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--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.
-- SPAD files for the integration world should be compiled in the
-- following order:
--
-- intaux rderf intrf curve curvepkg divisor pfo
-- intalg intaf EFSTRUC rdeef intef irexpand integrat
)abbrev package SYMFUNC SymmetricFunctions
++ The elementary symmetric functions
++ Author: Manuel Bronstein
++ Date Created: 13 Feb 1989
++ Date Last Updated: 28 Jun 1990
++ Description: Computes all the symmetric functions in n variables.
SymmetricFunctions(R:Ring): Exports == Implementation where
UP ==> SparseUnivariatePolynomial R
Exports ==> with
symFunc: List R -> Vector R
++ symFunc([r1,...,rn]) returns the vector of the
++ elementary symmetric functions in the \spad{ri's}:
++ \spad{[r1 + ... + rn, r1 r2 + ... + r(n-1) rn, ..., r1 r2 ... rn]}.
symFunc: (R, PositiveInteger) -> Vector R
++ symFunc(r, n) returns the vector of the elementary
++ symmetric functions in \spad{[r,r,...,r]} \spad{n} times.
Implementation ==> add
signFix: (UP, NonNegativeInteger) -> Vector R
symFunc(x, n) == signFix((monomial(1, 1)$UP - x::UP) ** n, 1 + n)
symFunc l ==
signFix(*/[monomial(1, 1)$UP - a::UP for a in l], 1 + #l)
signFix(p, n) ==
m := minIndex(v := vectorise(p, n)) + 1
for i in 0..((#v quo 2) - 1)::NonNegativeInteger repeat
qsetelt!(v, 2*i + m, - qelt(v, 2*i + m))
reverse! v
)abbrev package TANEXP TangentExpansions
++ Expansions of tangents of sums and quotients
++ Author: Manuel Bronstein
++ Date Created: 13 Feb 1989
++ Date Last Updated: 20 Apr 1990
++ Description: Expands tangents of sums and scalar products.
TangentExpansions(R:Field): Exports == Implementation where
PI ==> PositiveInteger
Z ==> Integer
UP ==> SparseUnivariatePolynomial R
QF ==> Fraction UP
Exports ==> with
tanSum: List R -> R
++ tanSum([a1,...,an]) returns \spad{f(a1,...,an)} such that
++ if \spad{ai = tan(ui)} then \spad{f(a1,...,an) = tan(u1 + ... + un)}.
tanAn : (R, PI) -> UP
++ tanAn(a, n) returns \spad{P(x)} such that
++ if \spad{a = tan(u)} then \spad{P(tan(u/n)) = 0}.
tanNa : (R, Z) -> R
++ tanNa(a, n) returns \spad{f(a)} such that
++ if \spad{a = tan(u)} then \spad{f(a) = tan(n * u)}.
Implementation ==> add
import SymmetricFunctions(R)
import SymmetricFunctions(UP)
m1toN : Integer -> Integer
tanPIa: PI -> QF
m1toN n == (odd? n => -1; 1)
tanAn(a, n) == a * denom(q := tanPIa n) - numer q
tanNa(a, n) ==
zero? n => 0
negative? n => - tanNa(a, -n)
(numer(t := tanPIa(n::PI)) a) / ((denom t) a)
tanSum l ==
m := minIndex(v := symFunc l)
+/[m1toN(i+1) * v(2*i - 1 + m) for i in 1..(#v quo 2)]
/ +/[m1toN(i) * v(2*i + m) for i in 0..((#v - 1) quo 2)]
-- tanPIa(n) returns P(a)/Q(a) such that
-- if a = tan(u) then P(a)/Q(a) = tan(n * u);
tanPIa n ==
m := minIndex(v := symFunc(monomial(1, 1)$UP, n))
+/[m1toN(i+1) * v(2*i - 1 + m) for i in 1..(#v quo 2)]
/ +/[m1toN(i) * v(2*i + m) for i in 0..((#v - 1) quo 2)]
)abbrev package EFSTRUC ElementaryFunctionStructurePackage
++ Risch structure theorem
++ Author: Manuel Bronstein
++ Date Created: 1987
++ Date Last Updated: 16 August 1995
++ Description:
++ ElementaryFunctionStructurePackage provides functions to test the
++ algebraic independence of various elementary functions, using the
++ Risch structure theorem (real and complex versions).
++ It also provides transformations on elementary functions
++ which are not considered simplifications.
++ Keywords: elementary, function, structure.
ElementaryFunctionStructurePackage(R,F): Exports == Implementation where
R : Join(IntegralDomain, RetractableTo Integer,
LinearlyExplicitRingOver Integer)
F : Join(AlgebraicallyClosedField, TranscendentalFunctionCategory,
FunctionSpace R)
B ==> Boolean
N ==> NonNegativeInteger
Z ==> Integer
Q ==> Fraction Z
SY ==> Symbol
K ==> Kernel F
UP ==> SparseUnivariatePolynomial F
SMP ==> SparseMultivariatePolynomial(R, K)
REC ==> Record(func:F, kers: List K, vals:List F)
U ==> Union(vec:Vector Q, func:F, fail: Boolean)
Exports ==> with
normalize: F -> F
++ normalize(f) rewrites \spad{f} using the least possible number of
++ real algebraically independent kernels.
normalize: (F, SY) -> F
++ normalize(f, x) rewrites \spad{f} using the least possible number of
++ real algebraically independent kernels involving \spad{x}.
rischNormalize: (F, SY) -> REC
++ rischNormalize(f, x) returns \spad{[g, [k1,...,kn], [h1,...,hn]]}
++ such that \spad{g = normalize(f, x)} and each \spad{ki} was
++ rewritten as \spad{hi} during the normalization.
realElementary: F -> F
++ realElementary(f) rewrites \spad{f} in terms of the 4 fundamental real
++ transcendental elementary functions: \spad{log, exp, tan, atan}.
realElementary: (F, SY) -> F
++ realElementary(f,x) rewrites the kernels of \spad{f} involving \spad{x}
++ in terms of the 4 fundamental real
++ transcendental elementary functions: \spad{log, exp, tan, atan}.
validExponential: (List K, F, SY) -> Union(F, "failed")
++ validExponential([k1,...,kn],f,x) returns \spad{g} if \spad{exp(f)=g}
++ and \spad{g} involves only \spad{k1...kn}, and "failed" otherwise.
rootNormalize: (F, K) -> F
++ rootNormalize(f, k) returns \spad{f} rewriting either \spad{k} which
++ must be an nth-root in terms of radicals already in \spad{f}, or some
++ radicals in \spad{f} in terms of \spad{k}.
tanQ: (Q, F) -> F
++ tanQ(q,a) is a local function with a conditional implementation.
Implementation ==> add
macro POWER == '%power
macro NTHR == 'nthRoot
import TangentExpansions F
import IntegrationTools(R, F)
import IntegerLinearDependence F
import AlgebraicManipulations(R, F)
import InnerCommonDenominator(Z, Q, Vector Z, Vector Q)
k2Elem : (K, List SY) -> F
realElem : (F, List SY) -> F
smpElem : (SMP, List SY) -> F
deprel : (List K, K, SY) -> U
rootDep : (List K, K) -> U
qdeprel : (List F, F) -> U
factdeprel : (List K, K) -> U
toR : (List K, F) -> List K
toY : List K -> List F
toZ : List K -> List F
toU : List K -> List F
toV : List K -> List F
ktoY : K -> F
ktoZ : K -> F
ktoU : K -> F
ktoV : K -> F
gdCoef? : (Q, Vector Q) -> Boolean
goodCoef : (Vector Q, List K, SY) ->
Union(Record(index:Z, ker:K), "failed")
tanRN : (Q, K) -> F
localnorm : F -> F
rooteval : (F, List K, K, Q) -> REC
logeval : (F, List K, K, Vector Q) -> REC
expeval : (F, List K, K, Vector Q) -> REC
taneval : (F, List K, K, Vector Q) -> REC
ataneval : (F, List K, K, Vector Q) -> REC
depeval : (F, List K, K, Vector Q) -> REC
expnosimp : (F, List K, K, Vector Q, List F, F) -> REC
tannosimp : (F, List K, K, Vector Q, List F, F) -> REC
rtNormalize : F -> F
rootNormalize0 : F -> REC
rootKernelNormalize: (F, List K, K) -> Union(REC, "failed")
tanSum : (F, List F) -> F
comb? := F has CombinatorialOpsCategory
mpiover2:F := pi()$F / (-2::F)
realElem(f, l) == smpElem(numer f, l) / smpElem(denom f, l)
realElementary(f, x) == realElem(f, [x])
realElementary f == realElem(f, variables f)
toY ker == [func for k in ker | (func := ktoY k) ~= 0]
toZ ker == [func for k in ker | (func := ktoZ k) ~= 0]
toU ker == [func for k in ker | (func := ktoU k) ~= 0]
toV ker == [func for k in ker | (func := ktoV k) ~= 0]
rtNormalize f == rootNormalize0(f).func
toR(ker, x) == select(is?(#1, NTHR) and first argument(#1) = x, ker)
if R has GcdDomain then
tanQ(c, x) ==
tanNa(rootSimp zeroOf tanAn(x, denom(c)::PositiveInteger), numer c)
else
tanQ(c, x) ==
tanNa(zeroOf tanAn(x, denom(c)::PositiveInteger), numer c)
-- tanSum(c, [a1,...,an]) returns f(c, a1,...,an) such that
-- if ai = tan(ui) then f(c, a1,...,an) = tan(c + u1 + ... + un).
-- MUST BE CAREFUL FOR WHEN c IS AN ODD MULTIPLE of pi/2
tanSum(c, l) ==
k := c / mpiover2 -- k = - 2 c / pi, check for odd integer
-- tan((2n+1) pi/2 x) = - 1 / tan x
(r := retractIfCan(k)@Union(Z, "failed")) case Z and odd?(r::Z) =>
- inv tanSum l
tanSum concat(tan c, l)
rootNormalize0 f ==
ker := select!(is?(#1, NTHR) and empty? variables first argument #1,
tower f)$List(K)
empty? ker => [f, empty(), empty()]
(n := (#ker)::Z - 1) < 1 => [f, empty(), empty()]
for i in 1..n for kk in rest ker repeat
(u := rootKernelNormalize(f, first(ker, i), kk)) case REC =>
rec := u::REC
rn := rootNormalize0(rec.func)
return [rn.func, concat(rec.kers, rn.kers), concat(rec.vals, rn.vals)]
[f, empty(), empty()]
deprel(ker, k, x) ==
is?(k, 'log) or is?(k, 'exp) =>
qdeprel([differentiate(g, x) for g in toY ker],
differentiate(ktoY k, x))
is?(k, 'atan) or is?(k, 'tan) =>
qdeprel([differentiate(g, x) for g in toU ker],
differentiate(ktoU k, x))
is?(k, NTHR) => rootDep(ker, k)
comb? and is?(k, 'factorial) =>
factdeprel([x for x in ker | is?(x,'factorial) and x~=k],k)
[true]
ktoY k ==
is?(k, 'log) => k::F
is?(k, 'exp) => first argument k
0
ktoZ k ==
is?(k, 'log) => first argument k
is?(k, 'exp) => k::F
0
ktoU k ==
is?(k, 'atan) => k::F
is?(k, 'tan) => first argument k
0
ktoV k ==
is?(k, 'tan) => k::F
is?(k, 'atan) => first argument k
0
smpElem(p, l) ==
map(k2Elem(#1, l), #1::F, p)$PolynomialCategoryLifting(
IndexedExponents K, K, R, SMP, F)
k2Elem(k, l) ==
ez, iez, tz2: F
kf := k::F
not(empty? l) and empty? [v for v in variables kf | member?(v, l)] => kf
empty?(args :List F := [realElem(a, l) for a in argument k]) => kf
z := first args
is?(k, POWER) => (zero? z => 0; exp(last(args) * log z))
is?(k, 'cot) => inv tan z
is?(k, 'acot) => atan inv z
is?(k, 'asin) => atan(z / sqrt(1 - z**2))
is?(k, 'acos) => atan(sqrt(1 - z**2) / z)
is?(k, 'asec) => atan sqrt(1 - z**2)
is?(k, 'acsc) => atan inv sqrt(1 - z**2)
is?(k, 'asinh) => log(sqrt(1 + z**2) + z)
is?(k, 'acosh) => log(sqrt(z**2 - 1) + z)
is?(k, 'atanh) => log((z + 1) / (1 - z)) / (2::F)
is?(k, 'acoth) => log((z + 1) / (z - 1)) / (2::F)
is?(k, 'asech) => log((inv z) + sqrt(inv(z**2) - 1))
is?(k, 'acsch) => log((inv z) + sqrt(1 + inv(z**2)))
is?(k, '%paren) or is?(k, '%box) =>
empty? rest args => z
kf
if has?(op := operator k, 'htrig) then iez := inv(ez := exp z)
is?(k, 'sinh) => (ez - iez) / (2::F)
is?(k, 'cosh) => (ez + iez) / (2::F)
is?(k, 'tanh) => (ez - iez) / (ez + iez)
is?(k, 'coth) => (ez + iez) / (ez - iez)
is?(k, 'sech) => 2 * inv(ez + iez)
is?(k, 'csch) => 2 * inv(ez - iez)
if has?(op, 'trig) then tz2 := tan(z / (2::F))
is?(k, 'sin) => 2 * tz2 / (1 + tz2**2)
is?(k, 'cos) => (1 - tz2**2) / (1 + tz2**2)
is?(k, 'sec) => (1 + tz2**2) / (1 - tz2**2)
is?(k, 'csc) => (1 + tz2**2) / (2 * tz2)
op args
--The next 5 functions are used by normalize, once a relation is found
depeval(f, lk, k, v) ==
is?(k, 'log) => logeval(f, lk, k, v)
is?(k, 'exp) => expeval(f, lk, k, v)
is?(k, 'tan) => taneval(f, lk, k, v)
is?(k, 'atan) => ataneval(f, lk, k, v)
is?(k, NTHR) => rooteval(f, lk, k, v(minIndex v))
[f, empty(), empty()]
rooteval(f, lk, k, n) ==
nv := nthRoot(x := first argument k, m := retract(n)@Z)
l := [r for r in concat(k, toR(lk, x)) |
retract(second argument r)@Z ~= m]
lv := [nv ** (n / (retract(second argument r)@Z::Q)) for r in l]
[eval(f, l, lv), l, lv]
ataneval(f, lk, k, v) ==
w := first argument k
s := tanSum [tanQ(qelt(v,i), x)
for i in minIndex v .. maxIndex v for x in toV lk]
g := +/[qelt(v, i) * x for i in minIndex v .. maxIndex v for x in toU lk]
h:F :=
zero?(d := 1 + s * w) => mpiover2
atan((w - s) / d)
g := g + h
[eval(f, [k], [g]), [k], [g]]
gdCoef?(c, v) ==
for i in minIndex v .. maxIndex v repeat
retractIfCan(qelt(v, i) / c)@Union(Z, "failed") case "failed" =>
return false
true
goodCoef(v, l, s) ==
for i in minIndex v .. maxIndex v for k in l repeat
is?(k, s) and
((r:=recip(qelt(v,i))) case Q) and
(retractIfCan(r::Q)@Union(Z, "failed") case Z)
and gdCoef?(qelt(v, i), v) => return([i, k])
"failed"
taneval(f, lk, k, v) ==
u := first argument k
fns := toU lk
c := u - +/[qelt(v, i) * x for i in minIndex v .. maxIndex v for x in fns]
(rec := goodCoef(v, lk, 'tan)) case "failed" =>
tannosimp(f, lk, k, v, fns, c)
v0 := retract(inv qelt(v, rec.index))@Z
lv := [qelt(v, i) for i in minIndex v .. maxIndex v |
i ~= rec.index]$List(Q)
l := [kk for kk in lk | kk ~= rec.ker]
g := tanSum(-v0 * c, concat(tanNa(k::F, v0),
[tanNa(x, - retract(a * v0)@Z) for a in lv for x in toV l]))
[eval(f, [rec.ker], [g]), [rec.ker], [g]]
tannosimp(f, lk, k, v, fns, c) ==
every?(is?(#1, 'tan), lk) =>
dd := (d := (cd := splitDenominator v).den)::F
newt := [tan(u / dd) for u in fns]$List(F)
newtan := [tanNa(t, d) for t in newt]$List(F)
h := tanSum(c, [tanNa(t, qelt(cd.num, i))
for i in minIndex v .. maxIndex v for t in newt])
lk := concat(k, lk)
newtan := concat(h, newtan)
[eval(f, lk, newtan), lk, newtan]
h := tanSum(c, [tanQ(qelt(v, i), x)
for i in minIndex v .. maxIndex v for x in toV lk])
[eval(f, [k], [h]), [k], [h]]
expnosimp(f, lk, k, v, fns, g) ==
every?(is?(#1, 'exp), lk) =>
dd := (d := (cd := splitDenominator v).den)::F
newe := [exp(y / dd) for y in fns]$List(F)
newexp := [e ** d for e in newe]$List(F)
h := */[e ** qelt(cd.num, i)
for i in minIndex v .. maxIndex v for e in newe] * g
lk := concat(k, lk)
newexp := concat(h, newexp)
[eval(f, lk, newexp), lk, newexp]
h := */[exp(y) ** qelt(v, i)
for i in minIndex v .. maxIndex v for y in fns] * g
[eval(f, [k], [h]), [k], [h]]
logeval(f, lk, k, v) ==
z := first argument k
c := z / (*/[x**qelt(v, i)
for x in toZ lk for i in minIndex v .. maxIndex v])
-- CHANGED log ktoZ x TO ktoY x SINCE WE WANT log exp f TO BE REPLACED BY f.
g := +/[qelt(v, i) * x
for i in minIndex v .. maxIndex v for x in toY lk] + log c
[eval(f, [k], [g]), [k], [g]]
rischNormalize(f, v) ==
empty?(ker := varselect(tower f, v)) => [f, empty(), empty()]
first(ker) ~= kernel(v)@K => error "Cannot happen"
ker := rest ker
(n := (#ker)::Z - 1) < 1 => [f, empty(), empty()]
for i in 1..n for kk in rest ker repeat
klist := first(ker, i)
-- NO EVALUATION ON AN EMPTY VECTOR, WILL CAUSE INFINITE LOOP
(c := deprel(klist, kk, v)) case vec and not empty?(c.vec) =>
rec := depeval(f, klist, kk, c.vec)
rn := rischNormalize(rec.func, v)
return [rn.func,
concat(rec.kers, rn.kers), concat(rec.vals, rn.vals)]
c case func =>
rn := rischNormalize(eval(f, [kk], [c.func]), v)
return [rn.func, concat(kk, rn.kers), concat(c.func, rn.vals)]
[f, empty(), empty()]
rootNormalize(f, k) ==
(u := rootKernelNormalize(f, toR(tower f, first argument k), k))
case "failed" => f
(u::REC).func
rootKernelNormalize(f, l, k) ==
(c := rootDep(l, k)) case vec =>
rooteval(f, l, k, (c.vec)(minIndex(c.vec)))
"failed"
localnorm f ==
for x in variables f repeat
f := rischNormalize(f, x).func
f
validExponential(twr, eta, x) ==
fns : List F
(c := solveLinearlyOverQ(construct([differentiate(g, x)
for g in (fns := toY twr)]$List(F))@Vector(F),
differentiate(eta, x))) case "failed" => "failed"
v := c::Vector(Q)
g := eta - +/[qelt(v, i) * yy
for i in minIndex v .. maxIndex v for yy in fns]
*/[exp(yy) ** qelt(v, i)
for i in minIndex v .. maxIndex v for yy in fns] * exp g
rootDep(ker, k) ==
empty?(ker := toR(ker, first argument k)) => [true]
[new(1,lcm(retract(second argument k)@Z,
"lcm"/[retract(second argument r)@Z for r in ker])::Q)$Vector(Q)]
qdeprel(l, v) ==
(u := solveLinearlyOverQ(construct(l)@Vector(F), v))
case Vector(Q) => [u::Vector(Q)]
[true]
expeval(f, lk, k, v) ==
y := first argument k
fns := toY lk
g := y - +/[qelt(v, i) * z for i in minIndex v .. maxIndex v for z in fns]
(rec := goodCoef(v, lk, 'exp)) case "failed" =>
expnosimp(f, lk, k, v, fns, exp g)
v0 := retract(inv qelt(v, rec.index))@Z
lv := [qelt(v, i) for i in minIndex v .. maxIndex v |
i ~= rec.index]$List(Q)
l := [kk for kk in lk | kk ~= rec.ker]
h :F := */[exp(z) ** (- retract(a * v0)@Z) for a in lv for z in toY l]
h := h * exp(-v0 * g) * (k::F) ** v0
[eval(f, [rec.ker], [h]), [rec.ker], [h]]
if F has CombinatorialOpsCategory then
normalize f == rtNormalize localnorm factorials realElementary f
normalize(f, x) ==
rtNormalize(rischNormalize(factorials(realElementary(f,x),x),x).func)
factdeprel(l, k) ==
((r := retractIfCan(n := first argument k)@Union(Z, "failed"))
case Z) and positive?(r::Z) => [factorial(r::Z)::F]
for x in l repeat
m := first argument x
((r := retractIfCan(n - m)@Union(Z, "failed")) case Z) and
positive?(r::Z) => return([*/[(m + i::F) for i in 1..r] * x::F])
[true]
else
normalize f == rtNormalize localnorm realElementary f
normalize(f, x) == rtNormalize(rischNormalize(realElementary(f,x),x).func)
)abbrev package ITRIGMNP InnerTrigonometricManipulations
++ Trigs to/from exps and logs
++ Author: Manuel Bronstein
++ Date Created: 4 April 1988
++ Date Last Updated: 9 October 1993
++ Description:
++ This package provides transformations from trigonometric functions
++ to exponentials and logarithms, and back.
++ F and FG should be the same type of function space.
++ Keywords: trigonometric, function, manipulation.
InnerTrigonometricManipulations(R,F,FG): Exports == Implementation where
R : IntegralDomain
F : Join(FunctionSpace R, RadicalCategory,
TranscendentalFunctionCategory)
FG : Join(FunctionSpace Complex R, RadicalCategory,
TranscendentalFunctionCategory)
Z ==> Integer
SY ==> Symbol
OP ==> BasicOperator
GR ==> Complex R
GF ==> Complex F
KG ==> Kernel FG
PG ==> SparseMultivariatePolynomial(GR, KG)
UP ==> SparseUnivariatePolynomial PG
Exports ==> with
GF2FG : GF -> FG
++ GF2FG(a + i b) returns \spad{a + i b} viewed as a function with
++ the \spad{i} pushed down into the coefficient domain.
FG2F : FG -> F
++ FG2F(a + i b) returns \spad{a + sqrt(-1) b}.
F2FG : F -> FG
++ F2FG(a + sqrt(-1) b) returns \spad{a + i b}.
explogs2trigs: FG -> GF
++ explogs2trigs(f) rewrites all the complex logs and
++ exponentials appearing in \spad{f} in terms of trigonometric
++ functions.
trigs2explogs: (FG, List KG, List SY) -> FG
++ trigs2explogs(f, [k1,...,kn], [x1,...,xm]) rewrites
++ all the trigonometric functions appearing in \spad{f} and involving
++ one of the \spad{xi's} in terms of complex logarithms and
++ exponentials. A kernel of the form \spad{tan(u)} is expressed
++ using \spad{exp(u)**2} if it is one of the \spad{ki's}, in terms of
++ \spad{exp(2*u)} otherwise.
Implementation ==> add
macro NTHR == 'nthRoot
ker2explogs: (KG, List KG, List SY) -> FG
smp2explogs: (PG, List KG, List SY) -> FG
supexp : (UP, GF, GF, Z) -> GF
GR2GF : GR -> GF
GR2F : GR -> F
KG2F : KG -> F
PG2F : PG -> F
ker2trigs : (OP, List GF) -> GF
smp2trigs : PG -> GF
sup2trigs : (UP, GF) -> GF
nth := R has RetractableTo(Integer) and F has RadicalCategory
GR2F g == real(g)::F + sqrt(-(1::F)) * imag(g)::F
KG2F k == map(FG2F, k)$ExpressionSpaceFunctions2(FG, F)
FG2F f == (PG2F numer f) / (PG2F denom f)
F2FG f == map(#1::GR, f)$FunctionSpaceFunctions2(R,F,GR,FG)
GF2FG f == (F2FG real f) + complex(0, 1)$GR ::FG * F2FG imag f
GR2GF gr == complex(real(gr)::F, imag(gr)::F)
-- This expects the argument to have only tan and atans left.
-- Does a half-angle correction if k is not in the initial kernel list.
ker2explogs(k, l, lx) ==
kf : FG
empty?([v for v in variables(kf := k::FG) |
member?(v, lx)]$List(SY)) => kf
empty?(args := [trigs2explogs(a, l, lx)
for a in argument k]$List(FG)) => kf
im := complex(0, 1)$GR :: FG
z := first args
is?(k,'tan) =>
e := (member?(k, l) => exp(im * z) ** 2; exp(2 * im * z))
- im * (e - 1) /$FG (e + 1)
is?(k,'atan) =>
im * log((1 -$FG im *$FG z)/$FG (1 +$FG im *$FG z))$FG / (2::FG)
(operator k) args
trigs2explogs(f, l, lx) ==
smp2explogs(numer f, l, lx) / smp2explogs(denom f, l, lx)
-- return op(arg) as f + %i g
-- op is already an operator with semantics over R, not GR
ker2trigs(op, arg) ==
"and"/[zero? imag x for x in arg] =>
complex(op [real x for x in arg]$List(F), 0)
a := first arg
is?(op,'exp) => exp a
is?(op,'log) => log a
is?(op,'sin) => sin a
is?(op,'cos) => cos a
is?(op,'tan) => tan a
is?(op,'cot) => cot a
is?(op,'sec) => sec a
is?(op,'csc) => csc a
is?(op,'asin) => asin a
is?(op,'acos) => acos a
is?(op,'atan) => atan a
is?(op,'acot) => acot a
is?(op,'asec) => asec a
is?(op,'acsc) => acsc a
is?(op,'sinh) => sinh a
is?(op,'cosh) => cosh a
is?(op,'tanh) => tanh a
is?(op,'coth) => coth a
is?(op,'sech) => sech a
is?(op,'csch) => csch a
is?(op,'asinh) => asinh a
is?(op,'acosh) => acosh a
is?(op,'atanh) => atanh a
is?(op,'acoth) => acoth a
is?(op,'asech) => asech a
is?(op,'acsch) => acsch a
is?(op,'abs) => sqrt(norm a)::GF
nth and is?(op, NTHR) => nthRoot(a, retract(second arg)@Z)
error "ker2trigs: cannot convert kernel to gaussian function"
sup2trigs(p, f) ==
map(smp2trigs, p)$SparseUnivariatePolynomialFunctions2(PG, GF) f
smp2trigs p ==
map(explogs2trigs(#1::FG),GR2GF, p)$PolynomialCategoryLifting(
IndexedExponents KG, KG, GR, PG, GF)
explogs2trigs f ==
(m := mainKernel f) case "failed" =>
GR2GF(retract(numer f)@GR) / GR2GF(retract(denom f)@GR)
op := operator(operator(k := m::KG))$F
arg := [explogs2trigs x for x in argument k]
num := univariate(numer f, k)
den := univariate(denom f, k)
is?(op,'exp) =>
e := exp real first arg
y := imag first arg
g := complex(e * cos y, e * sin y)$GF
gi := complex(cos(y) / e, - sin(y) / e)$GF
supexp(num,g,gi,b := (degree num)::Z quo 2)/supexp(den,g,gi,b)
sup2trigs(num, g := ker2trigs(op, arg)) / sup2trigs(den, g)
supexp(p, f1, f2, bse) ==
ans:GF := 0
while p ~= 0 repeat
g := explogs2trigs(leadingCoefficient(p)::FG)
if ((d := degree(p)::Z - bse) >= 0) then
ans := ans + g * f1 ** d
else ans := ans + g * f2 ** (-d)
p := reductum p
ans
PG2F p ==
map(KG2F, GR2F, p)$PolynomialCategoryLifting(IndexedExponents KG,
KG, GR, PG, F)
smp2explogs(p, l, lx) ==
map(ker2explogs(#1, l, lx), #1::FG, p)$PolynomialCategoryLifting(
IndexedExponents KG, KG, GR, PG, FG)
)abbrev package TRIGMNIP TrigonometricManipulations
++ Trigs to/from exps and logs
++ Author: Manuel Bronstein
++ Date Created: 4 April 1988
++ Date Last Updated: 14 February 1994
++ Description:
++ \spadtype{TrigonometricManipulations} provides transformations from
++ trigonometric functions to complex exponentials and logarithms, and back.
++ Keywords: trigonometric, function, manipulation.
TrigonometricManipulations(R, F): Exports == Implementation where
R : Join(GcdDomain, RetractableTo Integer,
LinearlyExplicitRingOver Integer)
F : Join(AlgebraicallyClosedField, TranscendentalFunctionCategory,
FunctionSpace R)
Z ==> Integer
SY ==> Symbol
K ==> Kernel F
FG ==> Expression Complex R
Exports ==> with
complexNormalize: F -> F
++ complexNormalize(f) rewrites \spad{f} using the least possible number
++ of complex independent kernels.
complexNormalize: (F, SY) -> F
++ complexNormalize(f, x) rewrites \spad{f} using the least possible
++ number of complex independent kernels involving \spad{x}.
complexElementary: F -> F
++ complexElementary(f) rewrites \spad{f} in terms of the 2 fundamental
++ complex transcendental elementary functions: \spad{log, exp}.
complexElementary: (F, SY) -> F
++ complexElementary(f, x) rewrites the kernels of \spad{f} involving
++ \spad{x} in terms of the 2 fundamental complex
++ transcendental elementary functions: \spad{log, exp}.
trigs : F -> F
++ trigs(f) rewrites all the complex logs and exponentials
++ appearing in \spad{f} in terms of trigonometric functions.
real : F -> F
++ real(f) returns the real part of \spad{f} where \spad{f} is a complex
++ function.
imag : F -> F
++ imag(f) returns the imaginary part of \spad{f} where \spad{f}
++ is a complex function.
real? : F -> Boolean
++ real?(f) returns \spad{true} if \spad{f = real f}.
complexForm: F -> Complex F
++ complexForm(f) returns \spad{[real f, imag f]}.
Implementation ==> add
import ElementaryFunctionSign(R, F)
import InnerTrigonometricManipulations(R,F,FG)
import ElementaryFunctionStructurePackage(R, F)
import ElementaryFunctionStructurePackage(Complex R, FG)
s1 := sqrt(-1::F)
ipi := pi()$F * s1
K2KG : K -> Kernel FG
kcomplex : K -> Union(F, "failed")
locexplogs : F -> FG
localexplogs : (F, F, List SY) -> FG
complexKernels: F -> Record(ker: List K, val: List F)
K2KG k == retract(tan F2FG first argument k)@Kernel(FG)
real? f == empty?(complexKernels(f).ker)
real f == real complexForm f
imag f == imag complexForm f
-- returns [[k1,...,kn], [v1,...,vn]] such that ki should be replaced by vi
complexKernels f ==
lk:List(K) := empty()
lv:List(F) := empty()
for k in tower f repeat
if (u := kcomplex k) case F then
lk := concat(k, lk)
lv := concat(u::F, lv)
[lk, lv]
-- returns f if it is certain that k is not a real kernel and k = f,
-- "failed" otherwise
kcomplex k ==
op := operator k
is?(k, 'nthRoot) =>
arg := argument k
even?(retract(n := second arg)@Z) and ((u := sign(first arg)) case Z)
and negative?(u::Z) => op(s1, n / 2::F) * op(- first arg, n)
"failed"
is?(k, 'log) and ((u := sign(a := first argument k)) case Z)
and negative?(u::Z) => op(- a) + ipi
"failed"
complexForm f ==
empty?((l := complexKernels f).ker) => complex(f, 0)
explogs2trigs locexplogs eval(f, l.ker, l.val)
locexplogs f ==
any?(has?(#1, 'rtrig),
operators(g := realElementary f))$List(BasicOperator) =>
localexplogs(f, g, variables g)
F2FG g
complexNormalize(f, x) ==
g : F
any?(has?(operator #1, 'rtrig),
[k for k in tower(g := realElementary(f, x))
| member?(x, variables(k::F))]$List(K))$List(K) =>
FG2F(rischNormalize(localexplogs(f, g, [x]), x).func)
rischNormalize(g, x).func
complexNormalize f ==
l := variables(g := realElementary f)
any?(has?(#1, 'rtrig), operators g)$List(BasicOperator) =>
h := localexplogs(f, g, l)
for x in l repeat h := rischNormalize(h, x).func
FG2F h
for x in l repeat g := rischNormalize(g, x).func
g
complexElementary(f, x) ==
g : F
any?(has?(operator #1, 'rtrig),
[k for k in tower(g := realElementary(f, x))
| member?(x, variables(k::F))]$List(K))$List(K) =>
FG2F localexplogs(f, g, [x])
g
complexElementary f ==
any?(has?(#1, 'rtrig),
operators(g := realElementary f))$List(BasicOperator) =>
FG2F localexplogs(f, g, variables g)
g
localexplogs(f, g, lx) ==
trigs2explogs(F2FG g, [K2KG k for k in tower f
| is?(k, 'tan) or is?(k, 'cot)], lx)
trigs f ==
real? f => f
g := explogs2trigs F2FG f
real g + s1 * imag g
)abbrev package CTRIGMNP ComplexTrigonometricManipulations
++ Real and Imaginary parts of complex functions
++ Author: Manuel Bronstein
++ Date Created: 11 June 1993
++ Date Last Updated: 14 June 1993
++ Description:
++ \spadtype{ComplexTrigonometricManipulations} provides function that
++ compute the real and imaginary parts of complex functions.
++ Keywords: complex, function, manipulation.
ComplexTrigonometricManipulations(R, F): Exports == Implementation where
R : Join(IntegralDomain, RetractableTo Integer)
F : Join(AlgebraicallyClosedField, TranscendentalFunctionCategory,
FunctionSpace Complex R)
SY ==> Symbol
FR ==> Expression R
K ==> Kernel F
Exports ==> with
complexNormalize: F -> F
++ complexNormalize(f) rewrites \spad{f} using the least possible number
++ of complex independent kernels.
complexNormalize: (F, SY) -> F
++ complexNormalize(f, x) rewrites \spad{f} using the least possible
++ number of complex independent kernels involving \spad{x}.
complexElementary: F -> F
++ complexElementary(f) rewrites \spad{f} in terms of the 2 fundamental
++ complex transcendental elementary functions: \spad{log, exp}.
complexElementary: (F, SY) -> F
++ complexElementary(f, x) rewrites the kernels of \spad{f} involving
++ \spad{x} in terms of the 2 fundamental complex
++ transcendental elementary functions: \spad{log, exp}.
real : F -> FR
++ real(f) returns the real part of \spad{f} where \spad{f} is a complex
++ function.
imag : F -> FR
++ imag(f) returns the imaginary part of \spad{f} where \spad{f}
++ is a complex function.
real? : F -> Boolean
++ real?(f) returns \spad{true} if \spad{f = real f}.
trigs : F -> F
++ trigs(f) rewrites all the complex logs and exponentials
++ appearing in \spad{f} in terms of trigonometric functions.
complexForm: F -> Complex FR
++ complexForm(f) returns \spad{[real f, imag f]}.
Implementation ==> add
import InnerTrigonometricManipulations(R, FR, F)
import ElementaryFunctionStructurePackage(Complex R, F)
rreal?: Complex R -> Boolean
kreal?: Kernel F -> Boolean
localexplogs : (F, F, List SY) -> F
real f == real complexForm f
imag f == imag complexForm f
rreal? r == zero? imag r
kreal? k == every?(real?, argument k)$List(F)
complexForm f == explogs2trigs f
trigs f ==
GF2FG explogs2trigs f
real? f ==
every?(rreal?, coefficients numer f)
and every?(rreal?, coefficients denom f) and every?(kreal?, kernels f)
localexplogs(f, g, lx) ==
trigs2explogs(g, [k for k in tower f
| is?(k, 'tan) or is?(k, 'cot)], lx)
complexElementary f ==
any?(has?(#1, 'rtrig),
operators(g := realElementary f))$List(BasicOperator) =>
localexplogs(f, g, variables g)
g
complexElementary(f, x) ==
g : F
any?(has?(operator #1, 'rtrig),
[k for k in tower(g := realElementary(f, x))
| member?(x, variables(k::F))]$List(K))$List(K) =>
localexplogs(f, g, [x])
g
complexNormalize(f, x) ==
g : F
any?(has?(operator #1, 'rtrig),
[k for k in tower(g := realElementary(f, x))
| member?(x, variables(k::F))]$List(K))$List(K) =>
(rischNormalize(localexplogs(f, g, [x]), x).func)
rischNormalize(g, x).func
complexNormalize f ==
l := variables(g := realElementary f)
any?(has?(#1, 'rtrig), operators g)$List(BasicOperator) =>
h := localexplogs(f, g, l)
for x in l repeat h := rischNormalize(h, x).func
h
for x in l repeat g := rischNormalize(g, x).func
g
|