// $G $D/$F.go && $L $F.$A && ./$A.out
// Copyright 2009 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
// Power series package
// A power series is a channel, along which flow rational
// coefficients. A denominator of zero signifies the end.
// Original code in Newsqueak by Doug McIlroy.
// See Squinting at Power Series by Doug McIlroy,
// http://www.cs.bell-labs.com/who/rsc/thread/squint.pdf
package main
import "os"
type rat struct {
num, den int64; // numerator, denominator
}
func (u rat) pr() {
if u.den==1 { print(u.num) }
else { print(u.num, "/", u.den) }
print(" ")
}
func (u rat) eq(c rat) bool {
return u.num == c.num && u.den == c.den
}
type dch struct {
req chan int;
dat chan rat;
nam int;
}
type dch2 [2] *dch
var chnames string
var chnameserial int
var seqno int
func mkdch() *dch {
c := chnameserial % len(chnames);
chnameserial++;
d := new(dch);
d.req = make(chan int);
d.dat = make(chan rat);
d.nam = c;
return d;
}
func mkdch2() *dch2 {
d2 := new(dch2);
d2[0] = mkdch();
d2[1] = mkdch();
return d2;
}
// split reads a single demand channel and replicates its
// output onto two, which may be read at different rates.
// A process is created at first demand for a rat and dies
// after the rat has been sent to both outputs.
// When multiple generations of split exist, the newest
// will service requests on one channel, which is
// always renamed to be out[0]; the oldest will service
// requests on the other channel, out[1]. All generations but the
// newest hold queued data that has already been sent to
// out[0]. When data has finally been sent to out[1],
// a signal on the release-wait channel tells the next newer
// generation to begin servicing out[1].
func dosplit(in *dch, out *dch2, wait chan int ) {
var t *dch;
both := false; // do not service both channels
select {
case <-out[0].req:
;
case <-wait:
both = true;
select {
case <-out[0].req:
;
case <-out[1].req:
t=out[0]; out[0]=out[1]; out[1]=t;
}
}
seqno++;
in.req <- seqno;
release := make(chan int);
go dosplit(in, out, release);
dat := <-in.dat;
out[0].dat <- dat;
if !both {
<-wait
}
<-out[1].req;
out[1].dat <- dat;
release <- 0;
}
func split(in *dch, out *dch2) {
release := make(chan int);
go dosplit(in, out, release);
release <- 0;
}
func put(dat rat, out *dch) {
<-out.req;
out.dat <- dat;
}
func get(in *dch) rat {
seqno++;
in.req <- seqno;
return <-in.dat;
}
// Get one rat from each of n demand channels
func getn(in []*dch) []rat {
n := len(in);
if n != 2 { panic("bad n in getn") };
req := new([2] chan int);
dat := new([2] chan rat);
out := make([]rat, 2);
var i int;
var it rat;
for i=0; i<n; i++ {
req[i] = in[i].req;
dat[i] = nil;
}
for n=2*n; n>0; n-- {
seqno++;
select {
case req[0] <- seqno:
dat[0] = in[0].dat;
req[0] = nil;
case req[1] <- seqno:
dat[1] = in[1].dat;
req[1] = nil;
case it = <-dat[0]:
out[0] = it;
dat[0] = nil;
case it = <-dat[1]:
out[1] = it;
dat[1] = nil;
}
}
return out;
}
// Get one rat from each of 2 demand channels
func get2(in0 *dch, in1 *dch) []rat {
return getn([]*dch{in0, in1});
}
func copy(in *dch, out *dch) {
for {
<-out.req;
out.dat <- get(in);
}
}
func repeat(dat rat, out *dch) {
for {
put(dat, out)
}
}
type PS *dch; // power series
type PS2 *[2] PS; // pair of power series
var Ones PS
var Twos PS
func mkPS() *dch {
return mkdch()
}
func mkPS2() *dch2 {
return mkdch2()
}
// Conventions
// Upper-case for power series.
// Lower-case for rationals.
// Input variables: U,V,...
// Output variables: ...,Y,Z
// Integer gcd; needed for rational arithmetic
func gcd (u, v int64) int64 {
if u < 0 { return gcd(-u, v) }
if u == 0 { return v }
return gcd(v%u, u)
}
// Make a rational from two ints and from one int
func i2tor(u, v int64) rat {
g := gcd(u,v);
var r rat;
if v > 0 {
r.num = u/g;
r.den = v/g;
} else {
r.num = -u/g;
r.den = -v/g;
}
return r;
}
func itor(u int64) rat {
return i2tor(u, 1);
}
var zero rat;
var one rat;
// End mark and end test
var finis rat;
func end(u rat) int64 {
if u.den==0 { return 1 }
return 0
}
// Operations on rationals
func add(u, v rat) rat {
g := gcd(u.den,v.den);
return i2tor(u.num*(v.den/g)+v.num*(u.den/g),u.den*(v.den/g));
}
func mul(u, v rat) rat {
g1 := gcd(u.num,v.den);
g2 := gcd(u.den,v.num);
var r rat;
r.num = (u.num/g1)*(v.num/g2);
r.den = (u.den/g2)*(v.den/g1);
return r;
}
func neg(u rat) rat {
return i2tor(-u.num, u.den);
}
func sub(u, v rat) rat {
return add(u, neg(v));
}
func inv(u rat) rat { // invert a rat
if u.num == 0 { panic("zero divide in inv") }
return i2tor(u.den, u.num);
}
// print eval in floating point of PS at x=c to n terms
func evaln(c rat, U PS, n int)
{
xn := float64(1);
x := float64(c.num)/float64(c.den);
val := float64(0);
for i:=0; i<n; i++ {
u := get(U);
if end(u) != 0 {
break;
}
val = val + x * float64(u.num)/float64(u.den);
xn = xn*x;
}
print(val, "\n");
}
// Print n terms of a power series
func printn(U PS, n int) {
done := false;
for ; !done && n>0; n-- {
u := get(U);
if end(u) != 0 { done = true }
else { u.pr() }
}
print(("\n"));
}
// Evaluate n terms of power series U at x=c
func eval(c rat, U PS, n int) rat {
if n==0 { return zero }
y := get(U);
if end(y) != 0 { return zero }
return add(y,mul(c,eval(c,U,n-1)));
}
// Power-series constructors return channels on which power
// series flow. They start an encapsulated generator that
// puts the terms of the series on the channel.
// Make a pair of power series identical to a given power series
func Split(U PS) *dch2 {
UU := mkdch2();
go split(U,UU);
return UU;
}
// Add two power series
func Add(U, V PS) PS {
Z := mkPS();
go func() {
var uv []rat;
for {
<-Z.req;
uv = get2(U,V);
switch end(uv[0])+2*end(uv[1]) {
case 0:
Z.dat <- add(uv[0], uv[1]);
case 1:
Z.dat <- uv[1];
copy(V,Z);
case 2:
Z.dat <- uv[0];
copy(U,Z);
case 3:
Z.dat <- finis;
}
}
}();
return Z;
}
// Multiply a power series by a constant
func Cmul(c rat,U PS) PS {
Z := mkPS();
go func() {
done := false;
for !done {
<-Z.req;
u := get(U);
if end(u) != 0 { done = true }
else { Z.dat <- mul(c,u) }
}
Z.dat <- finis;
}();
return Z;
}
// Subtract
func Sub(U, V PS) PS {
return Add(U, Cmul(neg(one), V));
}
// Multiply a power series by the monomial x^n
func Monmul(U PS, n int) PS {
Z := mkPS();
go func() {
for ; n>0; n-- { put(zero,Z) }
copy(U,Z);
}();
return Z;
}
// Multiply by x
func Xmul(U PS) PS {
return Monmul(U,1);
}
func Rep(c rat) PS {
Z := mkPS();
go repeat(c,Z);
return Z;
}
// Monomial c*x^n
func Mon(c rat, n int) PS {
Z:=mkPS();
go func() {
if(c.num!=0) {
for ; n>0; n=n-1 { put(zero,Z) }
put(c,Z);
}
put(finis,Z);
}();
return Z;
}
func Shift(c rat, U PS) PS {
Z := mkPS();
go func() {
put(c,Z);
copy(U,Z);
}();
return Z;
}
// simple pole at 1: 1/(1-x) = 1 1 1 1 1 ...
// Convert array of coefficients, constant term first
// to a (finite) power series
/*
func Poly(a []rat) PS {
Z:=mkPS();
begin func(a []rat, Z PS) {
j:=0;
done:=0;
for j=len(a); !done&&j>0; j=j-1)
if(a[j-1].num!=0) done=1;
i:=0;
for(; i<j; i=i+1) put(a[i],Z);
put(finis,Z);
}();
return Z;
}
*/
// Multiply. The algorithm is
// let U = u + x*UU
// let V = v + x*VV
// then UV = u*v + x*(u*VV+v*UU) + x*x*UU*VV
func Mul(U, V PS) PS {
Z:=mkPS();
go func() {
<-Z.req;
uv := get2(U,V);
if end(uv[0])!=0 || end(uv[1]) != 0 {
Z.dat <- finis;
} else {
Z.dat <- mul(uv[0],uv[1]);
UU := Split(U);
VV := Split(V);
W := Add(Cmul(uv[0],VV[0]),Cmul(uv[1],UU[0]));
<-Z.req;
Z.dat <- get(W);
copy(Add(W,Mul(UU[1],VV[1])),Z);
}
}();
return Z;
}
// Differentiate
func Diff(U PS) PS {
Z:=mkPS();
go func() {
<-Z.req;
u := get(U);
if end(u) == 0 {
done:=false;
for i:=1; !done; i++ {
u = get(U);
if end(u) != 0 { done=true }
else {
Z.dat <- mul(itor(int64(i)),u);
<-Z.req;
}
}
}
Z.dat <- finis;
}();
return Z;
}
// Integrate, with const of integration
func Integ(c rat,U PS) PS {
Z:=mkPS();
go func() {
put(c,Z);
done:=false;
for i:=1; !done; i++ {
<-Z.req;
u := get(U);
if end(u) != 0 { done= true }
Z.dat <- mul(i2tor(1,int64(i)),u);
}
Z.dat <- finis;
}();
return Z;
}
// Binomial theorem (1+x)^c
func Binom(c rat) PS {
Z:=mkPS();
go func() {
n := 1;
t := itor(1);
for c.num!=0 {
put(t,Z);
t = mul(mul(t,c),i2tor(1,int64(n)));
c = sub(c,one);
n++;
}
put(finis,Z);
}();
return Z;
}
// Reciprocal of a power series
// let U = u + x*UU
// let Z = z + x*ZZ
// (u+x*UU)*(z+x*ZZ) = 1
// z = 1/u
// u*ZZ + z*UU +x*UU*ZZ = 0
// ZZ = -UU*(z+x*ZZ)/u;
func Recip(U PS) PS {
Z:=mkPS();
go func() {
ZZ:=mkPS2();
<-Z.req;
z := inv(get(U));
Z.dat <- z;
split(Mul(Cmul(neg(z),U),Shift(z,ZZ[0])),ZZ);
copy(ZZ[1],Z);
}();
return Z;
}
// Exponential of a power series with constant term 0
// (nonzero constant term would make nonrational coefficients)
// bug: the constant term is simply ignored
// Z = exp(U)
// DZ = Z*DU
// integrate to get Z
func Exp(U PS) PS {
ZZ := mkPS2();
split(Integ(one,Mul(ZZ[0],Diff(U))),ZZ);
return ZZ[1];
}
// Substitute V for x in U, where the leading term of V is zero
// let U = u + x*UU
// let V = v + x*VV
// then S(U,V) = u + VV*S(V,UU)
// bug: a nonzero constant term is ignored
func Subst(U, V PS) PS {
Z:= mkPS();
go func() {
VV := Split(V);
<-Z.req;
u := get(U);
Z.dat <- u;
if end(u) == 0 {
if end(get(VV[0])) != 0 { put(finis,Z); }
else { copy(Mul(VV[0],Subst(U,VV[1])),Z); }
}
}();
return Z;
}
// Monomial Substition: U(c x^n)
// Each Ui is multiplied by c^i and followed by n-1 zeros
func MonSubst(U PS, c0 rat, n int) PS {
Z:= mkPS();
go func() {
c := one;
for {
<-Z.req;
u := get(U);
Z.dat <- mul(u, c);
c = mul(c, c0);
if end(u) != 0 {
Z.dat <- finis;
break;
}
for i := 1; i < n; i++ {
<-Z.req;
Z.dat <- zero;
}
}
}();
return Z;
}
func Init() {
chnameserial = -1;
seqno = 0;
chnames = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz";
zero = itor(0);
one = itor(1);
finis = i2tor(1,0);
Ones = Rep(one);
Twos = Rep(itor(2));
}
func check(U PS, c rat, count int, str string) {
for i := 0; i < count; i++ {
r := get(U);
if !r.eq(c) {
print("got: ");
r.pr();
print("should get ");
c.pr();
print("\n");
panic(str)
}
}
}
const N=10
func checka(U PS, a []rat, str string) {
for i := 0; i < N; i++ {
check(U, a[i], 1, str);
}
}
func main() {
Init();
if len(os.Args) > 1 { // print
print("Ones: "); printn(Ones, 10);
print("Twos: "); printn(Twos, 10);
print("Add: "); printn(Add(Ones, Twos), 10);
print("Diff: "); printn(Diff(Ones), 10);
print("Integ: "); printn(Integ(zero, Ones), 10);
print("CMul: "); printn(Cmul(neg(one), Ones), 10);
print("Sub: "); printn(Sub(Ones, Twos), 10);
print("Mul: "); printn(Mul(Ones, Ones), 10);
print("Exp: "); printn(Exp(Ones), 15);
print("MonSubst: "); printn(MonSubst(Ones, neg(one), 2), 10);
print("ATan: "); printn(Integ(zero, MonSubst(Ones, neg(one), 2)), 10);
} else { // test
check(Ones, one, 5, "Ones");
check(Add(Ones, Ones), itor(2), 0, "Add Ones Ones"); // 1 1 1 1 1
check(Add(Ones, Twos), itor(3), 0, "Add Ones Twos"); // 3 3 3 3 3
a := make([]rat, N);
d := Diff(Ones);
for i:=0; i < N; i++ {
a[i] = itor(int64(i+1))
}
checka(d, a, "Diff"); // 1 2 3 4 5
in := Integ(zero, Ones);
a[0] = zero; // integration constant
for i:=1; i < N; i++ {
a[i] = i2tor(1, int64(i))
}
checka(in, a, "Integ"); // 0 1 1/2 1/3 1/4 1/5
check(Cmul(neg(one), Twos), itor(-2), 10, "CMul"); // -1 -1 -1 -1 -1
check(Sub(Ones, Twos), itor(-1), 0, "Sub Ones Twos"); // -1 -1 -1 -1 -1
m := Mul(Ones, Ones);
for i:=0; i < N; i++ {
a[i] = itor(int64(i+1))
}
checka(m, a, "Mul"); // 1 2 3 4 5
e := Exp(Ones);
a[0] = itor(1);
a[1] = itor(1);
a[2] = i2tor(3,2);
a[3] = i2tor(13,6);
a[4] = i2tor(73,24);
a[5] = i2tor(167,40);
a[6] = i2tor(4051,720);
a[7] = i2tor(37633,5040);
a[8] = i2tor(43817,4480);
a[9] = i2tor(4596553,362880);
checka(e, a, "Exp"); // 1 1 3/2 13/6 73/24
at := Integ(zero, MonSubst(Ones, neg(one), 2));
for c, i := 1, 0; i < N; i++ {
if i%2 == 0 {
a[i] = zero
} else {
a[i] = i2tor(int64(c), int64(i));
c *= -1
}
}
checka(at, a, "ATan"); // 0 -1 0 -1/3 0 -1/5
/*
t := Revert(Integ(zero, MonSubst(Ones, neg(one), 2)));
a[0] = zero;
a[1] = itor(1);
a[2] = zero;
a[3] = i2tor(1,3);
a[4] = zero;
a[5] = i2tor(2,15);
a[6] = zero;
a[7] = i2tor(17,315);
a[8] = zero;
a[9] = i2tor(62,2835);
checka(t, a, "Tan"); // 0 1 0 1/3 0 2/15
*/
}
}
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