Journal of Universal Computer Science, vol. 1, no. 7 (1995), submitted: 15/12/94, accepted: 26/6/95, appeared: 28/7/95 Springer Pub. Co.

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1 Journal of Unversal Computer Scence, vol. 1, no. 7 (1995), submtted: 15/12/94, accepted: 26/6/95, appeared: 28/7/95 Sprnger Pub. Co. Round-o error propagaton n the soluton of the heat equaton by nte derences Fabenne Jezequel (Unverste Perre et Mare Cure, France Fabenne.Jezequel@mas.bp.fr) Abstract: The eect of round-o errors on the numercal soluton of the heat equaton by nte derences can be theoretcally determned by computng the mean error at each tme step. The oatng pont error propagaton s then theoretcally tme lnear. The expermental smulatons agree wth ths result for the towards zero roundng arthmetc. However the results are not so good for the roundng to the nearest arthmetc. The theoretcal formulas provde an approxmaton of the expermental round-o errors. In these formulas the mean value of the assgnment operator s used, and consequently, ther relablty depends on the arthmetc used. Key Words: Floatng pont arthmetc, numercal error propagaton, partal derental equatons, nte derence methods Category: G Introducton In the computatonal soluton of partal derental equatons, two types of errors are generated : the method error due to approxmatons nherent n the numercal method and the round-o error due to the oatng pont arthmetc of the computer used. Ths paper presents an analyss of the round-o error propagaton n the soluton of the heat equaton by nte derences. For each pont of the mesh, the soluton s approxmated by a scalar product wth three terms. Therefore prevous studes concernng round-o errors n arthmetcal operatons and n scalar products are presented. Ths analyss has been carred out for the towards zero roundng arthmetc and for the roundng to the nearest arthmetc, these two roundng modes respectng the 754-IEEE standard. Dfferent cases have been consdered dependng on whether ntal data and nte derence scheme coecents are exactly represented or not n the computer. To conclude, the man results obtaned are nally represented. 2 Prevous results concernng round-o errors 2.1 Assgnment error Let x be a real number and X ts oatng pont representaton. The relatve assgnment error on X s = (X,x). Let P be the set of all the possble X relatve assgnment errors. The mean value and the standard devaton 2 of P can be computed accordng to the roundng mode, the base and the number of bts n the mantssa n the oatng pont representaton [see Alt 76, Alt 78, Hammng 7, Knuth 69, La Porte, Vgnes, 74a and Vgnes 93]. Let b be the base 469

2 (usually b s 2 or 16) and p the number of dgts n the mantssa n the standard oatng pont representaton, for the towards zero roundng arthmetc : = b,p (1,b) 2 log b 2 = b,2p [ (b2,1) 6 log b, (b,1)2 (2 log b) 2 ] and for the roundng to the nearest arthmetc : = 2 = b,2p (b 2,1) 24 log b. 2.2 Error due to arthmetcal operators Let +,,,, = be the exact operators on real numbers and,,, the correspondng oatng pont operators (addton, subtracton, multplcaton and dvson) on F, whch s the set of all the values representable n the machne. The followng formulas have been obtaned by consderng only rst-order approxmatons n b,p : Let x and y be real numbers, X and Y ther representatons n F : X x(1 + ) and Y y(1 + ) X Y x + y + x + y + (x + y) X Y x, y + x, y + (x, y) X Y (x y)( ) X Y (x=y)(1 +, + ) wth,,, beng elements of P. 2.3 Error n the computaton of scalar products Let us consder a scalar product r = P n =1 x y, wth x and y real numbers. If t s computed usng ths cumulatve method : R := ; FOR I =1TON DO R := R X[I] Y [I], the absolute error on the exact scalar product r s dened by : = R, r. If X = x (1 + ) and Y = y (1 + ) the error can be estmated by the followng formula : P n =1 x y ( + + ) + 1 (x 1 y 1 + x 2 y 2 ) + 2 (x 1 y 1 + x 2 y 2 + x 3 y 3 ) +::: + n,1 (x 1 y 1 + x 2 y 2 + ::: + x n y n ) wth,,, beng elements of P. 47

3 We assume that the errors,,, are ndependent and have the same mean value and the same standard devaton 2. Under these hypotheses, the mean values of and 2 are gven by : =r (n2 +7n,2) 2n 2 =() 2 [( 3n3 +41n n,72 12n )r 2 +( (n,2)(n2 +3n,6) 12n )s 2 ]+ 2 [( n+1 3 )r2 +( n2 +19n,6 6n )s 2 ] wth s 2 = P n =1 (x y ) 2 [see Alt 78 and La Porte, Vgnes 74b]. 3 Error propagaton n the soluton of the heat equaton 3.1 Fnte derence scheme The one-dmensonal heat equaton descrbes the heat propagaton n a lnear bar. Let U(x; t) be the temperature on ths bar at pont x and tme t. The heat equaton n [a; b] [t ; +1[ s descrbed by the followng 2 =, wth K> 8t t, U(a; t) =U a (t) and U(b; t) =U b (t) 8x 2 [a; b],u(x; t )=U (x) The constant K represents the materal thermc dusvty. The doman s dscretzed wth space step x and tme step t : x = a + x; =; 1; :::n x = a and x n = b t j = t + jt; j =; 1; ::: Let U j be the soluton at pont x and tme t j. The explct nte derence method s used : then:,u j t = K( U j,1,2u j +U j +1 ) for (x) 2 =1; :::; n, 1,j =; 1; :::. = Kt (x) 2 U j Kt,1 +(1, 2 (x) 2 )U j + Kt (x) 2 U j +1 To ensure the stablty of ths scheme, the relaton Kt (x) 2 < 1 must be satsed. 2 If c 1 = Kt (x) 2 and c 2 =1, 2c 1, the nte derence scheme s : = c 1 U j,1 + c 2U j + c 1U j +1 for =1; :::; n, 1,j =; 1; :::. 471

4 3.2 Theoretcal round-o error Relatve round-o error To estmate the round-o error n the computaton of U j wth the nte derence scheme prevously proposed, several notatons are necessary. Let U j, c 1, c 2 be the algebrac values and U ~ j,~c 1,~c 2 the computed values. Then for =1; :::; n, 1,j =; 1; :::, ~ =((~c 1 ~ U j,1 ) (~c 2 ~ U j )) (~c 1 ~ U j +1 ), ths formula beng nether commutatve nor assocatve. Let j be the relatve error on U j due to the cumulaton of assgnment errors and round-o errors generated n prevous teratons : ~ U j = U j (1 + j ). merely represents the assgnment error on U and the mean value s equal to the mean value of the assgnment operator. Let 1 and 2 be the relatve errors on c 1 and c 2 : ~c 1 = c 1 (1 + 1 ) and ~c 2 = c 2 (1 + 2 ). If c 1 and c 2 are not results of computatons or are computed n nnte precson, then 1 and 2 are merely assgnment errors. s computed by a scalar product wth three terms. Therefore the formula provdng the round-o error n the computaton of scalar products can be appled : j+1 = 1 c1 U j,1 ( 1 + j +,1 1) +c 2 U j ( 2 + j + 2) +c 1 U j +1 ( 1 + j ) + 1 (c 1 U j,1 + c 2U j ) + 2 (c 1 U j,1 + c 2U j + c 1U j +1 ) wth, beng elements of P. The errors 1 and 2 are due to addtons, 1, 2 and 3 to multplcatons. Assgnment errors, are assumed to be ndependent and consequently : = = 2 = 2 =() k =() 2 etc

5 3.2.2 Frst moment As assgnment errors are assumed to be ndependent and have the same mean value, the rst moment s, for j =; 1; ::: and =1; :::; n, 1: j+1 = c 1U j,1 ( j, )+ c 2U j ( j +3+ 2)+ c 1U j +1 ( j ): The space step and the tme step are assumed to be low enough to allow the followng approxmaton : 8j, 8 =1; :::; n, 1, U j,1 U j U j +1 1: Therefore at a xed teraton, all relatve errors j 8 =1; :::n, 1; j = j. have the same mean value : then 8j, j+1 = j +(3, c 1 ) +2c c 2 2 therefore : 8j ; j = + ((3, c 1 ) +2c c 2 2 )j: The round-o error propagaton n the soluton of the heat equaton by nte derences s theoretcally tme lnear. The general formula above s smpled f the coecents c 1 and c 2 and the ntal data U are exactly represented. For the roundng to the nearest arthmetc, the mean value of the assgnment errors, s theoretcally zero. In ths case, f the coecents c 1 and c 2 and the ntal data U are exactly represented, 1 = 2 = =, and the rst moment remans theoretcally zero. Thus t s necessary to estmate the second moment Second moment The estmaton of the second moment has been carred out for the roundng to the nearest arthmetc under the followng assumptons : c 1 and c 2 are exactly represented : 1 = 2 =, ntal data are exactly represented : 8 =1; :::; n, 1; =. As for the estmaton of the rst moment, t s assumed that : 8j 1, 8 =1; :::; n, 1, U j,1,1 U j U j,1 U j U j,1 +1 U j 1: The estmaton of ( j+1 ) 2 nduces the emergence of terms j, beng a relatve assgnment error. It s assumed that j =,1 j = +1 j. 473

6 Therefore, ( j+1 ) 2 =(2c c 2 2) ( j )2 +(7c c 2 ) 2 +2, (1, c 1 ) 1 j + 2 j + c 1 1 j + c 2 2 j + c 1 3 j. As 8 =1; :::; n, 1, =, (1 ) 2 =(7c c 2 ) 2. j are estmated, beng a relatve assgnment error : 1 j =(1, c 1) j,1 2 j = j,1 1 j = c j,1 2 j = c j,1 3 j = c j,1 As =, nally : 1 j =(1, c 1)j 2 2 j = j2 1 j = 3 j = c 1j 2 2 j = c 2j 2. Therefore : 8j 1, ( j+1 ) 2 =(2c c 2 2) ( j )2 +(1+2j) (7c c 2 ) 2. Then ( ) 2 = and 8j 1, ( j ) 2 =(7c c 2 ) 2 P j,1 k= (1 + 2k) (2c2 1 + c 2 2) j,k,1. The evoluton of the second moment s thus of degree 2. Ths result remans coherent wth the lnear evoluton of the rst moment. 3.3 Expermental round-o error 3.4 Frst moment The expermental rst moment j s computed accordng to the followng formula :! j = 1 n,1 X ~ U j, U j n, 1 ~ =1 U j 474

7 where U j represents the algebrac value and U ~ j the computed value. The number of sgncant dgts lost n computatons does not depend on the precson of the oatng pont arthmetc [see Chesneaux 88 and Chesneaux 9]. Therefore U j, theoretcally computed n nnte precson, can be computed n double precson. Then U ~ j s the result of the same computaton carred out n sngle precson [see Hull, Swenson 66]. The theoretcal expresson of the rst moment has been valdated for the towards zero roundng arthmetc and the roundng to the nearest arthmetc, on a computer usng base 2 wth p = 24 and respectng the 754-IEEE standard. Four cases can occur dependng on whether the coecents c 1 et c 2 and the ntal data U are exactly represented or not. Each case has been studed for the roundng to the nearest arthmetc, where the mean value of the assgnment errors s zero, and for the towards zero roundng arthmetc, where s not zero. The number of space steps n s set to 1, the number of tme steps s set to 1. 1st case : If the coecents c 1, c 2 and the ntal data U the rst moments: are both not exactly represented, 8j ; j = + ((3, c 1 ) +2c c 2 2 )j The expermental moment s tme lnear as well. However the theoretcal moment s n absolute value slghtly greater than the expermental moment. Ths derence may be due to an overvaluaton of the theoretcal mean value. Results concernng the followng example are presented n the appendx : c 1 = 1 6, c 2 = =; 1; ;n; U = sn( n ) + log 2 2nd case : If the coecents c 1 and c 2 are exactly represented, but the ntal data U not exactly represented, the rst moment s: are 8j ; j = +(3, c 1 )j In the towards zero roundng arthmetc, the theoretcal moment and the expermental one are both lnear. The theoretcal moment overestmates agan slghtly n absolute value the expermental moment. In the roundng to the nearest arthmetc, as the mean value s zero, the rst moment remans theoretcally equal to the mean error on data. In opposton to the theoretcal moment, the expermental moment s not constant. However ts order of magntude (1,7 )svery satsfyng for sngle precson results. Graphcal results for the followng example are presented n the appendx : c 1 = 3 16, c 2 = =; 1; ;n; U = sn( n ) + log 2 475

8 3rd case : If the ntal data U are exactly represented, but the coecents c 1 and c 2 are not exactly represented, the rst moment s: 8j ; j = ((3, c 1 ) +2c c 2 2 )j The choce of ntal data whch are exactly represented s more problematc than the choce of exactly represented coecents. For nstance, f ntal data are of the form : 8 =; 1; ;n; U = 2 r, wth r beng a relatve nteger, the scheme does not perform evolutons n tme : 8j ; U j = U. The followng example s presented n the appendx : c 1 = 1 6, c 2 = =; 1; ;n; f s odd, U = =16 f s even,u = The theoretcal moment s lnear and remans greater than the expermental moment n absolute value. In the towards zero roundng arthmetc, the expermental moment remans lnear for all tme ntervals consdered. In the roundng to the nearest arthmetc, the expermental moment s not perfectly lnear (see graphcal results n the appendx). 4th case : If the ntal data U rst moments: and the coecents c 1 and c 2 are exactly represented, the 8j ; j =(3, c 1 )j In ths case, the expermental moment s compared wth the theoretcal moment only n the towards zero roundng arthmetc, because n the roundng to the nearest arthmetc the mean value s theoretcally zero. In the appendx, the followng example s presented : c 1 = 3 16, c 2 = =; 1; ;n; f s odd, U = =16 f s even,u = The expermental moment, as well as the theoretcal one, s lnear. The theoretcal moment remans slghtly greater than the expermental one n absolute value because of the overvaluaton of the mean value of the assgnment errors. 476

9 3.4.1 Second moment The second moment, ( j ) 2, can be expermentally computed accordng to the followng formula : ( j ) 2 = 1 n,1 X n, 1 =1 ~! U j, U j 2 ~ U j where U j s the algebrac value and U ~ j the computed one. As for the rst moment, U j, theoretcally computed n nnte precson, s computed n double precson and U ~ j s computed n sngle precson. The second moment has been computed usng the roundng to the nearest arthmetc, when the ntal data U and the coecents c 1 and c 2 are exactly represented. In the appendx, the evoluton of the rato of the expermental moment by the theoretcal one ( j ) exp=( j ) theo s presented. The example consdered s the same as for the study of the rst moment, when both ntal data and coecents are exactly represented. 4 Concluson In the towards zero roundng arthmetc, the round-o error generated n the soluton of the heat equaton s correctly modelled for all nte derence scheme coecents and ntal data. The round-o error propagaton s then tme lnear. The theoretcal error depends strongly on the mean value of the assgnment errors and, whle provdng the order of magntude of the expermental error, overestmates t slghtly. In the roundng to the nearest arthmetc, the round-o error generated s always smaller than n the towards zero roundng arthmetc. In the case where nether the coecents nor the ntal data are exactly represented, the round-o error s lnear and s correctly descrbed by the theoretcal formula. However f the nte derence scheme coecents or the ntal data are exactly represented, theoretcal formulas are not vered by the expermental study. The round-o error modellng s rather dcult n the roundng to the nearest arthmetc, where the mean value of the assgnment error, whch s theoretcally zero, s never practcally zero. Theoretcal formulas are much more robust n the towards zero roundng arthmetc than n the roundng to the nearest arthmetc. However from ths study t seems obvous that the round-o error generated n the soluton of the heat equaton s usually lnear. 477

10 Appendx : graphcal results 1st case : c 1 = 1 6, c 2 = 2 3, n = 1; 8 =; 1; ;n; U = sn( n ) + log 2-2e-5 Expermental Theoretcal -4e-5-6e-5-8e Tme Fgure 1: Frst moment, towards zero roundng arthmetc 3.5e-5 Expermental Theoretcal 3e-5 2.5e-5 2e-5 1.5e-5 1e-5 5e Tme Fgure 2: Frst moment, roundng to the nearest arthmetc 478

11 2nd case : c 1 = 3 16, c 2 = 5 8, n = 1; 8 =; 1; ;n; U = sn( n ) + log 2 Expermental Theoretcal -2e-5-4e-5-6e-5-8e Tme Fgure 3: Frst moment, towards zero roundng arthmetc 4e-8 2e-8 Expermental Theoretcal -2e-8-4e-8-6e-8-8e-8-1e-7-1.2e-7-1.4e-7-1.6e-7-1.8e Tme Fgure 4: Frst moment, roundng to the nearest arthmetc 479

12 3rd case : c 1 = 1, 6 c 2 = 2, 3 n = 1; 8 =; 1; ;n; f s odd, U = =16 f s even,u = -2e-5 Expermental Theoretcal -4e-5-6e-5-8e Tme Fgure 5: Frst moment, towards zero roundng arthmetc 3.5e-5 Expermental Theoretcal 3e-5 2.5e-5 2e-5 1.5e-5 1e-5 5e Tme Fgure 6: Frst moment, roundng to the nearest arthmetc 48

13 4th case : c 1 = 3, 16 c 2 = 5, 8 n = 1; 8 =; 1; ;n;f s odd, U = =16 f s even,u = towards zero roundng arthmetc Expermental Theoretcal -2e-5-4e-5-6e-5-8e Tme Fgure 7: Frst moment 481

14 4th case : c 1 = 3, 16 c 2 = 5, 8 n = 1; 8 =; 1; ;n;f s odd, U = =16 f s even,u = roundng to the nearest arthmetc.7 Rato Tme Fgure 8: Rato of the expermental 2nd moment by the theoretcal 2nd moment 482

15 References [Alt 76] Alt, R.: \Etude statstque de l'erreur numerque d'aectaton sur un ordnateur en base quelconque. Applcaton a l'erreur commse dans le calcul d'une somme de produts de nombres"; IFP Report 76-5 (1976). [Alt 78] Alt, R.: \Error propagaton n Fourer Transforms"; Mathematcs and Computers n Smulaton, 2 (1978), [Chesneaux 88] Chesneaux, J.-M.: \Etude theorque et mplementaton en ADA de la methode CESTAC"; Doctoral Thess, Unv. Pars VI, (1988). [Chesneaux 9] Chesneaux, J.-M.: \Study of the computng accuracy by usng probablstc approach. Contrbuton to computer arthmetc and self-valdatng numercal methods"; ed. C. Ulrch, J.C. Baltzer (199), [Hammng 7] Hammng, R. W.: \On the dstrbuton of numbers"; Bell System Techn. J. 49, 8 (197), [Hull, Swenson 66] Hull, T. E., Swenson, J. R.: \Test of probalstc model for propagaton of round-o errors"; A.C.M. 9, 2 (1966) [Knuth 69] Knuth, D. E.: \The art of computer programmng"; Add. Wesley. (1969). [La Porte, Vgnes 74a] La Porte, M., Vgnes, J.: \Etude statstque des erreurs dans l'arthmetque des ordnateurs; applcaton au contr^ole des resultats d'algorthmes numerques"; Numer. Math. 23 (1974) [La Porte, Vgnes 74b] La Porte, M., Vgnes, J.: \Algorthmes numerques, analyse et mse en uvre"; Vol. 1, Edtons Technp, Pars (1974). [Vgnes 93] Vgnes, J.: \A stochastc arthmetc for relable computaton"; Mathematcs and Computers n Smulaton, 35 (1993),