: string;
end;
{ TPollynomy }
constructor TPollynomy.Create(Terms: TArray<BigRational>);
begin
self.Terms := copy(Terms, 0, length(Terms));
end;
class operator TPollynomy.Divide(a, b: TPollynomy): TArray<TPollynomy>;
var
q, r: TPollynomy;
begin
var o: TArray<BigRational>;
SetLength(o, length(a.Terms));
for var i := 0 to High(a.Terms) do
o[i] := BigRational.Create(a.Terms[i]);
for var i := 0 to length(a.Terms) - length(b.Terms) do
begin
o[i] := BigRational.Create(o[i] div b.Terms[0]);
var coef := BigRational.Create(o[i]);
if coef.Sign <> 0 then
begin
var aa: BigRational := 0;
for var j := 1 to High(b.Terms) do
begin
aa := (-b.Terms[j]) * coef;
o[i + j] := o[i + j] + aa;
end;
end;
end;
var separator := length(o) - (length(b.Terms) - 1);
q := TPollynomy.Create(copy(o, 0, separator));
r := TPollynomy.Create(copy(o, separator, length(o)));
result := [q, r];
end;
function TPollynomy.ToString: string;
begin
Result := '[';
for var e in Terms do
Result := Result + e.ToString + ' ';
Result := Result + ']';
end;
var
p1, p2: TPollynomy;
begin
p1 := TPollynomy.Create([BigRational.Create(1, 1), BigRational.Create(-12, 1),
BigRational.Create(0, 1), BigRational.Create(-42, 1)]);
p2 := TPollynomy.Create([BigRational.Create(1, 1), BigRational.Create(-3, 1)]);
write(p1.ToString, ' / ', p2.ToString, ' = ');
var result := p1 / p2;
writeln(result[0].ToString, ' remainder ', result[1].ToString);
readln;
end.
- Output:
[1 -12 0 -42 ] / [1 -3 ] = [1 -9 -27 ] remainder [-123 ]
FreeBASIC
' Function to perform polynomial division
Function divrem(dividend() As Double, divisor() As Double) As String
Dim As Integer i, j
Dim As Integer dividendLen = Ubound(dividend) + 1
Dim As Integer divisorLen = Ubound(divisor) + 1
Dim As Double result(dividendLen - 1)
' Copy dividend to result
For i = 0 To dividendLen - 1
result(i) = dividend(i)
Next
Dim As Integer quotientlen = divisorLen - 1
For i = 0 To dividendLen - quotientlen - 1
If result(i) <> 0 Then
result(i) /= divisor(0)
For j = 0 To quotientlen - 1
result(i + j + 1) -= divisor(j + 1) * result(i)
Next
End If
Next
' Prepare the output string
Dim As String quotient = "{"
For i = 0 To dividendLen - quotientlen - 1
quotient &= Str(result(i)) & ", "
Next
If Len(quotient) > 1 Then quotient = Left(quotient, Len(quotient) - 2)
quotient &= "}"
Dim As String remainder = "{"
For i = dividendLen - quotientlen To dividendLen - 1
remainder &= Str(result(i)) & ", "
Next
If Len(remainder) > 1 Then remainder = Left(remainder, Len(remainder) - 2)
remainder &= "}"
Return quotient & ", remainder " & remainder
End Function
' Main program
Dim As Double n1(3) = {1, -12, 0, -42}
Dim As Double d1(1) = {1, -3}
Dim As Double n2(3) = {1, -12, 0, -42}
Dim As Double d2(2) = {1, 1, -3}
Dim As Double n3(4) = {1, 0, 0, 0, -2}
Dim As Double d3(3) = {1, 1, 1, 1}
Dim As Double n4(3) = {6, 5, 0, -7}
Dim As Double d4(2) = {3, -2, -1}
Print "Polynomial synthetic division"
Print "{1, -12, 0, -42} / {1, -3} = "; divrem(n1(), d1())
Print "{1, 0, 0, 0, -2} / {1, 1, 1, 1} = "; divrem(n2(), d2())
Print "{1, 0, 0, 0, -2} / {1, 1, 1, 1} = "; divrem(n3(), d3())
Print "{6, 5, 0, -7} / {3, -2, -1} = "; divrem(n4(), d4())
Sleep
- Output:
Polynomial synthetic division
{1, -12, 0, -42} / {1, -3} = {1, -9, -27}, remainder {-123}
{1, -12, 0, -42} / {1, 1, -3} = {1, -13}, remainder {16, -81}
{1, 0, 0, 0, -2} / {1, 1, 1, 1} = {1, -1}, remainder {0, 0, -1}
{6, 5, 0, -7} / {3, -2, -1} = {2, 3}, remainder {8, -4}
Go
package main
import (
"fmt"
"math/big"
)
func div(dividend, divisor []*big.Rat) (quotient, remainder []*big.Rat) {
out := make([]*big.Rat, len(dividend))
for i, c := range dividend {
out[i] = new(big.Rat).Set(c)
}
for i := 0; i < len(dividend)-(len(divisor)-1); i++ {
out[i].Quo(out[i], divisor[0])
if coef := out[i]; coef.Sign() != 0 {
var a big.Rat
for j := 1; j < len(divisor); j++ {
out[i+j].Add(out[i+j], a.Mul(a.Neg(divisor[j]), coef))
}
}
}
separator := len(out) - (len(divisor) - 1)
return out[:separator], out[separator:]
}
func main() {
N := []*big.Rat{
big.NewRat(1, 1),
big.NewRat(-12, 1),
big.NewRat(0, 1),
big.NewRat(-42, 1)}
D := []*big.Rat{big.NewRat(1, 1), big.NewRat(-3, 1)}
Q, R := div(N, D)
fmt.Printf("%v / %v = %v remainder %v\n", N, D, Q, R)
}
- Output:
[1/1 -12/1 0/1 -42/1] / [1/1 -3/1] = [1/1 -9/1 -27/1] remainder [-123/1]
Haskell
import Data.List
normalized :: (Eq a, Num a) => [a] -> [a]
normalized = dropWhile (== 0)
isZero :: (Eq a, Num a) => [a] -> Bool
isZero = null . normalized
shortDiv :: (Eq a, Fractional a) => [a] -> [a] -> ([a], [a])
shortDiv p1 p2
| isZero p2 = error "zero divisor"
| otherwise =
let go 0 p = p
go i (h:t) = (h/a) : go (i-1) (zipWith (+) (map ((h/a) *) ker) t)
in splitAt k $ go k p1
where
k = length p1 - length as
a:as = normalized p2
ker = negate <$> (as ++ repeat 0)
*Main> shortDiv [1,-12,0,-42] [1,1,-3] ([1.0,-13.0],[16.0,-81.0]) *Main> shortDiv [6,5,0,-7] [3,-2,-1] ([2.0,3.0],[8.0,-4.0])
For monic divisors it is possible to perform purely integral computations (without Fractional constraint):
isMonic :: (Eq a, Num a) => [a] -> Bool
isMonic = ([1] ==) . take 1 . normalized
shortDivMonic :: (Eq a, Num a) => [a] -> [a] -> ([a], [a])
shortDivMonic p1 p2
| isZero p2 = error "zero divisor"
| not (isMonic p2) = error "divisor is not monic"
| otherwise =
let go 0 p = p
go i (h:t) = h : go (i-1) (zipWith (+) (map (h *) ker) t)
in splitAt k $ go k p1
where
k = length p1 - length as
_:as = normalized p2
ker = negate <$> as ++ repeat 0
shortDivMonic [1,-12,0,-42] [1,1,-3 :: Int] ([1,-13],[16,-81])
J
Solving this the easy way:
psd=: [:(}. ;{.) ([ (] -/@,:&}. (* {:)) ] , %&{.~)^:(>:@-~&#)~
Task example:
(1, (-12), 0, -42) psd (1, -3)
┌────────┬────┐
│1 _9 _27│_123│
└────────┴────┘
Java
import java.util.Arrays;
public class Test {
public static void main(String[] args) {
int[] N = {1, -12, 0, -42};
int[] D = {1, -3};
System.out.printf("%s / %s = %s",
Arrays.toString(N),
Arrays.toString(D),
Arrays.deepToString(extendedSyntheticDivision(N, D)));
}
static int[][] extendedSyntheticDivision(int[] dividend, int[] divisor) {
int[] out = dividend.clone();
int normalizer = divisor[0];
for (int i = 0; i < dividend.length - (divisor.length - 1); i++) {
out[i] /= normalizer;
int coef = out[i];
if (coef != 0) {
for (int j = 1; j < divisor.length; j++)
out[i + j] += -divisor[j] * coef;
}
}
int separator = out.length - (divisor.length - 1);
return new int[][]{
Arrays.copyOfRange(out, 0, separator),
Arrays.copyOfRange(out, separator, out.length)
};
}
}
[1, -12, 0, -42] / [1, -3] = [[1, -9, -27], [-123]]
jq
Works with gojq, the Go implementation of jq
Works with jaq, the Rust implementation of jq
In this entry, the polynomial of degree n, SIGMA c[i] * x^(n-i), is represented by the JSON array c.
# Solution: {"quotient", "remainder"}
def extendedSyntheticDivision($dividend; $divisor):
{ out: $dividend,
normalizer: $divisor[0],
separator: (($dividend|length) - ($divisor|length) + 1) }
| reduce range(0; .separator) as $i (.;
.out[$i] = ((.out[$i] / .normalizer)|trunc)
| .out[$i] as $coef
| if $coef != 0
then reduce range(1; $divisor|length) as $j (.;
.out[$i + $j] -= $divisor[$j] * $coef )
else .
end )
| {quotient: .out[0:.separator], remainder: .out[.separator:]} ;
def task($n; $d):
def r: if length==1 then first else . end;
extendedSyntheticDivision($n; $d)
| "\($n) / \($d) = \(.quotient), remainder \(.remainder|r)" ;
task([1, -12, 0, -42]; [1, -3]),
task([1, 0, 0, 0, -2];[1, 1, 1, 1])- Output:
[1,-12,0,-42] / [1,-3] = [1,-9,-27], remainder -123 [1,0,0,0,-2] / [1,1,1,1] = [1,-1], remainder [0,0,-1]
Julia
function divrem(dividend::Vector, divisor::Vector)
result = copy(dividend)
quotientlen = length(divisor) - 1
for i in 1:length(dividend)-quotientlen
if result[i] != 0
result[i] /= divisor[1]
for j in 1:quotientlen
result[i + j] -= divisor[j + 1] * result[i]
end
end
end
return result[1:end-quotientlen], result[end-quotientlen+1:end]
end
testpolys = [([1, -12, 0, -42], [1, -3]), ([1, 0, 0, 0, -2], [1, 1, 1, 1])]
for (n, d) in testpolys
quotient, remainder = divrem(n, d)
println("[$n] / [$d] = [$quotient] with remainder [$remainder]")
end
- Output:
[[1, -12, 0, -42]] / [[1, -3]] = [[1, -9, -27]] with remainder [[-123]] [[1, 0, 0, 0, -2]] / [[1, 1, 1, 1]] = [[1, -1]] with remainder [[0, 0, -1]]
Kotlin
// version 1.1.2
fun extendedSyntheticDivision(dividend: IntArray, divisor: IntArray): Pair<IntArray, IntArray> {
val out = dividend.copyOf()
val normalizer = divisor[0]
val separator = dividend.size - divisor.size + 1
for (i in 0 until separator) {
out[i] /= normalizer
val coef = out[i]
if (coef != 0) {
for (j in 1 until divisor.size) out[i + j] += -divisor[j] * coef
}
}
return out.copyOfRange(0, separator) to out.copyOfRange(separator, out.size)
}
fun main(args: Array<String>) {
println("POLYNOMIAL SYNTHETIC DIVISION")
val n = intArrayOf(1, -12, 0, -42)
val d = intArrayOf(1, -3)
val (q, r) = extendedSyntheticDivision(n, d)
print("${n.contentToString()} / ${d.contentToString()} = ")
println("${q.contentToString()}, remainder ${r.contentToString()}")
println()
val n2 = intArrayOf(1, 0, 0, 0, -2)
val d2 = intArrayOf(1, 1, 1, 1)
val (q2, r2) = extendedSyntheticDivision(n2, d2)
print("${n2.contentToString()} / ${d2.contentToString()} = ")
println("${q2.contentToString()}, remainder ${r2.contentToString()}")
}
- Output:
POLYNOMIAL SYNTHETIC DIVISION [1, -12, 0, -42] / [1, -3] = [1, -9, -27], remainder [-123] [1, 0, 0, 0, -2] / [1, 1, 1, 1] = [1, -1], remainder [0, 0, -1]
Mathematica / Wolfram Language
MakePolynomial[l_List, x_] := FromCoefficientRules[Thread[List /@ Range[Length[l] - 1, 0, -1] -> l], {x}]
num = MakePolynomial[{1, -12, 0, -42}, x];
den = MakePolynomial[{1, -3}, x];
PolynomialQuotient[num, den, x]
PolynomialRemainder[num, den, x]
- Output:
-27 - 9 x + x^2 -123
Nim
import strformat
type Polynomial = seq[int]
func `$`(p: Polynomial): string = system.`$`(p)[1..^1]
func extendedSyntheticDivision(dividend, divisor: Polynomial): tuple[q, r: Polynomial] =
var res = dividend
let normalizer = divisor[0]
let separator = dividend.len - divisor.len
for i in 0..separator:
res[i] = res[i] div normalizer
let coef = res[i]
if coef != 0:
for j in 1..divisor.high:
res[i + j] += -divisor[j] * coef
result = (res[0..separator], res[(separator+1)..^1])
when isMainModule:
echo "Polynomial synthetic division"
let n1 = @[1, -12, 0, -42]
let d1 = @[1, -3]
let (q1, r1) = extendedSyntheticDivision(n1, d1)
echo &"{n1} / {d1} = {q1}, remainder {r1}"
let n2 = @[1, 0, 0, 0, -2]
let d2 = @[1, 1, 1, 1]
let (q2, r2) = extendedSyntheticDivision(n2, d2)
echo &"{n2} / {d2} = {q2}, remainder {r2}"
- Output:
Polynomial synthetic division [1, -12, 0, -42] / [1, -3] = [1, -9, -27], remainder [-123] [1, 0, 0, 0, -2] / [1, 1, 1, 1] = [1, -1], remainder [0, 0, -1]
Perl
sub synthetic_division {
my($numerator,$denominator) = @_;
my @result = @$numerator;
my $end = @$denominator-1;
for my $i (0 .. @$numerator-($end+1)) {
next unless $result[$i];
$result[$i] /= @$denominator[0];
$result[$i+$_] -= @$denominator[$_] * $result[$i] for 1 .. $end;
}
return join(' ', @result[0 .. @result-($end+1)]), join(' ', @result[-$end .. -1]);
}
sub poly_divide {
*n = shift; *d = shift;
my($quotient,$remainder)= synthetic_division( \@n, \@d );
"[@n] / [@d] = [$quotient], remainder [$remainder]\n";
}
print poly_divide([1, -12, 0, -42], [1, -3]);
print poly_divide([1, 0, 0, 0, -2], [1, 1, 1, 1]);
- Output:
[1 -12 0 -42] / [1 -3] = [1 -9 -27], remainder [-123] [1 0 0 0 -2] / [1 1 1 1] = [1 -1], remainder [0 0 -1]
Phix
with javascript_semantics function extendedSyntheticDivision(sequence dividend, divisor) sequence out = deep_copy(dividend) integer normalizer = divisor[1], separator = length(dividend) - length(divisor) + 1 for i=1 to separator do out[i] /= normalizer integer coef = out[i] if (coef != 0) then for j=2 to length(divisor) do integer odx = i+j-1 out[odx] += -divisor[j] * coef end for end if end for return {out[1..separator],out[separator+1..$]} end function constant tests = {{{1, -12, 0, -42},{1, -3}}, {{1, -12, 0, -42},{1, 1, -3}}, {{1, 0, 0, 0, -2},{1, 1, 1, 1}}, {{6, 5, 0, -7},{3, -2, -1}}} printf(1,"Polynomial synthetic division\n") for t=1 to length(tests) do sequence {n,d} = tests[t], {q,r} = extendedSyntheticDivision(n, d) printf(1,"%v / %v = %v, remainder %v\n",{n,d,q,r}) end for
- Output:
Polynomial synthetic division
{1,-12,0,-42} / {1,-3} = {1,-9,-27}, remainder {-123}
{1,-12,0,-42} / {1,1,-3} = {1,-13}, remainder {16,-81}
{1,0,0,0,-2} / {1,1,1,1} = {1,-1}, remainder {0,0,-1}
{6,5,0,-7} / {3,-2,-1} = {2,3}, remainder {8,-4}
Python
Here is an extended synthetic division algorithm, which means that it supports a divisor polynomial (instead of just a monomial/binomial). It also supports non-monic polynomials (polynomials which first coefficient is different than 1). Polynomials are represented by lists of coefficients with decreasing degree (left-most is the major degree , right-most is the constant).
from __future__ import print_function
from __future__ import division
#!/usr/bin/python
# coding=UTF-8
def extended_synthetic_division(dividend, divisor):
'''Fast polynomial division by using Extended Synthetic Division. Also works with non-monic polynomials.'''
# dividend and divisor are both polynomials, which are here simply lists of coefficients. Eg: x^2 + 3x + 5 will be represented as [1, 3, 5]
out = list(dividend) # Copy the dividend
normalizer = divisor[0]
for i in xrange(len(dividend)-(len(divisor)-1)):
out[i] /= normalizer # for general polynomial division (when polynomials are non-monic),
# we need to normalize by dividing the coefficient with the divisor's first coefficient
coef = out[i]
if coef != 0: # useless to multiply if coef is 0
for j in xrange(1, len(divisor)): # in synthetic division, we always skip the first coefficient of the divisor,
# because it's only used to normalize the dividend coefficients
out[i + j] += -divisor[j] * coef
# The resulting out contains both the quotient and the remainder, the remainder being the size of the divisor (the remainder
# has necessarily the same degree as the divisor since it's what we couldn't divide from the dividend), so we compute the index
# where this separation is, and return the quotient and remainder.
separator = -(len(divisor)-1)
return out[:separator], out[separator:] # return quotient, remainder.
if __name__ == '__main__':
print("POLYNOMIAL SYNTHETIC DIVISION")
N = [1, -12, 0, -42]
D = [1, -3]
print(" %s / %s =" % (N,D), " %s remainder %s" % extended_synthetic_division(N, D))
Sample output:
POLYNOMIAL SYNTHETIC DIVISION [1, -12, 0, -42] / [1, -3] = [1, -9, -27] remainder [-123]
Racket
#lang racket/base
(require racket/list)
;; dividend and divisor are both polynomials, which are here simply lists of coefficients.
;; Eg: x^2 + 3x + 5 will be represented as (list 1 3 5)
(define (extended-synthetic-division dividend divisor)
(define out (list->vector dividend)) ; Copy the dividend
;; for general polynomial division (when polynomials are non-monic), we need to normalize by
;; dividing the coefficient with the divisor's first coefficient
(define normaliser (car divisor))
(define divisor-length (length divisor)) ; } we use these often enough
(define out-length (vector-length out)) ; }
(for ((i (in-range 0 (- out-length divisor-length -1))))
(vector-set! out i (quotient (vector-ref out i) normaliser))
(define coef (vector-ref out i))
(unless (zero? coef) ; useless to multiply if coef is 0
(for ((i+j (in-range (+ i 1) ; in synthetic division, we always skip the first
(+ i divisor-length))) ; coefficient of the divisior, because it's
(divisor_j (in-list (cdr divisor)))) ; only used to normalize the dividend coefficients
(vector-set! out i+j (+ (vector-ref out i+j) (* coef divisor_j -1))))))
;; The resulting out contains both the quotient and the remainder, the remainder being the size of
;; the divisor (the remainder has necessarily the same degree as the divisor since it's what we
;; couldn't divide from the dividend), so we compute the index where this separation is, and return
;; the quotient and remainder.
;; return quotient, remainder (conveniently like quotient/remainder)
(split-at (vector->list out) (- out-length (sub1 divisor-length))))
(module+ main
(displayln "POLYNOMIAL SYNTHETIC DIVISION")
(define N '(1 -12 0 -42))
(define D '(1 -3))
(define-values (Q R) (extended-synthetic-division N D))
(printf "~a / ~a = ~a remainder ~a~%" N D Q R))
- Output:
POLYNOMIAL SYNTHETIC DIVISION (1 -12 0 -42) / (1 -3) = (1 -9 -27) remainder (-123)
Raku
(formerly Perl 6)
sub synthetic-division ( @numerator, @denominator ) {
my @result = @numerator;
my $end = @denominator.end;
for ^(@numerator-$end) -> $i {
@result[$i] /= @denominator[0];
@result[$i+$_] -= @denominator[$_] * @result[$i] for 1..$end;
}
'quotient' => @result[0 ..^ *-$end],
'remainder' => @result[*-$end .. *];
}
my @tests =
[1, -12, 0, -42], [1, -3],
[1, 0, 0, 0, -2], [1, 1, 1, 1];
for @tests -> @n, @d {
my %result = synthetic-division( @n, @d );
say "[{@n}] / [{@d}] = [%result<quotient>], remainder [%result<remainder>]";
}
- Output:
[1 -12 0 -42] / [1 -3] = [1 -9 -27], remainder [-123] [1 0 0 0 -2] / [1 1 1 1] = [1 -1], remainder [0 0 -1]
REXX
/* REXX Polynomial Division */
/* extended to support order of divisor >1 */
call set_dd '1 0 0 0 -1'
Call set_dr '1 1 1 1'
Call set_dd '1 -12 0 -42'
Call set_dr '1 -3'
q.0=0
Say list_dd '/' list_dr
do While dd.0>=dr.0
q=dd.1/dr.1
Do j=1 To dr.0
dd.j=dd.j-q*dr.j
End
Call set_q q
Call shift_dd
End
say 'Quotient:' mk_list_q() 'Remainder:' mk_list_dd()
Exit
set_dd:
Parse Arg list
list_dd='['
Do i=1 To words(list)
dd.i=word(list,i)
list_dd=list_dd||dd.i','
End
dd.0=i-1
list_dd=left(list_dd,length(list_dd)-1)']'
Return
set_dr:
Parse Arg list
list_dr='['
Do i=1 To words(list)
dr.i=word(list,i)
list_dr=list_dr||dr.i','
End
dr.0=i-1
list_dr=left(list_dr,length(list_dr)-1)']'
Return
set_q:
z=q.0+1
q.z=arg(1)
q.0=z
Return
shift_dd:
Do i=2 To dd.0
ia=i-1
dd.ia=dd.i
End
dd.0=dd.0-1
Return
mk_list_q:
list='['q.1''
Do i=2 To q.0
list=list','q.i
End
Return list']'
mk_list_dd:
list='['dd.1''
Do i=2 To dd.0
list=list','dd.i
End
Return list']'
- Output:
[1,-12,0,-42] / [1,-3] Quotient: [1,-9,-27] Remainder: -123 [1,0,0,0,-2] / [1,1,1,1] Quotient: [1,-1] Remainder: [0,0,-1]
Scala
Java Interoperability
- Output:
Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).
import java.util
object PolynomialSyntheticDivision extends App {
val N: Array[Int] = Array(1, -12, 0, -42)
val D: Array[Int] = Array(1, -3)
def extendedSyntheticDivision(dividend: Array[Int],
divisor: Array[Int]): Array[Array[Int]] = {
val out = dividend.clone
val normalizer = divisor(0)
for (i <- 0 until dividend.length - (divisor.length - 1)) {
out(i) /= normalizer
val coef = out(i)
if (coef != 0)
for (j <- 1 until divisor.length) out(i + j) += -divisor(j) * coef
}
val separator = out.length - (divisor.length - 1)
Array[Array[Int]](util.Arrays.copyOfRange(out, 0, separator),
util.Arrays.copyOfRange(out, separator, out.length))
}
println(f"${util.Arrays.toString(N)}%s / ${util.Arrays.toString(D)}%s = ${
util.Arrays
.deepToString(extendedSyntheticDivision(N, D).asInstanceOf[Array[AnyRef]])
}%s")
}
Sidef
func extended_synthetic_division(dividend, divisor) {
var end = divisor.end
var out = dividend.clone
var normalizer = divisor[0]
for i in ^(dividend.len - end) {
out[i] /= normalizer
var coef = out[i]
if (coef != 0) {
for j in (1 .. end) {
out[i+j] += -(divisor[j] * coef)
}
}
}
var remainder = out.splice(-end)
var quotient = out
return(quotient, remainder)
}
var (n, d) = ([1, -12, 0, -42], [1, -3])
print(" %s / %s =" % (n, d))
print(" %s remainder %s\n" % extended_synthetic_division(n, d))
- Output:
[1, -12, 0, -42] / [1, -3] = [1, -9, -27] remainder [-123]
Tcl
This uses a common utility proc range, and a less common one called lincr, which increments elements of lists. The routine for polynomial division is placed in a namespace ensemble, such that it can be conveniently shared with other commands for polynomial arithmetic (eg polynomial multiply).
# range ?start? end+1
# start defaults to 0: [range 5] = {0 1 2 3 4}
proc range {a {b ""}} {
if {$b eq ""} {
set b $a
set a 0
}
for {set r {}} {$a<$b} {incr a} {
lappend r $a
}
return $r
}
# lincr list idx ?...? increment
# By analogy with [lset] and [incr]:
# Adds incr to the item at [lindex list idx ?...?]. incr may be a float.
proc lincr {_ls args} {
upvar 1 $_ls ls
set incr [lindex $args end]
set idxs [lrange $args 0 end-1]
lset ls {*}$idxs [expr {$incr + [lindex $ls {*}$idxs]}]
}
namespace eval polynomial {
# polynomial division, returns [list $dividend $remainder]
proc divide {top btm} {
set out $top
set norm [lindex $btm 0]
foreach i [range [expr {[llength $top] - [llength $btm] + 1}]] {
lset out $i [set coef [expr {[lindex $out $i] * 1.0 / $norm}]]
if {$coef != 0} {
foreach j [range 1 [llength $btm]] {
lincr out [expr {$i+$j}] [expr {-[lindex $btm $j] * $coef}]
}
}
}
set terms [expr {[llength $btm]-1}]
list [lrange $out 0 end-$terms] [lrange $out end-[incr terms -1] end]
}
namespace export *
namespace ensemble create
}
proc test {} {
set top {1 -12 0 -42}
set btm {1 -3}
set div [polynomial divide $top $btm]
puts "$top / $btm = $div"
}
test
- Output:
1 -12 0 -42 / 1 -3 = {1.0 -9.0 -27.0} -123.0
Wren
import "./dynamic" for Tuple
var Solution = Tuple.create("Solution", ["quotient", "remainder"])
var extendedSyntheticDivision = Fn.new { |dividend, divisor|
var out = dividend.toList
var normalizer = divisor[0]
var separator = dividend.count - divisor.count + 1
for (i in 0...separator) {
out[i] = (out[i] / normalizer).truncate
var coef = out[i]
if (coef != 0) {
for (j in 1...divisor.count) out[i + j] = out[i + j] - divisor[j] * coef
}
}
return Solution.new(out[0...separator], out[separator..-1])
}
System.print("POLYNOMIAL SYNTHETIC DIVISION")
var n = [1, -12, 0, -42]
var d = [1, -3]
var sol = extendedSyntheticDivision.call(n, d)
System.write("%(n) / %(d) = ")
System.print("%(sol.quotient), remainder %(sol.remainder)")
System.print()
var n2 = [1, 0, 0, 0, -2]
var d2 = [1, 1, 1, 1]
var sol2 = extendedSyntheticDivision.call(n2, d2)
System.write("%(n2) / %(d2) = ")
System.print("%(sol2.quotient), remainder %(sol2.remainder)")
- Output:
POLYNOMIAL SYNTHETIC DIVISION [1, -12, 0, -42] / [1, -3] = [1, -9, -27], remainder [-123] [1, 0, 0, 0, -2] / [1, 1, 1, 1] = [1, -1], remainder [0, 0, -1]
zkl
fcn extended_synthetic_division(dividend, divisor){
# Fast polynomial division by using Extended Synthetic Division. Also works with non-monic polynomials.
# dividend and divisor are both polynomials, which are here simply lists of coefficients. Eg: x^2 + 3x + 5 will be represented as [1, 3, 5]
out,normalizer:=dividend.copy(), divisor[0];
foreach i in (dividend.len() - (divisor.len() - 1)){
out[i] /= normalizer; # for general polynomial division (when polynomials are non-monic),
# we need to normalize by dividing the coefficient with the divisor's first coefficient
coef := out[i];
if(coef != 0){ # useless to multiply if coef is 0
foreach j in ([1..divisor.len() - 1]){ # in synthetic division, we always skip the first coefficient of the divisior,
out[i + j] += -divisor[j] * coef; # because it's only used to normalize the dividend coefficients
}
}
}
# out contains the quotient and remainder, the remainder being the size of the divisor (the remainder
# has necessarily the same degree as the divisor since it's what we couldn't divide from the dividend), so we compute the index
# where this separation is, and return the quotient and remainder.
separator := -(divisor.len() - 1);
return(out[0,separator], out[separator,*]) # return quotient, remainder.
}println("POLYNOMIAL SYNTHETIC DIVISION");
N,D := T(1, -12, 0, -42), T(1, -3);
print(" %s / %s =".fmt(N,D));
println(" %s remainder %s".fmt(extended_synthetic_division(N,D).xplode()));- Output:
POLYNOMIAL SYNTHETIC DIVISION L(1,-12,0,-42) / L(1,-3) = L(1,-9,-27) remainder L(-123)