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.
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>
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 ${r2.contentToString()}")
}</lang>
- Output:
POLYNOMIAL SYNTHETIC DIVISION [1, -12, 0, -42] / [1, -3] = [1, -9, -27], remainder [-123] [1, 0, 0, 0, -2] / [1, 1, 1, 1] = [1, -1], remainder [0, 0, -1]
Perl
<lang perl>sub synthetic_division {
my($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]
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[*-$end .. *];
}
my @tests = [1, -12, 0, -42], [1, -3], [1, 0, 0, 0, -2], [1, 1, 1, 1];
for @tests -> @n, @d {
my %result = synthetic-division( @n, @d ); say "[{@n}] / [{@d}] = [%result<quotient>], remainder [%result<remainder>]";
}</lang>
- Output:
[1 -12 0 -42] / [1 -3] = [1 -9 -27], remainder [-123] [1 0 0 0 -2] / [1 1 1 1] = [1 -1], remainder [0 0 -1]
Phix
<lang Phix>function extendedSyntheticDivision(sequence dividend, divisor)
sequence out = dividend integer normalizer = divisor[1] integer separator = length(dividend) - length(divisor) + 1 for i=1 to separator do out[i] /= normalizer integer coef = out[i] if (coef != 0) then for j=2 to length(divisor) do out[i+j-1] += -divisor[j] * coef end for end if end for return {out[1..separator],out[separator+1..$]}
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)
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
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)