aboutsummaryrefslogtreecommitdiff
path: root/src/algebra/fs2ups.spad.pamphlet
blob: 5189d4d2edff42b92ded5cab581b2d8522a6e66e (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
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra fs2ups.spad}
\author{Clifton J. Williamson}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package FS2UPS FunctionSpaceToUnivariatePowerSeries}
<<package FS2UPS FunctionSpaceToUnivariatePowerSeries>>=
)abbrev package FS2UPS FunctionSpaceToUnivariatePowerSeries
++ Author: Clifton J. Williamson
++ Date Created: 21 March 1989
++ Date Last Updated: 2 December 1994
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords: elementary function, power series
++ Examples:
++ References:
++ Description:
++   This package converts expressions in some function space to power
++   series in a variable x with coefficients in that function space.
++   The function \spadfun{exprToUPS} converts expressions to power series
++   whose coefficients do not contain the variable x. The function
++   \spadfun{exprToGenUPS} converts functional expressions to power series
++   whose coefficients may involve functions of \spad{log(x)}.
FunctionSpaceToUnivariatePowerSeries(R,FE,Expon,UPS,TRAN,x):_
 Exports == Implementation where
  R     : Join(GcdDomain,RetractableTo Integer,_
               LinearlyExplicitRingOver Integer)
  FE    : Join(AlgebraicallyClosedField,TranscendentalFunctionCategory,_
               FunctionSpace R) with coerce: Expon -> %
  Expon : OrderedRing
  UPS   : Join(UnivariatePowerSeriesCategory(FE,Expon),Field,_
               TranscendentalFunctionCategory)
            with
              differentiate: % -> %
                ++ differentiate(x) returns the derivative of x since we 
                ++ need to be able to differentiate a power series
              integrate: % -> %
                ++ integrate(x) returns the integral of x since
                ++ we need to be able to integrate a power series
  TRAN  : PartialTranscendentalFunctions UPS
  x     : Symbol
  B       ==> Boolean
  BOP     ==> BasicOperator
  I       ==> Integer
  NNI     ==> NonNegativeInteger
  K       ==> Kernel FE
  L       ==> List
  RN      ==> Fraction Integer
  S       ==> String
  SY      ==> Symbol
  PCL     ==> PolynomialCategoryLifting(IndexedExponents K,K,R,SMP,FE)
  POL     ==> Polynomial R
  SMP     ==> SparseMultivariatePolynomial(R,K)
  SUP     ==> SparseUnivariatePolynomial Polynomial R
  Problem ==> Record(func:String,prob:String)
  Result  ==> Union(%series:UPS,%problem:Problem)
  SIGNEF  ==> ElementaryFunctionSign(R,FE)

  Exports ==> with
    exprToUPS : (FE,B,S) -> Result
      ++ exprToUPS(fcn,posCheck?,atanFlag) converts the expression
      ++ \spad{fcn} to a power series.  If \spad{posCheck?} is true,
      ++ log's of negative numbers are not allowed nor are nth roots of
      ++ negative numbers with n even.  If \spad{posCheck?} is false,
      ++ these are allowed.  \spad{atanFlag} determines how the case
      ++ \spad{atan(f(x))}, where \spad{f(x)} has a pole, will be treated.
      ++ The possible values of \spad{atanFlag} are \spad{"complex"},
      ++ \spad{"real: two sides"}, \spad{"real: left side"},
      ++ \spad{"real: right side"}, and \spad{"just do it"}.
      ++ If \spad{atanFlag} is \spad{"complex"}, then no series expansion
      ++ will be computed because, viewed as a function of a complex
      ++ variable, \spad{atan(f(x))} has an essential singularity.
      ++ Otherwise, the sign of the leading coefficient of the series
      ++ expansion of \spad{f(x)} determines the constant coefficient
      ++ in the series expansion of \spad{atan(f(x))}.  If this sign cannot
      ++ be determined, a series expansion is computed only when
      ++ \spad{atanFlag} is \spad{"just do it"}.  When the leading term
      ++ in the series expansion of \spad{f(x)} is of odd degree (or is a
      ++ rational degree with odd numerator), then the constant coefficient
      ++ in the series expansion of \spad{atan(f(x))} for values to the
      ++ left differs from that for values to the right.  If \spad{atanFlag}
      ++ is \spad{"real: two sides"}, no series expansion will be computed.
      ++ If \spad{atanFlag} is \spad{"real: left side"} the constant
      ++ coefficient for values to the left will be used and if \spad{atanFlag}
      ++ \spad{"real: right side"} the constant coefficient for values to the
      ++ right will be used.
      ++ If there is a problem in converting the function to a power series,
      ++ a record containing the name of the function that caused the problem
      ++ and a brief description of the problem is returned.
      ++ When expanding the expression into a series it is assumed that
      ++ the series is centered at 0.  For a series centered at a, the
      ++ user should perform the substitution \spad{x -> x + a} before calling
      ++ this function.

    exprToGenUPS : (FE,B,S) -> Result
      ++ exprToGenUPS(fcn,posCheck?,atanFlag) converts the expression
      ++ \spad{fcn} to a generalized power series.  If \spad{posCheck?}
      ++ is true, log's of negative numbers are not allowed nor are nth roots
      ++ of negative numbers with n even. If \spad{posCheck?} is false,
      ++ these are allowed.  \spad{atanFlag} determines how the case
      ++ \spad{atan(f(x))}, where \spad{f(x)} has a pole, will be treated.
      ++ The possible values of \spad{atanFlag} are \spad{"complex"},
      ++ \spad{"real: two sides"}, \spad{"real: left side"},
      ++ \spad{"real: right side"}, and \spad{"just do it"}.
      ++ If \spad{atanFlag} is \spad{"complex"}, then no series expansion
      ++ will be computed because, viewed as a function of a complex
      ++ variable, \spad{atan(f(x))} has an essential singularity.
      ++ Otherwise, the sign of the leading coefficient of the series
      ++ expansion of \spad{f(x)} determines the constant coefficient
      ++ in the series expansion of \spad{atan(f(x))}.  If this sign cannot
      ++ be determined, a series expansion is computed only when
      ++ \spad{atanFlag} is \spad{"just do it"}.  When the leading term
      ++ in the series expansion of \spad{f(x)} is of odd degree (or is a
      ++ rational degree with odd numerator), then the constant coefficient
      ++ in the series expansion of \spad{atan(f(x))} for values to the
      ++ left differs from that for values to the right.  If \spad{atanFlag}
      ++ is \spad{"real: two sides"}, no series expansion will be computed.
      ++ If \spad{atanFlag} is \spad{"real: left side"} the constant
      ++ coefficient for values to the left will be used and if \spad{atanFlag}
      ++ \spad{"real: right side"} the constant coefficient for values to the
      ++ right will be used.
      ++ If there is a problem in converting the function to a power
      ++ series, we return a record containing the name of the function
      ++ that caused the problem and a brief description of the problem.
      ++ When expanding the expression into a series it is assumed that
      ++ the series is centered at 0.  For a series centered at a, the
      ++ user should perform the substitution \spad{x -> x + a} before calling
      ++ this function.
    localAbs: FE -> FE
      ++ localAbs(fcn) = \spad{abs(fcn)} or \spad{sqrt(fcn**2)} depending
      ++ on whether or not FE has a function \spad{abs}.  This should be
      ++ a local function, but the compiler won't allow it.

  Implementation ==> add

    ratIfCan            : FE -> Union(RN,"failed")
    carefulNthRootIfCan : (UPS,NNI,B,B) -> Result
    stateProblem        : (S,S) -> Result
    polyToUPS           : SUP -> UPS
    listToUPS           : (L FE,(FE,B,S) -> Result,B,S,UPS,(UPS,UPS) -> UPS)_
                                            -> Result
    isNonTrivPower      : FE -> Union(Record(val:FE,exponent:I),"failed")
    powerToUPS          : (FE,I,B,S) -> Result
    kernelToUPS         : (K,B,S) -> Result
    nthRootToUPS        : (FE,NNI,B,S) -> Result
    logToUPS            : (FE,B,S) -> Result
    atancotToUPS        : (FE,B,S,I) -> Result
    applyIfCan          : (UPS -> Union(UPS,"failed"),FE,S,B,S) -> Result
    tranToUPS           : (K,FE,B,S) -> Result
    powToUPS            : (L FE,B,S) -> Result
    newElem             : FE -> FE
    smpElem             : SMP -> FE
    k2Elem              : K -> FE
    contOnReals?        : S -> B
    bddOnReals?         : S -> B
    iExprToGenUPS       : (FE,B,S) -> Result
    opsInvolvingX       : FE -> L BOP
    opInOpList?         : (SY,L BOP) -> B
    exponential?        : FE -> B
    productOfNonZeroes? : FE -> B
    powerToGenUPS       : (FE,I,B,S) -> Result
    kernelToGenUPS      : (K,B,S) -> Result
    nthRootToGenUPS     : (FE,NNI,B,S) -> Result
    logToGenUPS         : (FE,B,S) -> Result
    expToGenUPS         : (FE,B,S) -> Result
    expGenUPS           : (UPS,B,S) -> Result
    atancotToGenUPS     : (FE,FE,B,S,I) -> Result
    genUPSApplyIfCan    : (UPS -> Union(UPS,"failed"),FE,S,B,S) -> Result
    applyBddIfCan       : (FE,UPS -> Union(UPS,"failed"),FE,S,B,S) -> Result
    tranToGenUPS        : (K,FE,B,S) -> Result
    powToGenUPS         : (L FE,B,S) -> Result

    ZEROCOUNT : I := 1000
    -- number of zeroes to be removed when taking logs or nth roots

    ratIfCan fcn == retractIfCan(fcn)@Union(RN,"failed")

    carefulNthRootIfCan(ups,n,posCheck?,rightOnly?) ==
      -- similar to 'nthRootIfCan', but it is fussy about the series
      -- it takes as an argument.  If 'n' is EVEN and 'posCheck?'
      -- is truem then the leading coefficient of the series must
      -- be POSITIVE.  In this case, if 'rightOnly?' is false, the
      -- order of the series must be zero.  The idea is that the
      -- series represents a real function of a real variable, and
      -- we want a unique real nth root defined on a neighborhood
      -- of zero.
      n < 1 => error "nthRoot: n must be positive"
      deg := degree ups
      if (coef := coefficient(ups,deg)) = 0 then
        deg := order(ups,deg + ZEROCOUNT :: Expon)
        (coef := coefficient(ups,deg)) = 0 =>
          error "log of series with many leading zero coefficients"
      -- if 'posCheck?' is true, we do not allow nth roots of negative
      -- numbers when n in even
      if even?(n :: I) then
        if posCheck? and ((signum := sign(coef)$SIGNEF) case I) then
          (signum :: I) = -1 =>
            return stateProblem("nth root","negative leading coefficient")
          not rightOnly? and not zero? deg => -- nth root not unique
            return stateProblem("nth root","series of non-zero order")
      (ans := nthRootIfCan(ups,n)) case "failed" =>
        stateProblem("nth root","no nth root")
      [ans :: UPS]

    stateProblem(function,problem) ==
      -- records the problem which occured in converting an expression
      -- to a power series
      [[function,problem]]

    exprToUPS(fcn,posCheck?,atanFlag) ==
      -- converts a functional expression to a power series
      --!! The following line is commented out so that expressions of
      --!! the form a**b will be normalized to exp(b * log(a)) even if
      --!! 'a' and 'b' do not involve the limiting variable 'x'.
      --!!                         - cjw 1 Dec 94
      --not member?(x,variables fcn) => [monomial(fcn,0)]
      (poly := retractIfCan(fcn)@Union(POL,"failed")) case POL =>
        [polyToUPS univariate(poly :: POL,x)]
      (sum := isPlus fcn) case L(FE) =>
        listToUPS(sum :: L(FE),exprToUPS,posCheck?,atanFlag,0,#1 + #2)
      (prod := isTimes fcn) case L(FE) =>
        listToUPS(prod :: L(FE),exprToUPS,posCheck?,atanFlag,1,#1 * #2)
      (expt := isNonTrivPower fcn) case Record(val:FE,exponent:I) =>
        power := expt :: Record(val:FE,exponent:I)
        powerToUPS(power.val,power.exponent,posCheck?,atanFlag)
      (ker := retractIfCan(fcn)@Union(K,"failed")) case K =>
        kernelToUPS(ker :: K,posCheck?,atanFlag)
      error "exprToUPS: neither a sum, product, power, nor kernel"

    polyToUPS poly ==
      -- converts a polynomial to a power series
      zero? poly => 0
      -- we don't start with 'ans := 0' as this may lead to an
      -- enormous number of leading zeroes in the power series
      deg  := degree poly
      coef := leadingCoefficient(poly) :: FE
      ans  := monomial(coef,deg :: Expon)$UPS
      poly := reductum poly
      while not zero? poly repeat
        deg  := degree poly
        coef := leadingCoefficient(poly) :: FE
        ans  := ans + monomial(coef,deg :: Expon)$UPS
        poly := reductum poly
      ans

    listToUPS(list,feToUPS,posCheck?,atanFlag,ans,op) ==
      -- converts each element of a list of expressions to a power
      -- series and returns the sum of these series, when 'op' is +
      -- and 'ans' is 0, or the product of these series, when 'op' is *
      -- and 'ans' is 1
      while not null list repeat
        (term := feToUPS(first list,posCheck?,atanFlag)) case %problem =>
          return term
        ans := op(ans,term.%series)
        list := rest list
      [ans]

    isNonTrivPower fcn ==
      -- is the function a power with exponent other than 0 or 1?
      (expt := isPower fcn) case "failed" => "failed"
      power := expt :: Record(val:FE,exponent:I)
      one? power.exponent => "failed"
      power

    powerToUPS(fcn,n,posCheck?,atanFlag) ==
      -- converts an integral power to a power series
      (b := exprToUPS(fcn,posCheck?,atanFlag)) case %problem => b
      n > 0 => [(b.%series) ** n]
      -- check lowest order coefficient when n < 0
      ups := b.%series; deg := degree ups
      if (coef := coefficient(ups,deg)) = 0 then
        deg := order(ups,deg + ZEROCOUNT :: Expon)
        (coef := coefficient(ups,deg)) = 0 =>
          error "inverse of series with many leading zero coefficients"
      [ups ** n]

    kernelToUPS(ker,posCheck?,atanFlag) ==
      -- converts a kernel to a power series
      (sym := symbolIfCan(ker)) case Symbol =>
        (sym :: Symbol) = x => [monomial(1,1)]
        [monomial(ker :: FE,0)]
      empty?(args := argument ker) => [monomial(ker :: FE,0)]
      not member?(x, variables(ker :: FE)) => [monomial(ker :: FE,0)]
      empty? rest args =>
        arg := first args
        is?(ker,"abs" :: Symbol) =>
          nthRootToUPS(arg*arg,2,posCheck?,atanFlag)
        is?(ker,"%paren" :: Symbol) => exprToUPS(arg,posCheck?,atanFlag)
        is?(ker,"log" :: Symbol) => logToUPS(arg,posCheck?,atanFlag)
        is?(ker,"exp" :: Symbol) =>
          applyIfCan(expIfCan,arg,"exp",posCheck?,atanFlag)
        tranToUPS(ker,arg,posCheck?,atanFlag)
      is?(ker,"%power" :: Symbol) => powToUPS(args,posCheck?,atanFlag)
      is?(ker,"nthRoot" :: Symbol) =>
        n := retract(second args)@I
        nthRootToUPS(first args,n :: NNI,posCheck?,atanFlag)
      stateProblem(string name operator ker,"unknown kernel")

    nthRootToUPS(arg,n,posCheck?,atanFlag) ==
      -- converts an nth root to a power series
      -- this is not used in the limit package, so the series may
      -- have non-zero order, in which case nth roots may not be unique
      (result := exprToUPS(arg,posCheck?,atanFlag)) case %problem => result
      ans := carefulNthRootIfCan(result.%series,n,posCheck?,false)
      ans case %problem => ans
      [ans.%series]

    logToUPS(arg,posCheck?,atanFlag) ==
      -- converts a logarithm log(f(x)) to a power series
      -- f(x) must have order 0 and if 'posCheck?' is true,
      -- then f(x) must have a non-negative leading coefficient
      (result := exprToUPS(arg,posCheck?,atanFlag)) case %problem => result
      ups := result.%series
      not zero? order(ups,1) =>
        stateProblem("log","series of non-zero order")
      coef := coefficient(ups,0)
      -- if 'posCheck?' is true, we do not allow logs of negative numbers
      if posCheck? then
        if ((signum := sign(coef)$SIGNEF) case I) then
          (signum :: I) = -1 =>
            return stateProblem("log","negative leading coefficient")
      [logIfCan(ups) :: UPS]

    if FE has abs: FE -> FE then
      localAbs fcn == abs fcn
    else
      localAbs fcn == sqrt(fcn * fcn)

    signOfExpression: FE -> FE
    signOfExpression arg == localAbs(arg)/arg

    atancotToUPS(arg,posCheck?,atanFlag,plusMinus) ==
      -- converts atan(f(x)) to a power series
      (result := exprToUPS(arg,posCheck?,atanFlag)) case %problem => result
      ups := result.%series; coef := coefficient(ups,0)
      (ord := order(ups,0)) = 0 and coef * coef = -1 =>
        -- series involves complex numbers
        return stateProblem("atan","logarithmic singularity")
      cc : FE :=
        ord < 0 =>
          atanFlag = "complex" =>
            return stateProblem("atan","essential singularity")
          (rn := ratIfCan(ord :: FE)) case "failed" =>
            -- this condition usually won't occur because exponents will
            -- be integers or rational numbers
            return stateProblem("atan","branch problem")
          if (atanFlag = "real: two sides") and (odd? numer(rn :: RN)) then
            -- expansions to the left and right of zero have different
            -- constant coefficients
            return stateProblem("atan","branch problem")
          lc := coefficient(ups,ord)
          (signum := sign(lc)$SIGNEF) case "failed" =>
            -- can't determine sign
            atanFlag = "just do it" =>
              plusMinus = 1 => pi()/(2 :: FE)
              0
            posNegPi2 := signOfExpression(lc) * pi()/(2 :: FE)
            plusMinus = 1 => posNegPi2
            pi()/(2 :: FE) - posNegPi2
            --return stateProblem("atan","branch problem")
          left? : B := atanFlag = "real: left side"; n := signum :: Integer
          (left? and n = 1) or (not left? and n = -1) =>
            plusMinus = 1 => -pi()/(2 :: FE)
            pi()
          plusMinus = 1 => pi()/(2 :: FE)
          0
        atan coef
      [(cc :: UPS) + integrate(plusMinus * differentiate(ups)/(1 + ups*ups))]

    applyIfCan(fcn,arg,fcnName,posCheck?,atanFlag) ==
      -- converts fcn(arg) to a power series
      (ups := exprToUPS(arg,posCheck?,atanFlag)) case %problem => ups
      ans := fcn(ups.%series)
      ans case "failed" => stateProblem(fcnName,"essential singularity")
      [ans :: UPS]

    tranToUPS(ker,arg,posCheck?,atanFlag) ==
      -- converts ker to a power series for certain functions
      -- in trig or hyperbolic trig categories
      is?(ker,"sin" :: SY) =>
        applyIfCan(sinIfCan,arg,"sin",posCheck?,atanFlag)
      is?(ker,"cos" :: SY) =>
        applyIfCan(cosIfCan,arg,"cos",posCheck?,atanFlag)
      is?(ker,"tan" :: SY) =>
        applyIfCan(tanIfCan,arg,"tan",posCheck?,atanFlag)
      is?(ker,"cot" :: SY) =>
        applyIfCan(cotIfCan,arg,"cot",posCheck?,atanFlag)
      is?(ker,"sec" :: SY) =>
        applyIfCan(secIfCan,arg,"sec",posCheck?,atanFlag)
      is?(ker,"csc" :: SY) =>
        applyIfCan(cscIfCan,arg,"csc",posCheck?,atanFlag)
      is?(ker,"asin" :: SY) =>
        applyIfCan(asinIfCan,arg,"asin",posCheck?,atanFlag)
      is?(ker,"acos" :: SY) =>
        applyIfCan(acosIfCan,arg,"acos",posCheck?,atanFlag)
      is?(ker,"atan" :: SY) => atancotToUPS(arg,posCheck?,atanFlag,1)
      is?(ker,"acot" :: SY) => atancotToUPS(arg,posCheck?,atanFlag,-1)
      is?(ker,"asec" :: SY) =>
        applyIfCan(asecIfCan,arg,"asec",posCheck?,atanFlag)
      is?(ker,"acsc" :: SY) =>
        applyIfCan(acscIfCan,arg,"acsc",posCheck?,atanFlag)
      is?(ker,"sinh" :: SY) =>
        applyIfCan(sinhIfCan,arg,"sinh",posCheck?,atanFlag)
      is?(ker,"cosh" :: SY) =>
        applyIfCan(coshIfCan,arg,"cosh",posCheck?,atanFlag)
      is?(ker,"tanh" :: SY) =>
        applyIfCan(tanhIfCan,arg,"tanh",posCheck?,atanFlag)
      is?(ker,"coth" :: SY) =>
        applyIfCan(cothIfCan,arg,"coth",posCheck?,atanFlag)
      is?(ker,"sech" :: SY) =>
        applyIfCan(sechIfCan,arg,"sech",posCheck?,atanFlag)
      is?(ker,"csch" :: SY) =>
        applyIfCan(cschIfCan,arg,"csch",posCheck?,atanFlag)
      is?(ker,"asinh" :: SY) =>
        applyIfCan(asinhIfCan,arg,"asinh",posCheck?,atanFlag)
      is?(ker,"acosh" :: SY) =>
        applyIfCan(acoshIfCan,arg,"acosh",posCheck?,atanFlag)
      is?(ker,"atanh" :: SY) =>
        applyIfCan(atanhIfCan,arg,"atanh",posCheck?,atanFlag)
      is?(ker,"acoth" :: SY) =>
        applyIfCan(acothIfCan,arg,"acoth",posCheck?,atanFlag)
      is?(ker,"asech" :: SY) =>
        applyIfCan(asechIfCan,arg,"asech",posCheck?,atanFlag)
      is?(ker,"acsch" :: SY) =>
        applyIfCan(acschIfCan,arg,"acsch",posCheck?,atanFlag)
      stateProblem(string name operator ker,"unknown kernel")

    powToUPS(args,posCheck?,atanFlag) ==
      -- converts a power f(x) ** g(x) to a power series
      (logBase := logToUPS(first args,posCheck?,atanFlag)) case %problem =>
        logBase
      (expon := exprToUPS(second args,posCheck?,atanFlag)) case %problem =>
        expon
      ans := expIfCan((expon.%series) * (logBase.%series))
      ans case "failed" => stateProblem("exp","essential singularity")
      [ans :: UPS]

-- Generalized power series: power series in x, where log(x) and
-- bounded functions of x are allowed to appear in the coefficients
-- of the series.  Used for evaluating REAL limits at x = 0.

    newElem f ==
    -- rewrites a functional expression; all trig functions are
    -- expressed in terms of sin and cos; all hyperbolic trig
    -- functions are expressed in terms of exp
      smpElem(numer f) / smpElem(denom f)

    smpElem p == map(k2Elem,#1::FE,p)$PCL

    k2Elem k ==
    -- rewrites a kernel; all trig functions are
    -- expressed in terms of sin and cos; all hyperbolic trig
    -- functions are expressed in terms of exp
      null(args := [newElem a for a in argument k]) => k::FE
      iez  := inv(ez  := exp(z := first args))
      sinz := sin z; cosz := cos z
      is?(k,"tan" :: Symbol)  => sinz / cosz
      is?(k,"cot" :: Symbol)  => cosz / sinz
      is?(k,"sec" :: Symbol)  => inv cosz
      is?(k,"csc" :: Symbol)  => inv sinz
      is?(k,"sinh" :: Symbol) => (ez - iez) / (2 :: FE)
      is?(k,"cosh" :: Symbol) => (ez + iez) / (2 :: FE)
      is?(k,"tanh" :: Symbol) => (ez - iez) / (ez + iez)
      is?(k,"coth" :: Symbol) => (ez + iez) / (ez - iez)
      is?(k,"sech" :: Symbol) => 2 * inv(ez + iez)
      is?(k,"csch" :: Symbol) => 2 * inv(ez - iez)
      (operator k) args

    CONTFCNS : L S := ["sin","cos","atan","acot","exp","asinh"]
    -- functions which are defined and continuous at all real numbers

    BDDFCNS : L S := ["sin","cos","atan","acot"]
    -- functions which are bounded on the reals

    contOnReals? fcn == member?(fcn,CONTFCNS)
    bddOnReals? fcn  == member?(fcn,BDDFCNS)

    exprToGenUPS(fcn,posCheck?,atanFlag) ==
      -- converts a functional expression to a generalized power
      -- series; "generalized" means that log(x) and bounded functions
      -- of x are allowed to appear in the coefficients of the series
      iExprToGenUPS(newElem fcn,posCheck?,atanFlag)

    iExprToGenUPS(fcn,posCheck?,atanFlag) ==
      -- converts a functional expression to a generalized power
      -- series without first normalizing the expression
      --!! The following line is commented out so that expressions of
      --!! the form a**b will be normalized to exp(b * log(a)) even if
      --!! 'a' and 'b' do not involve the limiting variable 'x'.
      --!!                         - cjw 1 Dec 94
      --not member?(x,variables fcn) => [monomial(fcn,0)]
      (poly := retractIfCan(fcn)@Union(POL,"failed")) case POL =>
        [polyToUPS univariate(poly :: POL,x)]
      (sum := isPlus fcn) case L(FE) =>
        listToUPS(sum :: L(FE),iExprToGenUPS,posCheck?,atanFlag,0,#1 + #2)
      (prod := isTimes fcn) case L(FE) =>
        listToUPS(prod :: L(FE),iExprToGenUPS,posCheck?,atanFlag,1,#1 * #2)
      (expt := isNonTrivPower fcn) case Record(val:FE,exponent:I) =>
        power := expt :: Record(val:FE,exponent:I)
        powerToGenUPS(power.val,power.exponent,posCheck?,atanFlag)
      (ker := retractIfCan(fcn)@Union(K,"failed")) case K =>
        kernelToGenUPS(ker :: K,posCheck?,atanFlag)
      error "exprToGenUPS: neither a sum, product, power, nor kernel"

    opsInvolvingX fcn ==
      opList := [op for k in tower fcn | unary?(op := operator k) _
                 and member?(x,variables first argument k)]
      removeDuplicates opList

    opInOpList?(name,opList) ==
      for op in opList repeat
        is?(op,name) => return true
      false

    exponential? fcn ==
      -- is 'fcn' of the form exp(f)?
      (ker := retractIfCan(fcn)@Union(K,"failed")) case K =>
        is?(ker :: K,"exp" :: Symbol)
      false

    productOfNonZeroes? fcn ==
      -- is 'fcn' a product of non-zero terms, where 'non-zero'
      -- means an exponential or a function not involving x
      exponential? fcn => true
      (prod := isTimes fcn) case "failed" => false
      for term in (prod :: L(FE)) repeat
        (not exponential? term) and member?(x,variables term) =>
          return false
      true

    powerToGenUPS(fcn,n,posCheck?,atanFlag) ==
      -- converts an integral power to a generalized power series
      -- if n < 0 and the lowest order coefficient of the series
      -- involves x, we are careful about inverting this coefficient
      -- the coefficient is inverted only if
      -- (a) the only function involving x is 'log', or
      -- (b) the lowest order coefficient is a product of exponentials
      --     and functions not involving x
      (b := exprToGenUPS(fcn,posCheck?,atanFlag)) case %problem => b
      n > 0 => [(b.%series) ** n]
      -- check lowest order coefficient when n < 0
      ups := b.%series; deg := degree ups
      if (coef := coefficient(ups,deg)) = 0 then
        deg := order(ups,deg + ZEROCOUNT :: Expon)
        (coef := coefficient(ups,deg)) = 0 =>
          error "inverse of series with many leading zero coefficients"
      xOpList := opsInvolvingX coef
      -- only function involving x is 'log'
      (null xOpList) => [ups ** n]
      (null rest xOpList and is?(first xOpList,"log" :: SY)) =>
        [ups ** n]
      -- lowest order coefficient is a product of exponentials and
      -- functions not involving x
      productOfNonZeroes? coef => [ups ** n]
      stateProblem("inv","lowest order coefficient involves x")

    kernelToGenUPS(ker,posCheck?,atanFlag) ==
      -- converts a kernel to a generalized power series
      (sym := symbolIfCan(ker)) case Symbol =>
        (sym :: Symbol) = x => [monomial(1,1)]
        [monomial(ker :: FE,0)]
      empty?(args := argument ker) => [monomial(ker :: FE,0)]
      empty? rest args =>
        arg := first args
        is?(ker,"abs" :: Symbol) =>
          nthRootToGenUPS(arg*arg,2,posCheck?,atanFlag)
        is?(ker,"%paren" :: Symbol) => iExprToGenUPS(arg,posCheck?,atanFlag)
        is?(ker,"log" :: Symbol) => logToGenUPS(arg,posCheck?,atanFlag)
        is?(ker,"exp" :: Symbol) => expToGenUPS(arg,posCheck?,atanFlag)
        tranToGenUPS(ker,arg,posCheck?,atanFlag)
      is?(ker,"%power" :: Symbol) => powToGenUPS(args,posCheck?,atanFlag)
      is?(ker,"nthRoot" :: Symbol) =>
        n := retract(second args)@I
        nthRootToGenUPS(first args,n :: NNI,posCheck?,atanFlag)
      stateProblem(string name operator ker,"unknown kernel")

    nthRootToGenUPS(arg,n,posCheck?,atanFlag) ==
      -- convert an nth root to a power series
      -- used for computing right hand limits, so the series may have
      -- non-zero order, but may not have a negative leading coefficient
      -- when n is even
      (result := iExprToGenUPS(arg,posCheck?,atanFlag)) case %problem =>
        result
      ans := carefulNthRootIfCan(result.%series,n,posCheck?,true)
      ans case %problem => ans
      [ans.%series]

    logToGenUPS(arg,posCheck?,atanFlag) ==
      -- converts a logarithm log(f(x)) to a generalized power series
      (result := iExprToGenUPS(arg,posCheck?,atanFlag)) case %problem =>
        result
      ups := result.%series; deg := degree ups
      if (coef := coefficient(ups,deg)) = 0 then
        deg := order(ups,deg + ZEROCOUNT :: Expon)
        (coef := coefficient(ups,deg)) = 0 =>
          error "log of series with many leading zero coefficients"
      -- if 'posCheck?' is true, we do not allow logs of negative numbers
      if posCheck? then
        if ((signum := sign(coef)$SIGNEF) case I) then
          (signum :: I) = -1 =>
            return stateProblem("log","negative leading coefficient")
      -- create logarithmic term, avoiding log's of negative rationals
      lt := monomial(coef,deg)$UPS; cen := center lt
      -- check to see if lowest order coefficient is a negative rational
      negRat? : Boolean :=
        ((rat := ratIfCan coef) case RN) =>
          (rat :: RN) < 0 => true
          false
        false
      logTerm : FE :=
        mon : FE := (x :: FE) - (cen :: FE)
        pow : FE := mon ** (deg :: FE)
        negRat? => log(coef * pow)
        term1 : FE := (deg :: FE) * log(mon)
        log(coef) + term1
      [monomial(logTerm,0) + log(ups/lt)]

    expToGenUPS(arg,posCheck?,atanFlag) ==
      -- converts an exponential exp(f(x)) to a generalized
      -- power series
      (ups := iExprToGenUPS(arg,posCheck?,atanFlag)) case %problem => ups
      expGenUPS(ups.%series,posCheck?,atanFlag)

    expGenUPS(ups,posCheck?,atanFlag) ==
      -- computes the exponential of a generalized power series.
      -- If the series has order zero and the constant term a0 of the
      -- series involves x, the function tries to expand exp(a0) as
      -- a power series.
      (deg := order(ups,1)) < 0 =>
        stateProblem("exp","essential singularity")
      deg > 0 => [exp ups]
      lc := coefficient(ups,0); xOpList := opsInvolvingX lc
      not opInOpList?("log" :: SY,xOpList) => [exp ups]
      -- try to fix exp(lc) if necessary
      expCoef :=
        normalize(exp lc,x)$ElementaryFunctionStructurePackage(R,FE)
      opInOpList?("log" :: SY,opsInvolvingX expCoef) =>
        stateProblem("exp","logs in constant coefficient")
      result := exprToGenUPS(expCoef,posCheck?,atanFlag)
      result case %problem => result
      [(result.%series) * exp(ups - monomial(lc,0))]

    atancotToGenUPS(fe,arg,posCheck?,atanFlag,plusMinus) ==
      -- converts atan(f(x)) to a generalized power series
      (result := exprToGenUPS(arg,posCheck?,atanFlag)) case %problem =>
        trouble := result.%problem
        trouble.prob = "essential singularity" => [monomial(fe,0)$UPS]
        result
      ups := result.%series; coef := coefficient(ups,0)
      -- series involves complex numbers
      (ord := order(ups,0)) = 0 and coef * coef = -1 =>
        y := differentiate(ups)/(1 + ups*ups)
        yCoef := coefficient(y,-1)
        [monomial(log yCoef,0) + integrate(y - monomial(yCoef,-1)$UPS)]
      cc : FE :=
        ord < 0 =>
          atanFlag = "complex" =>
            return stateProblem("atan","essential singularity")
          (rn := ratIfCan(ord :: FE)) case "failed" =>
            -- this condition usually won't occur because exponents will
            -- be integers or rational numbers
            return stateProblem("atan","branch problem")
          if (atanFlag = "real: two sides") and (odd? numer(rn :: RN)) then
            -- expansions to the left and right of zero have different
            -- constant coefficients
            return stateProblem("atan","branch problem")
          lc := coefficient(ups,ord)
          (signum := sign(lc)$SIGNEF) case "failed" =>
            -- can't determine sign
            atanFlag = "just do it" =>
              plusMinus = 1 => pi()/(2 :: FE)
              0
            posNegPi2 := signOfExpression(lc) * pi()/(2 :: FE)
            plusMinus = 1 => posNegPi2
            pi()/(2 :: FE) - posNegPi2
            --return stateProblem("atan","branch problem")
          left? : B := atanFlag = "real: left side"; n := signum :: Integer
          (left? and n = 1) or (not left? and n = -1) =>
            plusMinus = 1 => -pi()/(2 :: FE)
            pi()
          plusMinus = 1 => pi()/(2 :: FE)
          0
        atan coef
      [(cc :: UPS) + integrate(differentiate(ups)/(1 + ups*ups))]

    genUPSApplyIfCan(fcn,arg,fcnName,posCheck?,atanFlag) ==
      -- converts fcn(arg) to a generalized power series
      (series := iExprToGenUPS(arg,posCheck?,atanFlag)) case %problem =>
        series
      ups := series.%series
      (deg := order(ups,1)) < 0 =>
        stateProblem(fcnName,"essential singularity")
      deg > 0 => [fcn(ups) :: UPS]
      lc := coefficient(ups,0); xOpList := opsInvolvingX lc
      null xOpList => [fcn(ups) :: UPS]
      opInOpList?("log" :: SY,xOpList) =>
        stateProblem(fcnName,"logs in constant coefficient")
      contOnReals? fcnName => [fcn(ups) :: UPS]
      stateProblem(fcnName,"x in constant coefficient")

    applyBddIfCan(fe,fcn,arg,fcnName,posCheck?,atanFlag) ==
      -- converts fcn(arg) to a generalized power series, where the
      -- function fcn is bounded for real values
      -- if fcn(arg) has an essential singularity as a complex
      -- function, we return fcn(arg) as a monomial of degree 0
      (ups := iExprToGenUPS(arg,posCheck?,atanFlag)) case %problem =>
        trouble := ups.%problem
        trouble.prob = "essential singularity" => [monomial(fe,0)$UPS]
        ups
      (ans := fcn(ups.%series)) case "failed" => [monomial(fe,0)$UPS]
      [ans :: UPS]

    tranToGenUPS(ker,arg,posCheck?,atanFlag) ==
      -- converts op(arg) to a power series for certain functions
      -- op in trig or hyperbolic trig categories
      -- N.B. when this function is called, 'k2elem' will have been
      -- applied, so the following functions cannot appear:
      -- tan, cot, sec, csc, sinh, cosh, tanh, coth, sech, csch
      is?(ker,"sin" :: SY) =>
        applyBddIfCan(ker :: FE,sinIfCan,arg,"sin",posCheck?,atanFlag)
      is?(ker,"cos" :: SY) =>
        applyBddIfCan(ker :: FE,cosIfCan,arg,"cos",posCheck?,atanFlag)
      is?(ker,"asin" :: SY) =>
        genUPSApplyIfCan(asinIfCan,arg,"asin",posCheck?,atanFlag)
      is?(ker,"acos" :: SY) =>
        genUPSApplyIfCan(acosIfCan,arg,"acos",posCheck?,atanFlag)
      is?(ker,"atan" :: SY) =>
        atancotToGenUPS(ker :: FE,arg,posCheck?,atanFlag,1)
      is?(ker,"acot" :: SY) =>
        atancotToGenUPS(ker :: FE,arg,posCheck?,atanFlag,-1)
      is?(ker,"asec" :: SY) =>
        genUPSApplyIfCan(asecIfCan,arg,"asec",posCheck?,atanFlag)
      is?(ker,"acsc" :: SY) =>
        genUPSApplyIfCan(acscIfCan,arg,"acsc",posCheck?,atanFlag)
      is?(ker,"asinh" :: SY) =>
        genUPSApplyIfCan(asinhIfCan,arg,"asinh",posCheck?,atanFlag)
      is?(ker,"acosh" :: SY) =>
        genUPSApplyIfCan(acoshIfCan,arg,"acosh",posCheck?,atanFlag)
      is?(ker,"atanh" :: SY) =>
        genUPSApplyIfCan(atanhIfCan,arg,"atanh",posCheck?,atanFlag)
      is?(ker,"acoth" :: SY) =>
        genUPSApplyIfCan(acothIfCan,arg,"acoth",posCheck?,atanFlag)
      is?(ker,"asech" :: SY) =>
        genUPSApplyIfCan(asechIfCan,arg,"asech",posCheck?,atanFlag)
      is?(ker,"acsch" :: SY) =>
        genUPSApplyIfCan(acschIfCan,arg,"acsch",posCheck?,atanFlag)
      stateProblem(string name operator ker,"unknown kernel")

    powToGenUPS(args,posCheck?,atanFlag) ==
      -- converts a power f(x) ** g(x) to a generalized power series
      (logBase := logToGenUPS(first args,posCheck?,atanFlag)) case %problem =>
        logBase
      expon := iExprToGenUPS(second args,posCheck?,atanFlag)
      expon case %problem => expon
      expGenUPS((expon.%series) * (logBase.%series),posCheck?,atanFlag)

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--Copyright (C) 2007-2009, Gabriel Dos Reis.
--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>>

<<package FS2UPS FunctionSpaceToUnivariatePowerSeries>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}