aboutsummaryrefslogtreecommitdiff
path: root/src/algebra/d01agents.spad.pamphlet
blob: eadc8a14c158defc0ebd195cb313f8baf01c59e1 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra d01agents.spad}
\author{Brian Dupee}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain INTFTBL IntegrationFunctionsTable}
<<domain INTFTBL IntegrationFunctionsTable>>=
)abbrev domain INTFTBL IntegrationFunctionsTable
++ Author: Brian Dupee
++ Date Created: March 1995
++ Date Last Updated: June 1995
++ Description:
++
IntegrationFunctionsTable(): E == I where
  EF2	==> ExpressionFunctions2
  EFI	==> Expression Fraction Integer
  FI	==> Fraction Integer
  LEDF	==> List Expression DoubleFloat
  KEDF	==> Kernel Expression DoubleFloat
  EEDF	==> Equation Expression DoubleFloat
  EDF	==> Expression DoubleFloat
  PDF	==> Polynomial DoubleFloat
  LDF	==> List DoubleFloat
  SDF	==> Stream DoubleFloat
  DF	==> DoubleFloat
  F	==> Float
  ST	==> String
  LST	==> List String
  SI	==> SingleInteger
  SOCDF	==> Segment OrderedCompletion DoubleFloat
  OCDF	==> OrderedCompletion DoubleFloat
  OCEDF	==> OrderedCompletion Expression DoubleFloat
  EOCEFI  ==> Equation OrderedCompletion Expression Fraction Integer
  OCEFI   ==> OrderedCompletion Expression Fraction Integer
  OCFI    ==> OrderedCompletion Fraction Integer
  NIA	==> Record(var:Symbol,fn:EDF,range:SOCDF,abserr:DF,relerr:DF)
  INT	==> Integer
  CTYPE	==> Union(continuous: "Continuous at the end points",
             lowerSingular: "There is a singularity at the lower end point",
              upperSingular: "There is a singularity at the upper end point",
               bothSingular: "There are singularities at both end points",
                notEvaluated: "End point continuity not yet evaluated")
  RTYPE	==> Union(finite: "The range is finite",
              lowerInfinite: "The bottom of range is infinite",
                upperInfinite: "The top of range is infinite",
                  bothInfinite: "Both top and bottom points are infinite",
                    notEvaluated: "Range not yet evaluated")
  STYPE	==> Union(str:SDF,
                   notEvaluated:"Internal singularities not yet evaluated")
  ATT	==> Record(endPointContinuity:CTYPE,
                    singularitiesStream:STYPE,range:RTYPE)
  ROA	==> Record(key:NIA,entry:ATT)

  E ==> with

    showTheFTable:() -> $
      ++ showTheFTable() returns the current table of functions.
    clearTheFTable : () -> Void
      ++ clearTheFTable() clears the current table of functions.
    keys : $ -> List(NIA)
      ++ keys(f) returns the list of keys of f
    fTable: List Record(key:NIA,entry:ATT) -> $
      ++ fTable(l) creates a functions table from the elements of l.
    insert!:Record(key:NIA,entry:ATT) -> $
      ++ insert!(r) inserts an entry r into theIFTable
    showAttributes:NIA -> Union(ATT,"failed")
      ++ showAttributes(x) \undocumented{}
    entries : $ -> List Record(key:NIA,entry:ATT)
      ++ entries(x) \undocumented{}
    entry:NIA -> ATT
      ++ entry(n) \undocumented{}
  I ==> add

    Rep := Table(NIA,ATT)
    import Rep

    theFTable:$ := empty()$Rep

    showTheFTable():$ ==
      theFTable

    clearTheFTable():Void ==
      theFTable := empty()$Rep

    fTable(l:List Record(key:NIA,entry:ATT)):$ ==
      theFTable := table(l)$Rep

    insert!(r:Record(key:NIA,entry:ATT)):$ ==
      insert!(r,theFTable)$Rep

    keys(t:$):List NIA ==
      keys(t)$Rep

    showAttributes(k:NIA):Union(ATT,"failed") ==
      search(k,theFTable)$Rep

    entries(t:$):List Record(key:NIA,entry:ATT) ==
      members(t)$Rep

    entry(k:NIA):ATT ==
      qelt(theFTable,k)$Rep

@
\section{package D01AGNT d01AgentsPackage}
<<package D01AGNT d01AgentsPackage>>=
)abbrev package D01AGNT d01AgentsPackage
++ Author: Brian Dupee
++ Date Created: March 1994
++ Date Last Updated: December 1997
++ Basic Operations: rangeIsFinite, functionIsContinuousAtEndPoints,
++ functionIsOscillatory
++ Description:
++ \axiomType{d01AgentsPackage} is a package of numerical agents to be used
++ to investigate attributes of an input function so as to decide the
++ \axiomFun{measure} of an appropriate numerical integration routine.
++ It contains functions \axiomFun{rangeIsFinite} to test the input range and
++ \axiomFun{functionIsContinuousAtEndPoints} to check for continuity at 
++ the end points of the range.


d01AgentsPackage(): E == I where
  EF2	==> ExpressionFunctions2
  EFI	==> Expression Fraction Integer
  FI	==> Fraction Integer
  LEDF	==> List Expression DoubleFloat
  KEDF	==> Kernel Expression DoubleFloat
  EEDF	==> Equation Expression DoubleFloat
  EDF	==> Expression DoubleFloat
  PDF	==> Polynomial DoubleFloat
  LDF	==> List DoubleFloat
  SDF	==> Stream DoubleFloat
  DF	==> DoubleFloat
  F	==> Float
  ST	==> String
  LST	==> List String
  SI	==> SingleInteger
  SOCDF	==> Segment OrderedCompletion DoubleFloat
  OCDF	==> OrderedCompletion DoubleFloat
  OCEDF	==> OrderedCompletion Expression DoubleFloat
  EOCEFI  ==> Equation OrderedCompletion Expression Fraction Integer
  OCEFI   ==> OrderedCompletion Expression Fraction Integer
  OCFI    ==> OrderedCompletion Fraction Integer
  NIA	==> Record(var:Symbol,fn:EDF,range:SOCDF,abserr:DF,relerr:DF)
  INT	==> Integer
  CTYPE	==> Union(continuous: "Continuous at the end points",
             lowerSingular: "There is a singularity at the lower end point",
              upperSingular: "There is a singularity at the upper end point",
               bothSingular: "There are singularities at both end points",
                notEvaluated: "End point continuity not yet evaluated")
  RTYPE	==> Union(finite: "The range is finite",
              lowerInfinite: "The bottom of range is infinite",
                upperInfinite: "The top of range is infinite",
                  bothInfinite: "Both top and bottom points are infinite",
                    notEvaluated: "Range not yet evaluated")
  STYPE	==> Union(str:SDF,
                   notEvaluated:"Internal singularities not yet evaluated")
  ATT	==> Record(endPointContinuity:CTYPE,
                    singularitiesStream:STYPE,range:RTYPE)
  ROA	==> Record(key:NIA,entry:ATT)

  E ==> with
    
    rangeIsFinite : NIA -> RTYPE
      ++ rangeIsFinite(args) tests the endpoints of \spad{args.range} for 
      ++ infinite end points. 
    functionIsContinuousAtEndPoints: NIA -> CTYPE
      ++ functionIsContinuousAtEndPoints(args) uses power series limits
      ++ to check for problems at the end points of the range of \spad{args}.
    getlo : SOCDF -> DF
      ++ getlo(x) gets the \axiomType{DoubleFloat} equivalent of
      ++ the first endpoint of the range \axiom{x}
    gethi : SOCDF -> DF
      ++ gethi(x) gets the \axiomType{DoubleFloat} equivalent of
      ++ the second endpoint of the range \axiom{x}
    functionIsOscillatory:NIA -> F
      ++ functionIsOscillatory(a) tests whether the function \spad{a.fn}
      ++ has many zeros of its derivative.
    problemPoints: (EDF, Symbol, SOCDF) -> List DF
      ++ problemPoints(f,var,range) returns a list of possible problem points
      ++ by looking at the zeros of the denominator of the function if it
      ++ can be retracted to \axiomType{Polynomial DoubleFloat}.
    singularitiesOf:NIA -> SDF
      ++ singularitiesOf(args) returns a list of potential 
      ++ singularities of the function within the given range
    df2st:DF -> String 
      ++ df2st(n) coerces a \axiomType{DoubleFloat} to \axiomType{String}
    ldf2lst:LDF -> LST
      ++ ldf2lst(ln) coerces a List of \axiomType{DoubleFloat} to \axiomType{List String}
    sdf2lst:SDF -> LST
      ++ sdf2lst(ln) coerces a Stream of \axiomType{DoubleFloat} to \axiomType{List String}
    commaSeparate:LST -> ST
      ++ commaSeparate(l) produces a comma separated string from a 
      ++ list of strings.
    changeName:(Symbol,Symbol,Result) -> Result
      ++ changeName(s,t,r) changes the name of item \axiom{s} in \axiom{r}
      ++ to \axiom{t}.

  I ==> ExpertSystemContinuityPackage add

    import ExpertSystemToolsPackage
    import ExpertSystemContinuityPackage

    -- local functions
    ocdf2ocefi : OCDF -> OCEFI
    rangeOfArgument : (KEDF, NIA) -> DF
    continuousAtPoint? : (EFI,EOCEFI) -> Boolean
    rand:(SOCDF,INT) -> LDF 
    eval:(EDF,Symbol,LDF) -> LDF
    numberOfSignChanges:LDF -> INT
    rangeIsFiniteFunction:NIA -> RTYPE
    functionIsContinuousAtEndPointsFunction:NIA -> CTYPE
 
    changeName(s:Symbol,t:Symbol,r:Result):Result ==
      a := remove!(s,r)$Result
      a case Any =>
        insert!([t,a],r)$Result
        r
      r

    commaSeparate(l:LST):ST ==
      empty?(l)$LST => ""
      one?(#(l)) => concat(l)$ST
      f := first(l)$LST
      t := [concat([", ",l.i])$ST for i in 2..#(l)]
      concat(f,concat(t)$ST)$ST

    rand(seg:SOCDF,n:INT):LDF ==
      -- produced a sorted list of random numbers in the given range
      l:DF := getlo seg
      s:DF := (gethi seg) - l
      seed:INT := random()$INT
      dseed:DF := seed :: DF
      r:LDF := [(((random(seed)$INT) :: DF)*s/dseed + l) for i in 1..n]
      sort(r)$LDF

    eval(f:EDF,var:Symbol,l:LDF):LDF ==
      empty?(l)$LDF => [0$DF]
      ve := var::EDF
      [retract(eval(f,equation(ve,u::EDF)$EEDF)$EDF)@DF for u in l]

    numberOfSignChanges(l:LDF):INT ==
      -- calculates the number of sign changes in a list
      a := 0$INT
      empty?(l)$LDF => 0
      for i in 2..# l repeat
        if negative?(l.i*l.(i-1))  then
          a := a + 1
      a

    rangeOfArgument(k: KEDF, args:NIA): DF ==
      Args := copy args
      Args.fn := arg := first(argument(k)$KEDF)$LEDF
      functionIsContinuousAtEndPoints(Args) case continuous =>
        r:SOCDF := args.range
        low:EDF := (getlo r) :: EDF
        high:EDF := (gethi r) :: EDF
        eql := equation(a := args.var :: EDF, low)$EEDF
        eqh := equation(a, high)$EEDF
        e1 := (numeric(eval(arg,eql)$EDF)$Numeric(DF)) :: DF
        e2 := (numeric(eval(arg,eqh)$EDF)$Numeric(DF)) :: DF
        e2-e1
      0$DF

    ocdf2ocefi(r:OCDF):OCEFI ==
      finite?(r)$OCDF => (edf2efi(((retract(r)@DF)$OCDF)::EDF))::OCEFI
      r pretend OCEFI

    continuousAtPoint?(f:EFI,e:EOCEFI):Boolean ==
      (l := limit(f,e)$PowerSeriesLimitPackage(FI,EFI)) case OCEFI =>
                       finite?(l :: OCEFI)
      -- if the left hand limit equals the right hand limit, or if neither
      -- side has a limit at this point, the return type of  limit() is
      -- Union(Ordered Completion Expression Fraction Integer,"failed")
      false

    -- exported functions
    
    rangeIsFiniteFunction(args:NIA): RTYPE ==
      -- rangeIsFinite(x) tests the endpoints of x.range for infinite
      -- end points. 
      --             [-inf,  inf]  =>  4
      --             [ x  ,  inf]  =>  3
      --             [-inf,  x  ]  =>  1
      --             [ x  ,  y  ]  =>  0
      fr:SI := (3::SI * whatInfinity(hi(args.range))$OCDF 
                      - whatInfinity(lo(args.range))$OCDF)
      fr = 0 => ["The range is finite"]
      fr = 1 => ["The bottom of range is infinite"]
      fr = 3 => ["The top of range is infinite"]
      fr = 4 => ["Both top and bottom points are infinite"]
      error("rangeIsFinite",["this is not a valid range"])$ErrorFunctions

    rangeIsFinite(args:NIA): RTYPE ==
      nia := copy args
      (t := showAttributes(nia)$IntegrationFunctionsTable) case ATT =>
        s := coerce(t)@ATT
        s.range case notEvaluated => 
          s.range := rangeIsFiniteFunction(nia)
          r:ROA := [nia,s]
          insert!(r)$IntegrationFunctionsTable
          s.range
        s.range
      a:ATT := [["End point continuity not yet evaluated"],
                  ["Internal singularities not yet evaluated"],
                      e:=rangeIsFiniteFunction(nia)]
      r:ROA := [nia,a]
      insert!(r)$IntegrationFunctionsTable
      e

    functionIsContinuousAtEndPointsFunction(args:NIA):CTYPE ==

      v := args.var :: EFI :: OCEFI
      high:OCEFI := ocdf2ocefi(hi(args.range))
      low:OCEFI := ocdf2ocefi(lo(args.range))
      f := edf2efi(args.fn)
      l:Boolean := continuousAtPoint?(f,equation(v,low)$EOCEFI)
      h:Boolean := continuousAtPoint?(f,equation(v,high)$EOCEFI)
      l and h => ["Continuous at the end points"]
      l => ["There is a singularity at the upper end point"]
      h => ["There is a singularity at the lower end point"]
      ["There are singularities at both end points"]

    functionIsContinuousAtEndPoints(args:NIA): CTYPE ==
      nia := copy args
      (t := showAttributes(nia)$IntegrationFunctionsTable) case ATT =>
        s := coerce(t)@ATT
        s.endPointContinuity case notEvaluated => 
          s.endPointContinuity := functionIsContinuousAtEndPointsFunction(nia)
          r:ROA := [nia,s]
          insert!(r)$IntegrationFunctionsTable
          s.endPointContinuity
        s.endPointContinuity
      a:ATT := [e:=functionIsContinuousAtEndPointsFunction(nia),
                 ["Internal singularities not yet evaluated"],
                   ["Range not yet evaluated"]]
      r:ROA := [nia,a]
      insert!(r)$IntegrationFunctionsTable
      e

    functionIsOscillatory(a:NIA):F ==

      args := copy a
      k := tower(numerator args.fn)$EDF
      p:F := pi()$F
      for i in 1..# k repeat
        is?(ker := k.i, sin :: Symbol) => 
          ra := convert(rangeOfArgument(ker,args))@F
          ra > 2*p => return (ra/p)
        is?(ker, cos :: Symbol) => 
          ra := convert(rangeOfArgument(ker,args))@F
          ra > 2*p => return (ra/p)
      l:LDF := rand(args.range,30)
      l := eval(args.fn,args.var,l)
      numberOfSignChanges(l) :: F   

    singularitiesOf(args:NIA):SDF ==
      nia := copy args
      (t := showAttributes(nia)$IntegrationFunctionsTable) case ATT =>
        s:ATT := coerce(t)@ATT
        p:STYPE := s.singularitiesStream
        p case str => p.str
        e:SDF := singularitiesOf(nia.fn,[nia.var],nia.range)
        if not empty?(e) then
          if less?(e,10)$SDF then extend(e,10)$SDF
        s.singularitiesStream := [e]
        r:ROA := [nia,s]
        insert!(r)$IntegrationFunctionsTable
        e
      e:=singularitiesOf(nia.fn,[nia.var],nia.range)
      if not empty?(e) then
        if less?(e,10)$SDF then extend(e,10)$SDF
      a:ATT := [["End point continuity not yet evaluated"],[e],
                          ["Range not yet evaluated"]]
      r:ROA := [nia,a]
      insert!(r)$IntegrationFunctionsTable
      e

@
\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>>

<<domain INTFTBL IntegrationFunctionsTable>>
<<package D01AGNT d01AgentsPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}