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<Text-field prompt="> " style="Maple Input" layout="Normal">restart;</Text-field>
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<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
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<Input>
<Text-field style="Normal" layout="Normal">###########################################################################</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal"><Font bold="true" size="14">intCompare v1.0 - Comparison Operators for Interval Arithmetic in Maple </Font>
intCompare provides the four comparison operators <,<=, >=, > for intervals. In case of a Don't know</Text-field>
<Text-field style="Normal" layout="Normal">we return FAIL.
###########################################################################
Module header
###########################################################################
</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">intCompare:=module()
export Interval_Compare, Interval_LT, `&<`, Interval_LE, `&<=`,
Interval_GT, `&>`, Interval_GE, `&>=`, Interval_abs, `&abs`,
Interval_Normalize, Interval_IsBorder;
option package;
description "Comparison Operators for Interval Arithmetic in Maple";
#uses intpakX; <Font style="Normal">
</Font></Text-field>
<Text-field style="Normal" layout="Normal">###########################################################################</Text-field>
<Text-field style="Normal" layout="Normal">Name definitions (see below for body of corresponding procedures)</Text-field>
<Text-field style="Normal" layout="Normal">###########################################################################
</Text-field>
<Text-field prompt="> " style="Normal" layout="Normal"><Font style="Maple Input">`&abs`:=Interval_abs;
`&<`:=Interval_LT;
`&<=`:=Interval_LE;
`&>`:=Interval_GT;
`&>=`:=Interval_GE;
</Font></Text-field>
<Text-field style="Normal" layout="Normal">###########################################################################</Text-field>
<Text-field style="Normal" layout="Normal">-----------------------------------------------------------------------------</Text-field>
<Text-field style="Normal" layout="Normal"> PROCEDURE -Interval_abs</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">This procedure computes the absolute value of an interval or number or symbol
</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">Interval_abs:=proc(x)</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if x = [] then x</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> elif type(x,interval) then</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if x[1] = FAIL or x[2] = FAIL then FAIL</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> elif x[1] = x[2] then [abs(x[1]), abs(x[2])]</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> elif is_in(0, x) then [0, max(abs(x[1]), x[2])]</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> elif x[1] > 0 then x</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> else [abs(x[2]), abs(x[1])]</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> elif type(x, numeric) or type(x, constant) then abs(x)</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> else 'Interval_abs(x)'</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">end:</Text-field>
<Text-field style="Normal" layout="Normal">
</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">###########################################################################</Text-field>
<Text-field style="Normal" layout="Normal">-----------------------------------------------------------------------------</Text-field>
<Text-field style="Normal" layout="Normal"> PROCEDURE -Interval_Normalize</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">This procedure takes some argument, and in case of an interval it normalizes a collapsing interval [a, a] to a.
</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">Interval_Normalize:=proc(x)</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if type(x,interval) and (not x = []) and x[1] = x[2] then</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> x[1];</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> else x;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">end:</Text-field>
<Text-field style="Normal" layout="Normal">
</Text-field>
<Text-field style="Normal" layout="Normal">-----------------------------------------------------------------------------</Text-field>
<Text-field style="Normal" layout="Normal"> PROCEDURE -Interval_IsBorder</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">This procedure checks if the first argument is the border of the second argument.
To be a border means for intervals to be one of the edge-points, in case a non-collapsing interval is given as border-candidate
we return FAIL.
</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">Interval_IsBorder:=proc(b,i)</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">local x, y;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> </Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> x := Interval_Normalize(b);</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> y := Interval_Normalize(i);</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if x=[] then false</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> elif y=[] then false</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> elif type(x,interval) then undefined</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> elif type(y,interval) and (x = y[1] or x = y[2]) then true</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> elif x = y then true</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> else false;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">end:</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">-----------------------------------------------------------------------------</Text-field>
<Text-field style="Normal" layout="Normal"> PROCEDURE -Interval_Compare</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">This procedure takes three parameters. The first parameter is a comparison-operator, <, <=, >, >= and</Text-field>
<Text-field style="Normal" layout="Normal">the second and third is a symbol, number or interval.</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">Interval_Compare:=proc(c,x,y)</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">local p,q,i,ib,symnum;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">description "Generic comparison function for intervals";</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> ib := Interval_IsBorder;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> symnum :=proc(a) type(a, symbol) or type(a, numeric) end;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> </Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> # check for intersection</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> i := Interval_Normalize(Interval_intersect(x,y));</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if symnum(x) and symnum(y) then c(x,y)</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> elif i = [] or (not type(i, interval) and ib(i, x) and ib(i,y)) then</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if type(x,interval) then</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> # check for empty intervals</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if x = [] then RETURN(undefined); fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> # compute the relevant border for x</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if c(x[1], x[2]) then p := x[2]; else p:=x[1] fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> else p := x;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if type(y,interval) then</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> # check for empty intervals</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if y = [] then RETURN(undefined); fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> # compute the relevant border for y</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if c(y[1], y[2]) then q := y[1]; else q:=y[2] fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> else q := y;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> # now compute the result of the comparison</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> if type(p, num_or_FAIL) and type(q, num_or_FAIL) then c(p, q)</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> else 'Interval_Compare(c, p, q)'</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> else RETURN(undefined);</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"> fi;</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">end:</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">-----------------------------------------------------------------------------</Text-field>
<Text-field style="Normal" layout="Normal"> PROCEDURES -Interval_<R>, just instances of the general Interval_Compare procedure</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">Interval_LT:=proc(x,y) Interval_Compare(`<`,x,y) end proc:</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">Interval_LE:=proc(x,y) Interval_Compare(`<=`,x,y) end proc:</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">Interval_GT:=proc(x,y) Interval_Compare(`>`,x,y) end proc:</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">Interval_GE:=proc(x,y) Interval_Compare(`>=`,x,y) end proc:</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">
<Font style="Normal">-----------------------------------------------------------------------------</Font></Text-field>
<Text-field style="Normal" layout="Normal"> KNOWN PROBLEMS</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">1. As the interval-package don't support symbolic intervals, e.g. construct(Pi) = [3.141592653, 3.141592655]</Text-field>
<Text-field style="Text" layout="Normal">> Interval_Compare(`<`, Pi, Pi); </Text-field>
<Text-field style="Text" layout="Normal"> FAIL</Text-field>
<Text-field style="Text" layout="Normal"> Because the intersection-check yields an non-empty result:</Text-field>
<Text-field style="Text" layout="Normal">> Interval_intersect( Pi, Pi); </Text-field>
<Text-field style="Text" layout="Normal"> [3.141592653, 3.141592655]</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">-----------------------------------------------------------------------------</Text-field>
<Text-field style="Normal" layout="Normal"> TEST-CASES</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">data:</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">i1 := 1;</Text-field>
<Text-field style="Normal" layout="Normal">i2 := 1.;</Text-field>
<Text-field style="Normal" layout="Normal">i3 := 3;</Text-field>
<Text-field style="Normal" layout="Normal">i4 := Pi;</Text-field>
<Text-field style="Normal" layout="Normal">i5 := [1., 1.1];</Text-field>
<Text-field style="Normal" layout="Normal">i6 := [3., 3.1];</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">checks:</Text-field>
<Text-field style="Text" layout="Normal">i4 &> i6;</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">###########################################################################</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">end module:</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">###########################################################################</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">end package intCompare</Text-field>
<Text-field style="Normal" layout="Normal">###########################################################################</Text-field>
<Text-field style="Normal" layout="Normal">Disclaimer:</Text-field>
<Text-field style="_pstyle256" layout="_pstyle256">While every effort has been made to validate the solutions in this package,</Text-field>
<Text-field style="_pstyle257" layout="_pstyle257">Waterloo Maple Inc. and the contributors are not responsible for any errors</Text-field>
<Text-field style="_pstyle258" layout="_pstyle258">contained and are not liable for any damages resulting from the use of this</Text-field>
<Text-field style="_pstyle259" layout="_pstyle259">material.</Text-field>
<Text-field style="_pstyle259" layout="_pstyle259"></Text-field>
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<Text-field style="Normal" layout="Normal">###########################################################################</Text-field>
<Text-field style="Normal" layout="Normal">Installation:</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">* execute this:</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">libname:="/home/ewaryst/data/Maple/lib", libname;</Text-field>
</Input>
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<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal">* load the int-lib</Text-field>
<Text-field prompt="> " style="Maple Input" layout="Normal">with(intpakX);</Text-field>
</Input>
</Group>
<Group labelreference="L26" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">* go to the module-definition and hit Enter</Text-field>
</Input>
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
</Input>
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<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"></Text-field>
</Input>
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<Input>
<Text-field style="Text" layout="Normal">* Execute the savelib-command</Text-field>
</Input>
<Input>
<Text-field prompt="> " style="Maple Input" layout="Normal">savelib('intCompare');</Text-field>
</Input>
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<Input>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" layout="Normal">You're done!</Text-field>
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