/ 
Math-NumSeq-73
River stage zero No dependents
/ Math::NumSeq::Fibonacci

NAME

Math::NumSeq::Fibonacci -- Fibonacci numbers

SYNOPSIS

use Math::NumSeq::Fibonacci;
my $seq = Math::NumSeq::Fibonacci->new;
my ($i, $value) = $seq->next;

DESCRIPTION

The Fibonacci numbers F(i) = F(i-1) + F(i-2) starting from 0,1,

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
starting i=0

FUNCTIONS

See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::Fibonacci->new ()

Create and return a new sequence object.

Iterating

($i, $value) = $seq->next()

Return the next index and value in the sequence.

When $value exceeds the range of a Perl unsigned integer the return is a Math::BigInt to preserve precision.

$seq->seek_to_i($i)

Move the current sequence position to $i. The next call to next() will return $i and corresponding value.

Random Access

$value = $seq->ith($i)

Return the $i'th Fibonacci number.

For negative <$i> the sequence is extended backwards as F[i]=F[i+2]-F[i+1]. The effect is the same Fibonacci numbers but negative at negative even i.

 i     F[i]
---    ----
 0       0
-1       1
-2      -1       <----+ negative at even i
-3       2            |
-4      -3       <----+

When $value exceeds the range of a Perl unsigned integer the return is a Math::BigInt to preserve precision.

$bool = $seq->pred($value)

Return true if $value occurs in the sequence, so is a positive Fibonacci number.

$i = $seq->value_to_i_estimate($value)

Return an estimate of the i corresponding to $value. See "Value to i Estimate" below.

FORMULAS

Ith

Fibonacci F[i] can be calculated by a powering procedure with two squares per step. A pair of values F[k] and F[k-1] are maintained and advanced according to bits of i from high to low

start k=1, F[k]=1, F[k-1]=0
add = -2       # 2*(-1)^k

loop
  F[2k+1] = 4*F[k]^2 - F[k-1]^2 + add
  F[2k-1] =   F[k]^2 + F[k-1]^2

  F[2k] = F[2k+1] - F[2k-1]

  bit = next bit of i, high to low, skip high 1 bit
  if bit == 1
     take F[2k+1], F[2k] as new F[k],F[k-1]
     add = -2 (for next loop)
  else bit == 0
     take F[2k], F[2k-1] as new F[k],F[k-1]
     add = 2 (for next loop)

For the last (least significant) bit of i an optimization can be made with a single multiple for that last step, instead of two squares.

bit = least significant bit of i
if bit == 1
   F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + add
else
   F[2k]   = F[k]*(F[k]+2F[k-1])

The "add" amount is 2*(-1)^k which means +2 or -2 according to k odd or even, which means whether the previous bit taken from i was 1 or 0. That can be easily noted from each bit, to be used in the following loop iteration or the final step F[2k+1] formula.

For small i it's usually faster to just successively add F[k+1]=F[k]+F[k-1], but when in bignums the doubling k->2k by two squares is faster than doing k many individual additions for the same thing.

Value to i Estimate

F[i] increases as a power of phi, the golden ratio. The exact value is

F[i] = (phi^i - beta^i) / (phi - beta)    # exactly

phi = (1+sqrt(5))/2 = 1.618
beta = -1/phi = -0.618

Since abs(beta)<1 the beta^i term quickly becomes small. So taking a log (natural logarithm) to get i,

log(F[i]) ~= i*log(phi) - log(phi-beta)
i ~= (log(F[i]) + log(phi-beta)) / log(phi)

Or the same using log base 2 which can be estimated from the highest bit position of a bignum,

log2(F[i]) ~= i*log2(phi) - log2(phi-beta)
i ~= (log2(F[i]) + log2(phi-beta)) / log2(phi)

SEE ALSO

Math::NumSeq, Math::NumSeq::LucasNumbers, Math::NumSeq::Fibbinary, Math::NumSeq::FibonacciWord, Math::NumSeq::Pell, Math::NumSeq::Tribonacci

Math::Fibonacci, Math::Fibonacci::Phi

HOME PAGE

http://user42.tuxfamily.org/math-numseq/index.html

LICENSE

Copyright 2010, 2011, 2012, 2013, 2014, 2016 Kevin Ryde

Math-NumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.