gfun[proctorec] - Compute a recurrence by guessing, using more and more terms of a sequence - given as a procedure.

Calling Sequence

proctorec( procu, u, n, <nmin>,<nmax>, <homogeneous=bool>, <operator=name>)

Parameters

procu - pocedure for computing u(n)

u,n - name and index of recurrence,

nmin - (optional) minimal number of terms to guess with,

nmax - (optional) maximal number of terms to guess with.

homogeneous - (optional) boolean selecting homogeneous or inhomogeneous guessing; the default is true.

operator - (optional) name used verbatim as the shift operator

Description

The procedure outputs a list made of a recurrence and a number k :
- the linear recurrence defines u(0), ..., u(k) - and we have nmin <= k <= nmax (with k=nmin if nmin>nmax). The recurrence is looked for on initial values. The values u(0), ..., u(j) are used for guessing, where j goes exponentially from nmin to nmax :
If nmax is infinite, then nmin, 2*nmin, 4*nmin,... initial values are used for guessing, until a recurrence can be guessed.
Otherwise, nmin, nmax/2^i, 2*nmax/2^i, ..., up to nmax/4, nmax/2, and finally nmax terms are used for guessing, with i minimal s.t. the sequence is strictly increasing.
The option homogeneous=bool is passed to the recurrence guesser. Its default is true, preserving the historical behavior of proctorec.
With operator=name, the first component of the returned list is an operator polynomial (or L(name)=-q for inhomogeneous guessing), and initial conditions are not computed. The second component remains the number of terms used.

Examples

Use nmin to guarantee that enough terms are taken into account.

> L:=[1$9,seq(i,i=1..20)];

\[[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]\]

> procu:=proc(i) L[i+1] end;

\[\textbf{proc} (i) \\ \mapleIndent{1} L[i + 1]\\ \textbf{end\ proc};\]

> gfun:-proctorec(procu,u,n);

\[[\{-u \! \left(n \right)+u \! \left(n +1\right), u \! \left(0\right) = 1\}, 8]\]

> gfun:-proctorec(procu,u,n,20);

\[[\{\left(n -8\right) u \! \left(n \right)+\left(-2 n +16\right) u \! \left(n +1\right)+\left(n -8\right) u \! \left(n +2\right), u \! \left(0\right) = 1, u \! \left(1\right) = 1, u \! \left(2\right) = 1, u \! \left(3\right) = 1, u \! \left(4\right) = 1, u \! \left(5\right) = 1, u \! \left(6\right) = 1, u \! \left(7\right) = 1, u \! \left(8\right) = 1, u \! \left(9\right) = 1, u \! \left(10\right) = 2\}, 20]\]

Conversely, set a finite nmax to restrict the domain of the procedure, and prevent from failure.

> L:=L[1..7]; procu:=proc(i) L[i+1] end:

> gfun:-proctorec(procu,u,n,7,7);

\[[\{-u \! \left(n \right)+u \! \left(n +1\right), u \! \left(0\right) = 1\}, 7]\]

Setting nmin < nmax will result in a FAIL recurrence, even if guessing with nmin terms may succeed.

> gfun:-proctorec(procu,u,n,8,7);

\[[\mathit{FAIL}, 7]\]

gfun[proctorec] - Compute a recurrence by guessing, using more and more terms of a sequence - given as a procedure.

Calling Sequence

Parameters

Description

Examples

See Also