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 // run // 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. // Test concurrency primitives: power series. // 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, // https://swtch.com/~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) { 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: out[0], out[1] = out[1], out[0] } } 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 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 */ } }