Error-Free Algorithms to Solve Special and General Discrete Systems of Linear Equations

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1 to Solve Specal and General Dscrete Systems of Lnear Equatons roslav orháč Insttute of Physcs Slova Academy of Scences Dúbravsa cesta 9 Bratslava 845 Slova republc E-mal:roslavorhac@savbas Abstract he paper presents a survey of error-free algorthms to solve varous systems of lnear equatons he presented algorthms do not ntroduce computatonal errors nto the soluton and thus they are well suted to solve ll-condtoned lnear systems he error-free algorthms are based on modulo arthmetc wo basc approaches have been nvestgated n the paper he frst one s based on teratve scheme usng one modulus only he other one s parallel and uses several modul and the Chnese theorem It s based on polynomal algebra operatons that allow to epress the operaton of deconvoluton as a sequence of convolutons of both response and output sgnals PoS(ACA7 XI Internatonal Worshop on Advanced Computng and Analyss echnques n Physcs Research Amsterdam the etherlands -7 Aprl 7 Speaer Copyrght owned by the author(s under the terms of the Creatve Commons Attrbuton-onCommercal-ShareAle Lcence

2 roslav orháč Introducton he soluton of lnear algebrac equatons s a very frequent tas n numercal mathematcs and epermental physcs at all When solvng such a set of lnear equatons one often meets the problem of ll-condtoned matr of the set For large dense sets of lnear equatons whch are as a rule ll-condtoned the stablty of the soluton cannot be guaranteed Roundng-off and truncaton errors durng the numercal computatons nvolved n obtanng the soluton to the problem cannot be tolerated Let us llustrate the stuaton usng a smple eample Illustratve eample: Let us have ll-condtoned set of two lnear equatons + y 4 ( + y 4 whch has the soluton y On the other hand let us consder the set + y 4 ( y 4 hs set has the soluton y - Very small change n the coeffcents ( and caused enormous changes n the soluton he nverse matr of ( contans elements 5 of the order of It demonstrates ts ll-condtonalty We are motvated by the fact that n scentfc computatons there are large classes of llcondtoned problems and there are also numercally unstable algorthms Dgtal computer s a fnte machne and therefore t s capable of representng nternally only a fnte set of numbers here est dffcultes assocated wth the attempts at appromatng arthmetc n the feld of real numbers (R + by usng the fnte set of so-called floatng-pont numbers F more approprately called the set of computer-representable numbers here s no possblty of representng the contnuum of real numbers n any detal A practcal soluton s to represent a real number by the closest computer-representable number thereby ntroducng the roundng error It s well nown that computers can perform certan arthmetc operatons eactly f the operands are ntegers oreover there are many stuatons n practce when we wor wth nteger nput data Processng of spectra (hstograms n nuclear physcs can serve as a good eample When solvng ll-condtoned problems durng the analyss of spectra t s not reasonable to leave the world of eact ntegers and thus ntroduce nstablty It motvated us to consder nteger arthmetc as a mean of avodng roundng errors n the hope that certan llcondtoned problems can be solved eactly Resdue number system s an eample of a number system wth whch we can do eact arthmetc he am of the contrbuton s to present errorfree algorthms to solve ll-condtoned systems of lnear equatons Let us summarze how we can proceed when the soluton of ll-condtoned lnear system of equatons can be epressed as ratonal fractons of ntegers we can ncrease the precson of floatng pont representaton - however t does not need to guarantee the correct result PoS(ACA7

3 roslav orháč we can wor wth long ntegers - from algorthmc pont of vew t s many tmes very cumbersome we can wor wth ntegers n fnte rngs usng resdue class or modulo arthmetc and at the end of the calculaton to convert the modulo representaton nto real numbers In the rest of the paper when speang about modulo arthmetc we shall consder prme modul Let us now llustrate modulo representaton of negatve numbers and nverse numbers for modulus 7 (see able I and II (mod able I odulo representaton of numbers from the range ( / ( / for (mod able II odulo nverse numbers for 7 One can observe that the assgnment between numbers and ther negatves and nverses s prme modul unquely unque By defnton for we have taen nverse number to be Convoluton systems In many applcatons the dynamc behavor of a lnear system can be descrbed by means of mpulse response functon h( n he perodcal convoluton of fnte sets (n and h( n s yn ( hn ( mm ( n ( m where s the length of both vectors Very frequently the advantageous property of factorzaton of the convoluton of two sgnals to the product of ther Fourer coeffcents s utlzed [] [] F[ y( n ] F[ h( n ] F[ ( n ] (4 Snce the Fourer transform cannot be computed wth absolute precson the number theoretcal transforms were defned wth the am to carry out the fast error-free convoluton of data hese transforms are defned n the rng of ntegers usng the operatons carred out n modulo arthmetc where s prme Drect and nverse umber heoretcal ransforms ( can be defned n X( ( n α (mod n (5 n n ( X( α (mod n (6 where a (mod PoS(ACA7

4 roslav orháč b α (mod and must dvde Among the most famous are erssene and Fermat transforms [] hey are frequently used for error-free calculatons of convolutons However the problem of precson s much more crtcal n the nverse operaton e deconvoluton Precse deconvoluton usng the Fermat number transform [] Convoluton system ( can be wrtten n matr form y( h( h( h( h( ( y( ( ( ( ( ( h h h h (7 y( h( h( h( h( ( y( h( h( h( h( ( or y H (8 he soluton of any regular system of lnear equatons (8 can be epressed as m m D (9 where D s determnant of the matr H s chosen modulus (n our case Fermat number and m s a fnte number Substtutng (9 nto matr equaton (8 gves m y D H + H + + H m ( or after some modfcatons ( D H H H m y ( From ( we can defne the vectors y ( D H ( H y + m ( If the vector y + equals the zero vector the calculaton can be fnshed It can serve as an ndcaton of the end of teratons o calculate modulo soluton of (8 we can utlze the Fermat number transform For transformed values t holds R( R ( h R ( y (mod ( Usng nverse Fermat transform we obtan modulo soluton of the vector D D (mod D (mod D (mod (4 where D + D D D H( (mod (5 hen PoS(ACA7 4

5 roslav orháč ( ( D + + m (6 Substtutng (6 to ( yelds ( y H + H + H m or n general y (mod H m (7 Hence to fnd partcular soluton we can apply agan Fermat transform hen we can proceed to the net teraton step y H y + (8 he fnal soluton can be calculated usng (9 In (9 ( ( ( (4 we assumed we now the eact value of the determnant D of the convoluton matr H In what follows we ntroduce the algorthm to calculate t [] [4] By employng the Fourer transform the determnant of the convoluton matr of the system (8 can be epressed ( ( ( ( + ( D h + hw + hw + + h W (9 ( π where W e and s a power of e the determnant of such a matr can be epressed as the product of the coeffcents of the Fourer transform of the response vector h he Fourer transform s used only formally For detals we refer to [] Let us llustrate the algorthm by an eample wth 8 Applyng DF to vector h we obtan the bt reversed vector h h W W W W h W W W W h W W W W W W h 4 W W W W W W h5 W W W W W W h6 W W W W W W h7 h + h+ h + h + h4 + h5 + h6 + h7 a X } group h h+ h h + h4 h5 + h6 h 7 b X } group ( h h + h4 h6 + ( h h + h5 h7 W c + cw X ( h h + h4 h6 ( h h + h5 h7 W group c cw X ( h h4 + ( h h5 W + ( h h6 W + ( h h7 W d + dw + dw + dw X 4 ( h h4 ( h h5 W + ( h h6 W ( h h7 W + X 5 d dw d W d W ( h h4 + ( hh5 W ( h h6 W + ( h h7 W 4 group d + dw dw + dw X 6 ( h h4 ( hh5 W ( h h6 W ( h h7 W 7 d dw dw dw X PoS(ACA7 5

6 roslav orháč he numbers n the frst and the second group are real In the thrd group we multply the 4 numbers X X and n the fourth group the numbers X 4X 5X 6X 7 he fact that W must be taen nto account hen ( ( ( ( ( d dd d ( d dd d W e ew ( ( ( d dd d ( d dd d W e ew ( ( e + ew e + e X X c + cw c cw c + c X X d + dw + d W + d W d dw + d W d W X X d + dw d W + d W d dw d W d W X4 X5 X6 X7 e ew We see that both products are real numbers Generally let us have the vectors X X n n X n where for Xn and + X n t holds Xn ( X ( n Xn ( Xn ( / X ( n / + + Xn X n Xn ( / Xn ( / X n ( X n ( X n ( X n ( n n log s the number of reducton / + s the number of cyclc shfts n wth a negatve transton to the topmost poston and / hen for the reduced vector X n + we get ( Xn ( Xn ( / + ( Xn ( + Xn ( Xn+ ( Xn ( Xn ( / + ( Xn ( + Xn ( Repeatng the above descrbed procedure untl the length of the vector X n + s greater than we can calculate the eact nteger value of the product of the approprate group ultplyng the products of all groups we can get the determnant of the convoluton matr whch s used n the above outlned deconvoluton algorthm Illustratve eample: Let us have the convoluton system PoS(ACA7 6

7 roslav orháč ( ( ( ( 5 ( where the determnant D 65 We shall use the modulus 7 whch s the Fermat number and 4 hen the forward and nverse Fermat transform matrces are as follows hen ht h( mod 7 ( mod Employng Eucldan algorthm we calculate vector of nverse modulo values h t ( [ ] 5 mod st teraton step We calculate the vector of the transformed output values Ym ym ( mod hen due to factorzaton property of the Fermat number transform ( we get X (mod Applyng the nverse Fermat transform we obtan D D 6 (mod 7 (mod D8 D where D was calculated usng relaton (5 Let us suppose that the postve numbers are represented n the range ( / and the negatve ones n the range ( / + hen we obtan the negatve values for numbers greater than ( / by subtractng the modulus PoS(ACA7 7

8 roslav orháč [ 86 7 ] 65 Further we calculate y H y ( y D y m [ ] y (mod 7 m y -nd teraton step In the net teraton step we obtan [ ] [ ] [ ] Ym ym(mod X (mod Usng the nverse Fermat transform we calculate the vector X(mod 7 [ 5 6] Adustng the elements greater than ( / we obtan [ 5 ] hen we calculate y y y H [ 9 ] y [ 9 ] [ 9 ] [ ] 7 he zero vector y ndcates the end of calculaton he resultng vector s then obtaned by [ 7] [ ] 7 [ 5 ] [ ] D hs s the eact soluton of our llustratve eample ( he perodcal -dmensonal convoluton of the fnte sets (n n n and h( n n n s defned by ( ( ( y n n n m m m h n m n m n m m m m Analogously to one-dmensonal case n [5] we have derved the error-free algorthm of multdmensonal deconvoluton as well as algorthm to calculate the determnant of the multdmensonal convoluton matr PoS(ACA7 Error-free deconvoluton usng polynomal algebra concept [6] [7] Polynomal algebra plays an mportant role n dgtal sgnal processng because convolutons can be epressed n terms of operatons on polynomals [] Polynomal algebra s the bass of one group of fast and error-free one- and multdmensonal convoluton algorthms 8

9 roslav orháč Let us suppose the elements of h( m ( l to be coeffcents of the polynomals H ( z and X ( z of the degree Hence we have m l ( ( ( ( ( H z h m z X z l z m l If we multply H ( z by X ( z the resultng polynomal wll be of degree hus n Y z H z X z y n z ( ( ( ( ( n hs means that the convoluton of two sequences can be treated as the product of two polynomals If the one-dmensonal convoluton defned by ( s cyclc the ndces nml are calculated modulo hs mples that z (mod and therefore the cyclc convoluton can be consdered as the multplcaton of two polynomals Y(z H ( z X ( z ( where the powers of the polynomal varable z are calculated modulo or by Y z H z X z mod z (4 ( ( ( ( Let us proceed to multdmensonal case of convoluton he -dmensonal cyclc convoluton of the dscrete nput sgnal and the response sgnal h s defned ( ( ( y n n h m m n m n m (5 m m where n n n and ndces n m are calculated modulo ndces n m are calculated modulo etc By consderatons analogous to those of the one-dmensonal case the polynomal epresson of the -dmensonal convoluton s obtaned Y ( z z z H ( z z z X ( z z z (6 where the powers of the varable z are calculated modulo and the powers of the varable z are calculated modulo In the case of deconvoluton e when we now the output sgnal y and the response sgnal h the nput sgnal can be formally epressed n polynomal form X ( z Y( z H ( z (7 and X ( z z z Y ( z z z H ( z z z (8 for one- and -dmensonal deconvoluton respectvely Let us llustrate the algorthm of calculaton of (7 and (8 by the followng eamples PoS(ACA7 Illustratve eample for one-dmensonal deconvoluton: Let the vectors of output sgnal and mpulse response be 9

10 roslav orháč [ ] [ 4 ] y h Accordng to ( t holds + z + z + z + 4z+ z X z ( ( ( Formally we can epress the sought + z + z + z X( z + 4z + z ultplyng the numerator and the denomnator by the polynomal H ( z(modz we obtan ( + z + z + z ( 4z+ z ( 9z+ 4z 5z X ( z ( + 4z+ z ( 4z+ z ( 5z In the followng sectons of the paper the product H ( z H( z wll be called the reducton of the polynomal H ( z wth respect to the varable z If we multply the numerator and the denomnator by the polynomal 5+ z the resultng polynomal s ( 9z+ 4z 5z ( 5+ z 6 5z + 56z z X ( z ( 5 z ( 5+ z 9 he coeffcents of the polynomal X(z represent eact soluton of the eample Illustratve eample for two-dmensonal deconvoluton: Let the matrces of two-dmensonal output sgnal and two-dmensonal mpulse response be y h 4 Usng (6 for we have ( + z + ( + 4z z ( + z + ( + z z X ( z z or ( + z + ( + 4z z X ( z z ( + z + ( + z z By multplyng the numerator and the denomnator by the polynomal H ( z z e by ts reducton wth respect to the varable z we get ( ( ( ( ( ( ( ( ( ( + z + + 4z z + z + z z + z z z X( z z + z + + z z + z + z z + 4z ow we multply ths result by the polynomal 4z ( + z + ( 4 + z z( 4z ( 9 5z + ( 4 + z z X ( z z ( + 4z( 4z 7 he resultng matr s PoS(ACA7

11 roslav orháč ow let us generalze the algorthm for -dmensonal data Let us assume that the dmensons of the convoluton system are powers of hen the algorthm of nverson of the polynomal H ( zz z and ts smultaneous multplcaton by the polynomal Y( z z z can be epressed as successve reductons of the response functon (autoconvolutons and convolutons of Y( z z z wth reduced response (crossconvolutons he (p + threducton of the response n the drecton (autoconvoluton s p+ p+ p p p p ( ( ( ( h l l l h h m m m (9 where p p m l mod m l mod m + l mod ( ( ( l l l p+ and the number of the reducton p log et we derve the algorthm of crossconvoluton Let ( ( n y where hen the calculaton of the (p + threducton of the sgnal n n the drecton s PoS(ACA7 p p+ p p p ( ( ( ( n l l l n h m m m ( where m l mod m l mod m l mod ( ( ( p l l l and the number of the reducton p log We repeat the algorthm (9 ( untl the number of reductons s log hen we contnue wth the reductons of the response and the sgnal n n the drecton (agan usng formulas (9 ( We proceed n ths way untl all the reductons of the response n all the drectons are carred out e untl the -dmensonal dscrete sgnal of the response s reduced to the only non-zero element It represents the value of the determnant of the system matr h hence ( r D h

12 roslav orháč where r log he fnal soluton s n D Specal lnear systems Error-free algorthm to solve nteger oepltz system [8] In ths secton we propose an algorthm to solve nteger nonsymmetrcal oepltz system whch s based on Levnson algorthm [9] It removes roundng off errors her accumulaton can when usng classc algorthms deterorate or destroy the soluton oepltz system of lnear equatons s defned or a a a a y + a a a a + y a a a a y a y ( For nteger regular oepltz system one can wrte a y (4 D In [8] we derved the algorthm for fast soluton of nteger oepltz system (4 usng recursve procedure solvng n each teraton step the system ( v a y ; (5 D o determne quanttes wth the nde + we get ( + ( y+ D a+ v ( + + ( ad a+ b+ D D a a b v D + ( + ( ( + ( + c a D a c b a D a b ( (6 PoS(ACA7

13 roslav orháč o determne remanng components of the vectors v b c we have ( + ( ( + ( v v D v b D ( + ( ( + ( c c D c b D ( + ( ( + ( b b D+ b+ c + D wth ntal values ( ( ( D av yc ab a (8 From the above gven formulas one can see that all quanttes are ntegers After some teraton steps however ther magntudes ncrease rapdly whch complcates the realzaton of the calculaton o overcome ths problem we can carry out the calculaton n modulo arthmetc n dfferent prme modulo classes m r so that t holds [] P mm mr (9 where P must satsfy the condton / ( / P > ma{ ( A ( ( A ( y } (4 (A ma a + ( y ma y he calculaton n the gven modulo class s ndependent of other modulo classes and thus ths model of mplementaton s well suted for parallel computng By nverse converson from resdual representaton employng Chnese theorem [] [9] one can calculate resultng vector and determnant D (7 PoS(ACA7 Illustratve eample: Let us fnd the eact soluton of the oepltz system Frst we ntalze the vectors and varables [ ] [ ] [ ] [ ] ap aa an a a y [ ] [ ] [ ] b c v D Wth respect to (9 and (4 we choose the modul m 5 m 7 m Carryng out the frst teraton step yelds [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] b(mod 5 ; b(mod 7 ; b(mod 4 ; c(mod 5 ; c(mod 7 5 ; c(mod 5 4 ; v(mod 5 ; v(mod 7 6 ; v(mod 6 ; D(mod 5 4; D(mod 7 4; D(mod 4

14 roslav orháč After the second teraton step ( the calculaton n ths eample s fnshed he resultng values of vectors and determnant are then as follows b(mod 5 ; b(mod 7 ; b(mod 4 4 ; [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] c(mod 5 4 ; c(mod ; c(mod 5 4 ; v(mod 5 ; v(mod 7 ; v(mod 5 4 ; D(mod 5 ; D(mod 7 ; D(mod By converson of the vector v and the determnant D from resdue class code one obtans v [ 6 67] D Postve numbers are represented n the range ( P / and negatve ones n the range ( P / + P hen the negatve values are calculated by subtractng P hen the fnal soluton s [ 6 8 ] An algorthm to solve Hlbert system of lnear equatons eactly [] ypcal eample of etremely ll-condtoned matrces are Hlbert matrces In lterature one can fnd the eamples of Hlbert matrces wth the elements a ; + We shall consder more general Hlbert systems wth elements a ; d + hen for we can wrte b b b y y b b b 4 y b b4 b5 or after echangng the elements of the vector b b b s s s z R R R z z R R R z b4 b b s s s z R R R z b5 b4 b s s s PoS(ACA7 4

15 roslav orháč However ths represents oepltz system of lnear equatons Assumng that all quanttes are ntegers the determnant of the matr can be wrtten n the form of ratonal fracton For one obtans C C D D D + + D s ss ss sss sss s ssss D where left upper nde C denotes numerator and D denomnator Wthout beng nterested n the numerator for the denomnator of the determnant n general case one can wrte D D s + s + s s s ss We ntroduce pars of epressons of all quanttes for both numerator and denomnator respectvely Frst let us determne quanttes wth the nde + C ( C D ( C ( D ( D b b+ s ; b+ b + D D ( ; s D s b + C C ( D ( c C ( D ( D c+ s ; c+ c + ; D D ( + s+ D s+ c + D y C ( C C v D (+ D ( + C ( + D ( + y+ D v D D D ( y D s v s + + C C ( D b C D D + D+ s ; D+ D + D D ( s D s+ b c ; v v ; hen we determne remanng components of the vectors b c v s + D ( + D C C ( C ( + C ( b D D b b c D ( + D C ( C D D ( D ( + D ( D D b b c s s+ D ( + D C C ( C ( + C ( c D ( D C ( D D c c+ b+ c D + ; c C D D ( D ( + D ( D D c c b s b D ; b ; v D D v v b v c ; b D ( + D ( + D C C ( C ( + C ( D ( + D ( + v + C ( D ( + C D D ( D ( + D ( c+ D D+ v v+ b+ he ntal values were set D C D ( D C ( C D ( C ( D ( C ( D s; D ; v y; v y; b s ; b ; c s; c ; PoS(ACA7 Error-free algorthm to solve Vandermonde system of lnear equatons [] Let us have Vandermonde system 5

16 roslav orháč y y a a a a y a a a a A y a a a a It s well nown that the determnant of the system matr s (4 + D (a a and the elements of nverse matr can be epressed usng polynomal coeffcents [] P ( B a a (4 a (4 Analogously to the algorthm gven n [] we determne coeffcents of the polynomal P( ( a (mod + c + + c + c (mod (44 Polynomals from (4 modulo can be epressed z P( (a P ( (mod (mod (a a (a a hen one can derve z z ( c + z a (mod (46 z ( c + z a (mod + + Resultng nverse matr modulo s where d d d z z z z z z z z z A (mod (mod B d ( a a (mod We can employ teratve scheme of error-free soluton of lnear equaton system from the secton (45 (47 PoS(ACA7 Illustratve eample: 6

17 roslav orháč Let us have the Vandermonde system he determnant of the matr accordng to (4 s -48 We shall use modulus 5 hen D Usng the relatons (45 (46 (47 yelds nverse matr n modulo class B (mod hen the procedure to calculate the eact soluton s as follows 4 ym [ ] B (mod 5 (mod 5 y ; ym ; 4 4 D (mod 5 et we calculate vectors [ 44] [ ] [ ] y A ; y ; ym -st teraton step PoS(ACA B (mod 5 (mod 5 ym Agan let us suppose that the postve numbers are represented n the range ( / and the negatve ones n the range ( / + hen we obtan the negatve values for numbers greater than ( / by subtractng the modulus [ ] Further we calculate vectors [ ] [ ] [ ] y A 4 ; y 5 9 ; ym 4 7

18 roslav orháč -nd teraton step We contnue n calculatons wth B ym (mod 5 [ 4 ] After adustment of elements greater than ( [ ] / we have Agan we calculate vectors y ; y 77 ; ym [ ] [ ] [ ] A -rd teraton step Fnally we calculate B ym (mod 5 [ ] We calculate vectors y A [ 77 ] ; y 4 [ ] he zero vector y 4 ndcates the end of teratons he result s General lnear systems 4 Parallel error-free algorthm [4 5] PoS(ACA7 Let us have system of lnear equatons where all the elements of the matr A and vector y are ntegers or y(n a(nm(m n m a a a a y a a a a y Ay a a a a y Let us splt the calculaton nto several resdue classes satsfyng condtons (9 (4 hen n each resdue class we can carry out separately the calculatons A (mod m y r Calculatons n resdue classes can be carred out n parallel Gauss-Jordan elmnaton n resdue class m can be descrbed by the followng procedure Let ( a a ( modm; { n} { n+ } For n ( denotes elmnaton step do: 8

19 roslav orháč search for such c ( f f ( ( ( + < that a c ( { } a a a ( mod m n+ (49 for l n and l do: ( ( ( ( l l + l { } a a a a ( mod m n+ (5 he determnant n the resdue class m s obtaned as the modulo product of the elements (see (49 n J ( D( modm Dm ( a ( modm (5 ( a where J s the total number of echanges of the elements hen the sought soluton of (48 n the resdue class m s gven by D ( mod m ( mod m [ Dm ]( mod m m D (5 m D m After the calculaton of the soluton of a lnear equatons set n the modular arthmetc s fnshed we have n + element vectors ( D( mod m ( mod m n( mod m r (5 he resultng nteger values D n can be obtaned by the nverse converson of the vectors (5 from the resdue class code Snce we use the nown algorthm (gven eg n [6] we descrbe t very brefly from algorthm pont of vew We suppose we have a vector of modul β ( m m mr (54 and a vector of the resdue representaton of the nteger ( t ( mod m ( mod m ( mod m (55 r Further let us denote m m m ( ( ( ( β + + r ( ( t t t + t r d t ( ( mod m t ( ( mod m r If we defne the vector t to be ( + t + t d m ( mod β then the resultng sought number s t ( + m ( t ( + m t ( + + mr t r( r (56 PoS(ACA7 4 Sequental algorthm to solve general system of lnear equatons [7] So far we have mentoned two basc algorthm classes to solve lnear equatons eactly 9

20 roslav orháč teratve algorthm (appled n secton Fermat transform secton Vandermonde system parallel algorthm (appled n secton oepltz system secton 4 general system of lnear equatons In ths secton we want to demonstrate that sequental teratve algorthm can be appled also for general systems of lnear nteger equatons Let us defne the followng algorthm: Algorthm A a Let y y m b Calculate vector y m ( c It holds D ( mod A mod d Let e Calculate vectors ' ' y A y ( ym D y ym y ( mod (57 f g Calculate A y m ( mod (58 h For ( s the sze of vectors y calculate ( ( + ( Calculate vectors ' y y ' y A y+ ym+ y +( mod If for all y+ ( fnsh the calculaton If not ncrement and repeat the algorthm from pont g on he resultng soluton of system (48 s smply calculated as hen determnant of the D system of lnear equatons can be calculated usng the formula D a( a( ( (59 where the coeffcent vectors are calculated by eact soluton of sets A A (6 usng the Algorthm A et problem n the algorthm A s calculaton of the nverse matr n modulo arthmetc Let us defne teratve procedure where A A B B A + + A ( mod + + A A B B (6 hen one can derve PoS(ACA7

21 roslav orháč ( ( ( ( B A A A A mod B A A B mod B B A A mod B A + B B B mod (6 4 Sequental error-free algorthm to solve system of polynomal equatons [8] Sequental teratve algorthm can be successvely appled also for polynomal systems of lnear nteger equatons As ( ( s y( s (6 a ( p p ( p p s a s + + a s + a y s y s + + y s + y Assumng the coeffcents beng ntegers the soluton of the system gven by (6 can be epressed as p p p ( s ( s ( s + ( s ( s + + ( s ( s D( s (64 and vectors m ( s ( s + ( s + + m( s (65 where s a prme modulus D s determnant of the matr A and m are fnte ntegers Let us suppose for the moment that the determnant D( s (polynomal s nown We denote polynomal modulus as P( s s p he algorthms for the determnant and nverse matr of polynomal system can be derved analogously to relatons (59-6 For detals we refer to [8] ew problem s the calculaton of nverse polynomal In what follows we outlne brefly ths algorthm We have to fnd polynomal bs ( to polynomal as ( so that ( ( ( ( a s b s mod P s (mod (66 Accordng to [] the cyclc convoluton usng polynomal algebra concept s p c a b (67 l l where ndces l are calculated modulo p From what s gven above t follows that soluton of equaton (66 represents the soluton of lnear system a ( a( p a( b( a ( a ( a( b( (mod (68 a( p a( p a( b( p n resdual class Inverse polynomal bs ( can be obtaned by successve reductons of polynomal as ( or vector a For r+ st reducton t holds PoS(ACA7

22 roslav orháč r ( / a l a a m (mod ( ( ( ( r+ r+ r r r where r+ r m ( l (mod ; l r+ and number of reducton r log Let n [ ] and r ( / nl n am ( mod ( ( ( ( r+ r r r where r m ( l (mod ; l n and number of reducton r log he resultng soluton of polynomal nverson s t n b (mod t a( where t log Illustratve eample: Let us have lnear system consstng of polynomals s + s s + ( s s s + P s s and numercal modulus 5 Further we We shall use polynomal modulus ( shall assume that we now the determnant ( resdue class P( s and the resdue class ( ( ( ( ( D s s 5s + s + the determnant n the Dpm s D s mod s mod 5 s + (69 and the nverse matr n the resdue class P( s and the resdue class ( ( 5 s + 4s + s+ B( s A ( s mod ( s mod s + s + s + s We calculate s + 4s + s+ s + s p ( s ( mod ( s ( mod 5 s + s + s + s Usng Dpm( s from (69 we get ( ( 5 s s ( s Dpm( s mod ( s mod We ntalze vectors and control varables PoS(ACA7

23 roslav orháč 5 4 s + s 5s + s 4s + s+ ( s y( s D( s ( s 5s + s + s s + s+ s + ( s We calculate vectors s s s s s s ' y ( s As ( ( s s s + 4s + s 5 4 s 5s 6s ' ( s ( s y ( s s s + ( s ( mod ( s 5s 5s s s y ( s 5 5 s s 4s + 4s yp ( s y ( s( mod 5 s he vector ( y s thus we set ( ( 5 ( ( y ( ( s B s s mod s mod p s + 4s + s+ 4s + 4s 4s s ( mod ( s ( mod 5 ( mod 5 s s s s s If any coeffcent of the soluton s greater than ( / t s consdered negatve and accordng to the rules of modulo arthmetc ts negatve value s calculated by subtractng modulus We update ntermedate result s + s s 5s+ ( s + 5 Further we calculate approprate vectors s s s s s ' + y ( s a( s ( s s s + s 5 4 s 5s s 5s ' + ( s ( s 5 y ( s s + ( s ( mod ( s y ( s yp ( s 5 he vectors yp ( s ( s herefore we set control varables hen we calculate ( s s 5s s ( s yp ( s ( s( mod 5 s Further we calculate net partcular soluton PoS(ACA7

24 roslav orháč s 4s s s s ( ( ( ( 5 ( s s mod s mod mod s s s s whch we add to ntermedate result 4 s 5s+ s s + s 6s+ ( s + ( s s + Agan we repeat and calculate ' s + s s s 5s y ( s As ( ( s s s + y y y ' ( s ( s ( s 5 ( s ( s Both vectors y ( s and ( s he resultng soluton s p equal the zero vectors It means that we fnsh the calculaton 4 s + s 6s+ ( s ( s D( s s 5s + s+ s + 5 Volterra systems Drect generalzaton of convoluton systems for nonlnear systems are Volterra systems y( h ( l ( l + h ( l l ( l ( l l l l y ( + y ( + Problems connected wth Volterra systems can be dvded nto several tems determnaton of Volterra ernels [9] calculaton of the output of the nonlnear Volterra flter [] determnaton of nverse ernels to the gven nonlnear Volterra system [] We shall focus on determnaton of Volterra ernels Let us agan employ polynomal algebra and epress the nput sgnal and th Volterra ernel l l l l l l l l X( z (l z H ( z z z h( z z z z z z Accordng to [9] for the Volterra ernel of the th degree we can wrte p Y(zz z Y(z z z H (z z z X(z X(z X(z (7 D We can agan employ the algorthm based on successve reductons of the denomnator of (7 (see secton he algorthm conssts n autoconvolutons and crossconvolutons Let us demonstrate the algorthm usng smple eample (7 PoS(ACA7 Illustratve eample: Let and 4

25 roslav orháč [ 4] [ ] y hen accordng to (7 and after the reductons we have + 4zz ( + 4zz ( z ( z H(zz ( + z ( + z ( + z ( z ( + z ( z 9 4z 4z + 6zz 9 4z 4z + 6zz ( 9 Hence the resultng ernel of the second degree s h Conclusons In the contrbuton we have presented several error-free algorthms to solve one- and multdmensonal convoluton systems based on Fermat number theoretcal transform he man sgnfcance of the use of modulo arthmetc s elmnatng roundng-off and truncaton errors In the contrbuton we have derved the error-free algorthm to solve convoluton systems that s based on polynomal algebra concept hs form of -dmensonal deconvoluton algorthm permts to epress t as a sequence of -dmensonal autoconvolutons of the response sgnal and crossconvolutons of the response and the output sgnal Further n the contrbuton we have etended the error-free algorthms to specal lnear systems eg oepltz Hlbert Vandermonde systems We contnued wth general systems of lnear equatons In prncple the algorthms proposed n the contrbuton can be dvded nto two groups teratve (usng one modulus one resdue class parallel (usng several modul several resdue classes and Chnese theorem he parallel algorthm s well suted for the mplementaton on parallel computers that allows the ncrease of the calculaton speed We have etended the teratve algorthm also for the lnear system of polynomal equatons At the end of the contrbuton we outlned the possble etenson of the applcaton of error-free algorthms for nonlnear Volterra systems PoS(ACA7 References [] Ahmed and KK Rao Orthogonal transforms for dgtal sgnal processng Sprnger-Verlag 975 [] H J ussbaumer Fast Fourer transform and convoluton algorthms Sprnger-Verlag 98 [] orháč Precse deconvoluton usng the Fermat number transform Computers and athematcs wth Applcatons A (986 9 [4] orháč Error-free deconvoluton based on cyclc determnant calculaton approach Computer athematcs 49 (99 4 5

26 roslav orháč [5] orháč K-dmensonal error-free deconvoluton usng the Fermat number transform Computers and athematcs wth Applcatons 8 (989 [6] orháč Precse multdmensonal deconvoluton usng the polynomal algebra concept Internatonal J Computer athematcs (99 [7] orháč and V atouše Eact algorthm of multdmensonal crculant deconvoluton Appled athematcs and Computaton 64 (5 55 [8] orháč An error-free Levnson algorthm to solve nteger oepltz system Appled athematcs and Computaton 6 (994 5 [9] R E Blahut Fast algorthms for dgtal sgnal processng IB Corporaton Owego Y 985 [] ewman Solvng equatons eactly at Bureau Standards 7B: [] orháč An algorthm to solve Hlbert systems of lnear equatons precsely Appled athematcs and Computaton 7 (995 9 [] orháč An teratve error-free algorthm to solve Vandermonde systems Appled athematcs and Computaton 7 ( 45 [] W H Press et al umercal recpes Cambrdge Unversty Press Cambrdge 986 [4] orháč R Lórencz A modular system for solvng lnear equatons eactly I Archtecture and numercal algorthms Computers and Artfcal Intellgence (99 5 [5] R Lórencz orháč A modular system for solvng lnear equatons eactly II Hardware realzaton Computers and Artfcal Intellgence ( [6] RGregory EVKrshnamurthy ethods and applcatons of error-free computaton Sprnger- Verlag ew Yor Berln Hedelberg oyo 984 PoS(ACA7 [7] orháč One-modulus resdue arthmetc algorthm to solve lnear equatons eactly athematcal and Computer odellng 9 ( [8] orháč An error-free algorthm to solve lnear system of polynomal equatons athematcal and Computer odellng 9 ( [9] orháč Fast error-free algorthm for the determnaton of ernels of the perodcal Volterra representaton Appled athematcs and Computaton 8 (99 87 [] orháč A fast algorthm of nonlnear Volterra flterng IEEE ransactons on Sgnal Processng 9 (99 5 [] orháč Determnaton of nverse Volterra ernels n nonlnear dscrete systems onlnear Analyss heory ethods & Applcatons 5 (