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diff --git a/src/algebra/interval.spad.pamphlet b/src/algebra/interval.spad.pamphlet new file mode 100644 index 00000000..ab9bfa9e --- /dev/null +++ b/src/algebra/interval.spad.pamphlet @@ -0,0 +1,547 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/algebra interval.spad} +\author{Mike Dewar} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{category INTCAT IntervalCategory} +<<category INTCAT IntervalCategory>>= +)abbrev category INTCAT IntervalCategory ++++ Author: Mike Dewar ++++ Date Created: November 1996 ++++ Date Last Updated: ++++ Basic Functions: ++++ Related Constructors: ++++ Also See: ++++ AMS Classifications: ++++ Keywords: ++++ References: ++++ Description: ++++ This category implements of interval arithmetic and transcendental ++++ functions over intervals. +IntervalCategory(R: Join(FloatingPointSystem,TranscendentalFunctionCategory)): + Category == Join(GcdDomain, OrderedSet, TranscendentalFunctionCategory, RadicalCategory, RetractableTo(Integer)) with + approximate + interval : (R,R) -> % + ++ interval(inf,sup) creates a new interval, either \axiom{[inf,sup]} if + ++ \axiom{inf <= sup} or \axiom{[sup,in]} otherwise. + qinterval : (R,R) -> % + ++ qinterval(inf,sup) creates a new interval \axiom{[inf,sup]}, without + ++ checking the ordering on the elements. + interval : R -> % + ++ interval(f) creates a new interval around f. + interval : Fraction Integer -> % + ++ interval(f) creates a new interval around f. + inf : % -> R + ++ inf(u) returns the infinum of \axiom{u}. + sup : % -> R + ++ sup(u) returns the supremum of \axiom{u}. + width : % -> R + ++ width(u) returns \axiom{sup(u) - inf(u)}. + positive? : % -> Boolean + ++ positive?(u) returns \axiom{true} if every element of u is positive, + ++ \axiom{false} otherwise. + negative? : % -> Boolean + ++ negative?(u) returns \axiom{true} if every element of u is negative, + ++ \axiom{false} otherwise. + contains? : (%,R) -> Boolean + ++ contains?(i,f) returns true if \axiom{f} is contained within the interval + ++ \axiom{i}, false otherwise. + +@ +\section{domain INTRVL Interval} +<<domain INTRVL Interval>>= +)abbrev domain INTRVL Interval ++++ Author: Mike Dewar ++++ Date Created: November 1996 ++++ Date Last Updated: ++++ Basic Functions: ++++ Related Constructors: ++++ Also See: ++++ AMS Classifications: ++++ Keywords: ++++ References: ++++ Description: ++++ This domain is an implementation of interval arithmetic and transcendental ++++ functions over intervals. +Interval(R:Join(FloatingPointSystem,TranscendentalFunctionCategory)): IntervalCategory(R) == add + + import Integer +-- import from R + + Rep := Record(Inf:R, Sup:R) + + roundDown(u:R):R == + if zero?(u) then float(-1,-(bits() pretend Integer)) + else float(mantissa(u) - 1,exponent(u)) + + roundUp(u:R):R == + if zero?(u) then float(1, -(bits()) pretend Integer) + else float(mantissa(u) + 1,exponent(u)) + + -- Sometimes the float representation does not use all the bits (e.g. when + -- representing an integer in software using arbitrary-length Integers as + -- your mantissa it is convenient to keep them exact). This function + -- normalises things so that rounding etc. works as expected. It is only + -- called when creating new intervals. + normaliseFloat(u:R):R == + zero? u => u + m : Integer := mantissa u + b : Integer := bits() pretend Integer + l : Integer := length(m) + if l < b then + BASE : Integer := base()$R pretend Integer + float(m*BASE**((b-l) pretend PositiveInteger),exponent(u)-b+l) + else + u + + interval(i:R,s:R):% == + i > s => [roundDown normaliseFloat s,roundUp normaliseFloat i] + [roundDown normaliseFloat i,roundUp normaliseFloat s] + + interval(f:R):% == + zero?(f) => 0 + one?(f) => 1 + -- This next part is necessary to allow e.g. mapping between Expressions: + -- AXIOM assumes that Integers stay as Integers! +-- import from Union(value1:Integer,failed:"failed") + fnew : R := normaliseFloat f + retractIfCan(f)@Union(Integer,"failed") case "failed" => + [roundDown fnew, roundUp fnew] + [fnew,fnew] + + qinterval(i:R,s:R):% == + [roundDown normaliseFloat i,roundUp normaliseFloat s] + + exactInterval(i:R,s:R):% == [i,s] + exactSupInterval(i:R,s:R):% == [roundDown i,s] + exactInfInterval(i:R,s:R):% == [i,roundUp s] + + inf(u:%):R == u.Inf + sup(u:%):R == u.Sup + width(u:%):R == u.Sup - u.Inf + + contains?(u:%,f:R):Boolean == (f > inf(u)) and (f < sup(u)) + + positive?(u:%):Boolean == inf(u) > 0 + negative?(u:%):Boolean == sup(u) < 0 + + _< (a:%,b:%):Boolean == + if inf(a) < inf(b) then + true + else if inf(a) > inf(b) then + false + else + sup(a) < sup(b) + + _+ (a:%,b:%):% == + -- A couple of blatent hacks to preserve the Ring Axioms! + if zero?(a) then return(b) else if zero?(b) then return(a) + if a = b then return qinterval(2*inf(a),2*sup(a)) + qinterval(inf(a) + inf(b), sup(a) + sup(b)) + + + _- (a:%,b:%):% == + if zero?(a) then return(-b) else if zero?(b) then return(a) + if a = b then 0 else qinterval(inf(a) - sup(b), sup(a) - inf(b)) + + + _* (a:%,b:%):% == + -- A couple of blatent hacks to preserve the Ring Axioms! + if one?(a) then return(b) else if one?(b) then return(a) + if zero?(a) then return(0) else if zero?(b) then return(0) + prods : List R := sort [inf(a)*inf(b),sup(a)*sup(b), + inf(a)*sup(b),sup(a)*inf(b)] + qinterval(first prods, last prods) + + + _* (a:Integer,b:%):% == + if (a > 0) then + qinterval(a*inf(b),a*sup(b)) + else if (a < 0) then + qinterval(a*sup(b),a*inf(b)) + else + 0 + + _* (a:PositiveInteger,b:%):% == qinterval(a*inf(b),a*sup(b)) + + _*_* (a:%,n:PositiveInteger):% == + contains?(a,0) and zero?((n pretend Integer) rem 2) => + interval(0,max(inf(a)**n,sup(a)**n)) + interval(inf(a)**n,sup(a)**n) + + + _^ (a:%,n:PositiveInteger):% == + contains?(a,0) and zero?((n pretend Integer) rem 2) => + interval(0,max(inf(a)**n,sup(a)**n)) + interval(inf(a)**n,sup(a)**n) + + _- (a:%):% == exactInterval(-sup(a),-inf(a)) + + _= (a:%,b:%):Boolean == (inf(a)=inf(b)) and (sup(a)=sup(b)) + _~_= (a:%,b:%):Boolean == (inf(a)~=inf(b)) or (sup(a)~=sup(b)) + + 1 == + one : R := normaliseFloat 1 + [one,one] + + 0 == [0,0] + + recip(u:%):Union(%,"failed") == + contains?(u,0) => "failed" + vals:List R := sort [1/inf(u),1/sup(u)]$List(R) + qinterval(first vals, last vals) + + + unit?(u:%):Boolean == contains?(u,0) + + _exquo(u:%,v:%):Union(%,"failed") == + contains?(v,0) => "failed" + one?(v) => u + u=v => 1 + u=-v => -1 + vals:List R := sort [inf(u)/inf(v),inf(u)/sup(v),sup(u)/inf(v),sup(u)/sup(v)]$List(R) + qinterval(first vals, last vals) + + + gcd(u:%,v:%):% == 1 + + coerce(u:Integer):% == + ur := normaliseFloat(u::R) + exactInterval(ur,ur) + + + interval(u:Fraction Integer):% == +-- import log2 : % -> % +-- coerce : Integer -> % +-- retractIfCan : % -> Union(value1:Integer,failed:"failed") +-- from Float + flt := u::R + + -- Test if the representation in R is exact + --den := denom(u)::Float + bin : Union(Integer,"failed") := retractIfCan(log2(denom(u)::Float)) + bin case Integer and length(numer u)$Integer < (bits() pretend Integer) => + flt := normaliseFloat flt + exactInterval(flt,flt) + + qinterval(flt,flt) + + + retractIfCan(u:%):Union(Integer,"failed") == + not zero? width(u) => "failed" + retractIfCan inf u + + + retract(u:%):Integer == + not zero? width(u) => + error "attempt to retract a non-Integer interval to an Integer" + retract inf u + + + coerce(u:%):OutputForm == + bracket([coerce inf(u), coerce sup(u)]$List(OutputForm)) + + characteristic():NonNegativeInteger == 0 + + + -- Explicit export from TranscendentalFunctionCategory + pi():% == qinterval(pi(),pi()) + + -- From ElementaryFunctionCategory + log(u:%):% == + positive?(u) => qinterval(log inf u, log sup u) + error "negative logs in interval" + + + exp(u:%):% == qinterval(exp inf u, exp sup u) + + _*_* (u:%,v:%):% == + zero?(v) => if zero?(u) then error "0**0 is undefined" else 1 + one?(u) => 1 + expts : List R := sort [inf(u)**inf(v),sup(u)**sup(v), + inf(u)**sup(v),sup(u)**inf(v)] + qinterval(first expts, last expts) + + -- From TrigonometricFunctionCategory + + -- This function checks whether an interval contains a value of the form + -- `offset + 2 n pi'. + hasTwoPiMultiple(offset:R,ipi:R,i:%):Boolean == + next : Integer := retract ceiling( (inf(i) - offset)/(2*ipi) ) + contains?(i,offset+2*next*ipi) + + + -- This function checks whether an interval contains a value of the form + -- `offset + n pi'. + hasPiMultiple(offset:R,ipi:R,i:%):Boolean == + next : Integer := retract ceiling( (inf(i) - offset)/ipi ) + contains?(i,offset+next*ipi) + + + sin(u:%):% == + ipi : R := pi()$R + hasOne? : Boolean := hasTwoPiMultiple(ipi/(2::R),ipi,u) + hasMinusOne? : Boolean := hasTwoPiMultiple(3*ipi/(2::R),ipi,u) + + if hasOne? and hasMinusOne? then + exactInterval(-1,1) + else + vals : List R := sort [sin inf u, sin sup u] + if hasOne? then + exactSupInterval(first vals, 1) + else if hasMinusOne? then + exactInfInterval(-1,last vals) + else + qinterval(first vals, last vals) + + + + cos(u:%):% == + ipi : R := pi() + hasOne? : Boolean := hasTwoPiMultiple(0,ipi,u) + hasMinusOne? : Boolean := hasTwoPiMultiple(ipi,ipi,u) + + if hasOne? and hasMinusOne? then + exactInterval(-1,1) + else + vals : List R := sort [cos inf u, cos sup u] + if hasOne? then + exactSupInterval(first vals, 1) + else if hasMinusOne? then + exactInfInterval(-1,last vals) + else + qinterval(first vals, last vals) + + + + tan(u:%):% == + ipi : R := pi() + if width(u) > ipi then + error "Interval contains a singularity" + else + -- Since we know the interval is less than pi wide, monotonicity implies + -- that there is no singularity. If there is a singularity on a endpoint + -- of the interval the user will see the error generated by R. + lo : R := tan inf u + hi : R := tan sup u + + lo > hi => error "Interval contains a singularity" + qinterval(lo,hi) + + + + csc(u:%):% == + ipi : R := pi() + if width(u) > ipi then + error "Interval contains a singularity" + else +-- import from Integer + -- singularities are at multiples of Pi + if hasPiMultiple(0,ipi,u) then error "Interval contains a singularity" + vals : List R := sort [csc inf u, csc sup u] + if hasTwoPiMultiple(ipi/(2::R),ipi,u) then + exactInfInterval(1,last vals) + else if hasTwoPiMultiple(3*ipi/(2::R),ipi,u) then + exactSupInterval(first vals,-1) + else + qinterval(first vals, last vals) + + + + sec(u:%):% == + ipi : R := pi() + if width(u) > ipi then + error "Interval contains a singularity" + else +-- import from Integer + -- singularities are at Pi/2 + n Pi + if hasPiMultiple(ipi/(2::R),ipi,u) then + error "Interval contains a singularity" + vals : List R := sort [sec inf u, sec sup u] + if hasTwoPiMultiple(0,ipi,u) then + exactInfInterval(1,last vals) + else if hasTwoPiMultiple(ipi,ipi,u) then + exactSupInterval(first vals,-1) + else + qinterval(first vals, last vals) + + + + + cot(u:%):% == + ipi : R := pi() + if width(u) > ipi then + error "Interval contains a singularity" + else + -- Since we know the interval is less than pi wide, monotonicity implies + -- that there is no singularity. If there is a singularity on a endpoint + -- of the interval the user will see the error generated by R. + hi : R := cot inf u + lo : R := cot sup u + + lo > hi => error "Interval contains a singularity" + qinterval(lo,hi) + + + + -- From ArcTrigonometricFunctionCategory + + asin(u:%):% == + lo : R := inf(u) + hi : R := sup(u) + if (lo < -1) or (hi > 1) then error "asin only defined on the region -1..1" + qinterval(asin lo,asin hi) + + + acos(u:%):% == + lo : R := inf(u) + hi : R := sup(u) + if (lo < -1) or (hi > 1) then error "acos only defined on the region -1..1" + qinterval(acos hi,acos lo) + + + atan(u:%):% == qinterval(atan inf u, atan sup u) + + acot(u:%):% == qinterval(acot sup u, acot inf u) + + acsc(u:%):% == + lo : R := inf(u) + hi : R := sup(u) + if ((lo <= -1) and (hi >= -1)) or ((lo <= 1) and (hi >= 1)) then + error "acsc not defined on the region -1..1" + qinterval(acsc hi, acsc lo) + + + asec(u:%):% == + lo : R := inf(u) + hi : R := sup(u) + if ((lo < -1) and (hi > -1)) or ((lo < 1) and (hi > 1)) then + error "asec not defined on the region -1..1" + qinterval(asec lo, asec hi) + + + -- From HyperbolicFunctionCategory + + tanh(u:%):% == qinterval(tanh inf u, tanh sup u) + + sinh(u:%):% == qinterval(sinh inf u, sinh sup u) + + sech(u:%):% == + negative? u => qinterval(sech inf u, sech sup u) + positive? u => qinterval(sech sup u, sech inf u) + vals : List R := sort [sech inf u, sech sup u] + exactSupInterval(first vals,1) + + + cosh(u:%):% == + negative? u => qinterval(cosh sup u, cosh inf u) + positive? u => qinterval(cosh inf u, cosh sup u) + vals : List R := sort [cosh inf u, cosh sup u] + exactInfInterval(1,last vals) + + + csch(u:%):% == + contains?(u,0) => error "csch: singularity at zero" + qinterval(csch sup u, csch inf u) + + + coth(u:%):% == + contains?(u,0) => error "coth: singularity at zero" + qinterval(coth sup u, coth inf u) + + + -- From ArcHyperbolicFunctionCategory + + acosh(u:%):% == + inf(u)<1 => error "invalid argument: acosh only defined on the region 1.." + qinterval(acosh inf u, acosh sup u) + + + acoth(u:%):% == + lo : R := inf(u) + hi : R := sup(u) + if ((lo <= -1) and (hi >= -1)) or ((lo <= 1) and (hi >= 1)) then + error "acoth not defined on the region -1..1" + qinterval(acoth hi, acoth lo) + + + acsch(u:%):% == + contains?(u,0) => error "acsch: singularity at zero" + qinterval(acsch sup u, acsch inf u) + + + asech(u:%):% == + lo : R := inf(u) + hi : R := sup(u) + if (lo <= 0) or (hi > 1) then + error "asech only defined on the region 0 < x <= 1" + qinterval(asech hi, asech lo) + + + asinh(u:%):% == qinterval(asinh inf u, asinh sup u) + + atanh(u:%):% == + lo : R := inf(u) + hi : R := sup(u) + if (lo <= -1) or (hi >= 1) then + error "atanh only defined on the region -1 < x < 1" + qinterval(atanh lo, atanh hi) + + + -- From RadicalCategory + _*_* (u:%,n:Fraction Integer):% == interval(inf(u)**n,sup(u)**n) + +@ +\section{License} +<<license>>= +--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +--All rights reserved. +-- +--Redistribution and use in source and binary forms, with or without +--modification, are permitted provided that the following conditions are +--met: +-- +-- - Redistributions of source code must retain the above copyright +-- notice, this list of conditions and the following disclaimer. +-- +-- - Redistributions in binary form must reproduce the above copyright +-- notice, this list of conditions and the following disclaimer in +-- the documentation and/or other materials provided with the +-- distribution. +-- +-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the +-- names of its contributors may be used to endorse or promote products +-- derived from this software without specific prior written permission. +-- +--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. +@ +<<*>>= +<<license>> + +<<category INTCAT IntervalCategory>> +<<domain INTRVL Interval>> + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} + + |