Realization of uniform approximation by applying mean-square approximation

Transcription

1 Computer Applcatons n Electrcal Engneerng Realzaton of unform approxmaton by applyng mean-square approxmaton Jan Purczyńsk West Pomeranan Unversty of Technology Szczecn, ul 26 Kwetna 1, e-mal: janpurczynsk@pspl In the paper the polynomal mean-square approxmaton method was appled, where the appled crteron was the value of the maxmum error of the obtaned approxmaton The value of ths error depends on the number of approxmaton ponts wthn the range By changng the number of ponts wthn the range, t can be notced that the value of the maxmum error has the mnmum value for a partcular value of L number of consdered ponts For a polynomal of N degree, the optmum number of equdstant ponts of approxmaton L and the maxmum error of approxmaton are determned The proposed method was compared wth a unform approxmaton method, namely the Chebyshev polynomal The examples ncluded n the paper show that the proposed method yelds smaller values of the maxmum error than Chebyshev polynomal KEYWORDS: mean-square approxmaton, unform approxmaton, Chebyshev polynomal 34 1 Mean-square approxmaton unform approxmaton It s assumed that the seres of ponts x, x1,, xl and the values of functon f(x) n these ponts are gven: f f x ;,1,, L (1) The followng polynomal approxmaton s appled: N fa j ( x ) a j x (2) j By replacng functon f(x) wth approxmatng functon fa(x) the followng error s obtaned: fa( x ) f (3) Mean-square approxmaton means mnmzaton of the followng expresson: L 2 mn (4) Usng the necessary condton of exstence of a multple-varable functon extremum, from equatons (1), (2), (3), (4), the followng system of equatons s obtaned:

2 where: c jk x L j k N j c jk L a ; bk x j b k f k ; k,1,, N (5) The soluton of the system of equatons (5) n the form of a matrx s descrbed by the formula: 1 A C B (6) A - vector of coeffcents of the polynomal fa(x) (equaton (2)), where a j C ; c jk B b k For the case when the number of ponts x s larger than the degree of polynomal N ( L N ), t s actually approxmaton However, for L N there s the case of nterpolaton The equatons provded above are applcable for both the approxmaton and the nterpolaton Apart from a mean-square approxmaton (equaton (4)) there s also a unform approxmaton, whch s defned by the condton: max mn (7) L where s descrbed by (3) In accordance wth equaton (7) the pont s to mnmze the maxmum value of error wthn the nterval under study Among varous methods appled n the unform approxmaton, there are: Remez algorthm [2, 3], Pade approxmatons, Maclaurn seres and Chebyshev polynomals [1, 3] The scope of ths paper s lmted to Chebyshev polynomals, where ther applcaton conssted n solvng the problem of nterpolaton (L = N), where the knots fulfll the condton: x cos where:,1,, N (8) N 1 e they are roots of the Chebyshev polynomal The soluton s stll determned by equaton (6) In the paper a method s proposed whch draws on the fact that the maxmum error of a soluton obtaned usng the mean-square approxmaton method s hghly dependent on the number of ncluded ponts L + 1 Assumng the degree of polynomal N, fgure L s changed and the value of the maxmum error s determned For a determned optmum value L, whch ensures the smallest value of the maxmum error, the values of polynomal (2) coeffcents a j are determned The method s based on equatons (5) and (6), whch refer to the mean-square approxmaton, but whch refers to mnmzaton of the maxmum o 35

3 error The results of the proposed method wll be compared wth the results obtaned for the Chebyshev polynomals 2 Calculaton examples Due to the applcaton of the Chebyshev polynomals, the followng varablty nterval s assumed x 1,1 (9) Functon f(x) s gven by equaton: f ( x) arctg(4x) (1) The proposed method conssts n analyzng the value of error: eq fq x f (11) where fq x - result of the mean-square approxmaton ncludng LA ponts of approxmaton Symbol EQ refers to the maxmum error for a gven value LA: EQ max eq (12) LA EQ n LA n Fg 1 Dependency of the value of maxmum error EQ on the number of ncluded ponts of approxmaton LA for a polynomal of degree fve (N = 5) Usng Fg 1 such a value LA s found whch corresponds to the mnmum value of the maxmum error EQ The optmum value of the number of approxmaton ponts equals LO = 15 36

4 In Fg 2 absolute values of the error obtaned wth the proposed method eq and the error of the Chebyshev polynomals method were compared: ec fcx f (13) where: fcx - approxmatng functon obtaned for the Chebyshev polynomals 1 ec eq x Fg 2 Absolute values for the polynomal of degree fve (N = 5): sold lne and dashed lne eq (equaton (11), LO = 15) ec (equaton (13)) On the bass of Fg 2 the maxmum values of errors are determned, whch equal EC = 963 for the Chebyshev method EQ = 789 for the proposed method The determned value EQ = 789 corresponds to the mnmum value n Fg 1 The descrbed process s repeated for the followng values N = 3, 4, 14 (degree of the polynomal) The results of these calculatons are presented n Fg 3, where EA refers to the maxmum value of the error for the classc meansquare approxmaton conducted for L = 5 ponts Fg 3 shows that n the case of the Chebyshev polynomals, the error for the polynomal of degree three (N = 3) s smaller than for degree four N = 4 Ths stuaton reoccurs for the followng pars: N = 5/N = 6; N = 7/N = 8 It results from the fact that functon f ( x) arctg(4x) s an odd functon n the nterval under study x 1, 1 However, for error EQ (EA) t can be observed that the value of error determned for N = 3 s dentcal to the one determned for N = 4 Therefore, t s advsable to nclude only odd values of a degree of polynomal N, whch s the case n Fg 4 Fg 4 proves that, when consderng all the three methods, the proposed method yelds the smallest value of the maxmum error: EQ < EC < EA It should be notced that the classc approxmaton (EA) s only slghtly worse than the Chebyshev polynomals method (EC) 37

5 25 2 EC m 15 EQ m EA m Fg 3 Maxmum absolute values of error n a functon of polynomal degree N Sold lne wth rectangles EC represents the Chebyshev polynomal, dashed lne wth crcles EQ shows the error of an optmum polynomal and dashed lne wth pluses EA shows the error of the classc approxmaton 2 15 EC m EQ m 1 EA m Fg 4 Maxmum absolute values of the error n a functon of a polynomal degree for odd N The labels used n ths fgure are dentcal to the ones n Fg 3

6 Fgure 5 presents the dependence of the number of optmum ponts of approxmaton LO on the degree of polynomal N Ths dependence can be descrbed by a lnear functon of a form: LTm (14) As another example the followng functon was examned: 1 f x (15) x LO m LT m LO m Fg 5 Dependence of the number of optmum ponts of approxmaton LO on the degree of polynomal N Sold lne wth crcles represents emprcal values and dotted lne wth x represents the results of the lnear approxmaton (theoretcal dependence, equaton (14)) Fg 6 ncludes the maxmum absolute values of the error n the functon of a degree of polynomal N The stuaton s smlar to the one n Fg 3 but the dfference s that the smaller values of error are obtaned for even values of the degree of polynomal N The reason for the dfference les n the fact that the functon descrbed by equaton (15) s even Fgure 7 shows that the results (EQ) of the proposed method are burdened wth the smallest values of the maxmum error Interestngly, the classc meansquare approxmaton method yelds a smaller maxmum error than the Chebyshev polynomals method (EA < EC)Perhaps the reason les n the fact that n the process, L = 5 ponts of approxmaton were taken nto account, whch approxmately corresponds to the approxmaton conducted for a contnuous functon Ths fact may be mportant, as n the case of nterpolaton conducted for functon (15) a strong Runge-Kutta effect can be observed As for the Chebyshev polynomals method, t s an example of nterpolaton wth unevenly spaced knots Fg 8 presents the dependence of the number of optmum ponts of approxmaton LO on the degree of polynomal, where N s an even number Yet another example conssts n determnng errors of approxmaton for a functon descrbed by the followng dependence f ( x) ln( x 11) (16) 39

7 6 5 4 EC m EQ m 3 EA m Fg 6 Maxmum absolute values of error n the functon of a degree of polynomal N The labels used n ths fgure are dentcal to the ones n Fg EC m 2 EQ m EA m Fg 7 Maxmum absolute values of error n the functon of a degree of polynomal N, where N s an even value The labels used n ths fgure are dentcal to the ones n Fg 3

8 LO m LO m Fg 8 Dependence of the number of optmum ponts of approxmaton LO on a degree of polynomal N Fgure 9 presents the curve of the functon descrbed by equaton (16) for a gven nterval, x 1, f x Fg 9 Curve of the functon descrbed by equaton (16) for a gven nterval, x 1, eq ec Fg 1 Absolute values of the error for the polynomal of degree 5 (N=5): sold lne and dashed lne x eq ( LO=31) ec 41

9 Fgure 1 ncludes examples of absolute values of the error for a polynomal of degree 5 (N = 5): sold lne ec and dashed lne eq (LO = 31) From Fg 1 maxmum values of errors can be dentfed, whch equal EC = 163 for the Chebyshev method and EQ = 474 for the proposed method Fg 11 presents the values of the maxmum error for partcular methods It clearly shows the advantage of the proposed method over the other methods Apart from that, the error of the Chebyshev method s smlar to the error of the classc mean-square approxmaton method 2 EC m 15 EQ m 1 EA m Fg 11 Values of the maxmum error for partcular methods Labels as n Fg LO m LO m Fg 12 Dependence of the number of optmum ponts of approxmaton LO on a degree of polynomal N By ntroducng a new varable dentfed as: LOm Lm (17) N m

10 for whch the lnear approxmaton was conducted, the followng was obtaned: LTm (18) Based on Fg13, whch llustrates the dependence of varable L on a degree of polynomal N, t can be observed that equaton (18) fts emprcal data 1 L m 8 LT m Fg 13 Dependence of varable L (equaton (17)) on the degree of polynomal N Sold lne wth crcles represents emprcal values and dotted lne wth x represents the results of the lnear approxmaton (theoretcal dependence, equaton(18)) As the fnal example, a followng functon s examned: f x cos x (19) 2 Takng nto account the evenness of cosne functon, only even values of a degree of polynomal N were consdered Fg 14 presents maxmum values of error for partcular methods Due to small values of the error, a logarthmc scale was appled Fg 14 ndcates close smlarty of error values for the methods under study Wth a vew of explanng ths problem, quotents of errors are ntroduced n the forms: ECm EAm ecm ; eam (2) EQm EQm Fgure 15 shows that the rato of the maxmum error for the Chebyshev method to the maxmum error for the proposed method falls wthn nterval ec 139,1 89 Smlarly, for the classc approxmaton method, m the rato falls wthn ea 174, 2 13 It clearly proves a consderable m advantage of the proposed method over the other methods, whch was not clearly vsble n Fg14 43

11 EC m EQ m EA m Fg 14 Values of the maxmum error for partcular methods Labels as n Fg ec m 19 ea m Fg 15 Values of quotents of maxmum errors (equaton (2)) Summary On the bass of the aforementoned examples, t can be concluded that the Chebyshev polynomals method has no consderable advantage over the meansquare approxmaton method: smaller values of the maxmum error EC than EA were observed for the functon descrbed by equaton (19), smlar values EC and EA were obtaned for functon (1) and (16), however for functon (15) EA < EC s the case

12 In all the examned cases, the proposed method yelds smaller values of the maxmum error, yet the most substantal dfference s observed for functon (16), where quotents derved from equaton (2) fulfll nequalty ec m 2 ; ea m 2 Whle analyzng partcular examples, the dependence of the optmum number of approxmaton ponts LO on a degree of polynomal N was provded However no theoretcal dependence, true for all the examples, could be found Hence the only way s to determne optmum LO emprcally The algorthm s as follows: 1 As the number of approxmaton ponts assume LA=L and LA=L+1, f EQ(L) > EQ(L+1), then (accordng to Fg1) assume LA=L+2; 2 f EQ(LA) < EQ(LA+1),then take LA as a result; 3 the algorthm can be made faster by enlargng the step sze, eg from one to fve References [1] Fortuna Z, Macukow B, Wąsowsk J, Numercal methods, WNT, Warsaw, 1993 (n Polsh) [2] Jankowscy J M, Overvew of numercal methods and algorthms Vol1, WNT, Warsaw, 1981 (n Polsh) [3] Ralston A R, Introducton to numercal analyss, PWN, Warsaw, 1983 (n Polsh) 45