Polynomial derivative
Polynomial derivative 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.
Given a polynomial, represented by an ordered list of its coefficients by increasing degree (e.g. [-1, 6, 5] represents 5x2+6x-1), calculate the polynomial representing the derivative. For example, the derivative of the aforementioned polynomial is 10x+6, represented by [6, 10]. Test cases: 5, -3x+4, 5x2+6x-1, x3-2x2+3x-4, -x4-x3+x+1
- Related task
ALGOL 68
BEGIN # find the derivatives of polynominals, given their coefficients #
# returns the derivative polynominal of the polynominal defined by #
# the array of coeficients, where the coefficients are in #
# order of ioncreasing power of x #
OP DERIVATIVE = ( []INT p )[]INT:
BEGIN
[ 1 : UPB p - 1 ]INT result;
FOR i FROM 2 TO UPB p DO
result[ i - 1 ] := ( i - 1 ) * p[ i ]
OD;
result
END # DERIVATIVE # ;
# prints the polynomial defined by the coefficients in p #
OP SHOW = ( []INT p )VOID:
BEGIN
BOOL first := TRUE;
FOR i FROM UPB p BY -1 TO LWB p DO
IF p[ i ] /= 0 THEN
IF first THEN
IF p[ i ] < 0 THEN print( ( "-" ) ) FI
ELSE
IF p[ i ] < 0
THEN print( ( " - " ) )
ELSE print( ( " + " ) )
FI
FI;
first := FALSE;
IF i = LWB p
THEN print( ( whole( ABS p[ i ], 0 ) ) )
ELSE
IF ABS p[ i ] > 1 THEN print( ( whole( ABS p[ i ], 0 ) ) ) FI;
print( ( "x" ) );
IF i > LWB p + 1 THEN print( ( "^", whole( i - 1, 0 ) ) ) FI
FI
FI
OD;
IF first THEN
# all coefficients were 0 #
print( ( "0" ) )
FI
END # SHOW # ;
# task test cases #
PROC test = ( []INT p )VOID: BEGIN SHOW p; print( ( " -> " ) ); SHOW DERIVATIVE p; print( ( newline ) ) END;
test( ( 5 ) ); test( ( 4, -3 ) ); test( ( -1, 6, 5 ) ); test( ( -4, 3, -2, 1 ) ); test( ( 1, 1, 0, -1, -1 ) )
END- Output:
5 -> 0 -3x + 4 -> -3 5x^2 + 6x - 1 -> 10x + 6 x^3 - 2x^2 + 3x - 4 -> 3x^2 - 4x + 3 -x^4 - x^3 + x + 1 -> -4x^3 - 3x^2 + 1
Delphi
const Poly1: array [0..1-1] of double = (5); {5}
const Poly2: array [0..2-1] of double = (4,-3); {-3x+4}
const Poly3: array [0..3-1] of double = (-1,6,5); {5x^2+6x-1}
const Poly4: array [0..4-1] of double = (-4,3,-2,1); {x^3-2x^2+3x-4}
const Poly5: array [0..5-1] of double = (1,1,0,-1,-1); {-x^4-x^3+x+1}
function GetDerivative(P: array of double): TDoubleDynArray;
var I,N: integer;
begin
SetLength(Result,Length(P)-1);
if Length(P)<1 then exit;
for I:=0 to High(Result) do
Result[I]:= (I+1)*P[I+1];
end;
function GetPolyStr(D: array of double): string;
{Get polynomial in standard math format string}
var I: integer;
var S: string;
function GetSignStr(Lead: boolean; D: double): string;
{Get the sign of coefficient}
begin
Result:='';
if D>0 then
begin
if not Lead then Result:=' + ';
end
else
begin
Result:='-';
if not Lead then Result:=' - ';
end;
end;
begin
Result:='';
{Get each coefficient}
for I:=High(D) downto 0 do
begin
{Ignore zero values}
if D[I]=0 then continue;
{Get sign and coefficient}
S:=GetSignStr(Result='',D[I]);
S:=S+FloatToStrF(abs(D[I]),ffFixed,18,0);
{Combine with exponents }
if I>1 then Result:=Result+Format('%SX^%d',[S,I])
else if I=1 then Result:=Result+Format('%SX',[S,I])
else Result:=Result+Format('%S',[S]);
end;
end;
procedure ShowDerivative(Memo: TMemo; Poly: array of double);
{Show polynomial and and derivative}
var D: TDoubleDynArray;
begin
D:=GetDerivative(Poly);
Memo.Lines.Add('Polynomial: '+GetPolyStr(Poly));
Memo.Lines.Add('Derivative: '+'['+GetPolyStr(D)+']');
Memo.Lines.Add('');
end;
procedure ShowPolyDerivative(Memo: TMemo);
var D: TDoubleDynArray;
begin
ShowDerivative(Memo,Poly1);
ShowDerivative(Memo,Poly2);
ShowDerivative(Memo,Poly3);
ShowDerivative(Memo,Poly4);
ShowDerivative(Memo,Poly5);
end;
- Output:
Polynomial: 5 Derivative: [] Polynomial: -3X + 4 Derivative: [-3] Polynomial: 5X^2 + 6X - 1 Derivative: [10X + 6] Polynomial: 1X^3 - 2X^2 + 3X - 4 Derivative: [3X^2 - 4X + 3] Polynomial: -1X^4 - 1X^3 + 1X + 1 Derivative: [-4X^3 - 3X^2 + 1] Elapsed Time: 17.842 ms.
F#
// Polynomial derivative. Nigel Galloway: January 4th., 2023
let n=[[5];[4;-3];[-1;6;5];[-4;3;-2;1];[1;1;0;-1;-1]]|>List.iter((List.mapi(fun n g->n*g)>>List.skip 1>>printfn "%A"))
- Output:
[] [-3] [6; 10] [3; -4; 3] [1; 0; -3; -4]
Factor
USING: generalizations kernel math.polynomials prettyprint ;
{ 5 }
{ 4 -3 }
{ -1 6 5 }
{ -4 3 -2 1 }
{ 1 1 0 -1 -1 }
[ pdiff ] 5 napply .s clear
- Output:
{ }
{ -3 }
{ 6 10 }
{ 3 -4 3 }
{ 1 0 -3 -4 }
The implementation of pdiff:
USING: kernel math.vectors sequences ;
IN: math.polynomials
: pdiff ( p -- p' ) dup length <iota> v* rest ;
FreeBASIC
sub polydiff( p() as integer )
'differentiates the polynomial
'p(0) + p(1)x + p(2)x^2 +... + p(n)x^n
'in place
dim as integer i, n = ubound(p)
if n=0 then
p(0)=0
return
end if
for i = 0 to n - 1
p(i) = (i+1)*p(i+1)
next i
redim preserve p(0 to n-1)
return
end sub
sub print_poly( p() as integer )
'quick and dirty display of the poly
if ubound(p)=0 and p(0)=0 then
print 0
return
end if
for i as integer = 0 to ubound(p)
if i = 0 then print p(i);" ";
if i = 1 and p(i)>0 then print using "+ #x";p(i);
if i = 1 and p(i)<0 then print using "- #x";-p(i);
if i > 1 and p(i)>0 then print using "+ #x^#";p(i);i;
if i > 1 and p(i)<0 then print using "- #x^#";-p(i);i;
next i
print
end sub
'test cases
redim as integer p(0)
p(0) = 5
print_poly(p())
print "Differentiates to "
polydiff(p())
print_poly(p()): print
redim as integer p(1)
p(0) = 4 : p(1) = -3
print_poly(p())
print "Differentiates to "
polydiff(p())
print_poly(p()): print
redim as integer p(2)
p(0) = -1 : p(1) = 6 : p(2) = 5
print_poly(p())
print "Differentiates to "
polydiff(p())
print_poly(p()): print
redim as integer p(3)
p(0) = 4 : p(1) = 3 : p(2) = -2 : p(3) = 1
print_poly(p())
print "Differentiates to "
polydiff(p())
print_poly(p()): print
redim as integer p(4)
p(0) = 1 : p(1) = 1 : p(2) = 0 : p(3) = -1 : p(4) = -1
print_poly(p())
print "Differentiates to "
polydiff(p())
print_poly(p()): print
- Output:
5 Differentiates to 0
4 - 3x Differentiates to -3
-1 + 6x+ 5x^2 Differentiates to 6 + %10x
4 + 3x- 2x^2+ 1x^3 Differentiates to 3 - 4x+ 3x^2
1 + 1x- 1x^3- 1x^4 Differentiates to
1 - 3x^2- 4x^3
Go
package main
import (
"fmt"
"strings"
)
func derivative(p []int) []int {
if len(p) == 1 {
return []int{0}
}
d := make([]int, len(p)-1)
copy(d, p[1:])
for i := 0; i < len(d); i++ {
d[i] = p[i+1] * (i + 1)
}
return d
}
var ss = []string{"", "", "\u00b2", "\u00b3", "\u2074", "\u2075", "\u2076", "\u2077", "\u2078", "\u2079"}
// for n <= 20
func superscript(n int) string {
if n < 10 {
return ss[n]
}
if n < 20 {
return ss[1] + ss[n-10]
}
return ss[2] + ss[0]
}
func abs(n int) int {
if n < 0 {
return -n
}
return n
}
func polyPrint(p []int) string {
if len(p) == 1 {
return fmt.Sprintf("%d", p[0])
}
var terms []string
for i := 0; i < len(p); i++ {
if p[i] == 0 {
continue
}
c := fmt.Sprintf("%d", p[i])
if i > 0 && abs(p[i]) == 1 {
c = ""
if p[i] != 1 {
c = "-"
}
}
x := "x"
if i <= 0 {
x = ""
}
terms = append(terms, fmt.Sprintf("%s%s%s", c, x, superscript(i)))
}
for i, j := 0, len(terms)-1; i < j; i, j = i+1, j-1 {
terms[i], terms[j] = terms[j], terms[i]
}
s := strings.Join(terms, "+")
return strings.Replace(s, "+-", "-", -1)
}
func main() {
fmt.Println("The derivatives of the following polynomials are:\n")
polys := [][]int{{5}, {4, -3}, {-1, 6, 5}, {-4, 3, -2, 1}, {1, 1, 0, -1, -1}}
for _, poly := range polys {
deriv := derivative(poly)
fmt.Printf("%v -> %v\n", poly, deriv)
}
fmt.Println("\nOr in normal mathematical notation:\n")
for _, poly := range polys {
deriv := derivative(poly)
fmt.Println("Polynomial : ", polyPrint(poly))
fmt.Println("Derivative : ", polyPrint(deriv), "\n")
}
}
- Output:
The derivatives of the following polynomials are: [5] -> [0] [4 -3] -> [-3] [-1 6 5] -> [6 10] [-4 3 -2 1] -> [3 -4 3] [1 1 0 -1 -1] -> [1 0 -3 -4] Or in normal mathematical notation: Polynomial : 5 Derivative : 0 Polynomial : -3x+4 Derivative : -3 Polynomial : 5x²+6x-1 Derivative : 10x+6 Polynomial : x³-2x²+3x-4 Derivative : 3x²-4x+3 Polynomial : -x⁴-x³+x+1 Derivative : -4x³-3x²+1
Haskell
deriv = zipWith (*) [1..] . tail
main = mapM_ (putStrLn . line) ps
where
line p = "\np = " ++ show p ++ "\np' = " ++ show (deriv p)
ps = [[5],[4,-3],[-1,6,5],[-4,3,-2,1],[1,1,0,-1,-1]]
main p = [5] p' = [] p = [4,-3] p' = [-3] p = [-1,6,5] p' = [6,10] p = [-4,3,-2,1] p' = [3,-4,3] p = [1,1,0,-1,-1] p' = [1,0,-3,-4]
With fancy output
{-# language LambdaCase #-}
showPoly [] = "0"
showPoly p = foldl1 (\r -> (r ++) . term) $
dropWhile null $
foldMap (\(c, n) -> [show c ++ expt n]) $
zip p [0..]
where
expt = \case 0 -> ""
1 -> "*x"
n -> "*x^" ++ show n
term = \case [] -> ""
'0':'*':t -> ""
'-':'1':'*':t -> " - " ++ t
'1':'*':t -> " + " ++ t
'-':t -> " - " ++ t
t -> " + " ++ t
main = mapM_ (putStrLn . line) ps
where
line p = "\np = " ++ showPoly p ++ "\np' = " ++ showPoly (deriv p)
ps = [[5],[4,-3],[-1,6,5],[-4,3,-2,1],[1,1,0,-1,-1]]
main p = 5 p' = 0 p = 4 - 3*x p' = -3 p = -1 + 6*x + 5*x^2 p' = 6 + 10*x p = -4 + 3*x - 2*x^2 + x^3 p' = 3 - 4*x + 3*x^2 p = 1 + 1*x - 1*x^3 - 1*x^4 p' = 1 - 3*x^2 - 4*x^
J
Implementation:
pderiv=: -@(1 >. _1+#) {. (* i.@#)
Task examples:
pderiv 5
0
pderiv 4 _3
_3
pderiv _1 6 5
6 10
pderiv _4 3 _2 1
3 _4 3
pderiv 1 1 _1 _1
1 _2 _3
Note also that J's p. can be used to apply one of these polynomials to an argument. For example:
5 p. 2 3 5 7
5 5 5 5
(pderiv 5) p. 2 3 5 7
0 0 0 0
4 _3 p. 2 3 5 7
_2 _5 _11 _17
(pderiv 4 _3) p. 2 3 5 7
_3 _3 _3 _3
jq
Adapted from Wren (with corrections)
Works with gojq, the Go implementation of jq
The following definition of polyPrint has no restriction on the degree of the polynomial.
# The input should be a non-empty array of integers representing a polynomial.
# The output likewise represents its derivative.
def derivative:
. as $p
| if length == 1 then [0]
else reduce range(0; length-1) as $i (.[1:];
.[$i] = $p[$i+1] * ($i + 1) )
end;
def polyPrint:
def ss: ["\u2070", "\u00b9", "\u00b2", "\u00b3", "\u2074", "\u2075", "\u2076", "\u2077", "\u2078", "\u2079"];
def digits: tostring | explode[] | [.] | implode | tonumber;
ss as $ss
| def superscript:
if . <= 1 then ""
else reduce digits as $d (""; . + $ss[$d] )
end;
. as $p
| if length == 1 then .[0] | tostring
else reduce range(0; length) as $i ([];
if $p[$i] != 0
then (if $i > 0 then "x" else "" end) as $x
| ( if $i > 0 and ($p[$i]|length) == 1
then (if $p[$i] == 1 then "" else "-" end)
else ($p[$i]|tostring)
end ) as $c
| . + ["\($c)\($x)\($i|superscript)"]
else . end )
| reverse
| join("+")
| gsub("\\+-"; "-")
end ;
def task:
def polys: [ [5], [4, -3], [-1, 6, 5], [-4, 3, -2, 1], [1, 1, 0, -1, -1] ];
"Example polynomials and their derivatives:\n",
( polys[] | "\(.) -> \(derivative)" ),
"\nOr in normal mathematical notation:\n",
( polys[]
| "Polynomial : \(polyPrint)",
"Derivative : \(derivative|polyPrint)\n" ) ;
task- Output:
Example polynomials and their derivatives: [5] -> [0] [4,-3] -> [-3] [-1,6,5] -> [6,10] [-4,3,-2,1] -> [3,-4,3] [1,1,0,-1,-1] -> [1,0,-3,-4] Or in normal mathematical notation: Polynomial : 5 Derivative : 0 Polynomial : -3x+4 Derivative : -3 Polynomial : 5x²+6x-1 Derivative : 10x+6 Polynomial : x³-2x²+3x-4 Derivative : 3x²-4x+3 Polynomial : -x⁴-x³+x+1 Derivative : -4x³-3x²+1
Julia
using Polynomials
testcases = [
("5", [5]),
("-3x+4", [4, -3]),
("5x2+6x-1", [-1, 6, 5]),
("x3-2x2+3x-4", [-4, 3, -2, 1]),
("-x4-x3+x+1", [1, 1, 0, -1, -1]),
]
for (s, coef) in testcases
println("Derivative of $s: ", derivative(Polynomial(coef)))
end
- Output:
Derivative of 5: 0 Derivative of -3x+4: -3 Derivative of 5x2+6x-1: 6 + 10*x Derivative of x3-2x2+3x-4: 3 - 4*x + 3*x^2 Derivative of -x4-x3+x+1: 1 - 3*x^2 - 4*x^3
Nim
import std/sequtils
type
Polynomial[T] = object
coeffs: seq[T]
Term = tuple[coeff, exp: int]
template `[]`[T](poly: Polynomial[T]; idx: Natural): T =
poly.coeffs[idx]
template `[]=`[T](poly: var Polynomial; idx: Natural; val: T) =
poly.coeffs[idx] = val
template degree(poly: Polynomial): int =
poly.coeffs.high
func newPolynomial[T](coeffs: openArray[T]): Polynomial[T] =
## Create a polynomial from a list of coefficients.
result.coeffs = coeffs.toSeq
func newPolynomial[T](degree: Natural = 0): Polynomial[T] =
## Create a polynomial with given degree.
## Coefficients are all zeroes.
result.coeffs = newSeq[T](degree + 1)
const Superscripts: array['0'..'9', string] = ["⁰", "¹", "²", "³", "⁴", "⁵", "⁶", "⁷", "⁸", "⁹"]
func superscript(n: Natural): string =
## Return the Unicode string to use to represent an exponent.
if n == 1:
return ""
for d in $n:
result.add Superscripts[d]
func `$`(term: Term): string =
## Return the string representation of a term.
if term.coeff == 0: "0"
elif term.exp == 0: $term.coeff
else:
let base = 'x' & superscript(term.exp)
if term.coeff == 1: base
elif term.coeff == -1: '-' & base
else: $term.coeff & base
func `$`[T](poly: Polynomial[T]): string =
## Return the string representation of a polynomial.
for idx in countdown(poly.degree, 0):
let coeff = poly[idx]
var term: Term = (coeff: coeff, exp: idx)
if result.len == 0:
result.add $term
elif coeff > 0:
result.add '+'
result.add $term
elif coeff < 0:
term.coeff = -term.coeff
result.add '-'
result.add $term
func derivative[T](poly: Polynomial[T]): Polynomial[T] =
## Return the derivative of a polynomial.
if poly.degree == 0: return newPolynomial[T]()
result = newPolynomial[T](poly.degree - 1)
for degree in 1..poly.degree:
result[degree - 1] = degree * poly[degree]
for coeffs in @[@[5], @[4, -3], @[-1, 6, 5], @[-4, 3, -2, 1], @[1, 1, 0, -1, -1]]:
let poly = newPolynomial(coeffs)
echo "Polynomial: ", poly
echo "Derivative: ", poly.derivative
echo()
- Output:
Polynomial: 5 Derivative: 0 Polynomial: -3x+4 Derivative: -3 Polynomial: 5x²+6x-1 Derivative: 10x+6 Polynomial: x³-2x²+3x-4 Derivative: 3x²-4x+3 Polynomial: -x⁴-x³+x+1 Derivative: -4x³-3x²+1
Perl
use strict;
use warnings;
use feature 'say';
use utf8;
binmode(STDOUT, ':utf8');
sub pp {
my(@p) = @_;
return 0 unless @p;
my @f = $p[0];
push @f, ($p[$_] != 1 and $p[$_]) . 'x' . ($_ != 1 and (qw<⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹>)[$_])
for grep { $p[$_] != 0 } 1 .. $#p;
( join('+', reverse @f) =~ s/-1x/-x/gr ) =~ s/\+-/-/gr
}
for ([5], [4,-3], [-1,3,-2,1], [-1,6,5], [1,1,0,-1,-1]) {
my @poly = @$_;
say 'Polynomial: ' . join(', ', @poly) . ' ==> ' . pp @poly;
$poly[$_] *= $_ for 0 .. $#poly;
shift @poly;
say 'Derivative: ' . (@poly ? join', ', @poly : 0) . ' ==> ' . pp(@poly) . "\n";
}
- Output:
Polynomial: 5 ==> 5 Derivative: 0 ==> 0 Polynomial: 4, -3 ==> -3x+4 Derivative: -3 ==> -3 Polynomial: -1, 3, -2, 1 ==> x³-2x²+3x-1 Derivative: 3, -4, 3 ==> 3x²-4x+3 Polynomial: -1, 6, 5 ==> 5x²+6x-1 Derivative: 6, 10 ==> 10x+6 Polynomial: 1, 1, 0, -1, -1 ==> -x⁴-x³+x+1 Derivative: 1, 0, -3, -4 ==> -4x³-3x²+1
Phix
-- -- demo\rosetta\Polynomial_derivative.exw -- with javascript_semantics function derivative(sequence p) if p={} then return {} end if sequence r = repeat(0,length(p)-1) for i=1 to length(r) do r[i] = i*p[i+1] end for return r end function function poly(sequence si) -- display helper, copied from demo\rosetta\Polynomial_long_division.exw string r = "" for t=length(si) to 1 by -1 do integer sit = si[t] if sit!=0 then if sit=1 and t>1 then r &= iff(r=""? "":" + ") elsif sit=-1 and t>1 then r &= iff(r=""?"-":" - ") else if r!="" then r &= iff(sit<0?" - ":" + ") sit = abs(sit) end if r &= sprintf("%d",sit) end if r &= iff(t>1?"x"&iff(t>2?sprintf("^%d",t-1):""):"") end if end for if r="" then r="0" end if return r end function constant tests = {{5},{4,-3},{-1,6,5},{-4,3,-2,1},{1,1,0,-1,-1}} for i=1 to length(tests) do sequence t = tests[i], r = derivative(t) printf(1,"%20s ==> %16s (internally %v -> %v)\n",{poly(t),poly(r),t,r}) end for ?"done" {} = wait_key()
- Output:
5 ==> 0 (internally {5} -> {})
-3x + 4 ==> -3 (internally {4,-3} -> {-3})
5x^2 + 6x - 1 ==> 10x + 6 (internally {-1,6,5} -> {6,10})
x^3 - 2x^2 + 3x - 4 ==> 3x^2 - 4x + 3 (internally {-4,3,-2,1} -> {3,-4,3})
-x^4 - x^3 + x + 1 ==> -4x^3 - 3x^2 + 1 (internally {1,1,0,-1,-1} -> {1,0,-3,-4})
Python
def polynomial_derivative(list_coeff):
derivative = []
for i, val in enumerate(list_coeff):
derivative.append(i*val)
return derivative[1:]
print(polynomial_derivative([5]))
print(polynomial_derivative([4,-3]))
print(polynomial_derivative([-1,6,5]))
print(polynomial_derivative([-4,3,-2,1]))
print(polynomial_derivative([1,1,0,-1,-1]))
Output
[] [-3] [6, 10] [3, -4, 3] [1, 0, -3, -4]
R
polynomial_derivative<- function(list_coeff){
derivative = c()
if (length(list_coeff)>1){
i <- 1
for (val in list_coeff[2:length(list_coeff)]){
derivative <- c(derivative,i*val)
i <- i+1
}
}
derivative
}
print(polynomial_derivative(c(5)))
print(polynomial_derivative(c(4,-3)))
print(polynomial_derivative(c(-1,6,5)))
print(polynomial_derivative(c(-4,3,-2,1)))
print(polynomial_derivative(c(1,1,0,-1,-1)))
Output
NULL [1] -3 [1] 6 10 [1] 3 -4 3 [1] 1 0 -3 -4
Raku
use Lingua::EN::Numbers:ver<2.8+>;
sub pretty (@poly) {
join( '+', (^@poly).reverse.map: { @poly[$_] ~ "x{.&super}" } )\
.subst(/['+'|'-']'0x'<[⁰¹²³⁴⁵⁶⁷⁸⁹]>*/, '', :g).subst(/'x¹'<?before <-[⁰¹²³⁴⁵⁶⁷⁸⁹]>>/, 'x')\
.subst(/'x⁰'$/, '').subst(/'+-'/, '-', :g).subst(/(['+'|'-'|^])'1x'/, {"$0x"}, :g) || 0
}
for [5], [4,-3], [-1,3,-2,1], [-1,6,5], [1,1,0,-1,-1] -> $test {
say "Polynomial: " ~ "[{$test.join: ','}] ➡ " ~ pretty $test;
my @poly = |$test;
(^@poly).map: { @poly[$_] *= $_ };
shift @poly;
say "Derivative: " ~ "[{@poly.join: ','}] ➡ " ~ pretty @poly;
say '';
}
- Output:
Polynomial: [5] ➡ 5 Derivative: [] ➡ 0 Polynomial: [4,-3] ➡ -3x+4 Derivative: [-3] ➡ -3 Polynomial: [-1,3,-2,1] ➡ x³-2x²+3x-1 Derivative: [3,-4,3] ➡ 3x²-4x+3 Polynomial: [-1,6,5] ➡ 5x²+6x-1 Derivative: [6,10] ➡ 10x+6 Polynomial: [1,1,0,-1,-1] ➡ -x⁴-x³+x+1 Derivative: [1,0,-3,-4] ➡ -4x³-3x²+1
RPL
RPL can symbolically derivate many functions, including polynoms.
Built-in derivation
Formal derivation can be performed by the ∂ operator directly from the interpreter command line. Invoking then the COLCT function allows to simplify the formula but sometimes changes the order of terms, as shown with the last example.
5 'x' ∂ COLCT '-3*x+4' 'x' ∂ COLCT '5x^2+6*x-1' 'x' ∂ COLCT 'x^3-2*x^2+3*x-4' 'x' ∂ COLCT '-x^4-x^3+x+1' 'x' ∂ DUP COLCT
- Output:
6: 0 5: -3 4: '10*x+6' 3: '3*x^2-4*x+3' 2: '-(4*x^3)-3*x^2+1' 1: '1-4*x^3-3*x^2'
Classical programming
Assuming we ignore the existence of the ∂ operator, here is a typical RPL program that handles polynoms as lists of scalars, to be completed by another program to display the list on a fancier way.
≪ → coeffs
≪ IF coeffs SIZE 1 -
THEN
{ } 1 LAST FOR j
coeffs j 1 + GET j 1 MAX * +
NEXT
ELSE { 0 }
END
≫ ≫
'DPDX' STO
{5} DPDX
{4 -3} DPDX
{-1 6 5} DPDX
{-4 3 -2 1} DPDX
{1 1 0 -1 -1} DPDX
- Output:
5: { 0 }
4: { -3 }
3: { 6 10 }
2: { 3 -4 3 }
1: { 1 0 -3 -4 }
Fancy output
≪ → coeffs
≪ coeffs 1 GET
coeffs SIZE 2 FOR j
coeffs j GET ‘x’ j 1 - ^ * SWAP +
-1 STEP
≫ COLCT
≫
‘→EQ’ STO
{1 1 0 -1 -1} DPDX
DUP →EQ
- Output:
2: { 1 0 -3 -4 }
1: '1-3*x^2-4*x^3'
Sidef
func derivative(f) {
Poly(f.coeffs.map_2d{|e,k| [e-1, k*e] }.flat...)
}
var coeffs = [
[5],
[4,-3],
[-1,6,5],
[-4,3,-2,1],
[-1, 6, 5],
[1,1,0,-1,-1],
]
for c in (coeffs) {
var poly = Poly(c.flip)
var derv = derivative(poly)
var d = { derv.coeff(_) }.map(0..derv.degree)
say "Polynomial : #{'%20s' % c} = #{poly}"
say "Derivative : #{'%20s' % d} = #{derv || 0}\n"
}
- Output:
Polynomial : [5] = 5 Derivative : [0] = 0 Polynomial : [4, -3] = -3*x + 4 Derivative : [-3] = -3 Polynomial : [-1, 6, 5] = 5*x^2 + 6*x - 1 Derivative : [6, 10] = 10*x + 6 Polynomial : [-4, 3, -2, 1] = x^3 - 2*x^2 + 3*x - 4 Derivative : [3, -4, 3] = 3*x^2 - 4*x + 3 Polynomial : [-1, 6, 5] = 5*x^2 + 6*x - 1 Derivative : [6, 10] = 10*x + 6 Polynomial : [1, 1, 0, -1, -1] = -x^4 - x^3 + x + 1 Derivative : [1, 0, -3, -4] = -4*x^3 - 3*x^2 + 1
Wren
var derivative = Fn.new { |p|
if (p.count == 1) return [0]
var d = p[1..-1].toList
for (i in 0...d.count) d[i] = p[i+1] * (i + 1)
return d
}
var ss = ["", "", "\u00b2", "\u00b3", "\u2074", "\u2075", "\u2076", "\u2077", "\u2078", "\u2079"]
// for n <= 20
var superscript = Fn.new { |n| (n < 10) ? ss[n] : (n < 20) ? ss[1] + ss[n - 10] : ss[2] + ss[0] }
var polyPrint = Fn.new { |p|
if (p.count == 1) return p[0].toString
var terms = []
for (i in 0...p.count) {
if (p[i] == 0) continue
var c = p[i].toString
if (i > 0 && p[i].abs == 1) c = (p[i] == 1) ? "" : "-"
var x = (i > 0) ? "x" : ""
terms.add("%(c)%(x)%(superscript.call(i))")
}
return terms[-1..0].join("+").replace("+-", "-")
}
System.print("The derivatives of the following polynomials are:\n")
var polys = [ [5], [4, -3], [-1, 6, 5], [-4, 3, -2, 1], [1, 1, 0, -1, -1] ]
for (poly in polys) {
var deriv = derivative.call(poly)
System.print("%(poly) -> %(deriv)")
}
System.print("\nOr in normal mathematical notation:\n")
for (poly in polys) {
var deriv = derivative.call(poly)
System.print("Polynomial : %(polyPrint.call(poly))")
System.print("Derivative : %(polyPrint.call(deriv))\n")
}
- Output:
The derivatives of the following polynomials are: [5] -> [0] [4, -3] -> [-3] [-1, 6, 5] -> [6, 10] [-4, 3, -2, 1] -> [3, -4, 3] [1, 1, 0, -1, -1] -> [1, 0, -3, -4] Or in normal mathematical notation: Polynomial : 5 Derivative : 0 Polynomial : -3x+4 Derivative : -3 Polynomial : 5x²+6x-1 Derivative : 10x+6 Polynomial : x³-2x²+3x-4 Derivative : 3x²-4x+3 Polynomial : -x⁴-x³+x+1 Derivative : -4x³-3x²+1
XPL0
int IntSize, Cases, Case, Len, Deg, Coef;
[IntSize:= @Case - @Cases;
Cases:=[[ 5],
[ 4, -3],
[-1, 6, 5],
[-4, 3, -2, 1],
[ 1, 1, 0, -1, -1],
[ 0]];
for Case:= 0 to 5-1 do
[Len:= (Cases(Case+1) - Cases(Case)) / IntSize;
for Deg:= 0 to Len-1 do
[Coef:= Cases(Case, Deg);
if Deg = 0 then Text(0, "[")
else [IntOut(0, Coef*Deg);
if Deg < Len-1 then
Text(0, ", ");
];
];
Text(0, "]^M^J");
];
]- Output:
[] [-3] [6, 10] [3, -4, 3] [1, 0, -3, -4]