fricas
(1) -> <spad>
fricas
)abbrev package SPI SumProdInt
SumProdInt(R): Exports == Implementation where
R:Join(Comparable,IntegralDomain,RTZZ,LEZZ)
ZZ ==> Integer
RTZZ ==> RetractableTo ZZ
LEZZ ==> LinearlyExplicitOver ZZ
X ==> Expression R
PI ==> PositiveInteger
UK ==> Union(Kernel X,"failed")
UX ==> Union(X,"failed")
KX ==> Kernel X
LX ==> List X
SX ==> Segment X
SBX ==> SegmentBinding X
EFSP ==> ElementaryFunctionStructurePackage(R,X)
SPIR ==> Record(typ:Symbol, term:X, var:X, sym:X, lb:X, ub:X)
NFPRD ==> Record(fac:X, nump:List X, denp:List X)
QUOTR ==> Record(quotient: X,remainder: X)
Exports == Join(OrderedSet) with
destruct : X -> Union(SPIR,"failed")
construct : SPIR -> X
splitOp : (X,X) -> UX
++ splitOp function splits the summation and product at a term
++ and definite integral at an interval.
++ splitOp(expr, c), expr - summation, product or definite integral,
++ c - value at which expr should be split.
multiplyOp : (X,X) -> UX
++ multiplyOp function exponentiates the base operator, "+" to
++ "*", "*" to "^". The multiplicand can contain the iterating
++ variable or index of summation, product or definite integral.
++ multiplyOp(expr, c), expr - summation, product or definite integral
++ c - value to be multiplied.
splitArgs : X -> UX
++ splitArgs function splits summation and definite integral based
++ on the additive term and product based on the multiplicative
++ term in the inner expression.
++ splitArgs(expr), expr - summation, product or integral.
takeHigh : (X,Integer) -> UX
++ takeHigh function gets the higher terms of summation, product and
++ definite integral.
++ takeHigh(expr, n), expr - summation, product or definite integral,
++ n - number of terms to be fetched if expr is summation
++ or product else the limit if expr is a definite integral.
takeLow : (X,Integer) -> UX
++ The takeLow function gets the lower terms of summation, product
++ and definite integral.
++ takeLow(expr, n), expr - summation, product or definite integral,
++ n - number of terms to be fetched if expr is summation or product
++ else the limit if expr is a definite integral.
shiftRange : (X,X) -> UX
++ The shifRange function shifts or rearranges summation, product
++ and definite integral.
++ shiftRange(expr, n), expr - summation, product or definite integral,
++ n - value by which expr should be shifted
shiftToZero : X -> UX
++ shiftToZero(expr) is the same as shiftRange(expr,-lowerBound),
++ but does not need to know the start value.
reverseOrder : X -> UX
++ reverseOrder function reverses the order of terms for summation
++ and product and intervals for definite integral.
++ reverseOrder(expr), expr - summation, product or definite integral.
getCoefficient : (X,X,X) -> UX
++ getCoefficient function gets the coefficient of a term in a
++ univariate polynomial which is represented as a summation.
++ getCoefficient(expr,k,invar), expr - summation, k - coefficient
++ for kth term, invar - variable for which coefficient needs to be
++ fetched.
evalOp : (X,X,X) -> UX
++ evalOp function gets the terms of summation and product and value
++ of the definite integral. If the input is a summation then it returns
++ the closed form of summation if it can be evaluated else it returns
++ the summation unevaluated.
++ evalOp(expr,start,end), expr - summation, product or definite
++ integral, start - lower limit of expr starting from which the terms
++ or value need to be fetched, end - upper limit of expr until which
++ the terms or value need to be fetched
combineSplitArgs : X -> UX
++ combineSplitArgs function combines the summations and definite
++ integrals split on additive terms and products split on
++ multiplicative terms. This performs reverse operation as in case
++ of splitArgs function.
++ combineSplitArgs(expr), expr - expression.
pullConstant : X -> UX
pushConstant : (X,X) -> UX
reduceQuoProd: (X,X) -> Union(X,"failed")
getLowerBound : X -> UX
getUpperBound : X -> UX
getOpTerm : X -> UX
getOpTermAs : (X,X) -> UX
getOpType : X -> Union(Symbol,"failed")
setLowerBound : (X,X) -> UX
setUpperBound : (X,X) -> UX
setOpTerm : (X,X->X) -> UX
setOpType : (X,Symbol) -> UX
splitTimesND: X -> List List X
splitFactorOp : (List X,Symbol) -> List List X
makeTimesFactor : List X -> X
invertProd: X -> UX
nfProd : X -> NFPRD
reduceProductND : NFPRD -> NFPRD
reduceProducts: X -> X
Implementation == X add
spi_ops:List Symbol:=['%defsum,'%defprod,'%defint]
sum?(r:SPIR):Boolean == r.typ = '%defsum
prd?(r:SPIR):Boolean == r.typ = '%defprod
int?(r:SPIR):Boolean == r.typ = '%defint
-- construct a SPI term from SPIR
construct(r:SPIR):X ==
t:X:=subst(r.term,r.var=r.sym)
v:Symbol:=retract(r.sym)@Symbol
s:SBX:=equation(v,segment(r.lb,r.ub)$SX)$SBX
sum? r => summation(t,s)$X
prd? r => product(t,s)$X
int? r => integral(t,s)$X
error "Type must be %defsum,%defprod or %defint."
-- destruct a SPI term to SPIR
destruct(x:X):Union(SPIR,"failed") ==
opt:List Symbol:= [o for o in spi_ops | is?(x,o)]
empty? opt => "failed"
mk:UK:=mainKernel(x)
mk case "failed" => "failed"
K:KX:=mk
A:List X:=argument K
t:Symbol:=first opt
[t,A.1,A.2,A.3,A.4,A.5]
-- term x is free of SPI
freeOfSPI?(x:X):Boolean ==
nop:List Symbol:=[name o for o in operators x]
empty? setIntersection(spi_ops,nop)
-- freeOf?($,'%def$) also possible
-- gives the type of SPI
checkSPI(x:X):Union(X,"summation","product","integral") ==
name last operators x = '%defsum => "summation"
name last operators x = '%defint => "integral"
name last operators x = '%defprod => "product"
x
---
splitOp(x,s) ==
r:=destruct x
r case "failed" => "failed"
r.sym=s => "failed" -- more checks?
rc:=copy r
r.ub := s
rc.lb := s+1
sum? r => construct(r) + construct(rc)
prd? r => construct(r) * construct(rc)
rc.lb := s -- integral
int? r => construct(r) + construct(rc)
multiplyOp(x,s) ==
r:=destruct x
r case "failed" => "failed"
r.term := s * r.term
sum? r => construct(r)
int? r => construct(r)
-- r.term := (r.term)^s ;;; better using a exptOp
prd? r => construct(r)
splitArgs(x) ==
r:=destruct x
r case "failed" => "failed"
if sum? r or int? r then
B:Union(List(X),"failed"):=isPlus(r.term)
B case "failed" => "failed"
v:X:=0
for i in 1..#B repeat
r.term := B.i
v:=v + construct(r)
return v
if prd? r then
B:Union(List(X),"failed"):=isTimes(r.term)
B case "failed" => "failed"
v:X:=1
for i in 1..#B repeat
r.term := B.i
v:=v * construct(r)
return v
return("failed")
takeHigh(x,n) ==
r:=destruct x
r case "failed" => "failed"
not positive? n => "failed"
r.lb:=r.ub - n::X + 1
construct(r)
takeLow(x,n) ==
r:=destruct x
r case "failed" => "failed"
not positive? n => "failed"
r.ub:=r.lb + n::X - 1
construct(r)
shiftRange(x,n) ==
r:=destruct x
r case "failed" => "failed"
r.term := subst(r.term,r.var=r.var-n)
r.lb := r.lb + n
r.ub := r.ub + n
construct(r)
shiftToZero(x) ==
r:=destruct x
r case "failed" => "failed"
r.term := subst(r.term,r.var=r.var+r.lb)
r.ub := r.ub - r.lb
r.lb := r.lb - r.lb
construct(r)
reverseOrder(x:X):UX ==
r:=destruct x
r case "failed" => "failed"
if sum? r or prd? r then
r.term := subst(r.term, r.var=r.ub - r.var)
r.ub := r.ub - r.lb -- b-a
r.lb := r.lb - r.lb -- 0
c:X :=1
else
a := r.lb
r.lb := r.ub
r.ub := a
c:X := -1
c * construct(r)
getCoefficient(x:X,k:X,invar:X):UX ==
r:=destruct x
r case "failed" => "failed"
y:X:=eval(r.term,r.var=k)/(invar)^k
freeOf?(y,invar) => y
"failed"
evalOp(x:X,a:X,b:X):UX ==
r:=destruct x
r case "failed" => "failed"
if R has RetractableTo(Integer) then
d:Union(Integer,"failed"):=retractIfCan(b-a)
if d case "failed" or int? r then
r.lb:=a
r.ub:=b
return construct(r)
else
if d case Integer then
if positive? d then
l:List X :=[eval(r.term,r.var=a+i::X) for i in 0..d]
else
return "failed"
sum? r => reduce(_+,l)@X
prd? r => reduce(_*,l)@X
else
r.lb:=a
r.ub:=b
return construct(r)
combineSplitArgs(x:X):UX ==
p:=isPlus x ; t:=isTimes x
p case "failed" and t case "failed" => "failed"
x
pullConstant(x:X):UX ==
r:=destruct x
r case "failed" => "failed"
tr:=isTimes r.term
tr case "failed" => "failed"
fs:LX:=[s for s in tr | freeOf?(s,r.var)]
if not empty? fs then
ff:X:=reduce(_*,fs)
r.term := r.term / ff
else
ff:X:=1::X
return(ff * construct r)
pushConstant(x:X,c:X):UX ==
r:=destruct x
r case "failed" => "failed"
--not freeOf?(r.term,c) => "failed"
not freeOf?(c,r.var) => "failed"
not freeOf?(c,r.sym) => "failed"
r.term := c * r.term
construct r
-----------------
getLowerBound x ==
r:=destruct x
r case "failed" => "failed"
r.lb
getUpperBound x ==
r:=destruct x
r case "failed" => "failed"
r.ub
getOpTerm x ==
r:=destruct x
r case "failed" => "failed"
subst(r.term,r.var=r.sym)
getOpTermAs(x,y) ==
r:=destruct x
r case "failed" => "failed"
subst(r.term,r.var=y)
getOpType x ==
r:=destruct x
r case "failed" => "failed"
r.typ
setLowerBound(x,a) ==
r:=destruct x
r case "failed" => "failed"
r.lb:=a
construct(r)
setUpperBound(x,b) ==
r:=destruct x
r case "failed" => "failed"
r.ub:=b
construct(r)
setOpTerm(x,f) ==
r:=destruct x
r case "failed" => "failed"
r.term:=f(r.var)
construct(r)
setOpType(x,s) ==
r:=destruct x
r case "failed" => "failed"
not member?(s,spi_ops) => "failed"
r.typ:=s
construct(r)
sfloor(x:X):X ==
f:BasicOperator := operator('floor,1)
if R has RetractableTo(Integer) then
rx:Union(Fraction Integer, "failed"):=retractIfCan(x)
if rx case Fraction(Integer) then return floor(rx)::X
f(x)
sceiling(x:X):X ==
c:BasicOperator := operator('ceiling,1)
if R has RetractableTo(Integer) then
rx:Union(Fraction Integer, "failed"):=retractIfCan(x)
if rx case Fraction(Integer) then return ceiling(rx)::X
c(x)
startsplit(x:X):X ==
ss:BasicOperator := operator('startSplit,1)
if R has RetractableTo(Integer) then
rx:Union(Integer, "failed"):=retractIfCan(x)
if rx case Integer then
q:Fraction Integer:=rx/2
if odd? rx then
return ceiling(q)::X
else
return (1+ceiling(q))::X
ss(x)
-----------------
-- x -> N/D -> N1*N2*...Nk / D1*D2...Dj -> ([N1,N2,...],[D1,D2,...])
splitTimesND(x:X):List List X ==
--x:=normalize x
nx:=numerator x
dx:=denominator x
lnx:=isTimes nx
ldx:=isTimes dx
qnx:List X:=[]
qdx:List X:=[]
if lnx case "failed" then
qnx:=[nx]
else
qnx:=lnx
if ldx case "failed" then
qdx:=[dx]
else
qdx:=ldx
return [qnx,qdx]
-- [A1,A2,...] -> ([Ai not op(s)],[Ai is op(s)])
splitFactorOp(x:List X,s:Symbol):List List X ==
b:=[t for t in x | is?(t,s)]
a:=[t for t in x | not is?(t,s)]
[a,b]
makeTimesFactor(l:List X):X ==
if not empty? l then
reduce(_*,[t::X for t in l])
else
l:X:=1
invertProd(x:X):UX ==
r:=destruct x
if r case SPIR then
qr:QUOTR:=divide(1::X,r.term)$X
r.term := qr.quotient
construct(r)
else
"failed"
-- prod(A*f(i),i=0..b)/prod(B*f(i),i=0..c) -> evaluated
-- assumes lb shifted to zero, or if same ub, then reverse
-- and shift to zero-
reduceQuoProd(x:X,y:X):Union(X,"failed") ==
rx:=destruct x
ry:=destruct y
rx case "failed" => "failed"
ry case "failed" => "failed"
xy:X:=new()$Symbol::X
tx:=subst(rx.term,rx.var=xy)
ty:=subst(ry.term,ry.var=xy)
fac:X:=tx/ty
not freeOf?(fac,xy) => "failed"
if rx.lb = ry.lb then
rx.term := subst(rx.term,rx.var=rx.var+rx.lb)
rx.ub := rx.ub - rx.lb
rx.lb := rx.lb - rx.lb
ry.term := subst(ry.term,ry.var=ry.var+ry.lb)
ry.ub := ry.ub - ry.lb
ry.lb := ry.lb - ry.lb
else
if rx.ub = ry.ub then -- reverse
rx.term := subst(rx.term, rx.var=rx.ub - rx.var)
rx.ub := rx.ub - rx.lb -- b-a
rx.lb := rx.lb - rx.lb -- 0
ry.term := subst(ry.term, ry.var=ry.ub - ry.var)
ry.ub := ry.ub - ry.lb -- b-a
ry.lb := ry.lb - ry.lb -- 0
else
return "failed"
a:=rx.lb - ry.lb -- should be 0
b:=rx.ub - ry.ub
c:Union(Integer,"failed"):=retractIfCan(b)
c case "failed" or a~=0 => "failed"
n:Integer:=c::Integer
r:=copy ry
exp:X:=rx.ub + 1::X
fac:=fac^exp --normalize(fac^exp)$EFSP
n=0 => fac
if positive? n then
r.term := subst(r.term,r.var=r.var+ry.ub+1)
r.lb:=0::X -- shift to zero
r.ub:=n::X-1
return fac * construct(r)
else
r.term := subst(r.term,r.var=r.var+rx.ub+1)
r.lb:=0::X
r.ub:=-n::X -1
return fac / construct(r)
return("failed")
-- prod(f,i=a..b) -> f(a)*f(a+1)*...*f(b) if b-a is integer
-- Ex. expandProductIfCan product(sin(k),k=N-3..N)
expandProductIfCan(x:X):UX ==
not is?(x,'%defprod) => "failed"
r:=destruct x
r case "failed" => "failed"
if R has RetractableTo(Integer) then
d:Union(Integer,"failed"):=retractIfCan(r.ub - r.lb)
d case "failed" => "failed"
if positive? d or d=0 then
l:List X :=[eval(r.term,r.var=r.lb+i::X) for i in 0..d]
else
return "failed"
reduce(_*,l)@X
else
return "failed"
nfProd(x:X):NFPRD ==
sx:=splitTimesND x
nsx:=splitFactorOp(first sx, '%defprod)
dsx:=splitFactorOp(second sx, '%defprod)
fn:=makeTimesFactor(first nsx)
fd:=makeTimesFactor(first dsx)
[fn/fd,second nsx, second dsx]$NFPRD
reduceProductND(x:NFPRD):NFPRD ==
r:=copy x
for a in r.nump repeat
for b in r.denp repeat
q:=reduceQuoProd(a,b)
if not (q case "failed") then
r.denp:=remove(b,r.denp)
r.nump:=remove(a,r.nump)
r.fac:=q*r.fac
return r
reduceProducts(x:X):X ==
r:=nfProd x
rr:NFPRD:=[0::X,[]::LX,[]::LX]$NFPRD
while true repeat
rr:=reduceProductND r
if empty? rr.nump or empty? rr.denp then break
if rr=r then break else r:=rr
rv:X:=rr.fac * makeTimesFactor(rr.nump) / makeTimesFactor(rr.denp)
normalize(rv)$EFSP
--------------------------------------
fricas
)abbrev package HYPGEOM HyperGeometric
--------------------------------------
HyperGeometric(): Exports == Implementation where
R ==> Fraction Integer -- better than Integer for this purpose
X ==> Expression R
NNI ==> NonNegativeInteger
SMP ==> SparseMultivariatePolynomial(R, Kernel X) -- if R has IntegralDomain
SUP ==> SparseUnivariatePolynomial SMP
EFSP ==> ElementaryFunctionStructurePackage(R,X)
FACSUP ==> Factored SUP
FACREC ==> Record(factor:SUP, exponent:NNI)
HYPER ==> Record(ap:List X, bq:List X, fac:X)
PARAM ==> Record(ab:List SUP, const:List SUP)
Exports == with
factoredForm : (X->X) -> Fraction(FACSUP)
getParameters : FACSUP -> PARAM
convert : SUP -> X
isPoly? :(X->X) -> Boolean
isQuoRat? :(X->X) -> Boolean
pFq : (X->X) -> HYPER
pFq0 : (X->X) -> HYPER
pFq00 : (X->X) -> HYPER
construct: (HYPER,Symbol) -> X
transformIfCan : Fraction FACSUP -> Union(HYPER,"failed")
display : HYPER -> OutputForm
hyperLookup : HYPER -> X
Implementation == add
factoredForm(f:X->X):Fraction(FACSUP) ==
ns:X:=new()$Symbol::X
q:=f(ns+1)/f(ns)
-- check free of sums/prods/ints dep on var
import from SumProdInt(R)
--nq:=reduceProducts(nq) --> NEW! trial only
--
nq:=normalize(reduceProducts q)$EFSP
dnq:=denom nq
nnq:=numer nq
unnq:=univariate(nnq,first tower(ns))
udnq:=univariate(dnq,first tower(ns))
funnq:=factor unnq
fudnq:=factor udnq
funnq/fudnq
-- check degree .... may be > 1
getParameters x ==
qm:=create()$SingletonAsOrderedSet
r:List FACREC:=factors x
l1:List SUP:=[]
l2:List SUP:=[]
for s in r repeat
p:SUP:=s.factor
if variables p = [qm]
then
lc:SMP:=leadingCoefficient p
q1:=reductum(p) --> /lc
xq1:X:=convert q1 ; xlc:X:=convert(lc::SUP)
--q1:=(xq1/xlc)@SUP -- need Fraction SUP ??????
for i in 1..s.exponent repeat
l1:=cons(q1,l1)
l2:=cons(lc::SUP, l2)
else
q2:=p
for i in 1..s.exponent repeat
l2:=cons(q2,l2)
return [l1,l2]$PARAM
convert(x:SUP):X ==
k:Kernel(X):=first tower('_%::X)
y:SMP:=multivariate(x,k)$SMP
coerce(y)$X
transformIfCan(x:Fraction FACSUP):Union(HYPER,"failed") ==
nx:FACSUP:=numer x
dx:FACSUP:=denom x
rnx:List FACREC:=factors nx
rdx:List FACREC:=factors dx
dn:List NNI:=[degree t.factor for t in rnx]
dd:List NNI:=[degree t.factor for t in rdx]
any?((n:NNI):Boolean+->(n>1),dn) => "failed"
any?((n:NNI):Boolean+->(n>1),dn) => "failed"
C:X:=1
A:List X:=[]
B:List X:=[]
for r in rnx repeat
cr:X:=convert(r.factor)--$HYPGEOM
crr:X:=convert(reductum r.factor)--$HYPGEOM
if degree(r.factor)=0 then
C:=C * cr^(r.exponent)
else -- degree=1
lc:X:=D(cr,'$) -- leadingCoeff (not 0)
rr:X:=crr
a:X:=rr/lc
for i in 1..r.exponent repeat
A:=cons(a,A)
C:=C*lc
-- same for B (*->/)
for r in rdx repeat
cr:X:=convert(r.factor)--$HYPGEOM
crr:X:=convert(reductum r.factor)--$HYPGEOM
if degree(r.factor)=0 then
C:=C / cr^(r.exponent)
else -- degree=1
lc:X:=D(cr,'$) -- leadingCoeff (not 0)
rr:X:=crr
b:X:=rr/lc
for i in 1..r.exponent repeat
B:=cons(b,B)
C:=C / lc
-- satisfy pFq condition (could be done later in pFq?)
if member?(1$X,B) then
B:=delete(B,position(1$X,B))
else
A:=cons(1$X,A)
[A,B,C]$HYPER
pFq(f:X->X):HYPER ==
ff:=factoredForm(f)
gpn:=getParameters(numer ff).ab
gpd:=getParameters(denom ff).ab
[[convert t for t in gpn],[convert t for t in gpd],1$X]$HYPER
-- 1 in bq => remove it otherwise add 1 to aq
-- coeff and f(0) todo
-- pFq(k+->cos(k)) , pre-filter f(n+1)/f(n) rational?
isPoly?(f:X->X):Boolean ==
nsym:Symbol:=new()$Symbol
ns:X:=nsym::X
q:=f(ns)
D(q,nsym)$X=0 => true
nq:=normalize(q)$EFSP
dnq:=denom nq
nnq:=numer nq
unnq:=univariate(nnq,first tower(ns))
udnq:=univariate(dnq,first tower(ns))
empty?(variables unnq) => false
not empty?(variables udnq) => false
true
isQuoRat?(f:X->X):Boolean ==
isPoly?(f) => true -- accelerates e.g. k^1234567
nsym:Symbol:=new()$Symbol
ns:X:=nsym::X
q:=f(ns+1)/f(ns)
D(q,nsym)$X=0 => true
nq:=normalize(q)$EFSP
dnq:=denom nq
nnq:=numer nq
unnq:=univariate(nnq,first tower(ns))
udnq:=univariate(dnq,first tower(ns))
empty?(variables unnq) and empty?(variables udnq) => false
true
buildConst(x:List SUP):X ==
empty?(x) => 1$X
p:SUP:=reduce(_*,x)
convert p
pFq00(f:X->X):HYPER ==
ff:=factoredForm(f)
gpn:=getParameters(numer ff)
gpd:=getParameters(denom ff)
a:=[convert t for t in gpn.ab]
b:=[convert t for t in gpd.ab]
if member?(1$X,b) then
b:=delete(b,position(1$X,b))
else
a:=cons(1$X,a)
c:=convert unit(numer ff)
ca:=buildConst(gpn.const)
cb:=buildConst(gpd.const)
[a,b,c*ca*cb]$HYPER
pFq0(f:X->X):HYPER ==
ff:=factoredForm f
hff:=transformIfCan ff
hff case "failed" => error "not hyer?"
c:=convert unit(numer ff) -- necessary !!!! factor
hff.fac := c * hff.fac
return hff
construct(x,s) ==
v:=s::X
a:=x.ap
b:=x.bq
c:=x.fac
av:=[paren(v+t) for t in a]
bv:=[paren(v+t) for t in b]
num:=1$X
den:=1$X
if not empty? av then
num:=reduce(_*,av)
if not empty? bv then
den:=reduce(_*,bv)
c*num/den/(v+1$X)
display(x:HYPER):OutputForm ==
OF ==> OutputForm
a:=[s::OF for s in x.ap]
b:=[s::OF for s in x.bq]
c:=(x.fac)::OF
p:=(#a)::OF
q:=(#b)::OF
A:=sub(presub('F::OF,p),q)
if #a < #b then
a:=append(a,['*::OF for i in 1..#b- #a])
else
b:=append(b,['*::OF for i in 1..#a- #b])
--B:=binomial(blankSeparate a, blankSeparate b)
B:=blankSeparate [A,matrix [a,b],bracket c]
--hconcat(A,B)
hyperLookup(x:HYPER):X ==
a:=x.ap
b:=x.bq
c:=x.fac
p:= #a
q:= #b
if p=1 then
if q=0 then
return (1-c)^(-a.1)
0$X</spad>
fricas
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8470101588931044298-25px001.spad
using old system compiler.
SPI abbreviates package SumProdInt
------------------------------------------------------------------------
initializing NRLIB SPI for SumProdInt
compiling into NRLIB SPI
****** Domain: R already in scope
****** Domain: R already in scope
****** Domain: R already in scope
compiling local sum? : Record(typ: Symbol,term: Expression R,var: Expression R,sym: Expression R,lb: Expression R,ub: Expression R) -> Boolean
Time: 0.02 SEC.
compiling local prd? : Record(typ: Symbol,term: Expression R,var: Expression R,sym: Expression R,lb: Expression R,ub: Expression R) -> Boolean
Time: 0 SEC.
compiling local int? : Record(typ: Symbol,term: Expression R,var: Expression R,sym: Expression R,lb: Expression R,ub: Expression R) -> Boolean
Time: 0 SEC.
compiling exported construct : Record(typ: Symbol,term: Expression R,var: Expression R,sym: Expression R,lb: Expression R,ub: Expression R) -> Expression R
Time: 0 SEC.
compiling exported destruct : Expression R -> Union(Record(typ: Symbol,term: Expression R,var: Expression R,sym: Expression R,lb: Expression R,ub: Expression R),failed)
Time: 0.02 SEC.
compiling local freeOfSPI? : Expression R -> Boolean
Time: 0 SEC.
compiling local checkSPI : Expression R -> Union(Expression R,summation,product,integral)
Time: 0 SEC.
compiling exported splitOp : (Expression R,Expression R) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported multiplyOp : (Expression R,Expression R) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported splitArgs : Expression R -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported takeHigh : (Expression R,Integer) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported takeLow : (Expression R,Integer) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported shiftRange : (Expression R,Expression R) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported shiftToZero : Expression R -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported reverseOrder : Expression R -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported getCoefficient : (Expression R,Expression R,Expression R) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported evalOp : (Expression R,Expression R,Expression R) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported combineSplitArgs : Expression R -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported pullConstant : Expression R -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported pushConstant : (Expression R,Expression R) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported getLowerBound : Expression R -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported getUpperBound : Expression R -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported getOpTerm : Expression R -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported getOpTermAs : (Expression R,Expression R) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported getOpType : Expression R -> Union(Symbol,failed)
Time: 0 SEC.
compiling exported setLowerBound : (Expression R,Expression R) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported setUpperBound : (Expression R,Expression R) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported setOpTerm : (Expression R,Expression R -> Expression R) -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported setOpType : (Expression R,Symbol) -> Union(Expression R,failed)
Time: 0 SEC.
compiling local sfloor : Expression R -> Expression R
Time: 0 SEC.
compiling local sceiling : Expression R -> Expression R
Time: 0 SEC.
compiling local startsplit : Expression R -> Expression R
Time: 0 SEC.
compiling exported splitTimesND : Expression R -> List List Expression R
Time: 0 SEC.
compiling exported splitFactorOp : (List Expression R,Symbol) -> List List Expression R
Time: 0 SEC.
compiling exported makeTimesFactor : List Expression R -> Expression R
Time: 0 SEC.
compiling exported invertProd : Expression R -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported reduceQuoProd : (Expression R,Expression R) -> Union(Expression R,failed)
Time: 0.02 SEC.
compiling local expandProductIfCan : Expression R -> Union(Expression R,failed)
Time: 0 SEC.
compiling exported nfProd : Expression R -> Record(fac: Expression R,nump: List Expression R,denp: List Expression R)
Time: 0 SEC.
compiling exported reduceProductND : Record(fac: Expression R,nump: List Expression R,denp: List Expression R) -> Record(fac: Expression R,nump: List Expression R,denp: List Expression R)
Time: 0.02 SEC.
compiling exported reduceProducts : Expression R -> Expression R
Time: 0 SEC.
(time taken in buildFunctor: 293)
;;; *** |SumProdInt| REDEFINED
;;; *** |SumProdInt| REDEFINED
Time: 0 SEC.
Warnings:
[1] sum?: typ has no value
[2] prd?: typ has no value
[3] int?: typ has no value
[4] construct: term has no value
[5] construct: var has no value
[6] construct: sym has no value
[7] construct: lb has no value
[8] construct: ub has no value
[9] splitOp: sym has no value
[10] splitOp: ub has no value
[11] splitOp: lb has no value
[12] multiplyOp: term has no value
[13] splitArgs: term has no value
[14] takeHigh: lb has no value
[15] takeHigh: ub has no value
[16] takeLow: ub has no value
[17] takeLow: lb has no value
[18] shiftRange: term has no value
[19] shiftRange: var has no value
[20] shiftRange: lb has no value
[21] shiftRange: ub has no value
[22] shiftToZero: term has no value
[23] shiftToZero: var has no value
[24] shiftToZero: lb has no value
[25] shiftToZero: ub has no value
[26] reverseOrder: term has no value
[27] reverseOrder: var has no value
[28] reverseOrder: ub has no value
[29] reverseOrder: lb has no value
[30] getCoefficient: term has no value
[31] getCoefficient: var has no value
[32] evalOp: lb has no value
[33] evalOp: ub has no value
[34] evalOp: term has no value
[35] evalOp: var has no value
[36] pullConstant: term has no value
[37] pullConstant: var has no value
[38] pushConstant: var has no value
[39] pushConstant: sym has no value
[40] pushConstant: term has no value
[41] getLowerBound: lb has no value
[42] getUpperBound: ub has no value
[43] getOpTerm: term has no value
[44] getOpTerm: var has no value
[45] getOpTerm: sym has no value
[46] getOpTermAs: term has no value
[47] getOpTermAs: var has no value
[48] getOpType: typ has no value
[49] setLowerBound: lb has no value
[50] setUpperBound: ub has no value
[51] setOpTerm: term has no value
[52] setOpTerm: var has no value
[53] setOpType: typ has no value
[54] makeTimesFactor: The conditional modes (Expression R) and (List (Expression R)) conflict
[55] invertProd: term has no value
[56] invertProd: quotient has no value
[57] reduceQuoProd: term has no value
[58] reduceQuoProd: var has no value
[59] reduceQuoProd: lb has no value
[60] reduceQuoProd: ub has no value
[61] expandProductIfCan: ub has no value
[62] expandProductIfCan: lb has no value
[63] expandProductIfCan: term has no value
[64] expandProductIfCan: var has no value
[65] reduceProductND: nump has no value
[66] reduceProductND: denp has no value
[67] reduceProductND: fac has no value
[68] reduceProducts: nump has no value
[69] reduceProducts: denp has no value
[70] reduceProducts: fac has no value
[71] reduceProducts: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce (% %)) (SIGNATURE number? ((Boolean) %)) (IF (has R (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
[72] reduceProducts: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has R (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace R)) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce (% %)) (SIGNATURE number? ((Boolean) %)) (IF (has R (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
Cumulative Statistics for Constructor SumProdInt
Time: 0.19 seconds
--------------non extending category----------------------
.. SumProdInt(#1) of cat
(|Join| (|OrderedSet|)
(CATEGORY |package|
(SIGNATURE |destruct|
((|Union|
(|Record| (|:| |typ| (|Symbol|)) (|:| |term| (|Expression| |#1|))
(|:| |var| (|Expression| |#1|))
(|:| |sym| (|Expression| |#1|))
(|:| |lb| (|Expression| |#1|))
(|:| |ub| (|Expression| |#1|)))
"failed")
(|Expression| |#1|)))
(SIGNATURE |construct|
((|Expression| |#1|)
(|Record| (|:| |typ| (|Symbol|)) (|:| |term| (|Expression| |#1|))
(|:| |var| (|Expression| |#1|))
(|:| |sym| (|Expression| |#1|))
(|:| |lb| (|Expression| |#1|))
(|:| |ub| (|Expression| |#1|)))))
(SIGNATURE |splitOp|
((|Union| (|Expression| |#1|) #1="failed") (|Expression| |#1|)
(|Expression| |#1|)))
(SIGNATURE |multiplyOp|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)
(|Expression| |#1|)))
(SIGNATURE |splitArgs|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)))
(SIGNATURE |takeHigh|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|) (|Integer|)))
(SIGNATURE |takeLow|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|) (|Integer|)))
(SIGNATURE |shiftRange|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)
(|Expression| |#1|)))
(SIGNATURE |shiftToZero|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)))
(SIGNATURE |reverseOrder|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)))
(SIGNATURE |getCoefficient|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)
(|Expression| |#1|) (|Expression| |#1|)))
(SIGNATURE |evalOp|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)
(|Expression| |#1|) (|Expression| |#1|)))
(SIGNATURE |combineSplitArgs|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)))
(SIGNATURE |pullConstant|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)))
(SIGNATURE |pushConstant|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)
(|Expression| |#1|)))
(SIGNATURE |reduceQuoProd|
((|Union| (|Expression| |#1|) "failed") (|Expression| |#1|)
(|Expression| |#1|)))
(SIGNATURE |getLowerBound|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)))
(SIGNATURE |getUpperBound|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)))
(SIGNATURE |getOpTerm|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)))
(SIGNATURE |getOpTermAs|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)
(|Expression| |#1|)))
(SIGNATURE |getOpType|
((|Union| (|Symbol|) "failed") (|Expression| |#1|)))
(SIGNATURE |setLowerBound|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)
(|Expression| |#1|)))
(SIGNATURE |setUpperBound|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)
(|Expression| |#1|)))
(SIGNATURE |setOpTerm|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)
(|Mapping| (|Expression| |#1|) (|Expression| |#1|))))
(SIGNATURE |setOpType|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|) (|Symbol|)))
(SIGNATURE |splitTimesND|
((|List| (|List| (|Expression| |#1|))) (|Expression| |#1|)))
(SIGNATURE |splitFactorOp|
((|List| (|List| (|Expression| |#1|))) (|List| (|Expression| |#1|))
(|Symbol|)))
(SIGNATURE |makeTimesFactor|
((|Expression| |#1|) (|List| (|Expression| |#1|))))
(SIGNATURE |invertProd|
((|Union| (|Expression| |#1|) #1#) (|Expression| |#1|)))
(SIGNATURE |nfProd|
((|Record| (|:| |fac| (|Expression| |#1|))
(|:| |nump| (|List| (|Expression| |#1|)))
(|:| |denp| (|List| (|Expression| |#1|))))
(|Expression| |#1|)))
(SIGNATURE |reduceProductND|
((|Record| (|:| |fac| (|Expression| |#1|))
(|:| |nump| (|List| (|Expression| |#1|)))
(|:| |denp| (|List| (|Expression| |#1|))))
(|Record| (|:| |fac| (|Expression| |#1|))
(|:| |nump| (|List| (|Expression| |#1|)))
(|:| |denp| (|List| (|Expression| |#1|))))))
(SIGNATURE |reduceProducts|
((|Expression| |#1|) (|Expression| |#1|))))) has no
(|FunctionSpace| |#1|) finalizing NRLIB SPI
Processing SumProdInt for Browser database:
--->-->SumProdInt(constructor): Not documented!!!!
--->-->SumProdInt((destruct ((Union (Record (: typ (Symbol)) (: term (Expression R)) (: var (Expression R)) (: sym (Expression R)) (: lb (Expression R)) (: ub (Expression R))) failed) (Expression R)))): Not documented!!!!
--->-->SumProdInt((construct ((Expression R) (Record (: typ (Symbol)) (: term (Expression R)) (: var (Expression R)) (: sym (Expression R)) (: lb (Expression R)) (: ub (Expression R)))))): Not documented!!!!
--------(splitOp ((Union (Expression R) failed) (Expression R) (Expression R)))---------
--------(multiplyOp ((Union (Expression R) failed) (Expression R) (Expression R)))---------
--------(splitArgs ((Union (Expression R) failed) (Expression R)))---------
--------(takeHigh ((Union (Expression R) failed) (Expression R) (Integer)))---------
--------(takeLow ((Union (Expression R) failed) (Expression R) (Integer)))---------
--->-->SumProdInt((takeLow ((Union (Expression R) failed) (Expression R) (Integer)))): Improper first word in comments: The
"The takeLow function gets the lower terms of summation,{} product and definite integral. takeLow(expr,{} \\spad{n}),{} expr - summation,{} product or definite integral,{} \\spad{n} - number of terms to be fetched if expr is summation or product else the limit if expr is a definite integral."
--------(shiftRange ((Union (Expression R) failed) (Expression R) (Expression R)))---------
--->-->SumProdInt((shiftRange ((Union (Expression R) failed) (Expression R) (Expression R)))): Improper first word in comments: The
"The shifRange function shifts or rearranges summation,{} product and definite integral. shiftRange(expr,{} \\spad{n}),{} expr - summation,{} product or definite integral,{} \\spad{n} - value by which expr should be shifted"
--------(shiftToZero ((Union (Expression R) failed) (Expression R)))---------
--------(reverseOrder ((Union (Expression R) failed) (Expression R)))---------
--------(getCoefficient ((Union (Expression R) failed) (Expression R) (Expression R) (Expression R)))---------
--------(evalOp ((Union (Expression R) failed) (Expression R) (Expression R) (Expression R)))---------
--------(combineSplitArgs ((Union (Expression R) failed) (Expression R)))---------
--->-->SumProdInt((pullConstant ((Union (Expression R) failed) (Expression R)))): Not documented!!!!
--->-->SumProdInt((pushConstant ((Union (Expression R) failed) (Expression R) (Expression R)))): Not documented!!!!
--->-->SumProdInt((reduceQuoProd ((Union (Expression R) failed) (Expression R) (Expression R)))): Not documented!!!!
--->-->SumProdInt((getLowerBound ((Union (Expression R) failed) (Expression R)))): Not documented!!!!
--->-->SumProdInt((getUpperBound ((Union (Expression R) failed) (Expression R)))): Not documented!!!!
--->-->SumProdInt((getOpTerm ((Union (Expression R) failed) (Expression R)))): Not documented!!!!
--->-->SumProdInt((getOpTermAs ((Union (Expression R) failed) (Expression R) (Expression R)))): Not documented!!!!
--->-->SumProdInt((getOpType ((Union (Symbol) failed) (Expression R)))): Not documented!!!!
--->-->SumProdInt((setLowerBound ((Union (Expression R) failed) (Expression R) (Expression R)))): Not documented!!!!
--->-->SumProdInt((setUpperBound ((Union (Expression R) failed) (Expression R) (Expression R)))): Not documented!!!!
--->-->SumProdInt((setOpTerm ((Union (Expression R) failed) (Expression R) (Mapping (Expression R) (Expression R))))): Not documented!!!!
--->-->SumProdInt((setOpType ((Union (Expression R) failed) (Expression R) (Symbol)))): Not documented!!!!
--->-->SumProdInt((splitTimesND ((List (List (Expression R))) (Expression R)))): Not documented!!!!
--->-->SumProdInt((splitFactorOp ((List (List (Expression R))) (List (Expression R)) (Symbol)))): Not documented!!!!
--->-->SumProdInt((makeTimesFactor ((Expression R) (List (Expression R))))): Not documented!!!!
--->-->SumProdInt((invertProd ((Union (Expression R) failed) (Expression R)))): Not documented!!!!
--->-->SumProdInt((nfProd ((Record (: fac (Expression R)) (: nump (List (Expression R))) (: denp (List (Expression R)))) (Expression R)))): Not documented!!!!
--->-->SumProdInt((reduceProductND ((Record (: fac (Expression R)) (: nump (List (Expression R))) (: denp (List (Expression R)))) (Record (: fac (Expression R)) (: nump (List (Expression R))) (: denp (List (Expression R))))))): Not documented!!!!
--->-->SumProdInt((reduceProducts ((Expression R) (Expression R)))): Not documented!!!!
--->-->SumProdInt(): Missing Description
; compiling file "/var/aw/var/LatexWiki/SPI.NRLIB/SPI.lsp" (written 28 NOV 2024 09:29:55 PM):
; wrote /var/aw/var/LatexWiki/SPI.NRLIB/SPI.fasl
; compilation finished in 0:00:00.176
------------------------------------------------------------------------
SumProdInt is now explicitly exposed in frame initial
SumProdInt will be automatically loaded when needed from
/var/aw/var/LatexWiki/SPI.NRLIB/SPI
HYPGEOM abbreviates package HyperGeometric
------------------------------------------------------------------------
initializing NRLIB HYPGEOM for HyperGeometric
compiling into NRLIB HYPGEOM
compiling exported factoredForm : Expression Fraction Integer -> Expression Fraction Integer -> Fraction Factored SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer)
Time: 0.04 SEC.
compiling exported getParameters : Factored SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer) -> Record(ab: List SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer),const: List SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer))
Time: 0.01 SEC.
compiling exported convert : SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer) -> Expression Fraction Integer
Time: 0 SEC.
compiling exported transformIfCan : Fraction Factored SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer) -> Union(Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer),failed)
Time: 0.04 SEC.
compiling exported pFq : Expression Fraction Integer -> Expression Fraction Integer -> Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer)
Time: 0 SEC.
compiling exported isPoly? : Expression Fraction Integer -> Expression Fraction Integer -> Boolean
Time: 0 SEC.
compiling exported isQuoRat? : Expression Fraction Integer -> Expression Fraction Integer -> Boolean
Time: 0 SEC.
compiling local buildConst : List SparseUnivariatePolynomial SparseMultivariatePolynomial(Fraction Integer,Kernel Expression Fraction Integer) -> Expression Fraction Integer
Time: 0 SEC.
compiling exported pFq00 : Expression Fraction Integer -> Expression Fraction Integer -> Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer)
Time: 0.01 SEC.
compiling exported pFq0 : Expression Fraction Integer -> Expression Fraction Integer -> Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer)
Time: 0 SEC.
compiling exported construct : (Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer),Symbol) -> Expression Fraction Integer
Time: 0 SEC.
compiling exported display : Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer) -> OutputForm
processing macro definition OF ==> OutputForm
Time: 0 SEC.
compiling exported hyperLookup : Record(ap: List Expression Fraction Integer,bq: List Expression Fraction Integer,fac: Expression Fraction Integer) -> Expression Fraction Integer
Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |HyperGeometric| REDEFINED
;;; *** |HyperGeometric| REDEFINED
Time: 0 SEC.
Warnings:
[1] factoredForm: not known that (AlgebraicallyClosedField) is of mode (CATEGORY domain (IF (has (Fraction (Integer)) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Fraction (Integer)))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce (% %)) (SIGNATURE number? ((Boolean) %)) (IF (has (Fraction (Integer)) (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
[2] factoredForm: not known that (TranscendentalFunctionCategory) is of mode (CATEGORY domain (IF (has (Fraction (Integer)) (IntegralDomain)) (PROGN (ATTRIBUTE (AlgebraicallyClosedFunctionSpace (Fraction (Integer)))) (ATTRIBUTE (TranscendentalFunctionCategory)) (ATTRIBUTE (CombinatorialOpsCategory)) (ATTRIBUTE (LiouvillianFunctionCategory)) (ATTRIBUTE (SpecialFunctionCategory)) (SIGNATURE reduce (% %)) (SIGNATURE number? ((Boolean) %)) (IF (has (Fraction (Integer)) (PolynomialFactorizationExplicit)) (ATTRIBUTE (PolynomialFactorizationExplicit)) noBranch) (SIGNATURE setSimplifyDenomsFlag ((Boolean) (Boolean))) (SIGNATURE getSimplifyDenomsFlag ((Boolean)))) noBranch))
[3] getParameters: exponent has no value
[4] getParameters: l1 has no value
[5] transformIfCan: exponent has no value
[6] transformIfCan: B has no value
[7] transformIfCan: A has no value
[8] pFq00: ab has no value
[9] pFq00: const has no value
[10] pFq0: fac has no value
[11] construct: ap has no value
[12] construct: bq has no value
[13] construct: fac has no value
[14] display: ap has no value
[15] display: bq has no value
[16] display: fac has no value
[17] hyperLookup: ap has no value
[18] hyperLookup: bq has no value
[19] hyperLookup: fac has no value
Cumulative Statistics for Constructor HyperGeometric
Time: 0.16 seconds
finalizing NRLIB HYPGEOM
Processing HyperGeometric for Browser database:
--->-->HyperGeometric(constructor): Not documented!!!!
--->-->HyperGeometric((factoredForm ((Fraction (Factored (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer)))))))) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((getParameters ((Record (: ab (List (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer)))))))) (: const (List (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer))))))))) (Factored (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer)))))))))): Not documented!!!!
--->-->HyperGeometric((convert ((Expression (Fraction (Integer))) (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer))))))))): Not documented!!!!
--->-->HyperGeometric((isPoly? ((Boolean) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((isQuoRat? ((Boolean) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((pFq ((Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((pFq0 ((Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((pFq00 ((Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Mapping (Expression (Fraction (Integer))) (Expression (Fraction (Integer))))))): Not documented!!!!
--->-->HyperGeometric((construct ((Expression (Fraction (Integer))) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) (Symbol)))): Not documented!!!!
--->-->HyperGeometric((transformIfCan ((Union (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer))))) failed) (Fraction (Factored (SparseUnivariatePolynomial (SparseMultivariatePolynomial (Fraction (Integer)) (Kernel (Expression (Fraction (Integer))))))))))): Not documented!!!!
--->-->HyperGeometric((display ((OutputForm) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer)))))))): Not documented!!!!
--->-->HyperGeometric((hyperLookup ((Expression (Fraction (Integer))) (Record (: ap (List (Expression (Fraction (Integer))))) (: bq (List (Expression (Fraction (Integer))))) (: fac (Expression (Fraction (Integer)))))))): Not documented!!!!
--->-->HyperGeometric(): Missing Description
; compiling file "/var/aw/var/LatexWiki/HYPGEOM.NRLIB/HYPGEOM.lsp" (written 28 NOV 2024 09:29:55 PM):
; wrote /var/aw/var/LatexWiki/HYPGEOM.NRLIB/HYPGEOM.fasl
; compilation finished in 0:00:00.084
------------------------------------------------------------------------
HyperGeometric is now explicitly exposed in frame initial
HyperGeometric will be automatically loaded when needed from
/var/aw/var/LatexWiki/HYPGEOM.NRLIB/HYPGEOMfricas
)set output tex off
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)set output algebra on
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)set userlevel development
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)set break resume
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)expose UnittestCount UnittestAux Unittest
UnittestCount is now explicitly exposed in frame initial
UnittestAux is now explicitly exposed in frame initial
Unittest is now explicitly exposed in frame initial
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)expose hypergeom
hypergeom is not a known constructor. You can make the constructor
known to the system by loading it.
R ==> Fraction Integer -- better than Integer for this purpose
Type: Void
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X ==> Expression R
Type: Void
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NNI ==> NonNegativeInteger
Type: Void
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SMP ==> SparseMultivariatePolynomial(R, Kernel X) -- if R has IntegralDomain
Type: Void
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SUP ==> SparseUnivariatePolynomial SMP
Type: Void
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EFSP ==> ElementaryFunctionStructurePackage(R,X)
Type: Void
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FACSUP ==> Factored SUP
Type: Void
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FACREC ==> Record(factor:SUP, exponent:NNI)
Type: Void
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HYPER ==> Record(ap:List X, bq:List X, fac:X)
Type: Void
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--------------------------------------------------------------
-- Examples from https://www2.math.upenn.edu/~wilf/Downld.html
-- A=B, Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger
--------------------------------------------------------------
testsuite "identHG"
All user variables and function definitions have been cleared.
Type: Void
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testcase "pFq0"
All user variables and function definitions have been cleared.
Type: Void
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T1(n) == 1/(2*n-1)/factorial(2*n+1); T1(n)
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Compiling function T1 with type Variable(n) -> Expression(Integer)
1
(2) -------------------
(2 n - 1)(2 n + 1)!
Type: Expression(Integer)
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T2(k) == binomial(n,k)*(-1)^k/factorial(k); T2(k)
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Compiling function T2 with type Variable(k) -> Expression(Integer)
k n
(- 1) ( )
k
(3) ---------
k!
Type: Expression(Integer)
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T3(k) == (-1)^k*(x/2)^(2*k+p)/factorial(k)/factorial(k+p); T3(k)
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Compiling function T3 with type Variable(k) -> Expression(Integer)
k x p + 2 k
(- 1) (-)
2
(4) ----------------
k!(p + k)!
Type: Expression(Integer)
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T4(k) == binomial(n,k)^2*binomial(n+k,k)^2;T4(k)
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Compiling function T4 with type Variable(k) -> Expression(Integer)
2 2
n n + k
(5) ( ) ( )
k k
Type: Expression(Integer)
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T5(n) == binomial(n,k)^2*binomial(n+k,k)^2;T2(n) -- 1-k > 0 => k=0 => not hg
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Compiling function T2 with type Variable(n) -> Expression(Integer)
n
(- 1)
(6) ------
n!
Type: Expression(Integer)
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T6(k) == n^k; T6(k)
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Compiling function T6 with type Variable(k) -> Expression(Integer)
k
(7) n
Type: Expression(Integer)
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T7(k) == factorial(k);T7(k)
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Compiling function T7 with type Variable(k) -> Expression(Integer)
(8) k!
Type: Expression(Integer)
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T8(k) == factorial(2*k+7)/factorial(k-3); T8(k)
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Compiling function T8 with type Variable(k) -> Expression(Integer)
(2 k + 7)!
(9) ----------
(k - 3)!
Type: Expression(Integer)
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T9(k) == (k^2-1)*factorial(3*k+1)/(2*k+7)/factorial(k+3); T9(k)
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Compiling function T9 with type Variable(k) -> Expression(Integer)
2
(k - 1)(3 k + 1)!
(10) ------------------
(2 k + 7)(k + 3)!
Type: Expression(Integer)
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T10(k) == z^k/factorial(k); T10(k)
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Compiling function T10 with type Variable(k) -> Expression(Integer)
k
z
(11) --
k!
Type: Expression(Integer)
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T11(k) == 2^k/factorial(k)^2; T11(k)
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Compiling function T11 with type Variable(k) -> Expression(Integer)
k
2
(12) ---
2
k!
Type: Expression(Integer)
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T12(k) == binomial(n,k); T12(k)
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Compiling function T12 with type Variable(k) -> Expression(Integer)
n
(13) ( )
k
Type: Expression(Integer)
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sym(x,d,n) == if n=0 then 1 else product(x-j*d,j=0..n-1); sym(x,d,n);
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Compiling function sym with type (Variable(x), Variable(d), Variable
(n)) -> Expression(Integer)
Type: Expression(Integer)
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T13(k) == binomial(n,k)*sym(x,d,k)*sym(y,d,n-k); T13(k)
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Compiling function sym with type (Variable(x), Variable(d), Variable
(k)) -> Expression(Integer)
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Compiling function sym with type (Variable(y), Variable(d),
Polynomial(Integer)) -> Expression(Integer)
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Compiling function T13 with type Variable(k) -> Expression(Integer)
k - 1 n - k - 1
n ++-++ ++-++
(15) ( ) | | x - d j | | y - d j
k | | | |
j = 0 j = 0
Type: Expression(Integer)
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testT(T) ==>
output "============================================================"
output("summand:", T(k))
output "------------------------------------------------------------"
-- output("value @0:", T(0))
output("factored form: ", ff:=factoredForm T)
output ""
output("hypergeom rec: ", rr:=pFq0(T))
output ""
output("hypergeom sym: ", display(rr))
output ""
output("reconstruct from rec :", cc:=construct(rr,m)$HYPGEOM)
output ""
output("check if ZERO: ", dd:=reduceProducts(T(m+1)/T(m)) - distribute cc)
output "=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~"
testEquals("normalize dd","0")
Type: Void
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--- macro doesn't work properly in sandbox!?
testT T1
============================================================
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Compiling function T1 with type Variable(k) -> Expression(Integer)
1
summand: -------------------
(2 k - 1)(2 k + 1)!
------------------------------------------------------------fricas
Compiling function T1 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
1 1
- (? - -)
4 2
factored form: ---------------------
1 3
(? + -)(? + 1)(? + -)
2 2
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
1 1
- (? - -)
4 2
(17) ---------------------
1 3
(? + -)(? + 1)(? + -)
2 2Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
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display rr
There are 4 exposed and 7 unexposed library operations named display
having 1 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op display
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named
display with argument type(s)
Variable(rr)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Continuing to read the file...
testT T2
============================================================
k n
(- 1) ( )
k
summand: ---------
k!
------------------------------------------------------------
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Compiling function T2 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
? - n
factored form: --------
2
(? + 1)
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
? - n
(18) --------
2
(? + 1)Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
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display rr
There are 4 exposed and 7 unexposed library operations named display
having 1 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op display
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named
display with argument type(s)
Variable(rr)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Continuing to read the file...
testT T3
============================================================
k x p + 2 k
(- 1) (-)
2
summand: ----------------
k!(p + k)!
------------------------------------------------------------
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Compiling function T3 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
1 2
- x
4
factored form: - ------------------
(? + 1)(? + p + 1)
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
1 2
- x
4
(19) - ------------------
(? + 1)(? + p + 1)Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
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display rr
There are 4 exposed and 7 unexposed library operations named display
having 1 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op display
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named
display with argument type(s)
Variable(rr)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Continuing to read the file...
testT T4
============================================================
2 2
n n + k
summand: ( ) ( )
k k
------------------------------------------------------------
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Compiling function T4 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
2 2
(? - n) (? + n + 1)
factored form: --------------------
4
(? + 1)
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
2 2
(? - n) (? + n + 1)
(20) --------------------
4
(? + 1)Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
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display rr
There are 4 exposed and 7 unexposed library operations named display
having 1 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op display
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named
display with argument type(s)
Variable(rr)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Continuing to read the file...
testT T5
============================================================
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Compiling function T5 with type Variable(k) -> Expression(Integer)
2
2 k
summand: ( )
k
------------------------------------------------------------fricas
Compiling function T5 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
2
(? + k + 1)
factored form: ------------
2
(? - k + 1)
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
2
(? + k + 1)
(21) ------------
2
(? - k + 1)Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
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display rr
There are 4 exposed and 7 unexposed library operations named display
having 1 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op display
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named
display with argument type(s)
Variable(rr)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Continuing to read the file...
testT T6
============================================================
k
summand: n
------------------------------------------------------------
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Compiling function T6 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
factored form: n
hypergeom rec: [ap = [1], bq = [], fac = n]
+1+
hypergeom sym: F | | [n]
1 0 +*+
n(m + 1)
reconstruct from rec : --------
m + 1fricas
Compiling function T6 with type Polynomial(Integer) -> Expression(
Integer)fricas
Compiling function T6 with type Variable(m) -> Expression(Integer)
check if ZERO: 0
=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~=~
Type: Void
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ff
(23) n
Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
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display rr
+1+
(24) F | | [n]
1 0 +*+
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testT T7
============================================================
summand: k!
------------------------------------------------------------
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Compiling function T7 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
factored form: ? + 1
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
(25) ? + 1Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
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display rr
+1+
(26) F | | [n]
1 0 +*+
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testT T8
============================================================
(2 k + 7)!
summand: ----------
(k - 3)!
------------------------------------------------------------
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Compiling function T8 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
9
4 (? + 4)(? + -)
2
factored form: ----------------
? - 2
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
9
4 (? + 4)(? + -)
2
(27) ----------------
? - 2Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
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display rr
+1+
(28) F | | [n]
1 0 +*+
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testT T9
============================================================
2
(k - 1)(3 k + 1)!
summand: ------------------
(2 k + 7)(k + 3)!
------------------------------------------------------------
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Compiling function T9 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
2 4 7
27 ?(? + -)(? + -)(? + 2)(? + -)
3 3 2
factored form: --------------------------------
9
(? - 1)(? + 4)(? + -)
2
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
2 4 7
27 ?(? + -)(? + -)(? + 2)(? + -)
3 3 2
(29) --------------------------------
9
(? - 1)(? + 4)(? + -)
2Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
fricas
display rr
+1+
(30) F | | [n]
1 0 +*+
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testT T10
============================================================
k
z
summand: --
k!
------------------------------------------------------------
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Compiling function T10 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
z
factored form: -----
? + 1
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
z
(31) -----
? + 1Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
fricas
display rr
+1+
(32) F | | [n]
1 0 +*+
fricas
testT T11
============================================================
k
2
summand: ---
2
k!
------------------------------------------------------------
fricas
Compiling function T11 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
2
factored form: --------
2
(? + 1)
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
2
(33) --------
2
(? + 1)Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
fricas
display rr
+1+
(34) F | | [n]
1 0 +*+
fricas
testT T12
============================================================
n
summand: ( )
k
------------------------------------------------------------
fricas
Compiling function T12 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
? - n
factored form: - -----
? + 1
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
? - n
(35) - -----
? + 1Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
fricas
display rr
+1+
(36) F | | [n]
1 0 +*+
fricas
testT T13
============================================================
k - 1 n - k - 1
n ++-++ ++-++
summand: ( ) | | x - d j | | y - d j
k | | | |
j = 0 j = 0
------------------------------------------------------------
fricas
Compiling function sym with type (Variable(x), Variable(d),
Expression(Fraction(Integer))) -> Expression(Fraction(Integer))
fricas
Compiling function sym with type (Variable(y), Variable(d),
Expression(Fraction(Integer))) -> Expression(Fraction(Integer))
fricas
Compiling function T13 with type Expression(Fraction(Integer)) ->
Expression(Fraction(Integer))
(? - n)(d ? - x)
factored form: --------------------------
(? + 1)(d ? + y - d n + d)
>> Error detected within library code:
catdef: division by zero
Continuing to read the file...
ff
(? - n)(d ? - x)
(37) --------------------------
(? + 1)(d ? + y - d n + d)Type: Fraction(Factored(SparseUnivariatePolynomial
?(SparseMultivariatePolynomial
?(Fraction(Integer),
Kernel(Expression(Fraction(Integer)))))))
fricas
display rr
+1+
(38) F | | [n]
1 0 +*+
fricas
-------------
-- Results --
-------------
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)version
"FriCAS 1.3.10 compiled at Wed 10 Jan 02:19:45 CET 2024"
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)lisp (lisp-implementation-type)
Value = "SBCL"
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)lisp (lisp-implementation-version)
Value = "2.2.9.debian"
statistics()
=============================================================================
General WARNINGS:
* do not use ')clear completely' before having used 'statistics()'
It clears the statistics without warning!
* do not forget to pass the arguments of the testXxxx functions as Strings!
Otherwise, the test will fail and statistics() will not notice!
=============================================================================
Testsuite: identHG
failed (total): 0 (1)
=============================================================================
testsuite | testcases: failed (total) | tests: failed (total)
identHG 0 (1) 0 (1)
=============================================================================
File summary.
unexpected failures: 0
expected failures: 0
unexpected passes: 0
total tests: 1
Type: Void