currently seriesSolve uses a power series Ansatz of the form
f(x) = f_0 + f_1 x + ... + f_k x^k + \sum_{n>k} a x^n
(assuming that f_0, f_1, ..., f_k are already known) and solves for a.
This works surprisingly well, but of course not always.
The patch attached fixes this, please give OK to commit.
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Martin
Index: Makefile.in
===================================================================
--- Makefile.in (revision 756)
+++ Makefile.in (working copy)
@@ -349,7 +349,7 @@
RPOLCAT RURPK SHP SOLVETRA SUPFRACF SYMFUNC TBCMPPK TEMUTL URAGG \
UTS WEIER WP ZDSOLVE ZMOD
-GUESSLIST= SUPEXPR FAMR2 NEWTON UFPS GOPT GUESSF1 GUESSP1\
+GUESSLIST= SMPEXPR FAMR2 NEWTON UFPS GOPT GUESSF1 GUESSP1\
UTSSOL FFFG UFPS1 GOPT0 EXPRSOL FFFGF \
RECOP STNSR GUESS GUESSINT GUESSF GUESSP GUESSPI GUESSAN
Index: Makenotes.tex
===================================================================
--- Makenotes.tex (revision 756)
+++ Makenotes.tex (working copy)
@@ -490,7 +490,7 @@
we add it here.
<<USERLAYER>>=
-GUESSLIST= SUPEXPR FAMR2 NEWTON UFPS GOPT GUESSF1 \
+GUESSLIST= SMPEXPR FAMR2 NEWTON UFPS GOPT GUESSF1 \
UTSSOL FFFG UFPS1 GOPT0 EXPRSOL FFFGF \
RECOP GUESS GUESSINT GUESSP GUESSF GUESSAN
Index: rec.spad.pamphlet
===================================================================
--- rec.spad.pamphlet (revision 756)
+++ rec.spad.pamphlet (working copy)
@@ -424,7 +424,7 @@
$ExpressionSolve(R, F,
UnivariateFormalPowerSeries F,
UnivariateFormalPowerSeries
- SparseUnivariatePolynomialExpressions F)
+ SparseMultivariatePolynomialExpressions F)
setINFOSER(info, [eq, argsym, op, v, params, values, _
"failed", fn, explicit?]$INFOSER)
Index: ssolve.spad.pamphlet
===================================================================
--- ssolve.spad.pamphlet (revision 756)
+++ ssolve.spad.pamphlet (working copy)
@@ -9,7 +9,7 @@
\begin{abstract}
\end{abstract}
\tableofcontents
-\section{domain SUPEXPR SparseUnivariatePolynomialExpressions}
+\section{domain SMPEXPR SparseMultivariatePolynomialExpressions}
This domain is a hack, in some sense. What I'd really like to do -
automatically - is to provide all operations supported by the coefficient
@@ -17,16 +17,16 @@
long as they are just constants. I don't see another way to do this,
unfortunately.
-<<domain SUPEXPR SparseUnivariatePolynomialExpressions>>=
-)abb domain SUPEXPR SparseUnivariatePolynomialExpressions
-SparseUnivariatePolynomialExpressions(R: Ring): Exports == Implementation where
+<<domain SMPEXPR SparseMultivariatePolynomialExpressions>>=
+)abb domain SMPEXPR SparseMultivariatePolynomialExpressions
+SparseMultivariatePolynomialExpressions(R: Ring): Exports == Implementation where
- Exports == UnivariatePolynomialCategory R with
+ Exports ==> PolynomialCategory(R, IndexedExponents NonNegativeInteger, NonNegativeInteger) with
if R has TranscendentalFunctionCategory
then TranscendentalFunctionCategory
- Implementation == SparseUnivariatePolynomial R add
+ Implementation ==> SparseMultivariatePolynomial(R, NonNegativeInteger) add
if R has TranscendentalFunctionCategory then
log(p: %): % ==
@@ -88,45 +88,25 @@
<<package UTSSOL TaylorSolve>>=
)abb package UTSSOL TaylorSolve
-TaylorSolve(F, UTSF, UTSSUPF): Exports == Implementation where
+TaylorSolve(F, UTSF, UTSSMPF): Exports == Implementation where
F: Field
- SUP ==> SparseUnivariatePolynomialExpressions
+ SMPF ==> SparseMultivariatePolynomialExpressions F
UTSF: UnivariateTaylorSeriesCategory F
- UTSSUPF: UnivariateTaylorSeriesCategory SUP F
+ UTSSMPF: UnivariateTaylorSeriesCategory SMPF
NNI ==> NonNegativeInteger
Exports == with
- seriesSolve: (UTSSUPF -> UTSSUPF, List F) -> UTSF
+ seriesSolve: (UTSSMPF -> UTSSMPF, List F) -> UTSF
Implementation == add
<<implementation: UTSSOL TaylorSolve>>
@
-<<implementation: UTSSOL TaylorSolve>>=
- seriesSolve(f, l) ==
- c1 := map((x : F) : SUP F +-> x::(SUP F),
- l)$ListFunctions2(F, SUP F)::(Stream SUP F)
- coeffs: Stream SUP F := concat(c1, generate(monomial(1$F,1$NNI)))
--- coeffs: Stream SUP F := concat(c1, monomial(1$F,1$NNI))
-@
-
[[coeffs]] is the stream of the already computed coefficients of the solution,
plus one which is so far undetermined. We store in [[st.2]] the complete stream
and in [[st.1]] the stream starting with the first coefficient that has
possibly not yet been computed.
-\begin{ToDo}
- The mathematics is not quite worked out. If [[coeffs]] is initialized as
- stream with all coefficients set to the \emph{same} transcendental value,
- and not enough initial values are given, then the missing ones are
- implicitely assumed to be all identical. It may well happen that a solution
- is produced, although it is not uniquely determined\dots
-\end{ToDo}
-
-<<implementation: UTSSOL TaylorSolve>>=
- st: List Stream SUP F := [coeffs, coeffs]
-@
-
Consider an arbitrary equation $f\big(x, y(x)\big)=0$. When setting $x=0$, we
obtain $f\big(0, y(0)\big)=0$. It is not necessarily the case that this
determines $y(0)$ uniquely, so we need one initial value that satisfies this
@@ -161,43 +141,50 @@
\end{ToDo}
<<implementation: UTSSOL TaylorSolve>>=
+ seriesSolve(f, l) ==
+ l1 := [e::SMPF for e in l]::Stream SMPF
+ s1: Stream Integer := generate(inc, 0)$Stream(Integer)
+ l2 := map(i +-> monomial(1, monomial(1, i::NNI)
+ $IndexedExponents(NNI))
+ $SMPF, s1)$StreamFunctions2(Integer, SMPF)
+
+ coeffs: Stream SMPF := concat(l1, l2)
+ st: List Stream SMPF := [coeffs, coeffs]
+
next: () -> F :=
nr := st.1
res: F
- if ground?(coeff: SUP F := nr.1)$(SUP F)
-@
-%$
+ if ground?(coeff: SMPF := nr.1)$SMPF then
+-- If the next element was already calculated, we can simply return it:
-If the next element was already calculated, we can simply return it:
-
-<<implementation: UTSSOL TaylorSolve>>=
- then
res := ground coeff
st.1 := rest nr
else
-@
-Otherwise, we have to find the first non-satisfied relation and solve it. It
-should be linear, or a single non-constant monomial. That is, the solution
-should be unique.
+-- Otherwise, we have to find the first non-satisfied relation and solve it. It
+-- should be linear, or a single non-constant monomial. That is, the solution
+-- should be unique.
-<<implementation: UTSSOL TaylorSolve>>=
ns := st.2
- eqs: Stream SUP F := coefficients f series ns
+ eqs: Stream SMPF := coefficients f series ns
while zero? first eqs repeat eqs := rest eqs
- eq: SUP F := first eqs
- if degree eq > 1 then
+ eq: SMPF := first eqs
+ var := variables eq
+ if not one?(# var) or degree(eq, first var) > 1 then
if monomial? eq then res := 0
else
output(hconcat("The equation is: ", eq::OutputForm))
$OutputPackage
error "seriesSolve: equation for coefficient not linear"
- else res := (-coefficient(eq, 0$NNI)$(SUP F)
- /coefficient(eq, 1$NNI)$(SUP F))
+ else
+-- output(hconcat("The equation is: ", eq::OutputForm))
+-- $OutputPackage
+
+ res := (-coefficient(eq, monomial(0$NNI, first var)$IndexedExponents(NNI))$(SMPF)
+ /coefficient(eq, monomial(1$NNI, first var)$IndexedExponents(NNI))$(SMPF))
- nr.1 := res::SUP F
--- concat!(st.2, monomial(1$F,1$NNI))
+ nr.1 := res::SMPF
st.1 := rest nr
res
@@ -219,16 +206,16 @@
<<package EXPRSOL ExpressionSolve>>=
)abb package EXPRSOL ExpressionSolve
-ExpressionSolve(R, F, UTSF, UTSSUPF): Exports == Implementation where
+ExpressionSolve(R, F, UTSF, UTSSMPF): Exports == Implementation where
R: Join(Comparable, IntegralDomain, ConvertibleTo InputForm)
F: FunctionSpace R
UTSF: UnivariateTaylorSeriesCategory F
- SUP ==> SparseUnivariatePolynomialExpressions
- UTSSUPF: UnivariateTaylorSeriesCategory SUP F
+ SMPF ==> SparseMultivariatePolynomialExpressions F
+ UTSSMPF: UnivariateTaylorSeriesCategory SMPF
OP ==> BasicOperator
SY ==> Symbol
NNI ==> NonNegativeInteger
- MKF ==> MakeBinaryCompiledFunction(F, UTSSUPF, UTSSUPF, UTSSUPF)
+ MKF ==> MakeBinaryCompiledFunction(F, UTSSMPF, UTSSMPF, UTSSMPF)
Exports == with
@@ -291,15 +278,15 @@
seriesSolve(expr, op, sy, l) ==
ex := replaceDiffs(expr, op, sy)
f := compiledFunction(ex, name op, sy)$MKF
- seriesSolve(x +-> f(x, monomial(1,1)$UTSSUPF),
- l)$TaylorSolve(F, UTSF, UTSSUPF)
+ seriesSolve(x +-> f(x, monomial(1,1)$UTSSMPF),
+ l)$TaylorSolve(F, UTSF, UTSSMPF)
@
%$
\section{Bugs}
<<inp: seriesSolve>>=
-seriesSolve(sin f x / cos x, f, x, [1])$EXPRSOL(INT, EXPR INT, UFPS EXPR INT, UFPS SUPEXPR EXPR INT)
+seriesSolve(sin f x / cos x, f, x, [1])$EXPRSOL(INT, EXPR INT, UFPS EXPR INT, UFPS SMPEXPR EXPR INT)
@
returns
\begin{verbatim}
@@ -308,8 +295,8 @@
but
<<inp: seriesSolve>>=
-U ==> UFPS SUPEXPR EXPR INT
-seriesSolve(s +-> sin s *((cos monomial(1,1)$U)^-1)$U, f, x, [0])$EXPRSOL(INT, EXPR INT, UFPS EXPR INT, UFPS SUPEXPR EXPR INT)
+U ==> UFPS SMPEXPR EXPR INT
+seriesSolve(s +-> sin s *((cos monomial(1,1)$U)^-1)$U, f, x, [0])$EXPRSOL(INT, EXPR INT, UFPS EXPR INT, UFPS SMPEXPR EXPR INT)
@
works. This is probably due to missing [[/]] in [[UFPS]].
@@ -347,7 +334,7 @@
<<*>>=
<<license>>
-<<domain SUPEXPR SparseUnivariatePolynomialExpressions>>
+<<domain SMPEXPR SparseMultivariatePolynomialExpressions>>
<<package UTSSOL TaylorSolve>>
