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
a0 + a1 * x + a2 * x2 + a3 * x3 + ...
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
sinx = ∫ cosx cosx = 1 - ∫ sinx
Goals: Demonstrate how the language handles new numeric types and delayed (or lazy) evaluation.
D
| <lang d>
module fps; import q ; // a module for Rational Number import std.format : sformat = format; import std.math, std.stdio ; interface Gene(U) { U coef(int n) ; } class Term(U) { U[] cache ;
Gene!(U) gene ;
this(Gene!(U) g) { gene = g ; }
U opIndex(int n) {
if(n < 0) return cast(U)0 ;
if(n >= cache.length)
for(int i = cache.length ; i <= n ; i++)
cache ~= gene.coef(i) ;
return cache[n] ;
}
} struct FPS(U) { alias Term!(U) UT ;
alias Gene!(U) UG ;
alias FPS!(U) UF ;
static int DispTerm = 12 ;
static string XVar = "x" ;
UT term ;
static UF opCall(UT t) { UF f ; f.term = t ; return f ;}
static UF opCall(U[] polynomial) {
UF f ;
f.term = new UT( new class() UG { U coef(int n) {
if (n < 0 || n >= polynomial.length )
return cast(U)0 ;
return polynomial[n] ;
};}) ;
return f ;
}
U inverseCoef(int n) {
U[] res = new U[n + 1] ;
res[0] = 1/term[0] ;
for(int i = 1 ; i <= n ; i++) {
res[i] = cast(U)0 ;
for(int j = 0 ; j < i ; j++) {
res[i] += term[i - j] * res[j] ;
}
res[i] = -res[0] * res[i] ;
}
return res[n] ;
}
UF opAdd(UF rhs) { return UF(new UT(new class() UG { U coef(int n) {
return term[n] + rhs.term[n] ;
}}));}
UF opSub(UF rhs) { return UF(new UT(new class() UG { U coef(int n) {
return term[n] - rhs.term[n] ;
}}));}
UF opMul(UF rhs) { return UF(new UT(new class() UG { U coef(int n) {
U res = cast(U) 0 ;
for(int i = 0 ; i <=n ; i++)
res += term[i]*rhs.term[n -i] ; // Cauchy product with rhs
return res ;
}}));}
UF opDiv(UF rhs) { return UF(new UT(new class() UG { U coef(int n) {
U res = cast(U) 0 ;
for(int i = 0 ; i <=n ; i++)
res += term[i]*rhs.inverseCoef(n -i) ; // Cauchy product with inverse of rhs
return res ;
}}));}
UF Diff() { return UF(new UT(new class() UG { U coef(int n) {
return term[n + 1] * (n + 1) ;
}}));}
UF Intg() { return UF(new UT(new class() UG { U coef(int n) {
if (n == 0) return cast(U)0 ;
return term[n-1]/n ;
}}));}
string toString(int dpTerm = -1) {
string s ;
if(dpTerm <= 0) dpTerm = DispTerm ;
U c = term[0] ;
if(c != cast(U)0) s ~= sformat("%s", c) ;
for(int i = 1 ; i < dpTerm ; i++)
if((c = term[i]) != cast(U)0) {
string t ;
if(c > 0 && s.length > 0)
t = "+" ;
if (c == cast(U)1)
t ~= XVar ;
else if ( c == cast(U)-1)
t ~= sformat("-%s", XVar) ;
else
t ~= sformat("%s%s", c, XVar) ;
if(i > 1) t ~= sformat("%s", i) ;
s ~= t ;
}
if(s.length == 0)
s = "0" ;
s ~= "+..." ;
return s ;
}
} alias Q!(long) QT ; alias FPS!(QT) RF ; void main() { QT.DenoLimit = 0 ; RF COS ;
RF SIN = COS.Intg ;
COS = RF([QT(1,1)]) - SIN.Intg ;
writefln("SIN(x) = ",SIN) ;
writefln("COS(x) = ",COS) ;
}</lang> |
Output:
SIN(x) = x-1/6x3+1/120x5-1/5040x7+1/362880x9-1/39916800x11+... COS(x) = 1-1/2x2+1/24x4-1/720x6+1/40320x8-1/3628800x10+...
The q module. Because of the limited precision of long, the denominator will quickly overflow when multiplied, but it is sufficient to show first few coefficients in this task.
| <lang d>
module q; import std.format : sformat = format; T gcd(T)(T u, T v) { // binary gcd algorithm if(u < 0) u = -u ;
if(v < 0) v = -v ;
if(u == 0 || v == 0) return u | v ;
int shift = 0 ;
for(; 0 == ((u|v) & 1) ; shift++) { u >>>= 1 ; v >>>= 1 ; }
while((u & 1) == 0) u >>>= 1 ;
while (v != 0) {
while((v & 1) == 0) v >>>= 1 ;
if (u <= v) v -= u ;
else { v = u - v ; u -= v ;}
v >>>= 1 ;
}
return u << shift ;
} struct Q(T) { static T DenoLimit = int.max ; alias Q!(T) QX ; T num, den ; static QX opCall(long n, long d) {
QX q ;
with(q) { num = n ; den = d; reduce ; }
return q ;
}
static QX opCall(real r) {
QX q ;
T multiple = 1_000_000 ;
with(q) { num = cast(T)(r*multiple) ; den = multiple; reduce ; }
return q ;
}
void reduce() {
T g = gcd(num, den) ;
if(g == 0 || den == 0)
throw new Exception(sformat("Zero denominator:%s/%s gcd:%s", num, den, g )) ;
if(num == 0) { den = 1 ; return ; }
num /= g ; den /= g ;
if(den < 0) { num = -num ; den = -den ;} // keep den positive
if(DenoLimit != 0 && den > DenoLimit) {
// prevent denominator overflow, buggy?
den /= 2;
num /= 2;
reduce ;
}
}
T toBaseType() {T result = num /den ; return result ; }
real toReal() { real result = num /cast(real)den ; return result ; }
real opCast() { return toReal ; }
T opCmp(QX rhs) { return num*rhs.den - den*rhs.num ; }
T opCmp(int rhs) { return num - den*rhs ; }
T opCmp(real rhs) { return cast(T)(num - den*rhs) ; }
bool opEquals(QX rhs) { return opCmp(rhs) == 0 ; }
bool opEquals(int rhs) { return opCmp(rhs) == 0 ; }
bool opEquals(real rhs) { return opCmp(rhs) == 0 ; }
QX opPostInc() { num += den ; reduce ; return *this ; }
QX opPostDec() { num -= den ; reduce ; return *this ; }
QX opAddAssign(QX rhs)
{ num = num*rhs.den + rhs.num*den ; den *= rhs.den ; reduce ; return *this ;}
QX opSubAssign(QX rhs)
{ num = num*rhs.den - rhs.num*den ; den *= rhs.den ; reduce ; return *this ;}
QX opMulAssign(QX rhs) { num *= rhs.num ; den *= rhs.den ; reduce ; return *this ;}
QX opDivAssign(QX rhs) { num *= rhs.den ; den *= rhs.num ; reduce ; return *this ;}
QX opAddAssign(int rhs) { num += rhs*den ; reduce ; return *this ; }
QX opSubAssign(int rhs) { num -= rhs*den ; reduce ; return *this ; }
QX opMulAssign(int rhs) { num *= rhs ; reduce ; return *this ; }
QX opDivAssign(int rhs) { den *= rhs ; reduce ; return *this ; }
QX opNeg() { return QX(-num, den) ;}
QX opPos() { return QX(num, den) ;}
QX opAdd(QX rhs) { return QX(num*rhs.den + rhs.num*den, den*rhs.den);}
QX opSub(QX rhs) { return QX(num*rhs.den - rhs.num*den, den*rhs.den);}
QX opMul(QX rhs) { return QX(num*rhs.num, den*rhs.den);}
QX opDiv(QX rhs) { return QX(num*rhs.den, den*rhs.num);}
QX opAdd(int rhs) { return QX(num + rhs*den, den);}
QX opSub(int rhs) { return QX(num - rhs*den, den);}
QX opSub_r(int lhs) { return QX(lhs*den - num, den);}
QX opMul(int rhs) { return QX(num*rhs, den);}
QX opDiv(int rhs) { return QX(num, den*rhs);}
QX opDiv_r(int lhs) { return QX(lhs*den, num);}
string toString() {
if(den == 1) return sformat("%s", num) ;
return sformat("%s/%s", num, den) ;
}
}</lang> |
Haskell
<lang haskell>newtype Series a = S { coeffs :: [a] } deriving (Eq, Show)
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 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
S (f:ft) / S (g:gt) = S qs where qs = f/g : map (/g) (coeffs (S ft - S qs * S gt))
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</lang>
Output (with manual interruption):
*Main> sinx S [0%1,1%1,0%1,(-1)%6,0%1,1%120,0%1,(-1)%5040,0%1,1%362880,Interrupted. *Main> cosx S [1%1,0%1,(-1)%2,0%1,1%24,0%1,(-1)%720,0%1,1%40320,0%1,(-1)%3628800,Interrupted. *Main> sinx / cosx -- tangent S [0%1,1%1,0%1,1%3,0%1,2%15,0%1,17%315,0%1,62%2835,0%1,1382%155925,Interrupted.
Java
Copied from the D example. Java has no generic numeric interface, and has no templates, so we cannot make it work for all numeric types at the same time. Because the Java library does not come with a Fraction class (and I am too lazy to implement one, although there is one in Apache Commons Math), here I will just hard-code it to use doubles (64-bit floating-point numbers) instead of fractions. It won't be as pretty, but it can be changed to Fraction or any other type trivially by substituting the types.
<lang java> import java.util.*;
interface Gene {
double coef(int n);
}
class Term {
private final List<Double> cache = new ArrayList<Double>(); private final Gene gene;
public Term(Gene g) { gene = g; }
public double get(int n) {
if (n < 0)
return 0;
else if (n >= cache.size())
for (int i = cache.size(); i <= n; i++)
cache.add(gene.coef(i));
return cache.get(n);
}
}
public class FormalPS {
private static final int DISP_TERM = 12; private static final String X_VAR = "x"; private Term term;
public FormalPS() { }
public void copyFrom(FormalPS foo) {
term = foo.term;
}
public FormalPS(Term t) {
term = t;
}
public FormalPS(final double[] polynomial) {
this(new Term(new Gene() {
public double coef(int n) {
if (n < 0 || n >= polynomial.length)
return 0;
else
return polynomial[n];
}
}));
}
public double inverseCoef(int n) {
double[] res = new double[n + 1];
res[0] = 1 / term.get(0);
for (int i = 1; i <= n; i++) {
res[i] = 0;
for (int j = 0; j < i; j++)
res[i] += term.get(i-j) * res[j];
res[i] *= -res[0];
}
return res[n];
}
public FormalPS add(final FormalPS rhs) {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
return term.get(n) + rhs.term.get(n);
}
}));
}
public FormalPS sub(final FormalPS rhs) {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
return term.get(n) - rhs.term.get(n);
}
}));
}
public FormalPS mul(final FormalPS rhs) {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
double res = 0;
for (int i = 0; i <= n; i++)
res += term.get(i) * rhs.term.get(n-i);
return res;
}
}));
}
public FormalPS div(final FormalPS rhs) {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
double res = 0;
for (int i = 0; i <= n; i++)
res += term.get(i) * rhs.term.get(n-i);
return res;
}
}));
}
public FormalPS diff() {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
return term.get(n+1) * (n+1);
}
}));
}
public FormalPS intg() {
return new FormalPS(new Term(new Gene() {
public double coef(int n) {
if (n == 0)
return 0;
else
return term.get(n-1) / n;
}
}));
}
public String toString() {
return toString(DISP_TERM);
}
public String toString(int dpTerm) {
StringBuffer s = new StringBuffer();
{
double c = term.get(0);
if (c != 0)
s.append(c);
}
for (int i = 1; i < dpTerm; i++) {
double c = term.get(i);
if (c != 0) {
if (c > 0 && s.length() > 0)
s.append("+");
if (c == 1)
s.append(X_VAR);
else if (c == -1)
s.append("-" + X_VAR);
else
s.append(c + X_VAR);
if (i > 1)
s.append(i);
}
}
if (s.length() == 0)
s.append("0");
s.append("+...");
return s.toString();
}
public static void main(String[] args) {
FormalPS cos = new FormalPS();
FormalPS sin = cos.intg();
cos.copyFrom(new FormalPS(new double[]{1}).sub(sin.intg()));
System.out.println("SIN(x) = " + sin);
System.out.println("COS(x) = " + cos);
}
} </lang> Output:
SIN(x) = x-0.16666666666666666x3+0.008333333333333333x5-1.984126984126984E-4x7+2.7557319223985893E-6x9-2.505210838544172E-8x11+... COS(x) = 1.0-0.5x2+0.041666666666666664x4-0.001388888888888889x6+2.48015873015873E-5x8-2.7557319223985894E-7x10+...