diff options
Diffstat (limited to 'src/algebra')
| -rw-r--r-- | src/algebra/sttaylor.spad.pamphlet | 109 |
1 files changed, 101 insertions, 8 deletions
diff --git a/src/algebra/sttaylor.spad.pamphlet b/src/algebra/sttaylor.spad.pamphlet index a16e32ce..6d2a2afd 100644 --- a/src/algebra/sttaylor.spad.pamphlet +++ b/src/algebra/sttaylor.spad.pamphlet @@ -401,6 +401,14 @@ StreamTaylorSeriesOperations(A): Exports == Implementation where cssa(frst ststa,mapsa((rst x) * #1,comps(rst ststa,x))) if A has Algebra RN then +@ + +The following defines lazy integration on streams interpreted as Taylor series. +I.e. if [[x]] is $c_0,c_1,c_2,\dots$, then [[integ x]] returns +$c_0,\frac{1}{2}c_1,\frac{1}{3}c_2,\dots$. [[integrate]] prepends a given +constant of integration. + +<<package STTAYLOR StreamTaylorSeriesOperations>>= integre: (ST A,I) -> ST A integre(x,n) == delay empty? x => zro() @@ -412,6 +420,9 @@ StreamTaylorSeriesOperations(A): Exports == Implementation where integrate(a,x) == concat(a,integ x) lazyIntegrate(s,xf) == concat(s,integ(delay xf)) +@ + +<<package STTAYLOR StreamTaylorSeriesOperations>>= nldere:(ST ST A,ST A) -> ST A nldere(lslsa,c) == lazyIntegrate(0,addiag(comps(lslsa,c))) nlde lslsa == YS(nldere(lslsa,#1)) @@ -420,10 +431,45 @@ StreamTaylorSeriesOperations(A): Exports == Implementation where smult: (RN,ST A) -> ST A smult(rn,x) == map(rn * #1,x) +@ + +The following helper function computes +\begin{equation*} + 1+\int (rn+1) c x' dz - c (x-a_0), +\end{equation*} +where $a_0$ is the constant term of [[x]]. + +<<package STTAYLOR StreamTaylorSeriesOperations>>= powerrn:(RN,ST A,ST A) -> ST A powerrn(rn,x,c) == delay concat(1,integ(smult(rn + 1,c * deriv x)) - rst x * c) - powern(rn,x) == +@ + +The following operation raises the power series [[x]] to a rational power +[[rn]]. We first outline the general strategy. + +Suppose the constant term of [[x]] equals one. In this case, we have +\begin{equation*} + x^{rn+1} = 1 + \int (rn+1) x^{rn} x' dz, +\end{equation*} +since $(x^{rn+1})'= (rn+1)x^{rn} x'$. Thus, $x^{rn}$ is the fixed point of % +[[g +-> powerrn(rn, x, g)]]. We use [[Y$ParadoxicalCombinatorsForStreams(A)]]%$ +to compute this fixed point lazily. + +If the constant term of [[x]] is neither zero nor one, we use the identity +\begin{equation*} + (c_0 + c_1*z + c_2 z^2\dots)^{rn} = c_0^{rn} (1 + \frac{c_1}{c_0}*z +\dots)^{rn}. +\end{equation*} + +Finally, if the constant term of [[x]] is zero, we use the identity +\begin{equation*} + (c_0*z^o + c_1*z^{o+1} +\dots)^{rn} = z^{o rn} (c_0 + c_1*z +\dots)^{rn}. +\end{equation*} + +This implementation should be compared with [[cRationalPower$ISUPS]].%$ + +<<package STTAYLOR StreamTaylorSeriesOperations>>= + powern(rn, x) == order : I := 0 for n in 0.. repeat empty? x => return zro() @@ -431,24 +477,71 @@ StreamTaylorSeriesOperations(A): Exports == Implementation where x := rst x n = 1000 => error "**: series with many leading zero coefficients" +@ +First we determine the order of [[x]], i.e., the index of the first non-zero +coefficient. + +Remarks: +\begin{itemize} +\item Note that usually [[leave]] takes no arguments. I don't know what it does + here. +\item [[empty? x]] tests whether the stream [[x]] has no elements. This is + mathematically the same as the corresponding power series being zero. +\end{itemize} + +<<package STTAYLOR StreamTaylorSeriesOperations>>= (ord := (order exquo denom(rn))) case "failed" => error "**: rational power does not exist" +@ + +[[ord*numer(rn)]] will be the order of the new power series. If it is not an +integer, we won't get a Taylor expansion and thus raise an error. + +<<package STTAYLOR StreamTaylorSeriesOperations>>= + if ord > 0 and rn < 0 then + error "**: negative power does not exist" +@ + +If [[order]] was non-zero, we may not raise to a negative power. This test +should really be done before the previous one. + +<<package STTAYLOR StreamTaylorSeriesOperations>>= co := frst x (invCo := recip co) case "failed" => error "** rational power of coefficient undefined" --- This error message is misleading, isn't it? see sups.spad/cRationalPower +@ + +We need to be able to invert the leading coefficient. The error message is +slightly misleading, see [[sups.spad/cRationalPower]]. + +<<package STTAYLOR StreamTaylorSeriesOperations>>= power := --- one? co => YS(powerrn(rn,x,#1)) - (co = 1) => YS(powerrn(rn,x,#1)) + one? co => YS(powerrn(rn, x, #1)) (denom rn) = 1 => - not negative?(num := numer rn) => + not negative?(num := numer rn) => -- It seems that this cannot happen, but I don't know why - (co**num::NNI) * YS(powerrn(rn,(invCo :: A) * x,#1)) - (invCo :: A)**((-num)::NNI) * YS(powerrn(rn,(invCo :: A) * x,#1)) + (co**num::NNI) * YS(powerrn(rn, (invCo :: A) * x, #1)) + (invCo::A)**((-num)::NNI) * YS(powerrn(rn, (invCo :: A) * x, #1)) - RATPOWERS => co**rn * YS(powerrn(rn,(invCo :: A) * x,#1)) + RATPOWERS => co**rn * YS(powerrn(rn,(invCo :: A) * x, #1)) error "** rational power of coefficient undefined" +@ + +We now invoke the fixed point computation as explained above. We can do the +computation if +\begin{itemize} +\item the constant term equals one, or +\item [[rn]] is an integer, or +\item we have rational powers in the coefficient ring. +\end{itemize} +<<package STTAYLOR StreamTaylorSeriesOperations>>= + monom(1, (ord :: I) * numer(rn)) * power +@ + +Finally, we return the result with the correct order. + +<<package STTAYLOR StreamTaylorSeriesOperations>>= if A has Field then mapdiv(x,y) == delay empty? y => error "stream division by zero" |