Approximations for Continued Fraction Computations (GeneralizedContinuedFraction.of)

Summary

This file contains useful approximations for the values involved in the continued fractions computation GeneralizedContinuedFraction.of. In particular, we derive the so-called determinant formula for GeneralizedContinuedFraction.of: Aₙ * Bₙ₊₁ - Bₙ * Aₙ₊₁ = (-1)^(n + 1).

Moreover, we derive some upper bounds for the error term when computing a continued fraction up a given position, i.e. bounds for the term |v - (GeneralizedContinuedFraction.of v).convergents n|. The derived bounds will show us that the error term indeed gets smaller. As a corollary, we will be able to show that (GeneralizedContinuedFraction.of v).convergents converges to v in Algebra.ContinuedFractions.Computation.ApproximationCorollaries.

Main Theorems

• GeneralizedContinuedFraction.of_part_num_eq_one: shows that all partial numerators aᵢ are equal to one.
• GeneralizedContinuedFraction.exists_int_eq_of_part_denom: shows that all partial denominators bᵢ correspond to an integer.
• GeneralizedContinuedFraction.of_one_le_get?_part_denom: shows that 1 ≤ bᵢ.
• GeneralizedContinuedFraction.succ_nth_fib_le_of_nth_denom: shows that the nth denominator Bₙ is greater than or equal to the n + 1th fibonacci number Nat.fib (n + 1).
• GeneralizedContinuedFraction.le_of_succ_get?_denom: shows that bₙ * Bₙ ≤ Bₙ₊₁, where bₙ is the nth partial denominator of the continued fraction.
• GeneralizedContinuedFraction.abs_sub_convergents_le: shows that |v - Aₙ / Bₙ| ≤ 1 / (Bₙ * Bₙ₊₁), where Aₙ is the nth partial numerator.

References

• [Hardy, GH and Wright, EM and Heath-Brown, Roger and Silverman, Joseph][hardy2008introduction]
• https://en.wikipedia.org/wiki/Generalized_continued_fraction#The_determinant_formula

We begin with some lemmas about the stream of IntFractPairs, which presumably are not of great interest for the end user.

theorem GeneralizedContinuedFraction.IntFractPair.nth_stream_fr_nonneg_lt_one {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {ifp_n : GeneralizedContinuedFraction.IntFractPair K} (nth_stream_eq : GeneralizedContinuedFraction.IntFractPair.stream v n = some ifp_n) : 0 ≤ ifp_n.fr ∧ ifp_n.fr < 1

Shows that the fractional parts of the stream are in [0,1).

theorem GeneralizedContinuedFraction.IntFractPair.nth_stream_fr_nonneg {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {ifp_n : GeneralizedContinuedFraction.IntFractPair K} (nth_stream_eq : GeneralizedContinuedFraction.IntFractPair.stream v n = some ifp_n) : 0 ≤ ifp_n.fr

Shows that the fractional parts of the stream are nonnegative.

theorem GeneralizedContinuedFraction.IntFractPair.nth_stream_fr_lt_one {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {ifp_n : GeneralizedContinuedFraction.IntFractPair K} (nth_stream_eq : GeneralizedContinuedFraction.IntFractPair.stream v n = some ifp_n) : ifp_n.fr < 1

Shows that the fractional parts of the stream are smaller than one.

theorem GeneralizedContinuedFraction.IntFractPair.one_le_succ_nth_stream_b {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {ifp_succ_n : GeneralizedContinuedFraction.IntFractPair K} (succ_nth_stream_eq : GeneralizedContinuedFraction.IntFractPair.stream v (n + 1) = some ifp_succ_n) : 1 ≤ ifp_succ_n.b

Shows that the integer parts of the stream are at least one.

theorem GeneralizedContinuedFraction.IntFractPair.succ_nth_stream_b_le_nth_stream_fr_inv {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {ifp_n : GeneralizedContinuedFraction.IntFractPair K} {ifp_succ_n : GeneralizedContinuedFraction.IntFractPair K} (nth_stream_eq : GeneralizedContinuedFraction.IntFractPair.stream v n = some ifp_n) (succ_nth_stream_eq : GeneralizedContinuedFraction.IntFractPair.stream v (n + 1) = some ifp_succ_n) : ↑ifp_succ_n.b ≤ ifp_n.fr⁻¹

Shows that the n + 1th integer part bₙ₊₁ of the stream is smaller or equal than the inverse of the nth fractional part frₙ of the stream. This result is straight-forward as bₙ₊₁ is defined as the floor of 1 / frₙ.

Next we translate above results about the stream of IntFractPairs to the computed continued fraction GeneralizedContinuedFraction.of.

theorem GeneralizedContinuedFraction.of_one_le_get?_part_denom {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {b : K} (nth_part_denom_eq : Stream'.Seq.get? (GeneralizedContinuedFraction.partialDenominators (GeneralizedContinuedFraction.of v)) n = some b) : 1 ≤ b

Shows that the integer parts of the continued fraction are at least one.

theorem GeneralizedContinuedFraction.of_part_num_eq_one_and_exists_int_part_denom_eq {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {gp : GeneralizedContinuedFraction.Pair K} (nth_s_eq : Stream'.Seq.get? (GeneralizedContinuedFraction.of v).s n = some gp) : gp.a = 1 ∧ ∃ (z : ℤ), gp.b = ↑z

Shows that the partial numerators aᵢ of the continued fraction are equal to one and the partial denominators bᵢ correspond to integers.

theorem GeneralizedContinuedFraction.of_part_num_eq_one {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {a : K} (nth_part_num_eq : Stream'.Seq.get? (GeneralizedContinuedFraction.partialNumerators (GeneralizedContinuedFraction.of v)) n = some a) : a = 1

Shows that the partial numerators aᵢ are equal to one.

theorem GeneralizedContinuedFraction.exists_int_eq_of_part_denom {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {b : K} (nth_part_denom_eq : Stream'.Seq.get? (GeneralizedContinuedFraction.partialDenominators (GeneralizedContinuedFraction.of v)) n = some b) : ∃ (z : ℤ), b = ↑z

Shows that the partial denominators bᵢ correspond to an integer.

One of our next goals is to show that bₙ * Bₙ ≤ Bₙ₊₁. For this, we first show that the partial denominators Bₙ are bounded from below by the fibonacci sequence Nat.fib. This then implies that 0 ≤ Bₙ and hence Bₙ₊₂ = bₙ₊₁ * Bₙ₊₁ + Bₙ ≥ bₙ₊₁ * Bₙ₊₁ + 0 = bₙ₊₁ * Bₙ₊₁.

theorem GeneralizedContinuedFraction.fib_le_of_continuantsAux_b {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] : n ≤ 1 ∨ ¬GeneralizedContinuedFraction.TerminatedAt (GeneralizedContinuedFraction.of v) (n - 2) → ↑(Nat.fib n) ≤ (GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) n).b

theorem GeneralizedContinuedFraction.succ_nth_fib_le_of_nth_denom {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] (hyp : n = 0 ∨ ¬GeneralizedContinuedFraction.TerminatedAt (GeneralizedContinuedFraction.of v) (n - 1)) : ↑(Nat.fib (n + 1)) ≤ GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) n

Shows that the nth denominator is greater than or equal to the n + 1th fibonacci number, that is Nat.fib (n + 1) ≤ Bₙ.

As a simple consequence, we can now derive that all denominators are nonnegative.

theorem GeneralizedContinuedFraction.zero_le_of_continuantsAux_b {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] : 0 ≤ (GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) n).b

theorem GeneralizedContinuedFraction.zero_le_of_denom {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] : 0 ≤ GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) n

Shows that all denominators are nonnegative.

theorem GeneralizedContinuedFraction.le_of_succ_succ_get?_continuantsAux_b {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {b : K} (nth_part_denom_eq : Stream'.Seq.get? (GeneralizedContinuedFraction.partialDenominators (GeneralizedContinuedFraction.of v)) n = some b) : b * (GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) (n + 1)).b ≤ (GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) (n + 2)).b

theorem GeneralizedContinuedFraction.le_of_succ_get?_denom {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {b : K} (nth_part_denom_eq : Stream'.Seq.get? (GeneralizedContinuedFraction.partialDenominators (GeneralizedContinuedFraction.of v)) n = some b) : b * GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) n ≤ GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) (n + 1)

Shows that bₙ * Bₙ ≤ Bₙ₊₁, where bₙ is the nth partial denominator and Bₙ₊₁ and Bₙ are the n + 1th and nth denominator of the continued fraction.

theorem GeneralizedContinuedFraction.of_denom_mono {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] : GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) n ≤ GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) (n + 1)

Shows that the sequence of denominators is monotone, that is Bₙ ≤ Bₙ₊₁.

Determinant Formula

Next we prove the so-called determinant formula for GeneralizedContinuedFraction.of: Aₙ * Bₙ₊₁ - Bₙ * Aₙ₊₁ = (-1)^(n + 1).

theorem GeneralizedContinuedFraction.determinant_aux {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] (hyp : n = 0 ∨ ¬GeneralizedContinuedFraction.TerminatedAt (GeneralizedContinuedFraction.of v) (n - 1)) : (GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) n).a * (GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) (n + 1)).b - (GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) n).b * (GeneralizedContinuedFraction.continuantsAux (GeneralizedContinuedFraction.of v) (n + 1)).a = (-1) ^ n

theorem GeneralizedContinuedFraction.determinant {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] (not_terminated_at_n : ¬GeneralizedContinuedFraction.TerminatedAt (GeneralizedContinuedFraction.of v) n) : GeneralizedContinuedFraction.numerators (GeneralizedContinuedFraction.of v) n * GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) (n + 1) - GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) n * GeneralizedContinuedFraction.numerators (GeneralizedContinuedFraction.of v) (n + 1) = (-1) ^ (n + 1)

The determinant formula Aₙ * Bₙ₊₁ - Bₙ * Aₙ₊₁ = (-1)^(n + 1).

Approximation of Error Term

Next we derive some approximations for the error term when computing a continued fraction up a given position, i.e. bounds for the term |v - (GeneralizedContinuedFraction.of v).convergents n|.

theorem GeneralizedContinuedFraction.sub_convergents_eq {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {ifp : GeneralizedContinuedFraction.IntFractPair K} (stream_nth_eq : GeneralizedContinuedFraction.IntFractPair.stream v n = some ifp) : let g := GeneralizedContinuedFraction.of v; let B := (GeneralizedContinuedFraction.continuantsAux g (n + 1)).b; let pB := (GeneralizedContinuedFraction.continuantsAux g n).b; v - GeneralizedContinuedFraction.convergents g n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB))

This lemma follows from the finite correctness proof, the determinant equality, and by simplifying the difference.

theorem GeneralizedContinuedFraction.abs_sub_convergents_le {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] (not_terminated_at_n : ¬GeneralizedContinuedFraction.TerminatedAt (GeneralizedContinuedFraction.of v) n) : |v - GeneralizedContinuedFraction.convergents (GeneralizedContinuedFraction.of v) n| ≤ 1 / (GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) n * GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) (n + 1))

Shows that |v - Aₙ / Bₙ| ≤ 1 / (Bₙ * Bₙ₊₁).

theorem GeneralizedContinuedFraction.abs_sub_convergents_le' {K : Type u_1} {v : K} {n : ℕ} [LinearOrderedField K] [FloorRing K] {b : K} (nth_part_denom_eq : Stream'.Seq.get? (GeneralizedContinuedFraction.partialDenominators (GeneralizedContinuedFraction.of v)) n = some b) : |v - GeneralizedContinuedFraction.convergents (GeneralizedContinuedFraction.of v) n| ≤ 1 / (b * GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) n * GeneralizedContinuedFraction.denominators (GeneralizedContinuedFraction.of v) n)

Shows that |v - Aₙ / Bₙ| ≤ 1 / (bₙ * Bₙ * Bₙ). This bound is worse than the one shown in GCF.abs_sub_convergents_le, but sometimes it is easier to apply and sufficient for one's use case.