Formal power series

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Task
Formal power series
You are encouraged to solve this task according to the task description, using any language you may know.

A power series is an infinite sum of the form

a_0 + a_1 \cdot x + a_2 \cdot x^2 + a_3 \cdot x^3 + \cdots

The ai are called the coefficients of the series. Such sums can be added, multiplied etc., where the new coefficients of the powers of x are calculated according to the usual rules.

If one is not interested in evaluating such a series for particular values of x, or in other words, if convergence doesn't play a role, then such a collection of coefficients is called formal power series. It can be treated like a new kind of number.

Task: Implement formal power series as a numeric type. Operations should at least include addition, multiplication, division and additionally non-numeric operations like differentiation and integration (with an integration constant of zero). Take care that your implementation deals with the potentially infinite number of coefficients.

As an example, define the power series of sine and cosine in terms of each other using integration, as in

\sin x = \int_0^x \cos t\, dt

\cos x = 1 - \int_0^x \sin t\, dt

Goals: Demonstrate how the language handles new numeric types and delayed (or lazy) evaluation.

Contents

[edit] Ada

The Taylor series package is generic to be instantiated with any rational type implementation provided by the task Rational Arithmetic:

with Generic_Rational;
 
generic
with package Rational_Numbers is new Generic_Rational (<>);
package Generic_Taylor_Series is
use Rational_Numbers;
type Taylor_Series is array (Natural range <>) of Rational;
 
function "+" (A : Taylor_Series) return Taylor_Series;
function "-" (A : Taylor_Series) return Taylor_Series;
 
function "+" (A, B : Taylor_Series) return Taylor_Series;
function "-" (A, B : Taylor_Series) return Taylor_Series;
function "*" (A, B : Taylor_Series) return Taylor_Series;
 
function Integral (A : Taylor_Series) return Taylor_Series;
function Differential (A : Taylor_Series) return Taylor_Series;
 
function Value (A : Taylor_Series; X : Rational) return Rational;
 
Zero : constant Taylor_Series := (0 => Rational_Numbers.Zero);
One  : constant Taylor_Series := (0 => Rational_Numbers.One);
end Generic_Taylor_Series;

The package implementation:

package body Generic_Taylor_Series is
function Normalize (A : Taylor_Series) return Taylor_Series is
begin
for Power in reverse A'Range loop
if A (Power) /= 0 then
return A (0..Power);
end if;
end loop;
return Zero;
end Normalize;
 
function "+" (A : Taylor_Series) return Taylor_Series is
begin
return A;
end "+";
 
function "-" (A : Taylor_Series) return Taylor_Series is
Result : Taylor_Series (A'Range);
begin
for Power in A'Range loop
Result (Power) := -A (Power);
end loop;
return Result;
end "-";
 
function "+" (A, B : Taylor_Series) return Taylor_Series is
begin
if A'Last > B'Last then
return B + A;
else
declare
Result : Taylor_Series (0..B'Last);
begin
for Power in A'Range loop
Result (Power) := A (Power) + B (Power);
end loop;
for Power in A'Last + 1..B'Last loop
Result (Power) := B (Power);
end loop;
return Normalize (Result);
end;
end if;
end "+";
 
function "-" (A, B : Taylor_Series) return Taylor_Series is
begin
return A + (-B);
end "-";
 
function "*" (A, B : Taylor_Series) return Taylor_Series is
Result : Taylor_Series (0..A'Last + B'Last);
begin
for I in A'Range loop
for J in B'Range loop
Result (I + J) := A (I) * B (J);
end loop;
end loop;
return Normalize (Result);
end "*";
 
function Integral (A : Taylor_Series) return Taylor_Series is
begin
if A = Zero then
return Zero;
else
declare
Result : Taylor_Series (0..A'Last + 1);
begin
for Power in A'Range loop
Result (Power + 1) := A (Power) / Number (Power + 1);
end loop;
Result (0) := Rational_Numbers.Zero;
return Result;
end;
end if;
end Integral;
 
function Differential (A : Taylor_Series) return Taylor_Series is
begin
if A'Length = 1 then
return Zero;
else
declare
Result : Taylor_Series (0..A'Last - 1);
begin
for Power in Result'Range loop
Result (Power) := A (Power + 1) * Number (Power);
end loop;
return Result;
end;
end if;
end Differential;
 
function Value (A : Taylor_Series; X : Rational) return Rational is
Sum : Rational := A (A'Last);
begin
for Power in reverse 0..A'Last - 1 loop
Sum := Sum * X + A (Power);
end loop;
return Sum;
end Value;
 
end Generic_Taylor_Series;

The procedure Normalize is used to truncate the series when the coefficients are zero. The summation of a series (function Value) uses Horner scheme.

[edit] Test task

with Ada.Text_IO;  use Ada.Text_IO;
 
with Generic_Taylor_Series;
with Generic_Rational;
 
procedure Test_Taylor_Series is
package Integer_Rationals is new Generic_Rational (Integer);
package Integer_Taylor_Series is new Generic_Taylor_Series (Integer_Rationals);
use Integer_Taylor_Series;
-- Procedure to print a series
procedure Put (A : Taylor_Series) is
use Integer_Rationals;
procedure Put (A : Rational) is
begin
if Numerator (A) = 1 then
Put (" 1");
else
Put (Integer'Image (Numerator (A)));
end if;
if Denominator (A) /= 1 then
Put (" /");
Put (Integer'Image (Denominator (A)));
end if;
end Put;
begin
if A (0) /= 0 then
Put (A (0));
end if;
for Power in 1..A'Last loop
if A (Power) > 0 then
Put (" +");
Put (A (Power));
Put (" X **" & Integer'Image (Power));
elsif A (Power) < 0 then
Put (" -");
Put (abs A (Power));
Put (" X **" & Integer'Image (Power));
end if;
end loop;
end Put;
-- Cosine generator
function Cos (N : Natural) return Taylor_Series is
begin
if N = 0 then
return One;
else
return One - Integral (Integral (Cos (N - 1)));
end if;
end Cos;
begin
Put ("Cos ="); Put (Cos (5)); Put_Line (" ...");
Put ("Sin ="); Put (Integral (Cos (5))); Put_Line (" ...");
end Test_Taylor_Series;

Sample output:

Cos = 1 - 1 / 2 X ** 2 + 1 / 24 X ** 4 - 1 / 720 X ** 6 + 1 / 40320 X ** 8 - 1 / 3628800 X ** 10 ...
Sin = + 1 X ** 1 - 1 / 6 X ** 3 + 1 / 120 X ** 5 - 1 / 5040 X ** 7 + 1 / 362880X ** 9 - 1 / 39916800 X ** 11 ...

[edit] Clojure

This version takes advantage of the laziness of most of Clojure's sequence functions, including map, for, take-while, concat, and drop. A formal power series (FPS) is represented as a sequence of coefficients; for example, [1 2 3] represents 1 + 2*x + 3*x*x.

First addition and subtraction. Note that most of the complication arises in allowing for finite and infinite FPSs; if only infinite power series were at issue, the function (defn ips+ [ips0 ips1] (map + ips0 ips1)) would suffice.

(defn ps+ [ps0 ps1]
(letfn [(+zs [ps] (concat ps (repeat :z)))
(notz? [a] (not= :z a))
(nval [a] (if (notz? a) a 0))
(z+ [a0 a1] (if (= :z a0 a1) :z (+ (nval a0) (nval a1))))]
(take-while notz? (map z+ (+zs ps0) (+zs ps1)))))
 
(defn ps- [ps0 ps1] (ps+ ps0 (map - ps1)))

Multiplication next; again most of the complication is dealing with both finite and infinite FPS. This function explicitly uses the standard function lazy-seq to define the product sequence.

(defn ps*
([ps0 ps1] (ps* [0] ps0 ps1))
([[a0 & resta] [p0 & rest0] [p1 & rest1 :as ps1]]
(lazy-seq
(cons
(+ a0 (* p0 p1))
(let [mrest1 (if (or (nil? rest1) (zero? p0)) nil, (map #(* p0 %) rest1))
accum (cond (nil? resta) mrest1, (nil? mrest1) resta, :else (ps+ resta mrest1))]
(if (nil? rest0) accum, (ps* (or accum [0]) rest0 ps1)))))))

As with most of the other examples on this page, there's no definition for division. Mathematically, FPS is a commutative ring (in fact a Euclidean domain), but not a field: the set of FPSs is not closed under division.

Now we can define integration and differentiation:

(defn indexed [ps] (map vector (iterate inc 0) ps))
 
(defn differentiate [ps]
(drop 1 (for [[n a] (indexed ps)] (* n a))))
 
(defn integrate [ps]
(cons 0 (for [[n a] (indexed ps)] (/ a (inc n)))))

Some examples of using these functions; in each case a println call (which forces the lazy sequence) is followed by a comment showing the output.

(println (ps+ [1 2] [3 4 5]))
; (4 6 5)
 
(println (ps* [1 2] [3 4 5]))
; (3 10 13 10)

And some examples using infinite FPSs. First define the sequence of factorials (facts), then define sin and cos Taylor series.

(def nfacts (iterate (fn [[f n]] [(* f n) (inc n)]) [1 1]))
(def facts (map first nfacts))
 
(def sin (map / (cycle [0 1 0 -1]) facts))
(def cos (map / (cycle [1 0 -1 0]) facts))
 
(println (take 10 sin))
; (0 1 0 -1/6 0 1/120 0 -1/5040 0 1/362880)
 
(println (take 10 (integrate cos)))
; (0 1 0 -1/6 0 1/120 0 -1/5040 0 1/362880)
 
(println (take 20 (ps+ (ps* sin sin) (ps* cos cos))))
; (1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)

By using letfn, which supports defining mutually recursive functions, we can define the sin and cos power series directly in terms of integrals of the other series:

(letfn [(fsin [] (lazy-seq (integrate (fcos))))
(fcos [] (ps- [1] (integrate (fsin))))]
(def sinx (fsin))
(def cosx (fcos)))
 
(println (take 10 sinx))
; (0 1 0 -1/6 0 1/120 0 -1/5040 0 1/362880)

[edit] Common Lisp

Common Lisp isn't lazy, and doesn't define the arithmetic operators as generic functions. As such, this implementation defines lazy primitives (delay and force), and a lazy list built on top of them (lons, lar, ldr). This implementation also defines a package #:formal-power-series which uses all the symbols of the "COMMON-LISP" package except for +, -, *, and /, which are shadowed. Shadowing these symbols allows for definitions of generic functions which can be specialized for the power series, can default to the normal CL arithmetic operations for other kinds of objects, and can do "the right thing" for mixes thereof.

(defpackage #:formal-power-series
(:nicknames #:fps)
(:use "COMMON-LISP")
(:shadow
#:+ #:- #:* #:/))
 
(in-package #:formal-power-series)

[edit] Lazy Primitives

(defstruct promise
thunk value)
 
(defmacro delay (form)
`(make-promise :thunk #'(lambda () ,form)))
 
(defun force (object)
(cond
((not (promise-p object))
object)
((null (promise-thunk object))
(promise-value object))
(t (let ((val (funcall (promise-thunk object))))
(setf (promise-thunk object) nil
(promise-value object) val)))))

[edit] Lazy Lists

(defstruct lons
lar
ldr)
 
(defun lar (lons)
(lons-lar lons))
 
(defun ldr (lons)
(if (not (promise-p (lons-ldr lons)))
(lons-ldr lons)
(setf (lons-ldr lons)
(force (lons-ldr lons)))))
 
(defmacro lons (lar ldr)
`(make-lons :lar ,lar :ldr (delay ,ldr)))

A few utilities to make working with lazy lists easier.

(defun invoke-with-lons (function lons)
(funcall function (lar lons) (ldr lons)))
 
(defmacro with-lons ((lar ldr) lons &body body)
`(invoke-with-lons #'(lambda (,lar ,ldr) ,@body) ,lons))
 
(defun maplar (function llist &rest llists)
(let ((llists (list* llist llists)))
(if (some 'null llists) nil
(lons (apply function (cl:mapcar 'lar llists))
(apply 'maplar function (cl:mapcar 'ldr llists))))))
 
(defun take (n llist)
(if (zerop n) '()
(lons (lar llist)
(take (1- n) (ldr llist)))))
 
(defun force-list (llist)
(do ((fl '() (cons (lar l) fl))
(l llist (ldr l)))
((null l) (nreverse fl))))
 
(defun repeat (x)
(lons x (repeat x)))
 
(defun up-from (n)
(lons n (up-from (1+ n))))

[edit] Formal Power Series

The mathematical operations here are translations of the Haskell code, but we specialize the operations in various ways so that behavior for normal numeric operations is preserved.

(defstruct (series (:constructor series (coeffs)) (:conc-name))
coeffs)
 
(defgeneric negate (f)
(:method (f)
(cl:- f))
(:method ((f series))
(series (maplar 'negate (coeffs f)))))
 
(defgeneric + (f g)
(:method (f g)
(cl:+ f g))
(:method (f (g series))
(series (lons (+ f (lar (coeffs g))) (ldr (coeffs g)))))
(:method ((f series) g)
(+ g f))
(:method ((f series) (g series))
(series (maplar '+ (coeffs f) (coeffs g)))))
 
(defun - (f g)
(+ f (negate g)))
 
(defun series-* (f g)
(with-lons (f ft) (coeffs f)
(with-lons (g gt) (coeffs g)
(series (lons (* f g)
(coeffs (+ (* (series ft)
(series gt))
(* f (series gt)))))))))
 
(defgeneric * (f g)
(:method (f g)
(cl:* f g))
(:method ((f series) g)
(series (maplar #'(lambda (x) (* x g)) (coeffs f))))
(:method (f (g series))
(* g f))
(:method ((f series) (g series))
(series-* f g)))
 
(defun series-/ (f g)
(with-lons (f ft) (coeffs f)
(with-lons (g gt) (coeffs g)
(let ((qs nil))
(setf qs (lons (/ f g)
(maplar #'(lambda (x) (/ x g))
(coeffs (- (series ft)
(* (series qs)
(series gt)))))))))))
 
(defgeneric / (f g)
(:method (f g)
(cl:/ f g))
(:method ((f series) g)
(series (maplar #'(lambda (x) (/ x g)) (coeffs f))))
(:method (f (g series))
(/ (series (lons f (repeat 0))) g))
(:method ((f series) (g series))
(series-/ f g)))
 
(defun int (f)
(series (lons 0 (maplar '/ (coeffs (force f)) (up-from 1)))))
 
(defun diff (f)
(series (maplar '* (ldr (coeffs f)) (up-from 1))))

[edit] Example

(defparameter *sinx*
(locally (declare (special *cosx*))
(delay (int (force *cosx*)))))
 
(defparameter *cosx*
(delay (- 1 (int *sinx*))))
FPS > (force-list (take 10 (coeffs (force *sinx*))))
(0 1 0 -1/6 0 1/120 0 -1/5040 0 1/362880)

FPS > (force-list (take 10 (coeffs (force *cosx*))))
(1 0 -1/2 0 1/24 0 -1/720 0 1/40320 0)

FPS > (force-list (take 10 (coeffs (+ (force *cosx*) (force *sinx*)))))
(1 1 -1/2 -1/6 1/24 1/120 -1/720 -1/5040 1/40320 1/362880)

FPS > (force-list (take 10 (coeffs (- (+ 2 (force *cosx*))
                                      (* 3 (force *sinx*))))))
(3 -3 -1/2 1/2 1/24 -1/40 -1/720 1/1680 1/40320 -1/120960)

[edit] C

Following is a simple implementation of formal power series in C. It's not "new datatype for the language" per se, but does demonstrate how lazy evaluation and infinite list generation can be done for this task. Note that, to be of real use, one should also cache terms looked up and free up memory. Both are trivially done (I actually had them, but removed them for simplicity).

#include <stdio.h>
#include <stdlib.h>
#include <math.h> /* for NaN */
 
enum fps_type {
FPS_CONST = 0,
FPS_ADD,
FPS_SUB,
FPS_MUL,
FPS_DIV,
FPS_DERIV,
FPS_INT,
};
 
typedef struct fps_t *fps;
typedef struct fps_t {
int type;
fps s1, s2;
double a0;
} fps_t;
 
fps fps_new()
{
fps x = malloc(sizeof(fps_t));
x->a0 = 0;
x->s1 = x->s2 = 0;
x->type = 0;
return x;
}
 
/* language limit of C; when self or mutual recursive definition is needed,
* one has to be defined, then defined again after it's used. See how
* sin and cos are defined this way below
*/

void fps_redefine(fps x, int op, fps y, fps z)
{
x->type = op;
x->s1 = y;
x->s2 = z;
}
 
fps _binary(fps x, fps y, int op)
{
fps s = fps_new();
s->s1 = x;
s->s2 = y;
s->type = op;
return s;
}
 
fps _unary(fps x, int op)
{
fps s = fps_new();
s->s1 = x;
s->type = op;
return s;
}
 
/* Taking the n-th term of series. This is where actual work is done. */
double term(fps x, int n)
{
double ret = 0;
int i;
 
switch (x->type) {
case FPS_CONST: return n > 0 ? 0 : x->a0;
case FPS_ADD:
ret = term(x->s1, n) + term(x->s2, n); break;
 
case FPS_SUB:
ret = term(x->s1, n) - term(x->s2, n); break;
 
case FPS_MUL:
for (i = 0; i <= n; i++)
ret += term(x->s1, i) * term(x->s2, n - i);
break;
 
case FPS_DIV:
if (! term(x->s2, 0)) return NAN;
 
ret = term(x->s1, n);
for (i = 1; i <= n; i++)
ret -= term(x->s2, i) * term(x, n - i) / term(x->s2, 0);
break;
 
case FPS_DERIV:
ret = n * term(x->s1, n + 1);
break;
 
case FPS_INT:
if (!n) return x->a0;
ret = term(x->s1, n - 1) / n;
break;
 
default:
fprintf(stderr, "Unknown operator %d\n", x->type);
exit(1);
}
 
return ret;
}
 
#define _add(x, y) _binary(x, y, FPS_ADD)
#define _sub(x, y) _binary(x, y, FPS_SUB)
#define _mul(x, y) _binary(x, y, FPS_MUL)
#define _div(x, y) _binary(x, y, FPS_DIV)
#define _integ(x) _unary(x, FPS_INT)
#define _deriv(x) _unary(x, FPS_DERIV)
 
fps fps_const(double a0)
{
fps x = fps_new();
x->type = FPS_CONST;
x->a0 = a0;
return x;
}
 
int main()
{
int i;
fps one = fps_const(1);
fps fcos = fps_new(); /* cosine */
fps fsin = _integ(fcos); /* sine */
fps ftan = _div(fsin, fcos); /* tangent */
 
/* redefine cos to complete the mutual recursion; maybe it looks
* better if I said
* *fcos = *( _sub(one, _integ(fsin)) );
*/

fps_redefine(fcos, FPS_SUB, one, _integ(fsin));
 
fps fexp = fps_const(1); /* exponential */
/* make exp recurse on self */
fps_redefine(fexp, FPS_INT, fexp, 0);
 
printf("Sin:"); for (i = 0; i < 10; i++) printf(" %g", term(fsin, i));
printf("\nCos:"); for (i = 0; i < 10; i++) printf(" %g", term(fcos, i));
printf("\nTan:"); for (i = 0; i < 10; i++) printf(" %g", term(ftan, i));
printf("\nExp:"); for (i = 0; i < 10; i++) printf(" %g", term(fexp, i));
 
return 0;
}
Output:
Sin: 0 1 0 -0.166667 0 0.00833333 0 -0.000198413 0 2.75573e-06

Cos: 1 0 -0.5 0 0.0416667 0 -0.00138889 0 2.48016e-05 0 Tan: 0 1 0 0.333333 0 0.133333 0 0.0539683 0 0.0218695

Exp: 1 1 0.5 0.166667 0.0416667 0.00833333 0.00138889 0.000198413 2.48016e-05 2.75573e-06

[edit] D

See Formal_power_series/D.

[edit] Go

package main
 
import (
"fmt"
"math"
)
 
// Task: Formal power series type
//
// Go does not have a concept of numeric types other than the built in
// integers, floating points, and so on. Nor does it have function or
// operator overloading, or operator defintion. The type use to implement
// fps here is an interface with a single method, extract.
// While not named in the task description, extract is described in the
// WP article as "important." In fact, by representing a way to index
// all of the coefficients of a fps, any type that implements the interface
// represents a formal power series.
 
type fps interface {
extract(int) float64
}
 
// Task: Operations on FPS
//
// Separate operations are implemented with separate extract methods.
// This requires each operation on the fps type to have a concrete type.
// Executing a fps operation is the act of instantiating the concrete type.
// This is implemented here with constructor functions that construct a
// new fps from fps arguments.
 
// Constructor functions are shown here as a group, followed by concrete
// type definitions and associated extract methods.
 
func one() fps {
return &oneFps{}
}
 
func add(s1, s2 fps) fps {
return &sum{s1: s1, s2: s2}
}
 
func sub(s1, s2 fps) fps {
return &diff{s1: s1, s2: s2}
}
 
func mul(s1, s2 fps) fps {
return &prod{s1: s1, s2: s2}
}
 
func div(s1, s2 fps) fps {
return &quo{s1: s1, s2: s2}
}
 
func differentiate(s1 fps) fps {
return &deriv{s1: s1}
}
 
func integrate(s1 fps) fps {
return &integ{s1: s1}
}
 
// Example: Mutually recursive defintion of sine and cosine.
// This is a constructor just as those above. It is nullary and returns
// two fps. Note sin and cos implemented as instances of other fps defined
// above, and so do not need new concrete types. Note also the constant
// term of the integration fps provides the case that terminates recursion
// of the extract function.
func sinCos() (fps, fps) {
sin := &integ{}
cos := sub(one(), integrate(sin))
sin.s1 = cos
return sin, cos
}
 
// Following are type definitions and extract methods for fps operators
// (constructor functions) just defined.
//
// Goal: lazy evaluation
//
// Go has no built in support for lazy evaluation, so we make it from
// scratch here. Types contain, at a minimum, their fps operands and
// representation neccessary to implement lazy evaluation. Typically
// this is a coefficient slice, although constant terms are not stored,
// so in the case of a constant fps, no slice is needed at all.
// Coefficients are generated only as they are requested. Computed
// coefficients are stored in the slice and if requested subsequently,
// are returned immediately rather than recomputed.
//
// Types can also contain any other intermediate values useful for
// computing coefficients.
 
// Constant one: A constant is a nullary function and no coefficent
// storage is needed so an empty struct is used for the type.
type oneFps struct{}
 
// The extract method implements the fps interface. It simply has to
// return 1 for the first term and return 0 for all other terms.
func (*oneFps) extract(n int) float64 {
if n == 0 {
return 1
}
return 0
}
 
// Addition is a binary function so the sum type stores its two fps operands
// and its computed terms.
type sum struct {
s []float64
s1, s2 fps
}
 
func (s *sum) extract(n int) float64 {
for i := len(s.s); i <= n; i++ {
s.s = append(s.s, s.s1.extract(i)+s.s2.extract(i))
}
return s.s[n]
}
 
// Subtraction and other binary operations are similar.
// (The common field definitions could be factored out with an embedded
// struct, but the clutter of the extra syntax required doesn't seem
// to be worthwhile.)
type diff struct {
s []float64
s1, s2 fps
}
 
func (s *diff) extract(n int) float64 {
for i := len(s.s); i <= n; i++ {
s.s = append(s.s, s.s1.extract(i)-s.s2.extract(i))
}
return s.s[n]
}
 
type prod struct {
s []float64
s1, s2 fps
}
 
func (s *prod) extract(n int) float64 {
for i := len(s.s); i <= n; i++ {
c := 0.
for k := 0; k <= i; k++ {
c += s.s1.extract(k) * s.s1.extract(n-k)
}
s.s = append(s.s, c)
}
return s.s[n]
}
 
// Note a couple of fields in addition to those of other binary operators.
// They simply optimize computations a bit.
type quo struct {
s1, s2 fps
inv float64 // optimizes a divide
c []float64 // saves multiplications
s []float64
}
 
// WP formula. Note the limitation s2[0] cannot be 0. In this case
// the function returns NaN for all terms. The switch statement catches
// this case and avoids storing a slice of all NaNs.
func (s *quo) extract(n int) float64 {
switch {
case len(s.s) > 0:
case !math.IsInf(s.inv, 1):
a0 := s.s2.extract(0)
s.inv = 1 / a0
if a0 != 0 {
break
}
fallthrough
default:
return math.NaN()
}
for i := len(s.s); i <= n; i++ {
c := 0.
for k := 1; k <= i; k++ {
c += s.s2.extract(k) * s.c[n-k]
}
c = s.s1.extract(i) - c*s.inv
s.c = append(s.c, c)
s.s = append(s.s, c*s.inv)
}
return s.s[n]
}
 
// Note differentiation and integration are unary so their types contain
// only a single fps operand.
 
type deriv struct {
s []float64
s1 fps
}
 
func (s *deriv) extract(n int) float64 {
for i := len(s.s); i <= n; {
i++
s.s = append(s.s, float64(i)*s.s1.extract(i))
}
return s.s[n]
}
 
type integ struct {
s []float64
s1 fps
}
 
func (s *integ) extract(n int) float64 {
if n == 0 {
return 0 // constant term C=0
}
// with constant term handled, s starts at 1
for i := len(s.s) + 1; i <= n; i++ {
s.s = append(s.s, s.s1.extract(i-1)/float64(i))
}
return s.s[n-1]
}
 
// Demonstrate working sin, cos.
func main() {
// Format several terms in a way that is easy to compare visually.
partialSeries := func(f fps) (s string) {
for i := 0; i < 6; i++ {
s = fmt.Sprintf("%s %8.5f ", s, f.extract(i))
}
return
}
sin, cos := sinCos()
fmt.Println("sin:", partialSeries(sin))
fmt.Println("cos:", partialSeries(cos))
}

Output:

sin:   0.00000   1.00000   0.00000  -0.16667   0.00000   0.00833 
cos:   1.00000   0.00000  -0.50000   0.00000   0.04167   0.00000 

[edit] Elisa

The generic component FormalPowerSeries is instantiated with the Rational type as provided by the task Arithmetic/Rational

component FormalPowerSeries(Number);
type PowerSeries;
PowerSeries(Size = integer) -> PowerSeries;
 
+ PowerSeries -> PowerSeries;
- PowerSeries -> PowerSeries;
 
PowerSeries + PowerSeries -> PowerSeries;
PowerSeries - PowerSeries -> PowerSeries;
PowerSeries * PowerSeries -> PowerSeries;
 
Integral(PowerSeries) -> PowerSeries;
Differential(PowerSeries) -> PowerSeries;
 
Zero -> PowerSeries;
One -> PowerSeries;
 
Array(PowerSeries) -> array(Number);
begin
PowerSeries(Size) = PowerSeries:[T = array(Number, Size); Size];
 
+ A = A;
 
- A = [ C = PowerSeries(A.Size);
[ i = 1 .. A.Size; C.T[i] := - A.T[i] ];
C];
 
A + B = [ if A.Size > B.Size then return(B + A);
C = PowerSeries(B.Size);
[ i = 1 .. A.Size; C.T[i] := A.T[i] + B.T[i] ];
[ i = (A.Size +1) .. B.Size; C.T[i] := B.T[i] ];
C];
 
A - B = A + (- B );
 
A * B = [ C = PowerSeries(A.Size + B.Size - 1);
[ i = 1 .. A.Size;
[j = 1.. B.Size;
C.T[i + j - 1] := C.T[i + j - 1] + A.T[i] * B.T[j] ] ];
C];
 
Integral(A) = [ if A.Size == 0 then return (A);
C = PowerSeries(A.Size + 1);
[ i = 1 .. A.Size; C.T[i +1] := A.T[i] / Number( i )];
C.T[1]:= Number(0);
C ];
 
Differential(A) = [ if A.Size == 1 then return (A);
C = PowerSeries(A.Size - 1);
[ i = 1 .. C.Size; C.T[i] := A.T[i + 1] * Number( i )];
C ];
 
Zero = [ C = PowerSeries (1); C.T[1]:= Number(0); C];
One = [ C = PowerSeries (1); C.T[1]:= Number(1); C];
 
Array(PowerSeries) -> array(Number);
Array(TS) = TS.T;
 
end component FormalPowerSeries;
 

Tests

use RationalNumbers;
use FormalPowerSeries(Rational);
 
X => symbol;
term + term => term;
term / term => term;
term * term => term;
symbol ** integer => term;
 
Output(text,PowerSeries) -> term;
Output(Name,PS) = [ E1 := term:symbol(Name); E2:= null(term);
[ i = 1..size(Array(PS));
Num = Numerator(Array(PS)[i]);
if Num <> 0 then
[ E2:= term: Num / term: Denominator(Array(PS)[i]) * X ** (i-1);
E1:= E1 + E2 ];
];
E1];
 
Cos(integer) -> PowerSeries;
Cos(Limit) = [ if Limit == 1 then return(One);
( One - Integral(Integral(Cos (Limit - 1)))) ];
 
Sin(integer) -> PowerSeries;
Sin(Limit) = Integral(Cos (Limit));
 
Output("cos = ",Cos(5))?
Output("sin = ",Sin(5))?
 

Output

cos =  + 1 / 1 * X ** 0 + -1 / 2 * X ** 2 + 1 / 24 * X ** 4 + -1 / 720 * X ** 6 + 1 / 40320 * X ** 8 
sin = + 1 / 1 * X ** 1 + -1 / 6 * X ** 3 + 1 / 120 * X ** 5 + -1 / 5040 * X ** 7 + 1 / 362880 * X ** 9

[edit] Haskell

It's simpler to assume we are always dealing with an infinite list of coefficients. Mathematically, a finite power series can be generalized to an infinite power series with trailing zeros.

newtype Series a = S { coeffs :: [a] } deriving (Eq, Show)
-- Invariant: coeffs must be an infinite list
 
instance Num a => Num (Series a) where
fromInteger n = S $ fromInteger n : repeat 0
negate (S fs) = S $ map negate fs
S fs + S gs = S $ zipWith (+) fs gs
S (f:ft) * S gs@(g:gt) = S $ f*g : coeffs (S ft * S gs + S (map (f*) gt))
 
instance Fractional a => Fractional (Series a) where
fromRational n = S $ fromRational n : repeat 0
S (f:ft) / S (g:gt) = S qs where qs = f/g : map (/g) (coeffs (S ft - S qs * S gt))
 
-- utility function to convert from a finite polynomial
fromFiniteList xs = S (xs ++ repeat 0)
 
int (S fs) = S $ 0 : zipWith (/) fs [1..]
 
diff (S (_:ft)) = S $ zipWith (*) ft [1..]
 
sinx,cosx :: Series Rational
sinx = int cosx
cosx = 1 - int sinx
 
fiboS = 1 / fromFiniteList [1,-1,-1]

Output:

*Main> take 11 $ coeffs sinx
[0 % 1,1 % 1,0 % 1,(-1) % 6,0 % 1,1 % 120,0 % 1,(-1) % 5040,0 % 1,1 % 362880,0 % 1]

*Main> take 11 $ coeffs cosx
[1 % 1,0 % 1,(-1) % 2,0 % 1,1 % 24,0 % 1,(-1) % 720,0 % 1,1 % 40320,0 % 1,(-1) % 3628800]

*Main> take 11 $ coeffs $ sinx / cosx -- tangent
[0 % 1,1 % 1,0 % 1,1 % 3,0 % 1,2 % 15,0 % 1,17 % 315,0 % 1,62 % 2835,0 % 1]

*Main> take 11 $ map truncate $ coeffs $ fiboS  -- some fibonaccis
[1,1,2,3,5,8,13,21,34,55,89]

*Main> take 11 $ coeffs $ fromFiniteList [1,5] * fromFiniteList [2,3,7] -- multiplying polynomials
[2,13,22,35,0,0,0,0,0,0,0]

[edit] J

J does not allow the definition of types. Also, J requires the programmer be lazy, rather than being lazy itself. With these "limitations" understood, here is an implementation in J:

Ai=: (i.@]  =/ i.@[ -/ i.@>:@-)&#
divide=: [ +/ .*~ [:%.&.x: ] +/ .* Ai
diff=: 1 }. ] * i.@#
intg=: 0 , ] % 1 + i.@#
mult=: +//.@(*/)
plus=: +/@,:
minus=: -/@,:

See also section 2 at http://www.jsoftware.com/papers/tot1.htm

Note that this approach to division will not be accurate when exact division is not possible (a best fit polynomial will be used). Note also that if extended precision results are expected the &.x: in the definition of divide should be replaced with &x: or removed entirely (which would then require extended precision arguments).

Example use:

   1 2 1 mult 1 3 3 1
1 5 10 10 5 1
   1 5 10 10 5 1 divide 1 3 3 1
1 2 1
   1 0 plus 1 2 1
2 2 1
   intg@(1 0 minus intg)^:20 i.0    NB. sine
0 1 0 _0.166667 0 0.00833333 0 _0.000198413 0 2.75573e_6 0 _2.50521e_8 0 1.6059e_10 0 _7.64716e_13 0 2.81146e_15 0 _8.22064e_18 ...
   (1 0 minus intg)@intg^:20 i.0    NB. cosine
1 0 _0.5 0 0.0416667 0 _0.00138889 0 2.48016e_5 0 _2.75573e_7 0 2.08768e_9 0 _1.14707e_11 0 4.77948e_14 0 _1.56192e_16 0 ...

These sine and cosine results can be compared with taylor series expansions:

   1&o. t. i. 20    NB. sine
0 1 0 _0.166667 0 0.00833333 0 _0.000198413 0 2.75573e_6 0 _2.50521e_8 0 1.6059e_10 0 _7.64716e_13 0 2.81146e_15 0 _8.22064e_18
   2&o. t. i. 20    NB. cosine
1 0 _0.5 0 0.0416667 0 _0.00138889 0 2.48016e_5 0 _2.75573e_7 0 2.08768e_9 0 _1.14707e_11 0 4.77948e_14 0 _1.56192e_16 0

[edit] Java

See Formal power series/Java

[edit] jq

Works with: jq version 1.4

[edit] Introduction and Examples

Since a formal power series can be viewed as a function from the non-negative integers onto a suitable range, we shall identify a jq filter that maps integers to the appropriate range as a power series. For example, the jq function

1/(1+.)

represents the power series 1 + x/2 + x/3 + ... because 1/(1+.) maps i to 1/(i+1).

Similarly, the jq filter 1 (i.e. the filter that always returns 1) represents the power series Σ x^i.

The exponential power series, Σ (x^i)/i!, can be represented in jq by the filter:

1/factorial

assuming "factorial" is defined in the usual way:

def factorial:
reduce range(1; . + 1) as $i
(1; . * $i);

For ease of reference, we shall also define ps_exp as 1/factorial:

def ps_exp: 1/factorial;

In a later subsection of this article, we will define another function, ps_evaluate(p), for evaluating the power series, p, at the value specified by the input, so for example:

1 | ps_evaluate(ps_exp)

should evaluate to the number e approximately; using the version of ps_evaluate defined below, we find:

1 | ps_evaluate(1/factorial)

evaluates to 2.7182818284590455.

The following function definitions are useful for other power series:

def pow(n):
. as $x | n as $n
| reduce range(0;$n) as $i (1; . * $x);

For example, the power series 1 + Σ ( (x/i)^n ) where the summation is over i>0 can be written:

1/pow(.)

The power series representation of ln(1 + x) is as follows:

# ln(1+x) = x - x^2 / 2 + ...
def ln_1px:
def c: if . % 2 == 0 then -1 else 1 end;
. as $i | if $i == 0 then 0 else ($i|c) / $i end;
 

jq numbers are currently implemented using IEEE 754 64-bit arithmetic, and therefore this article will focus on power series that can be adequately represented using jq numbers. However, the approach used here can be used for power series defined on other domains, e.g. rationals, complex numbers, and so on.

[edit] Finite power series

To make it easy to represent finite power series, we define poly(ary) as follows:

def poly(ary): ary[.] // 0;

For example, poly( [1,2,3] ) represents the finite power series: 1 + 2x + 3x^2.

(The "// 0" ensures that the result is 0 for integers that are out-of-range with respect to the array.)

[edit] Addition and Subtraction

jq's "+" operator can be used to add two power series with the intended semantics; for example:

(poly([1,2,3]) + poly([-1,-2,-3]))

is equal to poly([]), i.e. 0.

This is simply because in jq, (i | (poly([1,2,3]) + poly([-1,-2,-3]))) evaluates to (i | (poly([1,2,3])) + (i|poly([-1,-2,-3]))).

Subtraction works in the same way and for the same reason. The product of two power series, however, must be handled specially.

[edit] Multiplication, Differentiation and Integration

# Multiply two power series, s and t:
def M(s;t):
. as $i | reduce range(0; 1+$i) as $k
(0; . + ($k|s) * (($i - $k)|t));
 
# Derivative of the power series, s:
def D(s): (. + 1) as $i | $i * ($i|s);
 
# Integral of the power series, s,
# with an integration constant equal to 0:
def I(s):
. as $i
| if $i == 0 then 0 else (($i-1)|s) /$i end;
 

[edit] Equality and Evaluation

The following function, ps_equal(s;t;k;eps) will check whether the first k coefficients of the two power series agree to within eps:

def ps_equal(s; t; k; eps): 
def abs: if . < 0 then -. else . end;
reduce range(0;k) as $i
(true;
if . then ((($i|s) - ($i|t))|abs) <= eps
else .
end);

To evaluate a power series, P(x), at a particular point, say y, we can define a function, ps_evaluate(p), so that (y|ps_evaluate(p)) evaluates to P(y), assuming that P(x) converges sufficiently rapidly to a value that can be represented using IEEE 754 64-bit arithmetic.

# evaluate p(x) based on the first k terms of polynomial p, where x is the input
def ps_eval(p; k):
. as $x
| reduce range(0;k) as $i
# state: [sum, x^i]
([0, 1];
.[1] as $xn
| ($i|p) as $coeff
| [ .[0] + $coeff * $xn, $x * $xn])
| .[0];
 
# If |x| < 1 then ps_evaluate(x) will evaluate to p(x) with high precision
# if the coefficients of the polynomial are eventually bounded.
#
# WARNING: ps_evaluate(p) will not detect divergence and is not intended to
# produce accurate results unless the terms of p(x) are reasonably well-behaved.
# For |x| > 1, the result will be null if x^n overflows before convergence is achieved.
#
def ps_evaluate(p):
def abs: if . < 0 then -. else . end;
def eval(p;x):
# state: [i, x^i, sum of i terms, delta, prevdelta]
recurse(
.[0] as $i
| .[1] as $xi
| .[2] as $sum
| .[3] as $delta
| .[4] as $prevdelta
| if $delta < 1e-17 and $prevdelta < 1e-17
and ( $xi < 1e-100
or ( $sum != 0 and
(($delta/$sum) | abs) < 1e-10 and
(($prevdelta/$sum) | abs) < 1e-10) )
then empty
else
($xi * ($i|p)) as $newdelta
| [ $i + 1,
x*$xi,
$sum+$newdelta,
($newdelta|abs), $delta]
end ) ;
. as $x
| [0, 1, 0, 1, 1]
| reduce eval(p; $x) as $vector (0; $vector[2]);

[edit] Examples

# Utility functions:
 
def abs: if . < 0 then -. else . end;
 
# The power series whose only non-zero coefficient is 1 at x^i:
def ps_at(i): if . == i then 1 else 0 end;
 
# Create an array consisting of the first . coefficients of the power series, p:
def ps_to_array(p): . as $in | reduce range(0;$in) as $i ([]; . + [$i|p]);
 
def pi: 4 * (1|atan);
 

'cos == I(sin)'

 
# Verify that the first 100 terms of I(cos) and of sin are the same:
 
ps_equal( I(ps_cos); ps_sin; 100; 1e-15)
# => true
 
# Verify that the two power series agree when evaluated at pi:
 
((pi | ps_evaluate(I(ps_cos))) - (pi | ps_evaluate(ps_sin))) | abs < 1e-15
# => true

'cos == 1 - I(sin)'

# Verify that the first 100 terms of cos and (1 - I(sin)) are the same:
 
ps_equal( ps_cos; ps_at(0) - I(ps_sin); 100; 1e-5)
# => true
 
# Verify that the two power series agree at pi:
 
((pi | ps_evaluate(ps_cos)) - (pi | ps_evaluate(ps_at(0) - I(ps_sin)))) | abs < 1e-15
# => true

[edit] Maxima

deftaylor(f(x), sum(n! * x^n, n, 0, inf))$
 
taylor(f(x), x, 0, 10);
/ * 1 + x + 2 * x^2 + 6 * x^3 + 24 * x^4 + 120 * x^5 + 720 * x^6 + 5040 * x^7 + 40320 * x^8 + 362880 * x^9 + 3628800 * x^10 + ... * /
 
taylor(f(x)^2, x, 0, 10);
/ * 1 + 2 * x + 5 * x^2 + 16 * x^3 + 64 * x^4 + 312 * x^5 + 1812 * x^6 + 12288 * x^7 + 95616 * x^8 + 840960 * x^9 + 8254080 * x^10 + ... * /
 
 
deftaylor(fcos(x), sum((-1)^n * x^(2 * n) / (2 * n)!, n, 0, inf))$
deftaylor(fsin(x), sum((-1)^n * x^(2 * n + 1) / (2 * n + 1)!, n, 0, inf))$
 
taylor(fcos(x)^2 + fsin(x)^2, x, 0, 20);
/ * 1 + ... * /

[edit] Lua

Parts of this depend on the formula for integration of a power series: integral(sum(a_n x^n)) = sum(a_n / n * x(n+1))

powerseries = setmetatable({
__add = function(z1, z2) return powerseries(function(n) return z1.coeff(n) + z2.coeff(n) end) end,
__sub = function(z1, z2) return powerseries(function(n) return z1.coeff(n) - z2.coeff(n) end) end,
__mul = function(z1, z2) return powerseries(function(n)
local ret = 0
for i = 0, n do
ret = ret + z1.coeff(i) * z2.coeff(n-i)
end
return ret
end) end,
__div = function(z1, z2) return powerseries(function(n)
local ret = z1.coeff(n)
local function coeffs(a)
local c = z1.coeff(a)
for j = 0, a - 1 do c = c - coeffs(j) * z2.coeff(a-j) end
return c / z2.coeff(0)
end
for i = 0, n-1 do
ret = ret - coeffs(i) * z2.coeff(n-i)
end
return ret / z2.coeff(0)
end) end,
__pow = function(z1, p) -- for a series z, z^n returns the nth derivative of z. negative values take integrals.
if p == 0 then return z1
elseif p > 0 then return powerseries(function(i) return z1.coeff(i+1)*(i+1) end)^(p-1)
else return powerseries(function(i) return z1.coeff(i-1)/i end)^(p+1)
end
end,
__unm = function(z1) return powerseries(function(n) return -z1.coeff(n) end) end,
__index = function(z, n) return z.coeff(n) end,
__call = function(z, n)
local ret = 0
for i = 0, 15 do --we do 20 terms, which is simpler than trying to check error bounds
ret = ret + z[i]*(n^i)
end
return ret
end},
{__call = function(z, f) return setmetatable({coeff = f}, z) end})
 
cosine = powerseries(function(n)
if(n == 0) then return 1
else return -((sine^(-1))[n]) --defer to the integral of sine function
end
end)
 
sine = powerseries(function(n)
if(n == 0) then return 0
else return (cosine^(-1))[n] --defer to the integral of cosine function
end
end)
 
print(sine[1], sine[3], sine[5], sine[7], cosine[0], cosine[2], cosine[4], cosine[6])
print(sine(math.pi/3), sine(math.pi/2), cosine(math.pi/3), cosine(math.pi/2))
 
tangent = sine / cosine
print(tangent(math.pi/3), tangent(math.pi/4), tangent(math.pi/6)) --something like 30000 function calls!

[edit] PARI/GP

Uses the built-in power series handling. Change default(seriesprecision) to get more terms.

sin('x)
cos('x)

[edit] Perl

Although true Lazy Lists *can* be implemented using perl (using the builtin "tie" function), I felt that doing so would make the example harder to understand.

Instead, I chose to implement delayed evaluation with a generator function and a cache.

Creating a new arithmetic type in perl is relatively easy, using the "overload" module which comes with perl.

This was partly inspired by the perl6 example, but is far from being a direct translation.

 
package FPS;
use strict;
use warnings;
use Math::BigRat;
 
sub new {
my $class = shift;
return bless {@_}, $class unless @_ == 1;
my $arg = shift;
return bless { more => $arg }, $class if 'CODE' eq ref $arg;
return bless { coeff => $arg }, $class if 'ARRAY' eq ref $arg;
bless { coeff => [$arg] }, $class;
}
 
sub coeff {
my ($self, $i) = @_;
my $cache = ($self->{coeff} ||= []);
my $more = $self->{more};
for my $j ( @$cache .. $i ) {
last unless $more;
$cache->[$j] = $more->($j, $self);
}
$cache->[$i] or 0;
}
 
sub invert {
my $orig = shift;
ref($orig)->new( sub {
my ($i, $self) = @_;
unless( $i ) {
my $a0 = $orig->coeff(0);
die "Cannot invert power series with zero constant term."
unless $a0;
(Math::BigRat->new(1) / $a0);
} else {
my $sum = 0;
my $terms = $self->{coeff};
for my $j (1 .. $i) {
$sum += $orig->coeff($j) * $terms->[$i - $j];
}
-$terms->[0] * $sum;
}
} );
}
 
sub fixargs {
my ($x, $y, $swap) = @_;
my $class = ref $x;
$y = $class->new($y) unless UNIVERSAL::isa($y, $class);
($x, $y) = ($y, $x) if $swap;
($class, $x, $y);
}
 
use overload '+' => sub {
my ($class, $x, $y) = &fixargs;
$class->new( sub { $x->coeff($_[0]) + $y->coeff($_[0]) } );
}, '-' => sub {
my ($class, $x, $y) = &fixargs;
$class->new( sub { $x->coeff($_[0]) - $y->coeff($_[0]) } );
}, '*' => sub {
my ($class, $x, $y) = &fixargs;
$class->new( sub {
my $i = shift;
my $sum = 0;
$sum += $x->coeff($_) * $y->coeff($i-$_) for 0..$i;
$sum;
} );
}, '/' => sub {
my ($class, $x, $y) = &fixargs;
$x * $y->invert;
}, '""' => sub {
my $self = shift;
my $str = $self->coeff(0);
for my $i (1..10) {
my $c = $self->coeff($i);
next unless $c;
$str .= ($c < 0) ? (" - " . (-$c)) : (" + ".$c);
$str .= "x^$i";
}
$str;
};
 
sub differentiate {
my $orig = shift;
ref($orig)->new( sub {
my $i = shift;
($i+1) * $orig->coeff($i);
} );
}
 
sub integrate {
my $orig = shift;
ref($orig)->new( coeff => [0], more => sub {
my $i = shift;
$orig->coeff($i-1) / Math::BigRat->new($i);
} );
}
 
my $sin = __PACKAGE__->new;
my $cos = 1 - $sin->integrate;
%$sin = %{$cos->integrate};
my $tan = $sin / $cos;
my $exp = __PACKAGE__->new();
%$exp = (%{$exp->integrate}, coeff => [1]);
 
print "sin(x) ~= $sin\n";
print "cos(x) ~= $cos\n";
print "tan(x) ~= $tan\n";
print "exp(x) ~= $exp\n";
 
print "sin^2 + cos^s = ", $sin*$sin + $cos*$cos, "\n";
 
1;
__END__
 
Output:
sin(x) ~= 0 + 1x^1 - 1/6x^3 + 1/120x^5 - 1/5040x^7 + 1/362880x^9
cos(x) ~= 1 - 1/2x^2 + 1/24x^4 - 1/720x^6 + 1/40320x^8 - 1/3628800x^10
tan(x) ~= 0 + 1x^1 + 1/3x^3 + 2/15x^5 + 17/315x^7 + 62/2835x^9
exp(x) ~= 1 + 1x^1 + 1/2x^2 + 1/6x^3 + 1/24x^4 + 1/120x^5 + 1/720x^6 + 1/5040x^7 + 1/40320x^8 + 1/362880x^9 + 1/3628800x^10
sin^2 + cos^s = 1

For a version which *does* use proper lazy lists, see Formal power series/Perl

[edit] Perl 6

class DerFPS { ... }
class IntFPS { ... }
 
role FPS {
method coeffs { ... }
method differentiate { DerFPS.new(:x(self)) }
method integrate { IntFPS.new(:x(self)) }
 
method pretty($n) {
sub super($i) { $i.trans('0123456789' => '⁰¹²³⁴⁵⁶⁷⁸⁹') }
my $str = $.coeffs[0].perl;
for 1..$n Z $.coeffs[1..$n] -> $i, $_ {
when * > 0 { $str ~= " + {(+$_).perl}∙x{super($i)}" }
when * < 0 { $str ~= " - {(-$_).perl}∙x{super($i)}" }
}
$str;
}
}
 
class ExplicitFPS does FPS { has @.coeffs }
 
class SumFPS does FPS {
has FPS ($.x, $.y);
method coeffs { $.x.coeffs Z+ $.y.coeffs }
}
 
class DifFPS does FPS {
has FPS ($.x, $.y);
method coeffs { $.x.coeffs Z- $.y.coeffs }
}
 
class ProFPS does FPS {
has FPS ($.x, $.y);
method coeffs { (0..*).map: { [+] ($.x.coeffs[0..$_] Z* $.y.coeffs[$_...0]) } }
}
 
class InvFPS does FPS {
has FPS $.x;
method coeffs {
# see http://en.wikipedia.org/wiki/Formal_power_series#Inverting_series
gather {
my @a := $.x.coeffs;
@a[0] != 0 or fail "Cannot invert power series with zero constant term.";
take my @b = (1 / @a[0]);
take @b[$_] = -@b[0] * [+] (@a[1..$_] Z* @b[$_-1...0]) for 1..*;
}
}
}
 
class DerFPS does FPS {
has FPS $.x;
method coeffs { (1..*).map: { $_ * $.x.coeffs[$_] } }
}
 
class IntFPS does FPS {
has FPS $.x;
method coeffs { 0, (0..*).map: { $.x.coeffs[$_] / ($_+1) } }
}
 
class DeferredFPS does FPS {
has FPS $.realized is rw;
method coeffs { $.realized.coeffs }
}
 
# some arithmetic operations for formal power series
multi infix:<+>(FPS $x, FPS $y) { SumFPS.new(:$x, :$y) }
multi infix:<->(FPS $x, FPS $y) { DifFPS.new(:$x, :$y) }
multi infix:<*>(FPS $x, FPS $y) { ProFPS.new(:$x, :$y) }
multi infix:</>(FPS $x, FPS $y) { $x * InvFPS.new(:x($y)) }
 
# an example of a mixed-type operator:
multi infix:<->(Numeric $x, FPS $y) { ExplicitFPS.new(:coeffs($x, 0 xx *)) - $y }
 
# define sine and cosine in terms of each other
my $sin = DeferredFPS.new;
my $cos = 1 - $sin.integrate;
$sin.realized = $cos.integrate;
 
# define tangent in terms of sine and cosine
my $tan = $sin / $cos;
 
say 'sin(x) ≈ ', $sin.pretty(10);
say 'cos(x) ≈ ', $cos.pretty(10);
say 'tan(x) ≈ ', $tan.pretty(10);
Output:
sin(x) ≈ 0 + 1/1∙x¹ - 1/6∙x³ + 1/120∙x⁵ - 1/5040∙x⁷ + 1/362880∙x⁹
cos(x) ≈ 1 - 1/2∙x² + 1/24∙x⁴ - 1/720∙x⁶ + 1/40320∙x⁸ - 1/3628800∙x¹⁰
tan(x) ≈ 0/1 + 1/1∙x¹ + 1/3∙x³ + 2/15∙x⁵ + 17/315∙x⁷ + 62/2835∙x⁹

[edit] PicoLisp

With a 'lazy' function, as a frontend to 'cache',

(de lazy Args
(def (car Args)
(list (cadr Args)
(cons 'cache (lit (cons))
(caadr Args)
(cddr Args) ) ) ) )

we can build a formal power series functionality:

(scl 20)
 
(de fpsOne (N)
(if (=0 N) 1.0 0) )
 
(de fpsInverse (N X)
(last
(make
(let Res1 (- (link (*/ 1.0 1.0 (X 0))))
(for I N
(link
(*/
(sum '((Res J) (*/ (X J) Res 1.0))
(made)
(range I 1) )
Res1
1.0 ) ) ) ) ) ) )
 
(de fpsAdd (N X Y)
(+ (X N) (Y N)) )
 
(de fpsSub (N X Y)
(- (X N) (Y N)) )
 
(de fpsMul (N X Y)
(sum
'((I)
(*/ (X I) (Y (- N I)) 1.0) )
(range 0 N) ) )
 
(de fpsDiv (N X Y)
(sum
'((I)
(*/ (X I) (fpsInverse (- N I) Y) 1.0) )
(range 0 N) ) )
 
(de fpsDifferentiate (N)
(curry (X) (N)
(* (X (inc N)) N) ) )
 
(de fpsIntegrate (X)
(curry (X) (N)
(or
(=0 N)
(*/ (X (dec N)) N) ) ) )
 
(lazy fpsSin (N)
((fpsIntegrate fpsCos) N) )
 
(lazy fpsCos (N)
(fpsSub N fpsOne (fpsIntegrate fpsSin)) )
 
(lazy fpsTan (N)
(fpsDiv N fpsSin fpsCos) )
 
(lazy fpsExp (N)
(if (=0 N)
1.0
((fpsIntegrate fpsExp) N) ) )

Test:

(prin "SIN:")
(for N (range 1 11 2)
(prin " " (round (fpsSin N) 9)) )
(prinl)
 
(prin "COS:")
(for N (range 0 10 2)
(prin " " (round (fpsCos N) 9)) )
(prinl)
 
(prin "TAN:")
(for N (range 1 13 2)
(prin " " (round (fpsTan N) 7)) )
(prinl)
 
(prin "EXP:")
(for N (range 0 6)
(prin " " (round (fpsExp N) 7)) )
(prinl)

Output:

SIN: 1.000000000 -0.166666667 0.008333333 -0.000198413 0.000002756 -0.000000025
COS: 1.000000000 -0.500000000 0.041666667 -0.001388889 0.000024802 -0.000000276
TAN: 1.0000000 0.3333333 0.1333333 0.0539683 0.0218695 0.0088632 0.0035921
EXP: 1.0000000 1.0000000 0.5000000 0.1666667 0.0416667 0.0083333 0.0013889

[edit] Python

Works with: Python version 2.6, 3.x
''' \
For a discussion on pipe() and head() see
http://paddy3118.blogspot.com/2009/05/pipe-fitting-with-python-generators.html
'''

 
from itertools import islice
from fractions import Fraction
from functools import reduce
try:
from itertools import izip as zip # for 2.6
except:
pass
 
def head(n):
''' return a generator that passes through at most n items
'''

return lambda seq: islice(seq, n)
 
def pipe(gen, *cmds):
''' pipe(a,b,c,d, ...) -> yield from ...d(c(b(a)))
'''

return reduce(lambda gen, cmd: cmd(gen), cmds, gen)
 
def sinepower():
n = 0
fac = 1
sign = +1
zero = 0
yield zero
while True:
n +=1
fac *= n
yield Fraction(1, fac*sign)
sign = -sign
n +=1
fac *= n
yield zero
def cosinepower():
n = 0
fac = 1
sign = +1
yield Fraction(1,fac)
zero = 0
while True:
n +=1
fac *= n
yield zero
sign = -sign
n +=1
fac *= n
yield Fraction(1, fac*sign)
def pluspower(*powergenerators):
for elements in zip(*powergenerators):
yield sum(elements)
def minuspower(*powergenerators):
for elements in zip(*powergenerators):
yield elements[0] - sum(elements[1:])
def mulpower(fgen,ggen):
'From: http://en.wikipedia.org/wiki/Power_series#Multiplication_and_division'
a,b = [],[]
for f,g in zip(fgen, ggen):
a.append(f)
b.append(g)
yield sum(f*g for f,g in zip(a, reversed(b)))
def constpower(n):
yield n
while True:
yield 0
def diffpower(gen):
'differentiatiate power series'
next(gen)
for n, an in enumerate(gen, start=1):
yield an*n
def intgpower(k=0):
'integrate power series with constant k'
def _intgpower(gen):
yield k
for n, an in enumerate(gen, start=1):
yield an * Fraction(1,n)
return _intgpower
 
 
print("cosine")
c = list(pipe(cosinepower(), head(10)))
print(c)
print("sine")
s = list(pipe(sinepower(), head(10)))
print(s)
# integrate cosine
integc = list(pipe(cosinepower(),intgpower(0), head(10)))
# 1 - (integrate sine)
integs1 = list(minuspower(pipe(constpower(1), head(10)),
pipe(sinepower(),intgpower(0), head(10))))
 
assert s == integc, "The integral of cos should be sin"
assert c == integs1, "1 minus the integral of sin should be cos"

Sample output

cosine
[Fraction(1, 1), 0, Fraction(-1, 2), 0, Fraction(1, 24), 0, Fraction(-1, 720), 0, Fraction(1, 40320), 0]
sine
[0, Fraction(1, 1), 0, Fraction(-1, 6), 0, Fraction(1, 120), 0, Fraction(-1, 5040), 0, Fraction(1, 362880)]

[edit] Using cyclic iterators

Alternate version that uses a generator function to allow sine and cosine to be defined recursively, following the same method as Hamming numbers#Alternate version using "Cyclic Iterators":

Works with: Python version 2.6, 3.x
from itertools import islice, tee
from fractions import Fraction
try:
from itertools import izip as zip # for 2.6
except:
pass
 
def pluspower(*powergenerators):
for elements in zip(*powergenerators):
yield sum(elements)
def minuspower(*powergenerators):
for elements in zip(*powergenerators):
yield elements[0] - sum(elements[1:])
def mulpower(fgen,ggen):
'From: http://en.wikipedia.org/wiki/Power_series#Multiplication_and_division'
a,b = [],[]
for f,g in zip(fgen, ggen):
a.append(f)
b.append(g)
yield sum(f*g for f,g in zip(a, reversed(b)))
def constpower(n):
yield n
while True:
yield 0
def diffpower(gen):
'differentiatiate power series'
next(gen)
for n, an in enumerate(gen, start=1):
yield an*n
def intgpower(gen):
'integrate power series with bounds from 0 to x'
yield 0
for n, an in enumerate(gen, start=1):
yield an * Fraction(1,n)
 
 
def sine_cosine_series():
def deferred_sin():
for i in sinx_temp:
yield i
def deferred_cos():
for i in cosx_temp:
yield i
 
sinx_result, sinx_copy1 = tee(deferred_sin(), 2)
cosx_result, cosx_copy1 = tee(deferred_cos(), 2)
 
sinx_temp = intgpower(cosx_copy1)
cosx_temp = minuspower(constpower(1), intgpower(sinx_copy1))
 
return sinx_result, cosx_result
 
sinx, cosx = sine_cosine_series()
 
print("cosine")
print(list(islice(sinx, 10)))
print("sine")
print(list(islice(cosx, 10)))

Sample output

cosine
[0, Fraction(1, 1), Fraction(0, 1), Fraction(-1, 6), Fraction(0, 1), Fraction(1, 120), Fraction(0, 1), Fraction(-1, 5040), Fraction(0, 1), Fraction(1, 362880)]
sine
[1, Fraction(0, 1), Fraction(-1, 2), Fraction(0, 1), Fraction(1, 24), Fraction(0, 1), Fraction(-1, 720), Fraction(0, 1), Fraction(1, 40320), Fraction(0, 1)]

[edit] Racket

Using Lazy Racket:

 
#lang lazy
 
(require racket/match)
 
;; element-wise addition and subtraction
(define (<+> s1 s2) (map + s1 s2))
(define (<-> s1 s2) (map - s1 s2))
 
;; element-wise scaling
(define (scale a s) (map (λ (x) (* a x)) s))
 
;; series multiplication
(define (<*> fs gs)
(match-let ([(cons f ft) (! fs)]
[(cons g gt) (! gs)])
(cons (* f g) (<+> (scale f gt) (<*> ft gs)))))
 
;; series division
(define (</> fs gs)
(match-letrec ([(cons f ft) (! fs)]
[(cons g gt) (! gs)]
[qs (cons (/ f g) (scale (/ g) (<-> ft (<*> qs gt))))])
qs))
 
;; integration and differentiation
(define (int f) (map / f (enum 1)))
(define (diff f) (map * (cdr f) (enum 1)))
 
;; series of natural numbers greater then n
(define (enum n) (cons n (enum (+ 1 n ))))
 

Examples:

 
(define <sin> (cons 0 (int <cos>)))
(define <cos> (cons 1 (scale -1 (int <sin>))))
 
-> (!! (take 10 <sin>))
'(0 1 0 -1/6 0 1/120 0 -1/5040 0 1/362880)
 
-> (!! (take 10 <cos>))
'(1 0 -1/2 0 1/24 0 -1/720 0 1/40320 0)
 
-> (!! (take 10 (diff <sin>)))
'(1 0 -1/2 0 1/24 0 -1/720 0 1/40320 0)
 
; sin(x)² + cos(x)² = 1
-> (!! (take 10 (<+> (<*> <cos> <cos>) (<*> <sin> <sin>))))
'(1 0 0 0 0 0 0 0 0 0)
 
; series of (tan x)
-> (!! (take 10 (</> <sin> <cos>)))
'(0 1 0 1/3 0 2/15 0 17/315 0 62/2835)
 

[edit] Scheme

Definitions of operations on lazy lists:

(define-syntax lons
(syntax-rules ()
((_ lar ldr) (delay (cons lar (delay ldr))))))
 
(define (lar lons)
(car (force lons)))
 
(define (ldr lons)
(force (cdr (force lons))))
 
(define (lap proc . llists)
(lons (apply proc (map lar llists)) (apply lap proc (map ldr llists))))
 
(define (take n llist)
(if (zero? n)
(list)
(cons (lar llist) (take (- n 1) (ldr llist)))))
 
(define (iota n)
(lons n (iota (+ n 1))))
 
(define (repeat n)
(lons n (repeat n)))

Definitions of operations on formal power series:

(define (fps+ . llists)
(apply lap + llists))
 
(define (fps- . llists)
(apply lap - llists))
 
(define (fps* . llists)
(define (*fps* p q)
(let ((larp (lar p)) (larq (lar q)) (ldrp (ldr p)) (ldrq (ldr q)))
(lons (* larp larq)
(fps+ (lap (lambda (p) (* p larp)) ldrq)
(lap (lambda (p) (* p larq)) ldrp)
(lons 0 (*fps* ldrp ldrq))))))
(cond ((null? llists) (lons 1 (repeat 0)))
((null? (cdr llists)) (car llists))
(else
(apply fps* (cons (*fps* (car llists) (cadr llists)) (cddr llists))))))
 
(define (fps/ n . llists)
(define (*fps/ n d)
(let ((q (/ (lar n) (lar d))))
(lons q (*fps/ (fps- (ldr n) (lap (lambda (p) (* p q)) (ldr d))) d))))
(if (null? llists)
(*fps/ (lons 1 (repeat 0)) n)
(*fps/ n (apply fps* llists))))
 
(define (fpsint llist)
(lons 0 (lap * llist (lap / (iota 1)))))
 
(define (fpsdif llist)
(lap * (iota 1) (ldr llist)))

Now the sine and cosine functions can be defined in terms of eachother using integrals:

(define fpscos
(fps- (lons 1 (repeat 0)) (fpsint (delay (force fpssin)))))
 
(define fpssin
(fpsint (delay (force fpscos))))
 
(display (take 10 fpssin))
(newline)
 
(display (take 10 fpscos))
(newline)

Output:

(0 1 0 -1/6 0 1/120 0 -1/5040 0 1/362880)
(1 0 -1/2 0 1/24 0 -1/720 0 1/40320 0)

Now we can do some calculations with these, e.g. show that sin2x + cos2x = 1 or define \tan x = \frac{\sin x}{\cos x}:

(display (take 10 (fps+ (fps* fpssin fpssin) (fps* fpscos fpscos))))
(newline)
 
(define fpstan
(fps/ fpssin fpscos))
 
(display (take 10 fpstan))
(newline)

Output:

(1 0 0 0 0 0 0 0 0 0)
(0 1 0 1/3 0 2/15 0 17/315 0 62/2835)

[edit] Tcl

Works with: Tcl version 8.6
or
Library: TclOO

Tcl doesn't arbitrary definitions of numbers without extension packages, so we'll represent these formal power series as objects, which are really just wrappers around a pair of functions: one determining how many terms there are in the series (possibly including "infinitely many") and the other producing the factor for a particular term.

This code makes extensive use of the fact that objects can have methods and variables independent of their class. This greatly reduces the requirement for singleton classes.

package require TclOO
 
oo::class create PowerSeries {
variable name
constructor {{body {}} args} {
# Use the body to adapt the methods of the _object_
oo::objdefine [self] $body
# Use the rest to configure variables in the object
foreach {var val} $args {
set [my varname $var] $val
}
# Guess the name if not already set
if {![info exists [my varname name]]} {
set name [namespace tail [self]]
}
}
method name {} {
return $name
}
method term i {
return 0
}
method limit {} {
return inf
}
 
# A pretty-printer, that prints the first $terms non-zero terms
method print {terms} {
set result "${name}(x) == "
set limit [my limit]
if {$limit == 0} {
# Special case
return $result[my term 0]
}
set tCount 0
for {set i 0} {$tCount<$terms && $i<=$limit} {incr i} {
set t [my term $i]
if {$t == 0} continue
incr tCount
set t [format %.4g $t]
if {$t eq "1" && $i != 0} {set t ""}
if {$i == 0} {
append result "$t + "
} elseif {$i == 1} {
append result "${t}x + "
} else {
set p [string map {
0 \u2070 1 \u00b9 2 \u00b2 3 \u00b3 4 \u2074
5 \u2075 6 \u2076 7 \u2077 8 \u2078 9 \u2079
} $i]
append result "${t}x$p + "
}
}
return [string trimright $result "+ "]
}
 
# Evaluate (a prefix of) the series at a particular x
# The terms parameter gives the number; 5 is enough for show
method evaluate {x {terms 5}} {
set result 0
set limit [my limit]
set tCount 0
for {set i 0} {$tCount<$terms && $i<=$limit} {incr i} {
set t [my term $i]
if {$t == 0} continue
incr tCount
set result [expr {$result + $t * ($x ** $i)}]
}
return $result
}
 
# Operations to build new sequences from old ones
method add {s} {
PowerSeries new {
variable S1 S2
method limit {} {expr {max([$S1 limit],[$S2 limit])}}
method term i {
set t1 [expr {$i>[$S1 limit] ? 0 : [$S1 term $i]}]
set t2 [expr {$i>[$S2 limit] ? 0 : [$S2 term $i]}]
expr {$t1 + $t2}
}
} S1 [self] S2 $s name "$name+[$s name]"
}
method subtract {s} {
PowerSeries new {
variable S1 S2
method limit {} {expr {max([$S1 limit],[$S2 limit])}}
method term i {
set t1 [expr {$i>[$S1 limit] ? 0 : [$S1 term $i]}]
set t2 [expr {$i>[$S2 limit] ? 0 : [$S2 term $i]}]
expr {$t1 - $t2}
}
} S1 [self] S2 $s name "$name-[$s name]"
}
method integrate {{Name ""}} {
if {$Name eq ""} {set Name "Integrate\[[my name]\]"}
PowerSeries new {
variable S limit
method limit {} {
if {[info exists limit]} {return $limit}
try {
return [expr {[$S limit] + 1}]
} on error {} {
# If the limit spirals out of control, it's infinite!
return [set limit inf]
}
}
method term i {
if {$i == 0} {return 0}
set t [$S term [expr {$i-1}]]
expr {$t / double($i)}
}
} S [self] name $Name
}
method differentiate {{Name ""}} {
if {$Name eq ""} {set Name "Differentiate\[[my name]\]"}
PowerSeries new {
variable S
method limit {} {expr {[$S limit] ? [$S limit] - 1 : 0}}
method term i {expr {[incr i] * [$S term $i]}}
} S [self] name $Name
}
# Special constructor for making constants
self method constant n {
PowerSeries new {
variable n
method limit {} {return 0}
method term i {return $n}
} n $n name $n
}
}
 
# Define the two power series in terms of each other
PowerSeries create cos ;# temporary dummy object...
rename [cos integrate "sin"] sin
cos destroy ;# remove the dummy to make way for the real one...
rename [[PowerSeries constant 1] subtract [sin integrate]] cos

Demonstrating:

% sin print 7
sin(x) == x + -0.1667x³ + 0.008333x⁵ + -0.0001984x⁷ + 2.756e-06x⁹ + -2.505e-08x¹¹ + 1.606e-10x¹³
% cos print 7
1-Integrate[sin](x) == 1 + -0.5x² + 0.04167x⁴ + -0.001389x⁶ + 2.48e-05x⁸ + -2.756e-07x¹⁰ + 2.088e-09x¹²
% sin evaluate [expr acos(0)]
1.0000035425842861
% cos evaluate [expr acos(0)]
2.473727636463901e-5
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