Kevin Ryde > Math-NumSeq-64 > Math::NumSeq::LucasNumbers

Math-NumSeq-64.tar.gz

Dependencies

Annotate this POD

Website

# CPAN RT

 Open 0
View/Report Bugs
Module Version: 64   Source   Latest Release: Math-NumSeq-69

# NAME

Math::NumSeq::LucasNumbers -- Lucas numbers

# SYNOPSIS

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

# DESCRIPTION

The Lucas numbers, L(i) = L(i-1) + L(i-2) starting from values 1,3.

```    1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364,...
starting i=1```

This is the same recurrence as the Fibonacci numbers (Math::NumSeq::Fibonacci), but a different starting point.

`    L[i+1] = L[i] + L[i-1]`

Each Lucas number falls in between successive Fibonaccis, and in fact the distance is a further Fibonacci,

```    F[i+1] < L[i] < F[i+2]

L[i] = F[i+1] + F[i-1]      # above F[i+1]
L[i] = F[i+2] - F[i-2]      # below F[i+2]```

## Start

Optional `i_start => \$i` can start the sequence from somewhere other than the default i=1. For example starting at i=0 gives value 2 at i=0,

```    i_start => 0
2, 1, 3, 4, 7, 11, 18, ...```

# FUNCTIONS

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

`\$seq = Math::NumSeq::LucasNumbers->new ()`
`\$seq = Math::NumSeq::LucasNumbers->new (i_start => \$i)`

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 Lucas number.

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

Return true if `\$value` is a Lucas 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[k] and Lucas L[k] can be calculated together by a powering algorithm with two squares per doubling,

```    F[2k] = (F[k]+L[k])^2/2 - 3*F[k]^2 - 2*(-1)^k
L[2k] =                   5*F[k]^2 + 2*(-1)^k

F[2k+1] =    ((F[k]+L[k])/2)^2 + F[k]^2
L[2k+1] = 5*(((F[k]+L[k])/2)^2 - F[k]^2) - 4*(-1)^k```

The 2*(-1)^k terms mean adding or subtracting 2 according to k odd or even. This means add or subtract according to the previous bit handled.

At the last step, which is the lowest bit of i, only L[2k] or L[2k+1] is needed for the return, not the F[] too.

For any trailing zero bits of i the final doublings L[2k] can be done without the F[] and with just one square as

`    L[2k] = L[k]^2 - 2*(-1)^k`

So

```    main double/step L[],F[] until the lowest 1-bit of i is reached
then L[2k+1] formula for that bit
then L[2k] single squarings for any low 0-bits```

## Value to i Estimate

L[i] increases as a power of phi, the golden ratio,

```    L[i] = phi^i + beta^i    # exactly

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

So taking a log (natural logarithm) to get i, and ignoring beta^i which quickly becomes small since abs(beta)<1,

```    log(L[i]) ~= i*log(phi)
i ~= log(L[i]) * 1/log(phi)```

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

```    log2(L[i]) ~= i*log2(phi)
i ~= log2(L[i]) * 1/log2(phi)```

This is very close to the Fibonacci formula (see "Value to i Estimate" in Math::NumSeq::Fibonacci), being bigger by

```    Lestimate(value) - Festimate(value)
= log(value) / log(phi) - (log(value) + log(phi-beta)) / log(phi)
= -log(phi-beta) / log(phi)
= -1.67```

On that basis it could even be close enough to take Lestimate = Festimate-1 (or vice-versa).

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