Approaches to Tetration Gottfried Ultimate Fellow Posts: 898 Threads: 130 Joined: Aug 2007 07/17/2008, 10:43 AM (This post was last modified: 07/17/2008, 11:26 AM by Gottfried.) Code:- Approaches to tetration   We may classify the approaches to tetration in two classes: a "binary operator   approach" and an "iterative series approach"   -- the operator approach ----------------------------------   Here one tries to extend the hierarchy of binary operator    a+b : addition      a*b : multiplication    a^b : exponentiation    a^^b : tetration   in a meaningful way      Extensions of this are made in two ways   assigning an index to the operation and to evolve this   to higher or lesser indices    a[0]b : "zeration"       a[1]b : addition     a[2]b : multiplication    a[3]b : exponentiation   a[4]b : tetration    a[5]b : pentation     ... [see links]   * Ackermann-function   Closely related is the concept of the Ackermann-function   A(a,index,iterate)   [see: Ackermann-function, ]      * Reihenalgebra   Even an approach to extend this operator-hierarchy to fractional indices is known   where the index was adapted:                      a[1]x : "do nothing"    a[2]x : add one to a     a[1/2]b : subtract one // unary operator    a[3]b : addition         a[1/3]b : subtraction    a[4]b : multiplication   a[1/4]b : division    a[5]b : exponentiation   a[1/5,subscript]b : inverses root or logarithm    a[6]b : tetration        a[1/6,subscript]b : inverses depending on evaluation-precedence    ...   The author tried then to find meaningful interpolations for this indexing scheme.   [markus Müller, Reihenalgebra]   -- the iterative series approach -----------------------------------------      Here operations/functions are expressed as series; for instance the operation   of exponentiation as powerseries, such that       exp(x) = 1 + x + x^2/2 + ...   and then iteration    exp°2(x) = exp(exp(x)) = 1 + exp(x) + exp(x)^2/2! + ...                           = K + Ax + Bx^2 + Cx^3 + ...    --- powerseries       The most common approach, see for instance []     b^x     = 1 + log(b)x + log(b)^2 x^2/2! +...     (b,x)°1 = b^x     (b,x)°2 = b^(b^x) = 1 + log(b)b^x + log(b)^2 b^2x/2! + ...     (b,x)°h = (b,b^x)°(h-1)          The needed manipulations on powerseries are sometimes explicitely     expressed in matrix-notation [see links]    ---functions defined by general series ( dirichlet-like series,..)     The needed calculations are performed using derivatives of the functions     in the places where for powerseries we use their coefficients        [see Bell-matrix, Carleman-matrix]           --- series on integer iterates of the function itself      (b,x)°h = A x + B*(b,x)°1 + C*(b,x)°2 + ...      [for instance see : binomial expansion]        The iterative series approach implies also a different view: one assumes   an "initial-value" x, to which the operation is applied iteration-times   which then gives a final value.      So in     (b,x)°h we have a ternary operation where we apply the exponentiation     using base b h-times to the initial-value x.   This paradigm is followed in [see: dynamical systems, iteration theory, ... ]      The reverse-engineering of this paradigm leads then to the redefinition of   the common "binary operators" using the iteration-form (we use a local,   one-way notation here)                    b occurs h times   ---------------------------------  -------------------   (+,b,x)°h  :  x + b + b + ... b  : iterated addition   (+,b,x)°-h :  x - b - b - ... b  : iterated subtraction   (*,b,x)°h :  x * b * b * ... b   : iterated multiplication   (*,b,x)°-h : x / b / b / ... b   : iterated division   (^,b,x)°h :  b^ b ^ ... b^x      : iterated exponentiation, right associative   (^,b,x)°-h : Log_b(Log_b(...Log_b(x)))   Though for iterated addition and multiplication no powerseries is required,   they may be consistently expressed the same way using matrix-operators on   formal powerseries, which perform addition, multiplication and exponentiation   in the ring of formal powerseries according to this scheme [see link]      The latter form of notation (or some convenient adaptions [see "notations"])   like exp_b°h(x) for (^,b,x)°h allows then to write the iteration-scheme   for any function, for instance      (dxp,b,x)°h = dxp_b°h(x) :  where dxp_b(x):= b^x - 1   (sin,b,x)°h = sin_b°h(x) :  where sin_b(x):= i*(b^(ix) - b^(-ix))/2   --- Connection between series-iteration and operator paradigms ------------   The common binary operators can then be seen as reduced forms   of the series-iteration approach      a + b        == (+,b,a)°1   a + b + b      = a + 2*b == (+,b,a)°2   a - b        == (+,b,a)°-1   b - b  = 0   == (+,b,b)°-1   b + b  = 2*b == (+,b,b)°1 = (+,b,(+,b,b)°-1))°2 = (+,b,0)°2                == (*,b,2)°1         a * b      == (*,b,a)°1   a * b * b      = a*b^2 == (+,b,a)°2   a / b      == (*,b,a)°-1   b / b = 1  == (*,b,b)°-1   b * b      == (*,b,b)°1 = (*,b,(*,b,b)°-1))°2 = (*,b,1)°2              == (^,b,2)°1   b ^ a        == (^,b,a)°1   b ^(b ^ a)   == (^,b,a)°2   log_b(a)     == (^,b,a)°-1   log_b(b) = 1 == (^,b,b)°-1   b ^ b        == (^,b,b)°1 = (^,b,(^,b,b)°-1))°2 = (^,b,1)°2 Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Approaches to Tetration - by Gottfried - 07/17/2008, 10:43 AM RE: FAQ-discuss: Approaches to tetration - by bo198214 - 07/17/2008, 07:11 PM RE: FAQ-discuss: Approaches to tetration - by Gottfried - 07/17/2008, 07:41 PM RE: FAQ-discuss: Approaches to tetration - by bo198214 - 07/17/2008, 08:21 PM RE: FAQ-discuss: Approaches to tetration - by Gottfried - 07/18/2008, 09:09 AM RE: FAQ-discuss: Approaches to tetration - by bo198214 - 07/18/2008, 10:06 AM

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