Changes http://wiki.axiom-developer.org/GuessingFormulasForSequences/diff
--
??changed:
-Important Note 1
-
- This is an outdated version of the package. In particular, it uses only slow
rational interpolation and is unable to guess algebraic differential equations.
The current version is available upon request from the author and will be
installed here within the next days.
-
-Important Note 2
+Important Note
??changed:
Finally, please feel free to try this package in the
"SandBox":SandBoxGuessingSequence! If you would like to use
-this program at your own computer, you need the source of
-
- - RINTERPA and RINTERP from [Rational Interpolation]
-
- - PCDEN from [CommonDenominator for polynomials]
-
- - GUESS, GUESSINT and GUESSP from [Guess]
+this program at your own computer, you need the files
+
+ - [rec.spad] from RecurrenceRelationOperator
+
+ - [fffg.spad] from FractionFreeFastGaussianElimination and
+
+ - [mantepse.spad]
??changed:
\begin{axiom}
-)lib RINTERPA RINTERP PCDEN GUESS GUESSINT GUESSP
+)lib RECOP FAMR2 FFFG FFFGF NEWTON UFPS UFPS1 GOPT GOPT0 GUESS GUESSIN GUESSP
GUESSF1 GUESSF
\end{axiom}
??changed:
\end{axiom}
-The package provides currently four primitive guessing functions, namely:
+The package provides several guessing functions, for example:
??changed:
-Among these, 'guessRat' and 'guessPade' are very fast, 'guessPRec'
-is reasonably fast and 'guessExpRat' is rather slow.
+Note that, unfortunately, 'guessExpRat' is rather slow.
??changed:
\begin{axiom}
-guessPade(n, [1, 1, 2, 3, 5], n+->n)$GuessInteger
+guessPade([1, 1, 2, 3, 5])$GuessInteger
\end{axiom}
??changed:
Thus, we used the operation 'guessPade' from the package 'GuessInteger' to
-guess the $n^{th}$ term of the sequence 1,1,2,3,5. The meaning of 'n+->n'
-will be made clear later.
+guess the $n^{th}$ term of the sequence 1,1,2,3,5.
??changed:
\begin{axiom}
-guessPRec(n, [1, 1, 2, 3, 5, 8, 13, 21, 34], n+->n)$GuessInteger
+guessPRec([1, 1, 2, 3, 5, 8, 13, 21, 34])$GuessInteger
\end{axiom}
??changed:
\begin{axiom}
-guessRat(n, [3, 4, 7/2, 18/5, 11/3, 26/7], n+->n)$GuessInteger
+guessRat([3, 4, 7/2, 18/5, 11/3, 26/7])$GuessInteger
\end{axiom}
??changed:
-Here, $order=3$ indicates that the first two terms of the sequence *might* not
+Here, $order=2$ indicates that the zeroth and the first term of the sequence
*might* not
coincide with the value of the returned function. A similar situation occurs,
??changed:
that this is rather a feature than a "bug": most terms will be correct,
-just as in the example above, where the value at $n=1$ is indeed $3$.
+just as in the example above, where the value at $n=0$ is indeed $3$.
??changed:
\begin{axiom}
-guess(n, [0, 1, 3, 9, 33], n+->n, [guessRat],
- [guessSum, guessProduct, guessOne], 2)$GuessInteger
+guess([0, 1, 3, 9, 33], [guessRat], [guessSum, guessProduct])$GuessInteger
\end{axiom}
??changed:
-Thus, 'guess' takes six parameters. The first three parallel the parameters in
-the other guessing functions. The fourth parameter is a list of guessing
-functions to be tried, in the example 'guessRat' and 'guessPade'. The fifth
-parameter specifies which operators should be applied. Additionally the option
-'guessOne' can be specified, which tells the program to stop immediately if one
-fitting function has been found. Finally, the last parameter is a nonnegative
-integer that specifies how many levels of recursion will be tried. I.e., if the
-last parameter is zero, no operator will be applied, if it is one, the package
-applies one, and so on. In the example above, the sequence is first
differenced,
-then successive quotients are formed.
+Thus, 'guess' takes three parameters, the first one being the list of values.
+The second parameter is a list of guessing functions to be tried, in the
example
+'guessRat'. The fifth parameter specifies which operators should be applied.
??changed:
\begin{axiom}
-guess(n, [1, 1+q, 1+q+q^2, 1+q+q^2+q^3], n+->n, [guessRat],
- [guessSum, guessProduct], 2)$GuessPolynomial
+guess([1, 1+q, 1+q+q^2, 1+q+q^2+q^3], [guessRat],
+ [guessSum, guessProduct])$GuessPolynomial
\end{axiom}
--removed:
the methods described above. For example, they fail for Sequences (3) and (4).
-However, the functions demonstrated above all had an argument 'n+->n' we did
-not explain yet!
??changed:
The key idea is, that many sequences are not generated by a rational function
-in $n$, but rather by a rational function in $q^n$, which can be specified by
-'n+->q^n'. For example
+in $n$, but rather by a rational function in $q^n$. For example
??changed:
\begin{axiom}
-guessRat(n, [1,1+q+q^2,(1+q+q^2)*(1+q^2),
- (1+q^2)*(1+q+q^2+q^3+q^4)], n+->q^n)$GuessPolynomial
+guessqRat([1,1+q+q^2,(1+q+q^2)*(1+q^2),
+ (1+q^2)*(1+q+q^2+q^3+q^4)])$GuessPolynomial
\end{axiom}
??changed:
(1+q^4)*(1+q+q^2+q^3+q^4+q^5+2*q^6+2*q^7+2*q^8+2*q^9+q^10+q^11+q^12)];
-(guessPRec(n, l, n+->q^n)$GuessPolynomial).1
+guessqPRec(l)$GuessPolynomial
\end{axiom}
??changed:
- guess(n, s, n+->n, [guessPRec], [guessSum]).
+ guess(s, [guessPRec], [guessSum]).
??changed:
- guess(n, s, n+->n, [guessPRec], [guessSum, guessProduct])
+ guess(s, [guessPRec], [guessSum, guessProduct])
??changed:
- guess(n, s, n+->n, [guessPRec], [guessProduct])!
+ guess(s, [guessPRec], [guessProduct])!
--
forwarded from http://wiki.axiom-developer.org/[EMAIL PROTECTED]
