Polynomial synthetic division
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- 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
<lang 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))</lang>
- Output:
POLYNOMIAL SYNTHETIC DIVISION [1, -12, 0, -42] / [1, -3] = [1, -9, -27] remainder [-123]
C++
<lang cpp>/*
* 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";
} </lang>
C#
<lang csharp>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)
);
}
}
} </lang>
Delphi
Thanks Rudy Velthuis for the Velthuis.BigRationals library.
<lang Delphi>
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.</lang>
- Output:
[1 -12 0 -42 ] / [1 -3 ] = [1 -9 -27 ] remainder [-123 ]
Go
<lang 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)
}</lang>
- Output:
[1/1 -12/1 0/1 -42/1] / [1/1 -3/1] = [1/1 -9/1 -27/1] remainder [-123/1]
J
Solving this the easy way:
<lang J> psd=: [:(}. ;{.) ([ (] -/@,:&}. (* {:)) ] , %&{.~)^:(>:@-~&#)~</lang>
Task example:
<lang J> (1, (-12), 0, -42) psd (1, -3) ┌────────┬────┐ │1 _9 _27│_123│ └────────┴────┘ </lang>
Java
<lang 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)
};
}
}</lang>
[1, -12, 0, -42] / [1, -3] = [[1, -9, -27], [-123]]
Julia
<lang 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 [$remainder]")
end
</lang>
- Output:
[[1, -12, 0, -42]] / [[1, -3]] = [[1, -9, -27]] with remainder [[-123]] [[1, 0, 0, 0, -2]] / [[1, 1, 1, 1]] = [[1, -1]] with remainder [[0, 0, -1]]
Kotlin
<lang scala>// version 1.1.2
fun extendedSyntheticDivision(dividend: IntArray, divisor: IntArray): Pair<IntArray, IntArray> {
val out = dividend.copyOf()
val normalizer = divisor[0]
val separator = dividend.size - divisor.size + 1
for (i in 0 until separator) {
out[i] /= normalizer
val coef = out[i]
if (coef != 0) {
for (j in 1 until divisor.size) out[i + j] += -divisor[j] * coef
}
}
return out.copyOfRange(0, separator) to out.copyOfRange(separator, out.size)
}
fun main(args: Array<String>) {
println("POLYNOMIAL SYNTHETIC DIVISION")
val n = intArrayOf(1, -12, 0, -42)
val d = intArrayOf(1, -3)
val (q, r) = extendedSyntheticDivision(n, d)
print("${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 ${r2.contentToString()}")
}</lang>
- Output:
POLYNOMIAL SYNTHETIC DIVISION [1, -12, 0, -42] / [1, -3] = [1, -9, -27], remainder [-123] [1, 0, 0, 0, -2] / [1, 1, 1, 1] = [1, -1], remainder [0, 0, -1]
Mathematica / Wolfram Language
<lang Mathematica>MakePolynomial[l_List, x_] := FromCoefficientRules[Thread[List /@ Range[Length[l] - 1, 0, -1] -> l], {x}] num = MakePolynomial[{1, -12, 0, -42}, x]; den = MakePolynomial[{1, -3}, x]; PolynomialQuotient[num, den, x] PolynomialRemainder[num, den, x]</lang>
- Output:
-27 - 9 x + x^2 -123
Perl
<lang perl>sub synthetic_division {
my($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 [$remainder]\n";
}
print poly_divide([1, -12, 0, -42], [1, -3]); print poly_divide([1, 0, 0, 0, -2], [1, 1, 1, 1]);</lang>
- Output:
[1 -12 0 -42] / [1 -3] = [1 -9 -27], remainder [-123] [1 0 0 0 -2] / [1 1 1 1] = [1 -1], remainder [0 0 -1]
Phix
<lang Phix>function extendedSyntheticDivision(sequence dividend, divisor)
sequence out = dividend
integer normalizer = divisor[1]
integer separator = length(dividend) - length(divisor) + 1
for i=1 to separator do
out[i] /= normalizer
integer coef = out[i]
if (coef != 0) then
for j=2 to length(divisor) do out[i+j-1] += -divisor[j] * coef end for
end if
end for
return {out[1..separator],out[separator+1..$]}
end function
constant tests = {{{1, -12, 0, -42},{1, -3}},
{{1, 0, 0, 0, -2},{1, 1, 1, 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</lang>
- 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}
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).
<lang 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), print " %s remainder %s" % extended_synthetic_division(N, D)
</lang>
Sample output:
POLYNOMIAL SYNTHETIC DIVISION [1, -12, 0, -42] / [1, -3] = [1, -9, -27] remainder [-123]
Racket
<lang 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))</lang>
- Output:
POLYNOMIAL SYNTHETIC DIVISION (1 -12 0 -42) / (1 -3) = (1 -9 -27) remainder (-123)
Raku
(formerly Perl 6)
<lang perl6>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>]";
}</lang>
- 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
<lang 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']'
</lang>
- 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).
<lang Scala>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 = ${
util.Arrays
.deepToString(extendedSyntheticDivision(N, D).asInstanceOf[Array[AnyRef]])
}%s")
}</lang>
Sidef
<lang ruby>func extended_synthetic_division(dividend, divisor) {
var end = divisor.end var out = dividend.clone var normalizer = divisor[0]
for i in ^(dividend.len - end) {
out[i] /= normalizer
var coef = out[i]
if (coef != 0) {
for j in (1 .. end) {
out[i+j] += -(divisor[j] * coef)
}
}
}
var remainder = out.splice(-end) var quotient = out
return(quotient, remainder)
}
var (n, d) = ([1, -12, 0, -42], [1, -3]) print(" %s / %s =" % (n, d)) print(" %s remainder %s\n" % extended_synthetic_division(n, d))</lang>
- Output:
[1, -12, 0, -42] / [1, -3] = [1, -9, -27] remainder [-123]
Tcl
This uses a common utility proc range, and a less common one called lincr, which increments elements of lists. The routine for polynomial division is placed in a namespace ensemble, such that it can be conveniently shared with other commands for polynomial arithmetic (eg polynomial multiply).
<lang Tcl># range ?start? end+1
- start defaults to 0: [range 5] = {0 1 2 3 4}
proc range {a {b ""}} {
if {$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</lang>
- Output:
1 -12 0 -42 / 1 -3 = {1.0 -9.0 -27.0} -123.0
Wren
<lang ecmascript>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)")</lang>
- 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
<lang 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.
}</lang> <lang zkl>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()));</lang>
- Output:
POLYNOMIAL SYNTHETIC DIVISION L(1,-12,0,-42) / L(1,-3) = L(1,-9,-27) remainder L(-123)