On-line multiplication in real and complex base

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1 ) On-line multiplication in real complex base Christiane Frougny LIAFA, CNS UM place Jussieu, Paris Cedex 05, France Université Paris 8 ChristianeFrougny@liafajussieufr Athasit Surarerks Computer Engineering Department Faculty of Engineering Chulalongkorn University Phayathai road, Bangkok 100, Thail athasit@cpengchulaacth Abstract Multiplication of two numbers represented in base shown to be computable by an on-line algorithm when is a negative integer, a positive non-integer real number, or a complex number of the form, where is a positive integer 1 Introduction On-line arithmetic, introduced in [24], is a mode of computation where opers results flow through arithmetic units in a digit serial manner, starting with the most significant digit To generate the first digit of the result, the first digits of the opers are required The integer is called the delay of the algorithm This technique allows the pipelining of different operations such as addition, multiplication division It is also appropriate for the processing of real numbers having infinite expansions : it is well known that when multiplying two real numbers, only the left part of the result is significant On-line arithmetic is used for special circuits such as in signal processing, for very long precision arithmetic One of the interests of on-line computable functions is that they are continuous for the usual topology on the set of infinite sequences on a finite digit set To be able to perform on-line computation, it is necessary to use a redundant number system, where a number may have more than one representation (see [24, 5]) An example of such a system is the so-called signed-digit number system It is composed of an integer base a signed-digit set of the form, with In this sytem addition can be performed in constant time in parallel [1, 4] On-line multiplication is also feasible [24, 5] Parallel addition is used internally in the multiplication algorithm is On-line algorithms for addition, subtraction, multiplication, division square-root in integer base are well studied, see [2, 6, 1] In this paper we study the multiplication in base when is a negative integer, or a non-integer real number, or a complex number of the form It is known that any real number can be represented in negative integer base without a sign [12, 1, 15], that, with a signed-digit set, addition is computable in constant time in parallel, is computable by an on-line finite state automaton [7] We show that the on-line multiplication used in the signed-digit number system can be applied in negative base Negative base is related to some complex number systems, see below When the base is a real number, by the greedy algorithm of ényi [2], one can compute a representation in base of any real number belonging to the interval!", called its -expansion, where the digits +* are elements of the canonical digit set #&'(!) if is not an integer, or of #,-(! if is an integer In such a representation, when is not an integer, not all the patterns of digits are allowed (see [21] for instance) For instance in base the golden ratio, the - expansion of the number 8;:< 5 is (!7 = The pattern is forbidden Different -representations of 8 are >77 =, or 7!0?! 6 0? > 6 0@ > 6 for instance This system is thus naturally redundant epresentation of numbers in non-integer base is encountered for instance in coding theory, see [10], in the modelization of quasicrystals, see [] In [8] we have studied the problem of the conversion+* in a real base from a digit set AB&( > C! with C to the canonical digit set #, without changing the numerical value Addition multiplication by a fixed positive integer are particular cases of digit set conversion We have proved that the digit set conversion is on-line computable

2 & Moreover, if the base is a Pisot number 1 this conversion is realizable by an on-line finite automaton This means that in the Pisot case, digit set conversion needs only a finite storage memory, independent of the size of the data owever, addition is not computable in constant time in parallel In this work, we prove that multiplication of real numbers represented in real base is on-line computable with a certain delay explicitly computed Our algorithm can be applied to the particular case that is an integer With digit set >, this is the Carry- Save representation The delay of our on-line multiplication algorithm in the Carry-Save representation is greater than the delay of on-line multiplication in the signed-digit representation, but the Carry-Save representation takes less memory We then consider the Knuth complex number system, an in- which is composed of the base, with teger, canonical digit set (! Every complex number has a representation in this system [12] This allows a unified treatment of the real imaginary parts of a complex number We show that in the Knuth number system with a signed-digit set, with, multiplication is on-line computable It is known that addition is computable in constant time in parallel, is computable by an on-line finite state automaton [7] The case together with the signed-digit set 7, has been considered in [17, 16] with practical applications Other complex numeration systems have been considered For instance the Penney complex number system consists of the complex base the canonical digit set ( > Every complex number is representable in this system, the representation is unique finite for the Gaussian integers [22] It is shown in [25] that in base + with signed-digit set 7 > multiplication is online computable The techniques are different from the ones used in the Knuth number system 2 Preliminaries 21 On-line computability Let # be two finite digit sets, denote by # the set of infinite sequences of elements of # Let # 0? The function is said to be on-line computable with delay if there exists a natural number such that, for each there exists a function # such that 1 A Pisot number is an algebraic integer such that its algebraic conjugates are strictly less than in modulus The golden ratio the natural integers are Pisot numbers 0? 6, where # denotes the set of sequences of length of elements of # This definition extends readily to functions of several variables It is well known that some functions are not on-line computable, like addition in the binary system with canonical digit set ( > Addition is considered as a conversion from >7 to (! Denote by! the infinite concatenation "#, by the word concatenated & times Since 0?! ' 6 0?! 6! for any, one sees that the most significant digit of the result depends on the least significant digits of the input ecall that a distance ( can be defined on #) as follows: let 0* 6 +, 0*, 6 be in #, ( 0-, 6 10 where :; =<, if >, <, ( 0-, 6 otherwise The set #? is then a compact metric space This topology is equivalent to the product topology Then any function from # to which is on-line computable with delay is -Lipschitz, is thus uniformly continuous [8] Let A be a digit set We say that multiplication is on-line A A A A computable with delay in base on the digit set if there exists a CB 0?8 + 0-DE + 0GF such that + F + 8 B which is on-line computable with delay Note that apriori, because of redundancy, the result of such a process is not unique, but the algorithms we shall consider later on are deterministic, thus compute a function In the following, we will make the assumption that the opers begin with a run of zeroes This allows to ignore the delay inside the computation 22 Number representation A survey on numeration systems can be found in [14, Chapter 7] Let A be a finite digit set of real or complex digits let be a real or complex number such that A -representation of a real or complex number 8 with digits in A is a finite or a right +N infinite sequence I with 8 KJ A such that 8ML O 8 It is denoted by 0?8 8QP In this paper we consider only representations of the form 0?8 6 + J A5 representing a number 8 equal to L 8 When a representation ends with infinitely many zeroes, it is said to be finite, the zeroes are usually omitted + D

3 * * 2 Signed-digit number system The base is a positive integer With a signed-digit set of the form, the representation is redundant addition can be performed in constant time in parallel, is computable by an on-line finite automaton, [1, 4, 1] 24 Negative base numeration systems Let the base be a negative integer It is well known (see [12, 1, 15]) that any real number can be represented without a sign in base with digits from the canonical digit set #& > With a signed-digit set of the form,, with 2 the representation is redundant addition can be performed in constant time in parallel, is computable by an on-line finite automaton [7] 25 epresentations in real base Let be a real number, generally not an integer Any real number 8 J >" can be represented in base by the following greedy algorithm, [2] : denote by ) by 7 the integral part the fractional part of a number Let P,8 for let 8 ) Thus 8 L + 8, where the digits 8 +* are elements of the canonical digit set # '(!) if J, #= ( > otherwise The sequence of # is called the -expansion of 8 When is not an integer, a number 8 may have several different -representations on # : this system is naturally redundant The -expansion obtained by the greedy algorithm is the greatest one in the lexicographic order It is shown in [8] that addition in real base is on-line computable When the base is a Pisot number, addition is computable by an on-line finite automaton 26 Knuth number system ere the base is a complex number of the form, where is an integer Any complex number is representable in base with digits in the canonical digit set # B( >, (see [12, 11, ]) If is a square then every Gaussian integer has a unique finite representation of the form P, J # Thus, Since + &=, we have 02= P 0 = 6 02= D P 02= 6 So the -representation of can be obtained by intertwinning the 0 = 6 -representation of 8 the 0 = 6 - representation of D Most studied cases are # (! :, strongly related to base, & # B(! ([12, 1, 20, 17, 16]) With a signed-digit set of the form 7,,, addition is computable in constant time in parallel, is computable by an on-line finite state automaton [7] On-line multiplication algorithm in integer base 1 Classical on-line multiplication algorithm First we recall the classical algorithm for on-line multiplication in the signed-digit number system, see [24, 5] We give our own presentation TEOEM 1 Multiplication of two numbers represented in integer base with digits in 7,, is computable by an on-line algorithm with delay, where is the smallest positive integer such that 0 6 (1) Proof Denote by the partial sum L I*I 8 ( respectively ), denote by round( ) the closest integer to Classical on-line multiplication algorithm Input: two sequences 8 0?8 6 + D 0-D 6 of such that 8,8 D D, Output: a sequence F 0GF 6 + of such that L F ML + 8 B L D begin 1 F,, F 2 = 4 while, do 5 0 +)F 6 DE 8 6 F round0 6 7, end First let us prove by induction that for any

4 ) C C ) ) C * C By Line 5 of the algorithm, 0 F 6 0*D 8 6 By induction hypothesis, 1 F 0-D 8 6, the result follows from the fact that 8, the similar relations for Thus at step &, +0 F 6 Since F, the algorithm is convergent The sequencef F is a -representation of the most significant half of the product Now it remains to prove that the digits F s computed in Line 6 of the algorithm are in the digit set It is enough to show that " From Line 5 the fact that " are less than follows that 0 6 by (1) Note that additions multiplications by a digit in Line 5 are performed in constant time in parallel COOLLAY 1 The delay for Algorithm takes the following values If, If :, * If then If if, & 2 On-line multiplication in negative base Now we consider the case where the base is a negative integer the digit set is, POPOSITION 1 Multiplication of two numbers represented in negative base digit set 7, >, is computable by the classical on-line algorithm with delay, where is the smallest positive integer such that Proof At step & 0 6 (2) of the algorithm we get that +0 F 6 Since F, To show thatf is in, we have to show that From Line 5 the fact that are less than follows that by (2) On-line multiplication in real base Let+* AB > C> be a digit set containing #, that is, TEOEM 2 Multiplication of two numbers represented in base with digits in A is computable by an on-line algorithm with delay, where is the smallest positive integer such that 0 6 C4, () Proof +* Clearly a number satisfying () exists, because eal base on-line multiplication algorithm Input: two sequences D 0*D 6 + of A5 such that 8+,8Q D! D, 2 Output: a sequence F 0GF 6 + of A such that L F L + 8 B L D begin 1 F,, F 2 = 4 while, do 5 0 +)F 6 DE 8 6 F )C 7, end As above, at step & Since F, we get +0 F the algorithm is convergent It remains to prove that thef s are in A, ie F C It is enough to show that C4 From Line 5 the fact that are less than follows that by () 2 This implies that 0 6 C are in 6

5 EXAMPLE 1 Let ; be the golden ratio Then the canonical digit set is # ( > Multiplication on # is on-line computable with delay 5 by () This delay does not seem to be optimal, we conjecture that is optimal We give in Table 1 below the detail of a computation with 8 D&!! The numerical value of 8 is equal to 0 6 The result is F P >(7 >77 > Computations are represented in base, in a symbolic way 0 C 6 F 6 77 > 7 7! 8!7 >(! >(7 >7 > 10!7 >(7 > 11 >(!7! 12 7!7! 1 7 >77 > 14!7! 15 >(7 > > 17 7 > 18 7 > 1! 20 > Table 1 On-line multiplication in base with delay 5 Note that the result is not the greedy -expansion in general For instance with 8 ( >( >( >( >! D! >(! >( >( >!, Algorithm gives the result F >(7 >77 > (!, which contains the forbidden pattern 7 EXAMPLE 2 Let 0?: 56 golden ratio Then the canonical digit set is #4 (! Multiplication on # is on-line computable with delay 7 the delay is achievable, take 8 D 7 The result given by Algorithm would then be F!7: >( >7 >7 which is not on the alphabet # be the square of the : This delay is optimal for our algorithm : suppose that 5 Carry-Save versus Signed-Digit In this section is an integer Take A >, then the representation on A is redundant We call it the Carry-Save representation, because it is used in computer arithmetic under that name in the case that for internal additions in multipliers, see [18] By the real base algorithm, multiplication in base 2 on (! is on-line computable with delay : This delay is optimal, as shown by the following7 example Suppose that the delay is 2, take 8 D The result computed by the algorithm would be equal & to F &7 >(:! If :, multiplication on A ( > is on-line computable with the optimal delay elation () is never satisfied for integer C, which is not surprising because it is known that multiplication in integer base on the canonical digit set is not on-line computable Internal additions multiplications by a digit in Algorithm can be performed in parallel when is an integer This is well known when We give below the algorithm for addition in the general case POPOSITION 2 Addition in the Carry-Save representation can be performed in constant time in parallel Proof Input: 8 8 P D D'P with 8 D in AB > for & Output: P with in A such that P I -I P I*I begin 1 In parallel for 8 P I*I & do 2 8 D if then ; 4 if then if then ; else 7 ; 5 if ( ) then ; 6 if then 7 ; 7 if then if then ; else 7 ; 8 if then ; 10 end Clearly, P I -I P I*I 8 P I*I D D

6 * 6 6 One has to prove that the digits s are elements of A Since, then The case can happen only if,, which is impossible The case can happen only if, which is impossible as well The delay for multiplication in the signed-digit representation is better than in the Carry-Save representation, but note that the digit-set in the signed-digit representation has cardinality, to be compared to card 0@A 6, the digits in A being nonnegative take less memory to be stored 6 On-line multiplication in the Knuth complex number system TEOEM Multiplication of two complex numbers represented in base, with an integer, digit set,, is computable by an on-line algorithm with delay, where is the smallest odd integer such that 0? 6 (4) Proof On-line multiplication algorithm Input: two sequences 8 0?8 6 + D 0*D 6 + of ) such that 8,8, D D Output: a sequence F 0 F 6 of ) such that L + F L 8 B L D begin 1 F,, F 2 = 4 while do 5 0 +)F 6 D 8 6 F 4 # ) 0 6 ( 7 end The digit F will belong to if 0 6 By Line 6, for all, 0 F " 6 0 C CF Thus, by Line 5, ( First suppose that is odd Then the same holds true for Thus 0 6 0? 6 by (4) Suppose now that a better even delay could be achieved Then 0 6 thus C 6 This delay will work if 0? Suppose that the delay in (4) is of the form the delay in (5) is of the form Then 0 6 0,= 6 7 (5), set is the smallest positive integer such that is the smallest positive integer such that obviously Since for & 0 F 0 0, F 6 0 F the algorithm is convergent, F F is a - representation of the most significant half of

7 COOLLAY 2 The delay for Algorithm takes the following values If <, If or < then : If < ( then : In the other cases, for the delay is 5 EXAMPLE Let 7 7 > By Corollary 2 the delay is equal to 5 Let 8 > D 7 > The result is F P F 6 7! 7!7 8 >7 >7 ( ( > Table 2 On-line multiplication in base with delay 5 Acknowledgements We are pleased to thank the anonymous referee who indicated us references [17] [16] eferences [1] A Avizienis, Signed-digit number representations for fast parallel arithmetic IE Transactions on electronic computers 10 (161), [2] J-C Bajard, J Duprat, S Kla J-M Muller, Some operators for on-line radix 2 computations J of Parallel Distributed Computing 22 (14), 6-45 [] Č Burdík, Ch Frougny, J-P Gazeau Krejcar, Beta-integers as natural counting systems for quasicrystals J of Physics A: Math Gen 1 (18), [4] CY Chow JE obertson, Logical design of a redundant binary adder Proc 4th Symposium on Computer Arithmetic, IEEE Computer Society Press (178), [5] MD Ercegovac, On-line arithmetic: An overview eal time Signal Processing VII SPIE 45 (184), 86 [6] MD Ercegovac, An on-line square-rooting algorithm Proc 4th Symposium on Computer Arithmetic, IEEE Computer Society Press (178) [7] Ch Frougny, On-line finite automata for addition in some numeration systems Theoretical Informatics Applications (1), [8] Ch Frougny, On-line digit set conversion in real base Theoret Comp Sci 22 (200), [] W Gilbert, adix representations of quadratic fields J Math Anal Appl 8 (181), [10] W Kautz, Fibonacci codes for synchronization control IEEE Trans Infor Th 11 (165), [11] I Kátai J Szabó, Canonical number systems Acta Sci Math 7 (175), [12] DE Knuth, An imaginary number system CACM (160), [1] DE Knuth, The Art of Computer Programming, vol 2: Seminumerical Algorithms, 2nd ed, Addison- Wesley, 188 [14] M Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, 2002 [15] DW Matula, Basic Digit Sets for adix epresentation JACM 2 (182), [16] McIlhenny, Complex number on-line arithmetic for reconfigurable hardware : algorithms, implementations, applications PhD Dissertation, Computer Science Department, UCLA, 2002 [17] McIlhenny MD Ercegovac, On-line algorithms for complex number arithmetic Proc 2nd Asilomar Conference on Signals, Systems, Computers (18) [18] J-M Muller, Arithmétique des ordinateurs, Masson, 18 [1] J-M Muller, Some characterizations of functions computable in on-line arithmetic IEEE Trans on Computers 4 (14),

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