A Generalized Permanent Estimator and its Application in Computing Multi- Homogeneous Bézout Number

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1 Research Journal of Applied Sciences, Engineering and Technology 4(23): , 202 ISSN: Maxwell Scientific Organization, 202 Subitted: April 25, 202 Accepted: May 3, 202 Published: Deceber 0, 202 A Generalized Peranent Estiator and its Application in oputing Multi- Hoogeneous Bézout Nuber Hassan M.S. Bawazir, 2 Ali Abd Rahan and 2 Nor aini binti Aris Departent of Matheatics, Faculty of Education, Hadhraout University of Science and Technology, Seiyun, Hadhraout, Yeen 2 Departent of Matheatics, Faculty of Science, Universiti Teknologi Malaysia, Malaysia Abstract: The peranent of a atrix has any applications in any fields. Its coputation is #P-coplete. The coputation of exact peranent large-scale atrices is very costly in ters of eory and tie. There is a real need for an efficient ethod to deal well with such situations. This study designs a general algorith for estiating the peranents of the coplex square or non-square atrices. We prove that the Multi- Hoogeneous Bézout Nuber (MHBN) can be estiated efficiently using the new algorith. Further, a proposition that provides soe analytic results is presented and proved. The analytic results show the effectiveness and the efficiency of our algorith over soe recent ethods. Furtherore, with the new algorith we can control the accuracy as we need. A large aount of nuerical results are presented in this study. By applying the algorith that estiates MHBN we extend the applicability of the algorith. Keywords: Multi-hoogeneous bézout nuber, peranent, polynoial syste, rando path INTRODUTION The peranent of a atrix is very iportant and has any applications. Its coputation is #P-coplete (Liang et al., 2007). The coputation of exact peranent of large or even oderate atrices is very costly in ters of eory and tie. There is a real need for an efficient ethod to deal well with such situations. Liang et al. (2007) presented an estiator of peranents of square 0-atrices. In this study we present a general estiator, which can estiate the peranent of any atrix, square or non-square, integer atrix or even general coplex atrix, the focus will be on non-negative real atrices. With the new ethod we can deal very efficiently with large-scale atrices and we can control the accuracy as we need. An application of the new ethod is the estiation of MHBN. In this case we recognize the proble fro two sides. Firstly, MHBN of a ultivariate polynoial syste is an estiation of the nuber of isolated solutions of the syste and it is also the nuber of solution paths that can be traced to reach all isolated solutions. A polynoial syste can be hoogenized in any ways, each giving its own MHBN. The space of all ways of hoogenizing the syste (variable partitions) increases exponentially as the syste size grows, this proble is an NP-hard proble (Malaovich and Meer, 2007); we dealt with this proble in details in Bawazir and Abd Rahan (200). Secondly, the coputation of an MHBN at a fixed partition, especially when such a partition has a nuber of eleents close to the syste size, is equivalent to coputing the peranent of a atrix (Verschelde, 996). By Liang et al. (2007) such a proble is #P-coplete. In this study we will focus on coputing the MHBN at a fixed partition as an application of the general estiator of the peranents. The Multi-Hoogeneous Bézout Nuber (MHBN): onsider the ultivariate polynoial syste: F(x) = f (x), f 2 (x),, f n (x)) () where, x(x, x 2,, x n )0 n Let z = {z, z 2,, z } be an -partition of the unknowns X = {x, x 2,., x n } where, z = {z, z 2,, z k }, =, 2,,. =, 2,, Define the degree atrix of the syste F(x) = 0 as the following: d d2 K d d2 d22 K d2 D = d d K d n n2 n (2) orresponding Author: Hassan M.S. Bawazir, Departent of Matheatics, Faculty of Education, Hadhraout University of Science and Technology, Seiyun, Hadhraout, Yeen 5206

2 Res. J. Appl. Sci. Eng. Technol., 4(23): , 202 where, d i is the degree of polynoial f i w.r.t. the variable z. The degree polynoial of F w.r.t. the partition z is defined as: D i = i n f ( y) = d y The -Multi-Hoogeneous Bézout Nuber (or in short -MHBN) of F w.r.t. the partition z equals the coefficient of the onoial y k = y k y k2 2 y k in the degree polynoial f D (y), (3) and denoted by B, where k = (k, k 2,, k ) with k = #(z ), =, 2,, and k = n. = RELATIVE METHODOLOGY (3) Expanding the degree polynoial, (3), is called Basic Algorith (BA) (Wapler, 992). It is easy to find that BA has: N = ni k i ters (Also the nuber of addition operations), where n = n and n i = n i- -k i- for i = 2, 3,, n. This nuber N ranges fro N = for a -MHBN up to N = n for an n- MHBN, i.e., = n. In the following we describe two algoriths provided by Wapler (992). The first is the Row Expansion algorith (RE) and the second is the Row Expansion with Meory (REM). For RE, consider the degree atrix (2), D, first in row of D, suppose we choose eleent d, to coplete the degree product we ust choose only one eleent fro each of the reaining rows and only k eleents fro the th colun, so this procedure can be done by applying the sae steps on a inor atrix derived fro the atrix D by eliinating the row and the colun, denote this procedure by B(D, k-e, ), where e is the th row of the identity atrix of degree. Then, the row expansion algorith coputes the Bézout nuber as the su along the first row expansion procedure and can be expressed by the following recursive relation: BDki (,, ) = i =, k 0, if i = n + d B( D, k e, i + ), otherwise (4) where, -MHBN is B(D, k, ). This forula coputes directly the appropriate coefficient, i.e., -MHBN, so it saves the operations in coparison to expanding the degree polynoial (3). By Wapler (992), for the ost expensive case, = n, we have k i =, I =, 2,, n and the nuber of ultiplications coes to be (/!+/2!+ + /(n!)!)n!. As n increases, this rapidly approaches (e-)n!. Table : P tie for coputing MHBN using RE n Tie: t(n)(sec) Tie (in) t(n)/t(n-) Table 2: P tie for coputing MHBN using REM n Tie: t(n)(sec) Tie (in) t(n)/t(n-) For the row expansion with eory REM, Wapler (992) uses the sae recursive relation (4) with giving consideration to the repeated subtotals as shown fro the following exaple. Exaple : Let n = 4, = 2 and k = (2, 2). Using BA Bézout nuber B is: B = d d 2 d 32 d 42 +d d 22 d 3 d 42 +d d 22 d 32 d 4 +d 2 d 2 d 3 d 42 +d 2 d 2 d 32 d 4 +d 2 d 22 d 3 d 4 which uses 8 ultiplications. Using RE, the following is obtained: B = d (d 2 d 32 d 42 +d 22 (d 3 d 42 +d 32 d 4 ) +d 2 (d 2 (d 3 d 42 +d 32 d 4 )+d 22 d 3 d 4 ) which uses only 2 ultiplies. We note that the expression B(D, (, ), 3) = d 3 d 42 +d 32 d 4 appears twice by both B(D, (, 2), 2) and B(D, (2, ), 2), so REM,avoids such cases. Thus, REM requires a eory array of diension (k +) (k 2 +)... (k +). The nuber of ultiplications in the worst case = n is n2 n- which is uch saller than that of RE. This iproveent coes at the expense of a eory array that in the worst case has 2 n (Wapler, 992 ). We have done soe nuerical experients to show the cost of Wapler s ethods. Table and 2 show the P tie for coputing n!mhbn of the known cyclic n!roots proble by using RE and REM respectively. The algoriths are ipleented in MATLAB and executed on a personal coputer with Intel (R) Pentiu (R) Dual PU E GHz PU and 2.00 GB of RAM. Fro the fourth colun of the Table, we conclude that the tie devoted using RE at step n is about n ties the tie devoted in the step (n!), this is consistent with the analysis of the algorith RE. Fro this observation we can estiate the tie at n = 20 by h or onths. By the sae way, fro Table 2, the tie devoted at step n using REM is about double the tie at step n! by which we conclude that we need about 34 days to copute (n = 30) n-mhbn of such a proble. Further, the best recent algorith REM cannot be applicable for the case n>27 in such a P, because it cannot provide a eory array larger than 2 27 eleents. The recent ethods for iniizing Bézout nuber such as Genetic Algorith (Yan et al., 2008) used RE and 5207

3 Res. J. Appl. Sci. Eng. Technol., 4(23): , 202 REM algoriths for finding the iniu MHBN for oderate systes up to n =. Fro such cases, we conclude that there is a need to use efficient ethods to deal well with large-scale systes. Liang et al. (2007) presented an estiator of the peranents of the square 0-atrices; it is called rando path and shortly RP. This estiator is built up on the estiator of Rasussen (994) by adding the colun pivoting property. In this study we construct a general peranent estiator over RP. The new one is constructed for estiating the peranent of square or non-square atrices. Furtherore, it can be applicable for coputing MHBN. In the following we state in short about RP algorith. Rando path ethod: The peranent of an n n atrix A = (a i ) is defined by: Per ( A) = a i γ () i n γ (5) where, ( ranges over all perutations of {, 2,, n}. The definition of the peranent looks siilar to the definition of the deterinant of the atrix, but it is harder to copute. Using the Gauss eliination ethod, we convert the original atrix into an upper triangular atrix, as a result the deterinant of the original atrix is equal to the product of the diagonal eleents of the converted one and this property akes the coputation of the deterinant easier. There is no such property in the case of the peranent coputation. onsider an n n atrix A = (a i ) with entries 0, Define: M = {i: a i =, # i #n}, =, 2,, n. Naely, M is the set of the row indices of nonzero eleents in the th colun of the atrix A. Definition : Liang et al. (2007) A perutation ( = (i, i 2,, i n ) is called rando path of the atrix A, if there exists ##n, such that = = for t =, 2,, and t = a it t M + \ {i t } = i. Rando path with = n is called a feasible path; while that with < n is called a broken path. Definition 2: Liang et al. (2007) The path value of a feasible path ( = (i, i 2,,, i n ) is defined as M ( = M M 2 \{i } M n \{i, i 2,,, i n- } where M i is the nuber of the eleents in the set M i. Here is the RP algorith. Algorith : Liang et al. (2007) Rando Path algorith (RP). Input: A- an n n 0- atrix, M =, I = i. Output: X A - the estiate for Per(A). Step : For = to n t = Select c such that M c \ {a t } = in s0{,, n = }\I{ M s \ t {at} }; I = I c{c } hoose a fro Mc\c - t = {a t } uniforly at rando; M = M M c \c - t = {a t }. End Step 2: Generate X A = M. General Rando Path Algorith (GRPA): To establish our algorith we first present soe necessary concepts. Definition 3: Let A = (a i ) any coplex atrix of degree n, where ##n, let k = (k, k 2,, k )0N where ki = n. We define the peranent of the atrix A with respect to the vector k as the following: ki Per ( A) = a ( ) γ = γ i i (6) where, ( = (( (), ( (2),, ( () ) with ( (i) = (( (i), ( 2 (i),, ( (i) ki) ranges over all perutations of {, 2,, n} such that ( (i) ( () = i for all i,#i, #. If > nthen for suitable vector k we define Per(A) as the peranent of its transpose. We find that the peranent of a square atrix is a special case fro Definition 3 with = n and k = (,,,). Furtherore, this definition is consistent with the definition of -MHBN, B, so B is considered as a special peranent as following: Let A = D the degree atrix with k = (k, k 2,, k ) the vector that holds the cardinalities of the partition sets of the variables, so B = Per(D, k). onsider an n coplex atrix A = (a i ). Define: M = {i: a i 0, #i #n}, =, 2,,. Naely, M is the set of the row indices of nonzero eleents in the th colun of the atrix A. Let k = (k, k 2,, k )0N with ki = n. For s >, M s = Ø. Definition 4: A perutation ( = (( (), ( (2),, ( () ) of {, 2,, n} with ( (i) = (( (i), ( (i) 2,, ( (i) ki) is called a general rando path of the atrix A, if there exists # #, such that a ( ) 0 for t =, 2,, k i, i =,, and γ i t i M + \{( (i) t: t =,, k i, i =,, }=i. Rando path with = is called a feasible path; while that with < is called a broken path. 5208

4 Res. J. Appl. Sci. Eng. Technol., 4(23): , 202 Definition 5: The path value of a feasible path ( = (( (), ( i ) ( k ) i ( (2),, ( () ) with γ () i γ γ () i,,..., γ () i is defined s k i = γ () i = 2 i ki as s = a where s i = M i \{( (r) i t: t =,, k r, ( ) si si! r =,, i!} and =. ki ( ki)(! si ki)! For the illustration of the previous concepts we consider the following exaple. Exaple 2: onsider the following atrix: A = Suppose we need to find Per(A, k) with k = (2, 2, ) then by the definitions 4, 5 the following are two general rando paths and their values: (( (i), ( () 2, ( (2), ( (2) 2, ( (3) ) = (, 5, 2, 3, 4) and its value is: s s2 s3 a a a a a 2 2 k k 2 k3 4 ( )( ) = 2 ()() () = [()( 6 2)() ][()()() 5 ][( )( 2) ] = 20 ( ) ( ) ( ) ( ) ( 3) γ γ γ γ γ (( (),( () 2, ( (2), ( (2) 2,( (3) ) = (, 2, 3, 5, 4, ) and its value is: s () k a a s2 s3 γ ( ) a 2 a 2 a 3 2 ( ) ( ) k 2 ( ) γ 2 k γ γ 3 γ = 2 ( )( ) ( )( ) ( ) = [( 6) ( 2)( ) ][( )( )( 2) ][( )( 2) ] = 48 In the following we establish our algorith. Algorith 2: General Rando Path Algorith (GRPA). Input: -A -an n coplex atrix: k = (k, k 2, k )0ù with k i = n N 0 ù and M = 0 Output: E (A, k) - the estiate for Per(A, k). Step : Rearrange the coluns of A such that: M # M 2 # # M Step 2: For i = to N For = to hoose {( () ( ),, γ k } fro M \{( (r) w :w =,, k r, r =,,!} uniforly at rando. k Set a = 0, then a = a t = γ ( t ) s Set b = 0, then b = where s k as in Definition 5. End M = M + a b = End M = M/N Step 3: E (A, k) = M Proposition : Algorith 2 requires n + n eory variables, which can be reduced to n + n Proof: A is an array of n diension, the vector ( = (( (),( (2),, ( () ), with ( (i) = (( (i),( (i) () i 2,, ( ) for i =, 2,,, is of diension k + k2 + + k = n, (k, k 2,, k ) is of diension. Further, (s, s 2,,s ), (a, a 2,, a ) and (b, b 2,, b ) have a total of 3 eory variables, in this case we can use ust 3 variables s, a and b instead of 3 variables by accuulating the product in the second For loop of the algorith. Furtherore we have +2 variables (M, M 2,, M ), M and N. Therefore we have in total n + n n = n eory variables which can be reduced to n + n eory variables. We note that in the worst case = n, Algorith 2 requires n n + 5 eory variables. RESULTS AND DISUSSION Nuerical results are given for twelve different polynoial systes ranges fro size 0 up to 20. For each syste we pick up 00 variable partitions, taken at rando. The algorith 2 (GRPA) is used to estiate their MHBNs. The algorith is ipleented for two levels of accuracy N = 00 and N = 000. The results are copared with exact solutions, which are coputed firstly using REM. The algoriths are ipleented in MATLAB and executed on a personal coputer with Intel (R) Pentiu (R) Dual PU E GHz PU and 2.00 GB of RAM. ki 5209

5 Res. J. Appl. Sci. Eng. Technol., 4(23): , 202 We considered four types of polynoial systes as following: cyclic n-roots proble (Börck and Fröberg, 99), econoic n proble (Morgan, 987), noon n proble (Noonberg, 989) and reduced cyclic n proble (Verschelde, 996). The Table 3, 4, 5, 6, 7, 8, 9 and 0 show the percentages that relative errors of the coputational results fall into in the specific ranges. Each table shows the results for three systes of sizes 0, 5 and 20. The last colun of each table, which is denoted by % represents the percentage of the P tie using GRPA to the P tie using REM. The Table 3 and 4 are for the cyclic n polynoial syste; they are for two accuracy levels N=00, 000 respectively, while the Table 5 and 6 are for the econoic n polynoial syste, which are for two accuracy levels N=00, 000, respectively. The results of the noon n polynoial syste are in the Table 7 and 8 for two levels of accuracy, N = 00,000. Finally, the Table 9 and 0 are for the reduced cyclic n polynoial syste for two accuracy levels N = 00,000, respectively. We have three notes here. Firstly, in the case of the size of the syste being sall, say 0, using GRPA with accuracy N = 000, we note that the consuing tie by GRPA is bigger than that of REM, refer to the first row in the Table 4, 6, 8 and 0, but soeties in sall systes it is enough to take the level of accuracy as N = 00, refer to the first row in the Table 3, 5, 7 and 9. Secondly, for those systes with size larger than 0 say 5, 20 as in our exaples, the consuing tie by GRPA is uch saller than that of REM even with using high accuracy N = 000, refer to the second and third rows in the Table 3, 4, 5, 6, 7, 8, 9 and 0. Thirdly, for large systes, we note that the tie consued by GRPA becoes saller and saller Table 3: GRPA results for cyclic n proble with N=00 < n tie tie % s 43 s s 5 2 s s 6 h 3 34 s 0.9 Table 4: GRPA results for cyclic n proble with N=000 n tie tie % s 43 s s 5 2 s s 6 h 3 34 s.90 Table 5: GRPA results for econoic n proble with N=00. n tie tie % s 40 s s 7 5 s s 5 h s 0.2 Table 6: GRPA results for econoic n proble with N = 000. n tie tie % s 40 s s s 5 h s 2.06 Table 7: GRPA results for noon n proble with N = 00. n tie tie % s 43 s s 6 29 s s 5 h s 0.22 Table 8: GRPA results for noon n proble with N=000. n tie tie % s 43 s s 6 29 s s 5 h s 2.2 Table 9: GRPA results for reduced cyclic n proble with N = 00. n tie tie % s 40 s s 5 6 s s 5 h 56 9 s 0.2 Table 0: GRPA results for reduced cyclic n proble with N = 000 < n tie tie % s 40 s s 5 6 s s 5 h 56 9 s 2.03 copared with that of REM as syste size increases, refer to the last colun in the Table 3, 4, 5, 6, 7, 8, 9 and 0. This colun reflects the percentage of GRPA Tie to REM Tie. The nuerical result shows that with GRPA we can save uch tie copared with REM. The euciency increases exponentially as the syste size grows; for exaple, for a syste of size n = 0, GRPA estiates MHBN in about 53% of the tie needed by REM which is the best of the recent ethods; while for a syste of size n = 20 we need ust about 2% of that of REM. onsequently, the speed up of GRPA ipleentation becoes higher and higher as the syste size grows. Further, in the worst case when = n, GRPA requires only n 2 + 3n + 5eory variables, this is by Proposition, while REM requires 2 n eory variables, in other words, the eory storage needed by REM increases exponentially while that of GRPA increases in polynoial tie. Therefore GRPA is an eucient tool for handling the coputation of MHBNs for large systes and when REM becoes intractable or ipossible. 520

6 Res. J. Appl. Sci. Eng. Technol., 4(23): , 202 ONLUSION A General Rando Path Algorith (GRPA) for approxiating the peranent of a general coplex atrix is proposed in this study. The GRPA is constructed as a general algorith (ethod) over RP algorith. RP algorith is used for approxiating the peranent of square 0- atrices while GRPA estiates the peranent of square or non-square coplex atrices. The new algorith has been applied successfully in a new application field, that is the coputation of MHBN, which is considered as a special case of the general peranent. The large aount of nuerical results in the previous section shows considerable accuracy of estiating MHBN. The ethod is flexible in controlling the accuracy. Further, the presented proposition proves analytically the effectiveness and the efficiency of our ethod over soe recent ethods. AKNOWLEDGMENT All praise is due only to ALLAH, the lord of the worlds. Ultiately, only ALLAH has given us the strength and courage to proceed with our entire life. We would like to thank the Ministry of Higher Education, Malaysia for granting us FRGS Vot and Vot We also would like to thank Hudraout University of Science and Technology for their generous support. REFERENES Bawazir, H.M.S. and A. Abd Rahan, 200. A tabu search ethod for finding inial ultihoogeneous bézout nuber. J. Math. Stat., 6(2): Börck, G. and R. Fröberg, 99. A faster way to ount the solutions of inhoogeneous systes of algebraic equations with applications to cyclic n-roots. J. Sybolic op., 2: Liang, H., S. Linsong, B. Fengshan and L. Xiaoyan, Rando path ethod with pivoting for coputing peranents of atrices. Appl. Math. oput., 85: Malaovich, G. and K. Meer, oputing ultihoogeneous bézout nubers is hard. Theory oput. Syst., 40: Morgan, A.P., 987. Solving Polynoial Systes Using ontinuation for Engineering and Scientific Probles. st Edn., Prentice-Hall, Englewood liffs, New Jersey, pp: 48. Noonberg, V.W., 989. A neural network odeled by an adaptive Lotka-Volterra. SIAM J. Appl. Math., 49: Rasussen, L.E., 994. Approxiating the peranent: A siple approach. Rando Struct. Algoriths., 5: Verschelde, J., 996. Hootopy continuation ethods for solving polynoial systes using continuation for solving polynoial systes. Ph.D. Thesis. Katholieke Universiteit Leuven, pp: 78, Retrieved fro: 047&sid = 2&Ft = 2&clientId = 2690&RQT = 309&Vnae = PQD, Wapler,.W., 992. Bézout nuber calculations for ulti-hoogeneous polynoial systes. Appl. Math. oput., 5: Yan, D., J. Zhang, B. Yu,. Luo and S. Zhang, A genetic algorith for finding inial ultihoogeneous Bézout nuber. pp: Proceeding of the 7th IEEE/AIS International onference on oputer and Inforation Science, IEEE oputer Society. 52