On-line multiplication and division in real and complex bases

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1 1 / 14 On-line multiplication and division in real and complex bases Marta Brzicová, Christiane Frougny, Edita Pelantová, Milena Svobodová FNSPE, CTU Prague, Czech Republic IRIF, UMR 8243 CNRS & Université Paris-Diderot, France ARITH 23 July 10-13, 2016 Silicon Valley, USA

2 Contents 2 / 14 Positional numeration systems Parallel and on-line algorithms On-line and by algorithms of Trivedi & Ercegovac On-line property On-line and for R and C numeration systems Pre-processing of divisor Linear time complexity

3 Positional Numeration Systems 3 / 14 Positional numeration system (β, A) Base β, β > 1: not just β Z +, Z, but also β R, C; finite set of digits - Alphabet A 0: not just A Z - set of contiguous integers, but also A Z[β] = { n k=0 z kβ k z k Z} R, C. Finite representation of a number X C in the system (β, A): X = x n x 0 x 1 x m stands for X = n k=m x kβ k, with x k A. Ideal properties of a numeration system (β, A) A desired numeration system (β, A) has: operations of Addition and Subtraction doable in parallel, operations of Multiplication and Division doable on-line, Divisor Pre-processing algorithm available, and all these operations of (at most) linear time complexity.

4 Parallel and on-line algorithms For numeration systems (β, A) and (β, B), mapping φ : B A, where the digits of result V = φ(u) are obtained: in parallel: via sliding block code, or p-local function (p = r + t + 1) v j = ϕ(u j t,..., u j,..., u j+r ) z j = ϕ((x j t, y j t ),..., (x j, y j ),..., (x j+r, y j+r )) on-line: with a delay δ v j = ϕ(u 1,..., u j,..., u j+δ ) p j = ϕ((x 1, y 1 ),..., (x j, y j ),..., (x j+δ, y j+δ )) Both parallel and on-line algorithms require redundancy of (β, A). 4 / 14

5 Algorithms of K.Trivedi & M.Ercegovac For on-line and, denote Z k = z 1 z 2... z k for Z = z 1 z 2... in (β, A): On-line multiplication : X Y = P from X = x 1x 2 and Y = y 1y 2, with x j, y j = 0 for j = 1,..., δ, iterate for k = 1, 2,...: W k := β(w k 1 p k 1 ) + (x k Y k 1 + y k X k ) p k := Select M (W k ) A ensuring (X k Y k P k 1 ) = β k W k On-line division : N/D = Q from N = n 1n 2 and D = d 1d 2, with n j = 0 for j = 1,..., δ iterate for k = 1, 2,...: W k := β(w k 1 q k 1 D k 1+δ ) + β δ (n k+δ Q k 1 d k+δ ) q k := Select D (W k, D k+δ ) A ensuring N k+δ /D k+δ Q k 1 = β k W k /D k+δ (P k ), (Q k ) converge to P = XY and Q = N/D if: the sequences (W k ) are bounded, for both and, and each divisor D on input fulfils D k D min > 0 for any k Z +. 5 / 14

6 On-line property 6 / 14 Originally: on-line, algo s proposed for integer bases β Z + and symmetric integer alphabets A = { M,..., 0,..., M} Z. Newly: extension to a broader set of numeration systems: On-line (OL) property of (β, A) Numeration system (β, A) is said to possess the (OL) property if there exist ε > 0 and a bounded set I C (or R) satisfying: 0 I, and Z ε neighborhood of (βi ) a A such that B(Z, ε) I + a, where B(Z, ε) is the (real or complex) ball of center Z and radius ε. (OL) property sequences (W k ) bounded on-line algo s converge: On-line, due to (OL) property Let (β, A) possess the (OL) property and the D min > 0. Then and can be performed on-line by the Trivedi-Ercegovac algorithms.

7 On-line property? Given (β, A): How to assess the (OL) property, and find ε and I fulfilling: Z ε neighborhood of (βi ) a A such that B(Z, ε) I + a Note: The ε-wide overlaps between neighboring copies (I + a j ) and (I + a k ) allow to reach linear time complexity of the algorithms. 7 / 14

8 (OL) property - results: β R and A Z 8 / 14 Theorem: (OL) property for real bases Let β R have β > 1, and {m,..., 0,..., M} = A Z. If #A > β, then (β, A) has the (OL) property. Consequently, multiplication and division in (β, A) are performable on-line by the Trivedi Ercegovac algorithms, if D min > 0 for division.

9 (OL) property - results: β C and A Z 9 / 14 Theorem: (OL) property for complex bases Let β C \ R have β > 1, and { M,..., 0,..., M} = A Z. If #A > ββ + β + β, then (β, A) has the (OL) property. Consequently, multiplication and division in (β, A) are performable on-line by the Trivedi Ercegovac algorithms, if D min > 0 for division. This way, we find integer alphabets to fulfil the (OL) property for: Penney base β = ı 1: #A > ββ + β + β = 4 Eisenstein base β = ω 1 = exp 2πı 3 1: #A > ββ + β + β = 6

(OL) property - results: β C and A C 10 / 14 For any given base β C, with β > 1, we can always set an alphabet A C allowing on-line and, e.g. as follows: A = {a Z[ı] B(a, 1) B(0, β ) } with I = B(0, 1).

10 (OL) property - results: β C and A C 10 / 14 For any given base β C, with β > 1, we can always set an alphabet A C allowing on-line and, e.g. as follows: A = {a Z[ı] B(a, 1) B(0, β ) } with I = B(0, 1)... or with other sets I = B(0, r) and lattices Z[β] A for specific bases: But the alphabets selected in this way are too big!

11 (OL) property - results: β C and A C 11 / 14 In the specific cases, we find the set I C for (OL) property with much smaller alphabets, by individual approach: Eisenstein base β = ω 1, where ω = exp 2πı 3, with alphabet {0, ±1, ±ω, ±ω 2 } = A Z[ω] of size #A = 7; Penney base β = ı 1, with alphabet {0, ±1, ±ı} = A Z[ı] of size #A = 5.

12 Pre-processing of divisor D in (β, A) 12 / 14 (β, A) must have a constant D min > 0 such that each divisor D on input fulfils D k = k j=1 d jβ j D min, after pre-processing by: shifting the fractional point to the most significant digit, or applying convenient rewriting rules from a pre-defined set. (β, A) allows pre-processing if D min > 0 such that d 1 d 2 d 3 A N : 1 either d 1 d 2 d 3 d k D min for all k N, 2 or there exists j N such that for some string d 2 d3 d j A d 1 d 2 d 3 d j = 0 d 2 d3 d j rewriting rule Examples: rewriting rules for pre-processing β = 4, A = {2, 1, 0, 1, 2}: no rewriting rules needed β = 2, A = {1, 0, 1}: rewriting rules ±{ 1( 1) = 01} β 2 = β + 1, A = {1, 0, 1}: rewriting rules ±{ 10( 1) = 010, 1( 1)0 = 001, 1( 1)( 1) = 000}

13 Linear time complexity 13 / 14 Goal: obtain the n-th digit of the result with O(n) time complexity. W k := β(w k 1 p k 1 ) + (x k Y k 1 + y k X k ) p k := Select M (W k ) A W k := β(w k 1 q k 1 D k 1+δ ) + β δ (n k+δ Q k 1 d k+δ ) q k := Select D (W k, D k+δ ) A Besides the properties already required from numeration system (β, A): parallel + and, on-line and, and pre-processing of divisor, we also need to process the Select functions in constant time: normalize representations of interim variables (W k ) from left side: by applying the same rewriting rules as for divisor pre-processing; evaluate just truncated representations of (W k ), (D k ) from right side: as of a suitably fixed position.

14 Linear time complexity 14 / 14 Additionally, an alphabet A = A A closed under multiplication would greatly speed up calculation of (W k ) - such as: base β = 2 with alphabet A = {0, ±1} base β 2 = β + 1 with alphabet A = {0, ±1} Penney base β = ı 1 with alphabet A = {0, ±1, ±ı} Eisenstein base β = ω 1 with alphabet A = {0, ±1, ±ω, ±ω 2 }

Minimal Digit Sets for Parallel Addition in Non-Standard Numeration Systems

1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.17 Minimal Digit Sets for Parallel Addition in Non-Standard Numeration Systems Christiane Frougny LIAFA, CNRS UMR 7089 Case 7014