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--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.
)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.
)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 == positive? inf(u)
negative?(u:%):Boolean == negative? sup(u)
a:% < b:% ==
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 positive? a then
qinterval(a*inf(b),a*sup(b))
else if negative? a 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:%) == exactInterval(-sup(a),-inf(a))
a:% = b:% == (inf(a)=inf(b)) and (sup(a)=sup(b))
a:% ~= b:% == (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)
(u:% exquo 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)
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