Adv Sci Computing I
Adv Sci Computing I CS 6210
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Marian Kertzmann DVM
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This 24 page Class Notes was uploaded by Marian Kertzmann DVM on Monday October 26, 2015. The Class Notes belongs to CS 6210 at University of Utah taught by Christopher Sikorski in Fall. Since its upload, it has received 23 views. For similar materials see /class/229979/cs-6210-university-of-utah in ComputerScienence at University of Utah.
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Date Created: 10/26/15
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quota 3 N mm man2quot NM wV M39WxM u 01 442 ii Mp ovalw 1 f 5 32 4 39 Juice as expensive as Gaussian EL MEMa ou um 2 464 v 59 50 v 455 4 mm 39 5quotquot quot a m d ram 4 Q 9 2 V x gt0 39 2 PH 4 Fm i JYOua Iad vc Scat t39bn of aching radius nal Pm I 117 V e P Pumdukon mamx f PH P 13 60 c quot quot vcd of 3 dovdk quotaf x 6 Sine we 3a mt X r s 33gc x Bx c x quot 39 3Rg quotKN 1 3quot B x x HBI Max 62 59 FM 3 X 5 X comavac 9 a 3 Max i i A 939 Gianvalue 4 3 Nowww V8 0 3 5 e M v a gt Mu s stewequot Mun smaller 35 Ju I I 6 3 b V Iquot X J D p S 6 R M nr 3 06 J 4 395 J U x 4 L J 6am 3 Set 1 J I 5 If 41150 3 5 tag 1 O sca den 65 50R successive ww dunked In 6aan Sadat ne39 but 66 33 D xi 3 I 9 Dxi 399 w Na UK 6 yuL iH UD U Xe 605 QM r v X130 3 DHnU 03963 U I UCD QU b p minimizing 38 g lemma V OJ 3 5 M wcc c we a 2 67 68 3 K PFO C355 A 9 1 2 mquot39391 g 1 E lt 12 mquotquot1 242 and although the computed value will depend on the order in which the multiplications and divisions are performed the bounds are again indepen dent of that order The computed value g has a low relative error for any reasonable values of m and n 25 For extended sequences of additions it is no longer true that the computed value necessarily has a low relative error even though this was true for the sum of two numbers with the more accurate round o 39 rule In fact there is no useful bound for 1 flx1x2 xnx1x2 xnf 251 i3 39 j since the oatingpoint sum may be Zero when the true sum is not and versa Note that we use flx1xz x to denote the computed sum when additions take place in the order in which they are written quot f It is still true however that 9n flx1x2 xn as x x x if vvhere each x differs from x by a factor which is close to unity We de ne the quantities s recursively by the relations 91 x1 252 s Efl8r1xr s1x1egt 253 I 16 lt 2 4 254 Combining these equations we have quot sn 5 x1l 171 x21 772 xnl 77 255 12 162l6316n 256 1 77 1e1e391 1en r 2 n 257 12 quot 1 ltt l 771 lt 1 2 quotquot1 258 1 2n1r lt 1 7 s 12 n1 r r 2 n 259 Note that x1 and 62 have the same factor which is not surprising since they quot1play identical roles It is inconvenient that the factor associated with x1 is 53 jg39 not de ned by the same formula as the others and it is often simpler to write 1 2tn1r s 1 17 lt 12 n1r r 1 n 2510 which is true a fortiorz39 Note that the bounds themselves are dependent on the order of summation The upper bound for the error is smallest if the terms are added in order of s39increasing absolute magnitude since then the largest factor 1171 is quot filquotquotquotlassociated with the smallest xi Although this gives the smallest upper 39gbound it does not necessarily give the smallest error in practice Consider gffOr example the sum 1040 1025 1030 9123 102 0399663 39100399315 If we add them in the natural order we haVe 17 5210301127 3310201607 s4391006755 and there are no rounding errors at any stage If we add the numbers in the opposite order then we have 32 wit 01059 33 104 01018 s4 1007000 and there are rounding errors at each of the rst two stages It will be appreciated that this example is rather special The more normal case is illustrated by the following example Consider the sum 10804462 10 306412 10302413 1000 1234 If the additions take place in the order in which the terms are given the computed sum is 1000 39 1247 and the errors made in the three additions are 10 04 10 6 03 and 10 4 028 If they are added in the reverse order the computed sum is 100039124396 and the errors in the three additiOns ii are 104 0413 104 0412 and 104 0462 Adding the terms of smallest modulus rst ensures that the rounding errors made in the early stages are small Clearly if we include a large number of terms of order 10quot3 in a sum of the above type then the advantage of adding the small I terms rst can be very considerable a 26 We now consider the error made when an innerproduct is Computed in oatingpoint We write 5 f1a161a262 anbn 261 The notation implies that a and 6 are standard oating point numbers The products are computed rst and then added in the order in which they are written We de ne the quantities 3 and t recursively by the relations t flab 262 31 t1 3 fls139 t 263 From these relations we deduce tr 5 arbr1 r IErl lt 2quot 264 s a s1tgt1n lan lt 24 265 and hence Sn a161161 02621 62 anbn1 6 266 1 61 1El1 172 1nn 267 16 161n 141 r2 n 268 Hence I 1 24quot ltlt 161 lt 12tn 269 t6lt12t r2 r2n 2610 18 1 z tn r2 g 1
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