mirror of
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4f61fc96b2
The ken directory is untouched so we have some examples with explicit semis. R=gri CC=golang-dev https://golang.org/cl/2157041
723 lines
13 KiB
Go
723 lines
13 KiB
Go
// $G $D/$F.go && $L $F.$A && ./$A.out
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// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// Power series package
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// A power series is a channel, along which flow rational
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// coefficients. A denominator of zero signifies the end.
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// Original code in Newsqueak by Doug McIlroy.
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// See Squinting at Power Series by Doug McIlroy,
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// http://www.cs.bell-labs.com/who/rsc/thread/squint.pdf
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// Like powser1.go but uses channels of interfaces.
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// Has not been cleaned up as much as powser1.go, to keep
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// it distinct and therefore a different test.
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package main
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import "os"
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type rat struct {
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num, den int64 // numerator, denominator
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}
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type item interface {
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pr()
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eq(c item) bool
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}
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func (u *rat) pr(){
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if u.den==1 {
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print(u.num)
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} else {
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print(u.num, "/", u.den)
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}
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print(" ")
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}
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func (u *rat) eq(c item) bool {
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c1 := c.(*rat)
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return u.num == c1.num && u.den == c1.den
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}
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type dch struct {
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req chan int
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dat chan item
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nam int
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}
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type dch2 [2] *dch
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var chnames string
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var chnameserial int
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var seqno int
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func mkdch() *dch {
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c := chnameserial % len(chnames)
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chnameserial++
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d := new(dch)
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d.req = make(chan int)
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d.dat = make(chan item)
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d.nam = c
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return d
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}
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func mkdch2() *dch2 {
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d2 := new(dch2)
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d2[0] = mkdch()
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d2[1] = mkdch()
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return d2
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}
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// split reads a single demand channel and replicates its
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// output onto two, which may be read at different rates.
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// A process is created at first demand for an item and dies
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// after the item has been sent to both outputs.
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// When multiple generations of split exist, the newest
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// will service requests on one channel, which is
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// always renamed to be out[0]; the oldest will service
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// requests on the other channel, out[1]. All generations but the
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// newest hold queued data that has already been sent to
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// out[0]. When data has finally been sent to out[1],
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// a signal on the release-wait channel tells the next newer
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// generation to begin servicing out[1].
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func dosplit(in *dch, out *dch2, wait chan int ){
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both := false // do not service both channels
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select {
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case <-out[0].req:
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case <-wait:
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both = true
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select {
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case <-out[0].req:
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case <-out[1].req:
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out[0],out[1] = out[1], out[0]
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}
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}
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seqno++
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in.req <- seqno
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release := make(chan int)
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go dosplit(in, out, release)
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dat := <-in.dat
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out[0].dat <- dat
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if !both {
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<-wait
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}
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<-out[1].req
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out[1].dat <- dat
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release <- 0
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}
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func split(in *dch, out *dch2){
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release := make(chan int)
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go dosplit(in, out, release)
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release <- 0
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}
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func put(dat item, out *dch){
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<-out.req
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out.dat <- dat
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}
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func get(in *dch) *rat {
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seqno++
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in.req <- seqno
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return (<-in.dat).(*rat)
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}
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// Get one item from each of n demand channels
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func getn(in []*dch) []item {
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n:=len(in)
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if n != 2 { panic("bad n in getn") }
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req := make([] chan int, 2)
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dat := make([] chan item, 2)
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out := make([]item, 2)
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var i int
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var it item
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for i=0; i<n; i++ {
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req[i] = in[i].req
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dat[i] = nil
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}
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for n=2*n; n>0; n-- {
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seqno++
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select{
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case req[0] <- seqno:
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dat[0] = in[0].dat
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req[0] = nil
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case req[1] <- seqno:
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dat[1] = in[1].dat
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req[1] = nil
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case it = <-dat[0]:
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out[0] = it
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dat[0] = nil
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case it = <-dat[1]:
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out[1] = it
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dat[1] = nil
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}
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}
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return out
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}
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// Get one item from each of 2 demand channels
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func get2(in0 *dch, in1 *dch) []item {
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return getn([]*dch{in0, in1})
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}
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func copy(in *dch, out *dch){
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for {
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<-out.req
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out.dat <- get(in)
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}
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}
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func repeat(dat item, out *dch){
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for {
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put(dat, out)
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}
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}
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type PS *dch // power series
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type PS2 *[2] PS // pair of power series
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var Ones PS
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var Twos PS
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func mkPS() *dch {
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return mkdch()
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}
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func mkPS2() *dch2 {
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return mkdch2()
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}
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// Conventions
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// Upper-case for power series.
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// Lower-case for rationals.
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// Input variables: U,V,...
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// Output variables: ...,Y,Z
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// Integer gcd; needed for rational arithmetic
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func gcd (u, v int64) int64{
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if u < 0 { return gcd(-u, v) }
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if u == 0 { return v }
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return gcd(v%u, u)
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}
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// Make a rational from two ints and from one int
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func i2tor(u, v int64) *rat{
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g := gcd(u,v)
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r := new(rat)
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if v > 0 {
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r.num = u/g
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r.den = v/g
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} else {
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r.num = -u/g
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r.den = -v/g
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}
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return r
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}
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func itor(u int64) *rat{
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return i2tor(u, 1)
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}
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var zero *rat
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var one *rat
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// End mark and end test
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var finis *rat
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func end(u *rat) int64 {
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if u.den==0 { return 1 }
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return 0
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}
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// Operations on rationals
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func add(u, v *rat) *rat {
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g := gcd(u.den,v.den)
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return i2tor(u.num*(v.den/g)+v.num*(u.den/g),u.den*(v.den/g))
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}
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func mul(u, v *rat) *rat{
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g1 := gcd(u.num,v.den)
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g2 := gcd(u.den,v.num)
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r := new(rat)
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r.num =(u.num/g1)*(v.num/g2)
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r.den = (u.den/g2)*(v.den/g1)
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return r
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}
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func neg(u *rat) *rat{
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return i2tor(-u.num, u.den)
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}
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func sub(u, v *rat) *rat{
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return add(u, neg(v))
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}
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func inv(u *rat) *rat{ // invert a rat
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if u.num == 0 { panic("zero divide in inv") }
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return i2tor(u.den, u.num)
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}
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// print eval in floating point of PS at x=c to n terms
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func Evaln(c *rat, U PS, n int) {
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xn := float64(1)
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x := float64(c.num)/float64(c.den)
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val := float64(0)
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for i:=0; i<n; i++ {
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u := get(U)
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if end(u) != 0 {
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break
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}
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val = val + x * float64(u.num)/float64(u.den)
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xn = xn*x
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}
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print(val, "\n")
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}
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// Print n terms of a power series
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func Printn(U PS, n int){
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done := false
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for ; !done && n>0; n-- {
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u := get(U)
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if end(u) != 0 {
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done = true
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} else {
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u.pr()
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}
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}
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print(("\n"))
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}
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func Print(U PS){
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Printn(U,1000000000)
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}
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// Evaluate n terms of power series U at x=c
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func eval(c *rat, U PS, n int) *rat{
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if n==0 { return zero }
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y := get(U)
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if end(y) != 0 { return zero }
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return add(y,mul(c,eval(c,U,n-1)))
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}
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// Power-series constructors return channels on which power
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// series flow. They start an encapsulated generator that
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// puts the terms of the series on the channel.
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// Make a pair of power series identical to a given power series
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func Split(U PS) *dch2{
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UU := mkdch2()
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go split(U,UU)
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return UU
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}
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// Add two power series
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func Add(U, V PS) PS{
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Z := mkPS()
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go func(U, V, Z PS){
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var uv [] item
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for {
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<-Z.req
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uv = get2(U,V)
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switch end(uv[0].(*rat))+2*end(uv[1].(*rat)) {
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case 0:
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Z.dat <- add(uv[0].(*rat), uv[1].(*rat))
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case 1:
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Z.dat <- uv[1]
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copy(V,Z)
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case 2:
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Z.dat <- uv[0]
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copy(U,Z)
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case 3:
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Z.dat <- finis
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}
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}
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}(U, V, Z)
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return Z
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}
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// Multiply a power series by a constant
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func Cmul(c *rat,U PS) PS{
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Z := mkPS()
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go func(c *rat, U, Z PS){
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done := false
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for !done {
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<-Z.req
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u := get(U)
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if end(u) != 0 {
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done = true
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} else {
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Z.dat <- mul(c,u)
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}
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}
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Z.dat <- finis
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}(c, U, Z)
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return Z
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}
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// Subtract
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func Sub(U, V PS) PS{
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return Add(U, Cmul(neg(one), V))
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}
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// Multiply a power series by the monomial x^n
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func Monmul(U PS, n int) PS{
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Z := mkPS()
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go func(n int, U PS, Z PS){
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for ; n>0; n-- { put(zero,Z) }
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copy(U,Z)
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}(n, U, Z)
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return Z
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}
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// Multiply by x
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func Xmul(U PS) PS{
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return Monmul(U,1)
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}
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func Rep(c *rat) PS{
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Z := mkPS()
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go repeat(c,Z)
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return Z
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}
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// Monomial c*x^n
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func Mon(c *rat, n int) PS{
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Z:=mkPS()
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go func(c *rat, n int, Z PS){
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if(c.num!=0) {
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for ; n>0; n=n-1 { put(zero,Z) }
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put(c,Z)
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}
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put(finis,Z)
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}(c, n, Z)
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return Z
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}
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func Shift(c *rat, U PS) PS{
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Z := mkPS()
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go func(c *rat, U, Z PS){
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put(c,Z)
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copy(U,Z)
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}(c, U, Z)
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return Z
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}
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// simple pole at 1: 1/(1-x) = 1 1 1 1 1 ...
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// Convert array of coefficients, constant term first
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// to a (finite) power series
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/*
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func Poly(a [] *rat) PS{
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Z:=mkPS()
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begin func(a [] *rat, Z PS){
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j:=0
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done:=0
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for j=len(a); !done&&j>0; j=j-1)
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if(a[j-1].num!=0) done=1
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i:=0
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for(; i<j; i=i+1) put(a[i],Z)
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put(finis,Z)
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}()
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return Z
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}
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*/
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// Multiply. The algorithm is
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// let U = u + x*UU
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// let V = v + x*VV
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// then UV = u*v + x*(u*VV+v*UU) + x*x*UU*VV
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func Mul(U, V PS) PS{
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Z:=mkPS()
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go func(U, V, Z PS){
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<-Z.req
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uv := get2(U,V)
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if end(uv[0].(*rat))!=0 || end(uv[1].(*rat)) != 0 {
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Z.dat <- finis
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} else {
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Z.dat <- mul(uv[0].(*rat),uv[1].(*rat))
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UU := Split(U)
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VV := Split(V)
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W := Add(Cmul(uv[0].(*rat),VV[0]),Cmul(uv[1].(*rat),UU[0]))
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<-Z.req
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Z.dat <- get(W)
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copy(Add(W,Mul(UU[1],VV[1])),Z)
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}
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}(U, V, Z)
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return Z
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}
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// Differentiate
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func Diff(U PS) PS{
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Z:=mkPS()
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go func(U, Z PS){
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<-Z.req
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u := get(U)
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if end(u) == 0 {
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done:=false
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for i:=1; !done; i++ {
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u = get(U)
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if end(u) != 0 {
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done=true
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} else {
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Z.dat <- mul(itor(int64(i)),u)
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<-Z.req
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}
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}
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}
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Z.dat <- finis
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}(U, Z)
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return Z
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}
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// Integrate, with const of integration
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func Integ(c *rat,U PS) PS{
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Z:=mkPS()
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go func(c *rat, U, Z PS){
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put(c,Z)
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done:=false
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for i:=1; !done; i++ {
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<-Z.req
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u := get(U)
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if end(u) != 0 { done= true }
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Z.dat <- mul(i2tor(1,int64(i)),u)
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}
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Z.dat <- finis
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}(c, U, Z)
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return Z
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}
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// Binomial theorem (1+x)^c
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func Binom(c *rat) PS{
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Z:=mkPS()
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go func(c *rat, Z PS){
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n := 1
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t := itor(1)
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for c.num!=0 {
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put(t,Z)
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t = mul(mul(t,c),i2tor(1,int64(n)))
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c = sub(c,one)
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n++
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}
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put(finis,Z)
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}(c, Z)
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return Z
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}
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// Reciprocal of a power series
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// let U = u + x*UU
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// let Z = z + x*ZZ
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// (u+x*UU)*(z+x*ZZ) = 1
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// z = 1/u
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// u*ZZ + z*UU +x*UU*ZZ = 0
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// ZZ = -UU*(z+x*ZZ)/u
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func Recip(U PS) PS{
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Z:=mkPS()
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go func(U, Z PS){
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ZZ:=mkPS2()
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<-Z.req
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z := inv(get(U))
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Z.dat <- z
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split(Mul(Cmul(neg(z),U),Shift(z,ZZ[0])),ZZ)
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copy(ZZ[1],Z)
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}(U, Z)
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return Z
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}
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// Exponential of a power series with constant term 0
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// (nonzero constant term would make nonrational coefficients)
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// bug: the constant term is simply ignored
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// Z = exp(U)
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// DZ = Z*DU
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// integrate to get Z
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func Exp(U PS) PS{
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ZZ := mkPS2()
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split(Integ(one,Mul(ZZ[0],Diff(U))),ZZ)
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return ZZ[1]
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}
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// Substitute V for x in U, where the leading term of V is zero
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// let U = u + x*UU
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// let V = v + x*VV
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// then S(U,V) = u + VV*S(V,UU)
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// bug: a nonzero constant term is ignored
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func Subst(U, V PS) PS {
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Z:= mkPS()
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go func(U, V, Z PS) {
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VV := Split(V)
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<-Z.req
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u := get(U)
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Z.dat <- u
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|
if end(u) == 0 {
|
|
if end(get(VV[0])) != 0 {
|
|
put(finis,Z)
|
|
} else {
|
|
copy(Mul(VV[0],Subst(U,VV[1])),Z)
|
|
}
|
|
}
|
|
}(U, V, 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(U, Z PS, c0 *rat, n int) {
|
|
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
|
|
}
|
|
}
|
|
}(U, Z, c0, n)
|
|
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
|
|
*/
|
|
}
|
|
}
|