aboutsummaryrefslogtreecommitdiff
path: root/src/algebra/opalg.spad.pamphlet
blob: cad0a61c72e5d18e3d83831f40e81f31eb74481e (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
\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra opalg.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain MODOP ModuleOperator}
<<domain MODOP ModuleOperator>>=
)abbrev domain MODOP ModuleOperator
++ Author: Manuel Bronstein
++ Date Created: 15 May 1990
++ Date Last Updated: 17 June 1993
++ Description:
++ Algebra of ADDITIVE operators on a module.
ModuleOperator(R: Ring, M:LeftModule(R)): Exports == Implementation where
  O    ==> OutputForm
  OP   ==> BasicOperator
  FG   ==> FreeGroup OP
  RM   ==> Record(coef:R, monom:FG)
  TERM ==> List RM
  FAB  ==> FreeAbelianGroup TERM
  OPADJ   ==> "%opAdjoint"
  OPEVAL  ==> "%opEval"
  INVEVAL ==> "%invEval"

  Exports ==> Join(Ring, RetractableTo R, RetractableTo OP,
                   Eltable(M, M)) with
    if R has CharacteristicZero then CharacteristicZero
    if R has CharacteristicNonZero then CharacteristicNonZero
    if R has CommutativeRing then
      Algebra(R)
      adjoint: $ -> $
        ++ adjoint(op) returns the adjoint of the operator \spad{op}.
      adjoint: ($, $) -> $
        ++ adjoint(op1, op2) sets the adjoint of op1 to be op2.
        ++ op1 must be a basic operator
      conjug  : R -> R
        ++ conjug(x)should be local but conditional
    evaluate: ($, M -> M) -> $
      ++ evaluate(f, u +-> g u) attaches the map g to f.
      ++ f must be a basic operator
      ++ g MUST be additive, i.e. \spad{g(a + b) = g(a) + g(b)} for
      ++ any \spad{a}, \spad{b} in M.
      ++ This implies that \spad{g(n a) = n g(a)} for
      ++ any \spad{a} in M and integer \spad{n > 0}.
    evaluateInverse: ($, M -> M) -> $
	++ evaluateInverse(x,f) \undocumented
    **: (OP, Integer) -> $
	++ op**n \undocumented
    **: ($, Integer) -> $
	++ op**n \undocumented
    opeval  : (OP, M) -> M
      ++ opeval should be local but conditional
    makeop   : (R, FG) -> $
      ++ makeop should be local but conditional

  Implementation ==> FAB add
    import NoneFunctions1($)
    import BasicOperatorFunctions1(M)

    Rep := FAB

    inv      : TERM -> $
    termeval : (TERM, M) -> M
    rmeval   : (RM, M) -> M
    monomeval: (FG, M) -> M
    opInvEval: (OP, M) -> M
    mkop     : (R, FG) -> $
    termprod0: (Integer, TERM, TERM) -> $
    termprod : (Integer, TERM, TERM) -> TERM
    termcopy : TERM -> TERM
    trm2O    : (Integer, TERM) -> O
    term2O   : TERM -> O
    rm2O     : (R, FG) -> O
    nocopy   : OP -> $

    1                   == makeop(1, 1)
    coerce(n:Integer):$ == n::R::$
    coerce(r:R):$       == (zero? r => 0; makeop(r, 1))
    coerce(op:OP):$     == nocopy copy op
    nocopy(op:OP):$     == makeop(1, op::FG)
    elt(x:$, r:M)       == +/[t.exp * termeval(t.gen, r) for t in terms x]
    rmeval(t, r)        == t.coef * monomeval(t.monom, r)
    termcopy t          == [[rm.coef, rm.monom] for rm in t]
    characteristic == characteristic$R
    mkop(r, fg)         == [[r, fg]$RM]$TERM :: $
    evaluate(f, g)   == nocopy setProperty(retract(f)@OP,OPEVAL,g pretend None)

    if R has OrderedSet then
      makeop(r, fg) == (r >= 0 => mkop(r, fg); - mkop(-r, fg))
    else makeop(r, fg) == mkop(r, fg)

    inv(t:TERM):$ ==
      empty? t => 1
      c := first(t).coef
      m := first(t).monom
      inv(rest t) * makeop(1, inv m) * (recip(c)::R::$)

    x:$ ** i:Integer ==
      i = 0 => 1
      i > 0 => expt(x,i pretend PositiveInteger)$RepeatedSquaring($)
      (inv(retract(x)@TERM)) ** (-i)

    evaluateInverse(f, g) ==
      nocopy setProperty(retract(f)@OP, INVEVAL, g pretend None)

    coerce(x:$):O ==
      zero? x => (0$R)::O
      reduce(_+, [trm2O(t.exp, t.gen) for t in terms x])$List(O)

    trm2O(c, t) ==
      one? c => term2O t
      c = -1 => - term2O t
      c::O * term2O t

    term2O t ==
      reduce(_*, [rm2O(rm.coef, rm.monom) for rm in t])$List(O)

    rm2O(c, m) ==
      one? c => m::O
      one? m => c::O
      c::O * m::O

    x:$ * y:$ ==
      +/[ +/[termprod0(t.exp * s.exp, t.gen, s.gen) for s in terms y]
          for t in terms x]

    termprod0(n, x, y) ==
      n >= 0 => termprod(n, x, y)::$
      - (termprod(-n, x, y)::$)

    termprod(n, x, y) ==
      lc := first(xx := termcopy x)
      lc.coef := n * lc.coef
      rm := last xx
      one?(first(y).coef) =>
        rm.monom := rm.monom * first(y).monom
        concat_!(xx, termcopy rest y)
      one?(rm.monom) =>
        rm.coef := rm.coef * first(y).coef
        rm.monom := first(y).monom
        concat_!(xx, termcopy rest y)
      concat_!(xx, termcopy y)

    if M has ExpressionSpace then
      opeval(op, r) ==
        (func := property(op, OPEVAL)) case "failed" => kernel(op, r)
        ((func::None) pretend (M -> M)) r

    else
      opeval(op, r) ==
        (func := property(op, OPEVAL)) case "failed" =>
          error "eval: operator has no evaluation function"
        ((func::None) pretend (M -> M)) r

    opInvEval(op, r) ==
      (func := property(op, INVEVAL)) case "failed" =>
         error "eval: operator has no inverse evaluation function"
      ((func::None) pretend (M -> M)) r

    termeval(t, r)  ==
      for rm in reverse t repeat r := rmeval(rm, r)
      r

    monomeval(m, r) ==
      for rec in reverse_! factors m repeat
        e := rec.exp
        g := rec.gen
        e > 0 =>
          for i in 1..e repeat r := opeval(g, r)
        e < 0 =>
          for i in 1..(-e) repeat r := opInvEval(g, r)
      r

    recip x ==
      (r := retractIfCan(x)@Union(R, "failed")) case "failed" => "failed"
      (r1 := recip(r::R)) case "failed" => "failed"
      r1::R::$

    retractIfCan(x:$):Union(R, "failed") ==
      (r:= retractIfCan(x)@Union(TERM,"failed")) case "failed" => "failed"
      empty?(t := r::TERM) => 0$R
      empty? rest t =>
        rm := first t
        one?(rm.monom) => rm.coef
        "failed"
      "failed"

    retractIfCan(x:$):Union(OP, "failed") ==
      (r:= retractIfCan(x)@Union(TERM,"failed")) case "failed" => "failed"
      empty?(t := r::TERM) => "failed"
      empty? rest t =>
        rm := first t
        one?(rm.coef) => retractIfCan(rm.monom)
        "failed"
      "failed"

    if R has CommutativeRing then
      termadj  : TERM -> $
      rmadj    : RM -> $
      monomadj : FG -> $
      opadj    : OP -> $

      r:R * x:$        == r::$ * x
      x:$ * r:R        == x * (r::$)
      adjoint x        == +/[t.exp * termadj(t.gen) for t in terms x]
      rmadj t          == conjug(t.coef) * monomadj(t.monom)
      adjoint(op, adj) == nocopy setProperty(retract(op)@OP, OPADJ, adj::None)

      termadj t ==
        ans:$ := 1
        for rm in t repeat ans := rmadj(rm) * ans
        ans

      monomadj m ==
        ans:$ := 1
        for rec in factors m repeat ans := (opadj(rec.gen) ** rec.exp) * ans
        ans

      opadj op ==
        (adj := property(op, OPADJ)) case "failed" =>
           error "adjoint: operator does not have a defined adjoint"
        (adj::None) pretend $

      if R has conjugate:R -> R then conjug r == conjugate r else conjug r == r

@
\section{domain OP Operator}
<<domain OP Operator>>=
)abbrev domain OP Operator
++ Author: Manuel Bronstein
++ Date Created: 15 May 1990
++ Date Last Updated: 12 February 1993
++ Description:
++ Algebra of ADDITIVE operators over a ring.
Operator(R: Ring) == ModuleOperator(R,R)

@
\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 MODOP ModuleOperator>>
<<domain OP Operator>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}