mathlib3 documentation

algebra.continued_fractions.computation.basic

Computable Continued Fractions #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

Summary #

We formalise the standard computation of (regular) continued fractions for linear ordered floor fields. The algorithm is rather simple. Here is an outline of the procedure adapted from Wikipedia:

Take a value v. We call ⌊v⌋ the integer part of v and v - ⌊v⌋ the fractional part of v. A continued fraction representation of v can then be given by [⌊v⌋; b₀, b₁, b₂,...], where [b₀; b₁, b₂,...] recursively is the continued fraction representation of 1 / (v - ⌊v⌋). This process stops when the fractional part hits 0.

In other words: to calculate a continued fraction representation of a number v, write down the integer part (i.e. the floor) of v. Subtract this integer part from v. If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will terminate if and only if v is rational.

For an example, refer to int_fract_pair.stream.

Main definitions #

Implementation Notes #

There is an intermediate definition generalized_continued_fraction.int_fract_pair.seq1 between generalized_continued_fraction.int_fract_pair.stream and generalized_continued_fraction.of to wire up things. User should not (need to) directly interact with it.

The computation of the integer and fractional pairs of a value can elegantly be captured by a recursive computation of a stream of option pairs. This is done in int_fract_pair.stream. However, the type then does not guarantee the first pair to always be some value, as expected by a continued fraction.

To separate concerns, we first compute a single head term that always exists in generalized_continued_fraction.int_fract_pair.seq1 followed by the remaining stream of option pairs. This sequence with a head term (seq1) is then transformed to a generalized continued fraction in generalized_continued_fraction.of by extracting the wanted integer parts of the head term and the stream.

References #

Tags #

numerics, number theory, approximations, fractions

source
structure generalized_continued_fraction.int_fract_pair (K : Type u_1) :
Type u_1
  • b : ℤ
  • fr : K

We collect an integer part b = ⌊v⌋ and fractional part fr = v - ⌊v⌋ of a value v in a pair ⟨b, fr⟩.

Instances for generalized_continued_fraction.int_fract_pair

Interlude: define some expected coercions and instances.

source
@[protected, instance]
def generalized_continued_fraction.int_fract_pair.has_repr {K : Type u_1} [has_repr K] :
has_repr (generalized_continued_fraction.int_fract_pair K)

Make an int_fract_pair printable.

Equations
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@[protected, instance]
def generalized_continued_fraction.int_fract_pair.inhabited {K : Type u_1} [inhabited K] :
inhabited (generalized_continued_fraction.int_fract_pair K)
Equations
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def generalized_continued_fraction.int_fract_pair.mapFr {K : Type u_1} {β : Type u_2} (f : K → β) (gp : generalized_continued_fraction.int_fract_pair K) :
generalized_continued_fraction.int_fract_pair β

Maps a function f on the fractional components of a given pair.

Equations

Interlude: define some expected coercions.

source
@[protected, instance]
def generalized_continued_fraction.int_fract_pair.has_coe_to_int_fract_pair {K : Type u_1} {β : Type u_2} [has_coe K β] :
has_coe (generalized_continued_fraction.int_fract_pair K) (generalized_continued_fraction.int_fract_pair β)

Coerce a pair by coercing the fractional component.

Equations
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@[simp, norm_cast]
theorem generalized_continued_fraction.int_fract_pair.coe_to_int_fract_pair {K : Type u_1} {β : Type u_2} [has_coe K β] {b : ℤ} {fr : K} :
↑{b := b, fr := fr} = {b := b, fr := ↑fr}
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@[protected]
def generalized_continued_fraction.int_fract_pair.of {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) :
generalized_continued_fraction.int_fract_pair K

Creates the integer and fractional part of a value v, i.e. ⟨⌊v⌋, v - ⌊v⌋⟩.

Equations
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@[protected]
def generalized_continued_fraction.int_fract_pair.stream {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) :
stream (option (generalized_continued_fraction.int_fract_pair K))

Creates the stream of integer and fractional parts of a value v needed to obtain the continued fraction representation of v in generalized_continued_fraction.of. More precisely, given a value v : K, it recursively computes a stream of option ℤ × K pairs as follows:

  • stream v 0 = some ⟨⌊v⌋, v - ⌊v⌋⟩
  • stream v (n + 1) = some ⟨⌊frₙ⁻¹⌋, frₙ⁻¹ - ⌊frₙ⁻¹⌋⟩, if stream v n = some ⟨_, frₙ⟩ and frₙ ≠ 0
  • stream v (n + 1) = none, otherwise

For example, let (v : ℚ) := 3.4. The process goes as follows:

  • stream v 0 = some ⟨⌊v⌋, v - ⌊v⌋⟩ = some ⟨3, 0.4⟩
  • stream v 1 = some ⟨⌊0.4⁻¹⌋, 0.4⁻¹ - ⌊0.4⁻¹⌋⟩ = some ⟨⌊2.5⌋, 2.5 - ⌊2.5⌋⟩ = some ⟨2, 0.5⟩
  • stream v 2 = some ⟨⌊0.5⁻¹⌋, 0.5⁻¹ - ⌊0.5⁻¹⌋⟩ = some ⟨⌊2⌋, 2 - ⌊2⌋⟩ = some ⟨2, 0⟩
  • stream v n = none, for n ≥ 3
Equations
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theorem generalized_continued_fraction.int_fract_pair.stream_is_seq {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) :
(generalized_continued_fraction.int_fract_pair.stream v).is_seq

Shows that int_fract_pair.stream has the sequence property, that is once we return none at position n, we also return none at n + 1.

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@[protected]
def generalized_continued_fraction.int_fract_pair.seq1 {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) :
stream.seq1 (generalized_continued_fraction.int_fract_pair K)

Uses int_fract_pair.stream to create a sequence with head (i.e. seq1) of integer and fractional parts of a value v. The first value of int_fract_pair.stream is never none, so we can safely extract it and put the tail of the stream in the sequence part.

This is just an intermediate representation and users should not (need to) directly interact with it. The setup of rewriting/simplification lemmas that make the definitions easy to use is done in algebra.continued_fractions.computation.translations.

Equations
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@[protected]
def generalized_continued_fraction.of {K : Type u_1} [linear_ordered_field K] [floor_ring K] (v : K) :
generalized_continued_fraction K

Returns the generalized_continued_fraction of a value. In fact, the returned gcf is also a continued_fraction that terminates if and only if v is rational (those proofs will be added in a future commit).

The continued fraction representation of v is given by [⌊v⌋; b₀, b₁, b₂,...], where [b₀; b₁, b₂,...] recursively is the continued fraction representation of 1 / (v - ⌊v⌋). This process stops when the fractional part v - ⌊v⌋ hits 0 at some step.

The implementation uses int_fract_pair.stream to obtain the partial denominators of the continued fraction. Refer to said function for more details about the computation process.

Equations