[Axiom-developer] [ContinuedFractions] (new)

[Bill Page]

Changes http://wiki.axiom-developer.org/ContinuedFractions/diff
--
Description:

'ContinuedFraction' implements general continued fractions.
This version is not restricted to simple, finite fractions and
uses the 'Stream' as a representation.  The arithmetic functions
assume that the approximants alternate below/above the convergence
point. This is enforced by ensuring the partial numerators and
partial denominators are greater than 0 in the Euclidean domain
view of 'R' (i.e. 'sizeLess?(0, x)'). 

\begin{spad}
)abbrev domain CONTFRAC ContinuedFraction
++ Author: Stephen M. Watt
++ Date Created: January 1987
++ Change History:
++   11 April   1990
++    7 October 1991 -- SMW: Treat whole part specially.  Added comments.
++ Basic Operations:
++   (Field), (Algebra),
++   approximants, complete, continuedFraction, convergents, denominators,
++   extend, numerators, partialDenominators, partialNumerators,
++   partialQuotients, reducedContinuedFraction, reducedForm, wholePart
++ Related Constructors:
++ Also See: Fraction
++ AMS Classifications: 11A55 11J70 11K50 11Y65 30B70 40A15
++ Keywords: continued fraction, convergent
++ References:


ContinuedFraction(R): Exports == Implementation where
  R :     EuclideanDomain
  Q   ==> Fraction R
  MT  ==> MoebiusTransform Q
  OUT ==> OutputForm

  Exports ==> Join(Algebra R,Algebra Q,Field) with
    continuedFraction:        Q -> %
      ++ continuedFraction(r) converts the fraction \spadvar{r} with
      ++ components of type \spad{R} to a continued fraction over
      ++ \spad{R}.

    continuedFraction:        (R, Stream R, Stream R) -> %
      ++ continuedFraction(b0,a,b) constructs a continued fraction in
      ++ the following way:  if \spad{a = [a1,a2,...]} and \spad{b =
      ++ [b1,b2,...]} then the result is the continued fraction
      ++ \spad{b0 + a1/(b1 + a2/(b2 + ...))}.

    reducedContinuedFraction: (R, Stream R) -> %
      ++ reducedContinuedFraction(b0,b) constructs a continued
      ++ fraction in the following way:  if \spad{b = [b1,b2,...]}
      ++ then the result is the continued fraction \spad{b0 + 1/(b1 +
      ++ 1/(b2 + ...))}.  That is, the result is the same as
      ++ \spad{continuedFraction(b0,[1,1,1,...],[b1,b2,b3,...])}.

    partialNumerators:   % -> Stream R
      ++ partialNumerators(x) extracts the numerators in \spadvar{x}.
      ++ That is, if \spad{x = continuedFraction(b0, [a1,a2,a3,...],
      ++ [b1,b2,b3,...])}, then \spad{partialNumerators(x) =
      ++ [a1,a2,a3,...]}.

    partialDenominators: % -> Stream R
      ++ partialDenominators(x) extracts the denominators in
      ++ \spadvar{x}.  That is, if \spad{x = continuedFraction(b0,
      ++ [a1,a2,a3,...], [b1,b2,b3,...])}, then
      ++ \spad{partialDenominators(x) = [b1,b2,b3,...]}.

    partialQuotients:    % -> Stream R
      ++ partialQuotients(x) extracts the partial quotients in
      ++ \spadvar{x}.  That is, if \spad{x = continuedFraction(b0,
      ++ [a1,a2,a3,...], [b1,b2,b3,...])}, then
      ++ \spad{partialQuotients(x) = [b0,b1,b2,b3,...]}.

    wholePart:           % -> R
      ++ wholePart(x) extracts the whole part of \spadvar{x}.  That
      ++ is, if \spad{x = continuedFraction(b0, [a1,a2,a3,...],
      ++ [b1,b2,b3,...])}, then \spad{wholePart(x) = b0}.

    reducedForm:         % -> %
      ++ reducedForm(x) puts the continued fraction \spadvar{x} in
      ++ reduced form, i.e.  the function returns an equivalent
      ++ continued fraction of the form
      ++ \spad{continuedFraction(b0,[1,1,1,...],[b1,b2,b3,...])}.

    approximants:        % -> Stream Q
      ++ approximants(x) returns the stream of approximants of the
      ++ continued fraction \spadvar{x}. If the continued fraction is
      ++ finite, then the stream will be infinite and periodic with
      ++ period 1.

    convergents:         % -> Stream Q
      ++ convergents(x) returns the stream of the convergents of the
      ++ continued fraction \spadvar{x}. If the continued fraction is
      ++ finite, then the stream will be finite.

    numerators:          % -> Stream R
      ++ numerators(x) returns the stream of numerators of the
      ++ approximants of the continued fraction \spadvar{x}. If the
      ++ continued fraction is finite, then the stream will be finite.

    denominators:        % -> Stream R
      ++ denominators(x) returns the stream of denominators of the
      ++ approximants of the continued fraction \spadvar{x}. If the
      ++ continued fraction is finite, then the stream will be finite.

    extend:              (%,Integer) -> %
      ++ extend(x,n) causes the first \spadvar{n} entries in the
      ++ continued fraction \spadvar{x} to be computed.  Normally
      ++ entries are only computed as needed.

    complete:            % -> %
      ++ complete(x) causes all entries in \spadvar{x} to be computed.
      ++ Normally entries are only computed as needed.  If \spadvar{x}
      ++ is an infinite continued fraction, a user-initiated interrupt is
      ++ necessary to stop the computation.

  Implementation ==> add

 -- isOrdered  ==> R is Integer
    isOrdered  ==> R has OrderedRing and R has multiplicativeValuation
    canReduce? ==> isOrdered or R has additiveValuation

    Rec ==> Record(num: R, den: R)
    Str ==> Stream Rec
    Rep :=  Record(value: Record(whole: R, fract: Str), reduced?: Boolean)

    import Str

    genFromSequence:     Stream Q -> %
    genReducedForm:      (Q, Stream Q, MT)    -> Stream Rec
    genFractionA:        (Stream R,Stream R)  -> Stream Rec
    genFractionB:        (Stream R,Stream R)  -> Stream Rec
    genNumDen:           (R,R, Stream Rec)    -> Stream R

    genApproximants:     (R,R,R,R,Stream Rec) -> Stream Q
    genConvergents:      (R,R,R,R,Stream Rec) -> Stream Q
    iGenApproximants:    (R,R,R,R,Stream Rec) -> Stream Q
    iGenConvergents:     (R,R,R,R,Stream Rec) -> Stream Q

    reducedForm c == 
        c.reduced? => c
        explicitlyFinite? c.value.fract =>
                      continuedFraction last complete convergents c
        canReduce? => genFromSequence approximants c
        error "Reduced form not defined for this continued fraction."

    eucWhole(a: Q): R == numer a quo denom a

    eucWhole0(a: Q): R ==
        isOrdered =>
            n := numer a
            d := denom a
            q := n quo d
            r := n - q*d
            if r < 0 then q := q - 1
            q
        eucWhole a

    x = y ==
        x := reducedForm x
        y := reducedForm y

        x.value.whole ^= y.value.whole => false

        xl := x.value.fract; yl := y.value.fract

        while not empty? xl and not empty? yl repeat
            frst.xl.den ^= frst.yl.den => return false
            xl := rst xl; yl := rst yl
        empty? xl and empty? yl

    continuedFraction q == q :: %

    if isOrdered then
        continuedFraction(wh,nums,dens) == [[wh,genFractionA(nums,dens)],false]

        genFractionA(nums,dens) ==
            empty? nums or empty? dens => empty()
            n := frst nums
            d := frst dens
            n < 0 => error "Numerators must be greater than 0."
            d < 0 => error "Denominators must be greater than 0."
            concat([n,d]$Rec, delay genFractionA(rst nums,rst dens))
    else
        continuedFraction(wh,nums,dens) == [[wh,genFractionB(nums,dens)],false]

        genFractionB(nums,dens) ==
            empty? nums or empty? dens => empty()
            n := frst nums
            d := frst dens
            concat([n,d]$Rec, delay genFractionB(rst nums,rst dens))

    reducedContinuedFraction(wh,dens) ==
        continuedFraction(wh, repeating [1], dens)

    coerce(n:Integer):% == [[n::R,empty()], true]
    coerce(r:R):%       == [[r,   empty()], true]

    coerce(a: Q): % ==
      wh := eucWhole0 a
      fr := a - wh::Q
      zero? fr => [[wh, empty()], true]

      l : List Rec := empty()
      n := numer fr
      d := denom fr
      while not zero? d repeat
        qr := divide(n,d)
        l  := concat([1,qr.quotient],l)
        n  := d
        d  := qr.remainder
      [[wh, construct rest reverse_! l], true]

    characteristic() == characteristic()$Q


    genFromSequence apps ==
        lo := first apps; apps := rst apps
        hi := first apps; apps := rst apps
        while eucWhole0 lo ^= eucWhole0 hi repeat
            lo := first apps; apps := rst apps
            hi := first apps; apps := rst apps
        wh := eucWhole0 lo
        [[wh, genReducedForm(wh::Q, apps, moebius(1,0,0,1))], canReduce?]

    genReducedForm(wh0, apps, mt) ==
        lo: Q := first apps - wh0; apps := rst apps
        hi: Q := first apps - wh0; apps := rst apps
        lo = hi and zero? eval(mt, lo) => empty()
        mt  := recip mt
        wlo := eucWhole eval(mt, lo)
        whi := eucWhole eval(mt, hi)
        while wlo ^= whi repeat
            wlo := eucWhole eval(mt, first apps - wh0); apps := rst apps
            whi := eucWhole eval(mt, first apps - wh0); apps := rst apps
        concat([1,wlo], delay genReducedForm(wh0, apps, shift(mt, -wlo::Q)))

    wholePart c           == c.value.whole
    partialNumerators c   == map(#1.num, c.value.fract)$StreamFunctions2(Rec,R)
    partialDenominators c == map(#1.den, c.value.fract)$StreamFunctions2(Rec,R)
    partialQuotients c    == concat(c.value.whole, partialDenominators c)

    approximants c ==
      empty? c.value.fract => repeating [c.value.whole::Q]
      genApproximants(1,0,c.value.whole,1,c.value.fract)
    convergents c ==
      empty? c.value.fract => concat(c.value.whole::Q, empty())
      genConvergents (1,0,c.value.whole,1,c.value.fract)
    numerators c ==
      empty? c.value.fract => concat(c.value.whole, empty())
      genNumDen(1,c.value.whole,c.value.fract)
    denominators c ==
      genNumDen(0,1,c.value.fract)

    extend(x,n) == (extend(x.value.fract,n); x)
    complete(x) == (complete(x.value.fract); x)

    iGenApproximants(pm2,qm2,pm1,qm1,fr) == delay
      nd := frst fr
      pm := nd.num*pm2 + nd.den*pm1
      qm := nd.num*qm2 + nd.den*qm1
      genApproximants(pm1,qm1,pm,qm,rst fr)

    genApproximants(pm2,qm2,pm1,qm1,fr) ==
      empty? fr => repeating [pm1/qm1]
      concat(pm1/qm1,iGenApproximants(pm2,qm2,pm1,qm1,fr))

    iGenConvergents(pm2,qm2,pm1,qm1,fr) == delay
      nd := frst fr
      pm := nd.num*pm2 + nd.den*pm1
      qm := nd.num*qm2 + nd.den*qm1
      genConvergents(pm1,qm1,pm,qm,rst fr)

    genConvergents(pm2,qm2,pm1,qm1,fr) ==
      empty? fr => concat(pm1/qm1, empty())
      concat(pm1/qm1,iGenConvergents(pm2,qm2,pm1,qm1,fr))

    genNumDen(m2,m1,fr) ==
      empty? fr => concat(m1,empty())
      concat(m1,delay genNumDen(m1,m2*frst(fr).num + m1*frst(fr).den,rst fr))

    gen  ==> genFromSequence
    apx  ==> approximants

    c, d: %
    a: R
    q: Q
    n: Integer

    0 == (0$R) :: %
    1 == (1$R) :: %

    c + d   == genFromSequence map(#1 + #2, apx c, apx d)
    c - d   == genFromSequence map(#1 - #2, apx c, rest apx d)
    - c     == genFromSequence map(   - #1, rest apx c)
    c * d   == genFromSequence map(#1 * #2, apx c, apx d)
    a * d   == genFromSequence map( a * #1, apx d)
    q * d   == genFromSequence map( q * #1, apx d)
    n * d   == genFromSequence map( n * #1, apx d)
    c / d   == genFromSequence map(#1 / #2, apx c, rest apx d)
    recip c ==(c = 0 => "failed";
               genFromSequence map( 1 / #1, rest apx c))

    showAll?: () -> Boolean
    showAll?() ==
      NULL(_$streamsShowAll$Lisp)$Lisp => false
      true

    zagRec(t:Rec):OUT == zag(t.num :: OUT,t.den :: OUT)

    coerce(c:%): OUT ==
      wh := c.value.whole
      fr := c.value.fract
      empty? fr => wh :: OUT
      count : NonNegativeInteger := _$streamCount$Lisp
      l : List OUT := empty()
      for n in 1..count while not empty? fr repeat
        l  := concat(zagRec frst fr,l)
        fr := rst fr
      if showAll?() then
        for n in (count + 1).. while explicitEntries? fr repeat
          l  := concat(zagRec frst fr,l)
          fr := rst fr
      if not explicitlyEmpty? fr then l := concat("..." :: OUT,l)
      l := reverse_! l
      e := reduce("+",l)
      zero? wh => e
      (wh :: OUT) + e
\end{spad}

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