Outline. Bits. Bits. Computational Projects. Lecture 3: Numbers and numerical errors. Dr Rob Jack, DAMTP

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1 Comutational Projects Outline Lecture 3: Numbers and numerical errors Dr Rob Jack, DAMTP How does the comuter store and maniulate numbers? (and imlications for numerical comutation) How to quantify the uncertainty in numerical results (imortant for all comutational rojects) htt:// 1 Bits Almost all modern comuters store information using binary digits, which are called bits Bits Any ositive integer can be reresented as nx = a i i i=0 for some integers (n, a 0,a 1,a,...,a n ), with a i {0, 1}. Within the comuter, a bit is a hysical device that has two ossible states (think of a switch that can be on or off) [Practical bits: small magnets (hard disks), stored electric charge (most comuter memory), magnetic tae...] The a i form a binary string, for eamle 14 is reresented as 1110, that means n =3,a 3 =1,a =1,a 1 =1,a 0 =0 Tyically one works at fied n, and each a i is a bit To allow for negative numbers let nx = n + a i i i=0 3 4

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With 3-bits, ok u to ~10 digits.... this is usually not a roblem but we have to kee it in mind... What subset does the comuter use, in ractice? (we call this the "set of reresentable numbers") 5 6 Real numbers Floating oint reresentation What would be a good subset? Smallest ossible ositive number [ schematic icture not to scale... ] To turn the icture into mathematics, consider the real number = a 10 b where a, b are integers. For eamle = corresonding set of negative numbers 0 1 Largest ossible ositive number Eg, for ositive numbers, take A 1 ale a ale A and B ale b<b +. This allows for a good range of numbers: the largest ossible number is A 10 B + 1 ), the smallest ossible number is A 1 10 B. closely-saced numbers for small, wider sacing for larger 7 The idea is that for any real number y in the usable range, there y should be a reresentable number such that y is small (eg no larger than 10/A ). <lateit sha1_base64="bzmfnytojp0l3ust5zlimvwufs=">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</lateit> 8

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sha1_base64="zrgyigrty1dkybn04rrh04jv18=">aaab7nicbvbns8naej3ur1q/qh69lbbbiyurqu9s8okgvantndtiu3wzc7kzaqn+efw+kepx3eppfug1z0nyha4/3zizfysca+o6305hbx1jc6u4xdrz3ds/kb8enxwckoynfotytqoquxcjdconwhaikeabwfywuv5rsdumsfy0uws9cm6kdzkjbortcy9ri7iufeuufv3drjkvjuiee9v/7q9mowrigne1trjucms+omwjnja6qcaeshedymdssspufjy/d0rorninyassupm6u+jjezat6ladkbudpwynp/8zqcw/8jmsknsjzylgycmjimvud9llczsteesout7csnqskmmmtktkqvowxv0nzsuq5ve/hqlk7zemowgmcwjl4ca01uic6nidbcj7hfd6chl3prwvbyweo4q+cz9k5i7b</lateit> Floating oint reresentation In ractice the comuter uses where a, b are integers. = a b For double recision (a common choice), a is a 5-bit integer (u to 16 decimal digits) and b is 11 bits (values u to 300) This means that if we have two real numbers, y with y y , then the comuter can t resolve the difference between these numbers.... no eact reresentation of 1 3,,, etc... root-finding can t work with a tolerance less than (or at least, it s more comlicated... ) <lateit 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sha1_base64="mvp7rx54gw5vmkbdg6r49g7lz0=">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</lateit> <lateit sha1_base64="mvp7rx54gw5vmkbdg6r49g7lz0=">aaac7xicbvfnb9naelxnvzefdxdksijflrjecyuej1qeujyjnjuiqnovb7bq+yhtttoicz/dg6ik7+jg7+cm5s0b9iy7fvzezmvmlqkrwnh7/c6nbto3fvhdyphz89piw9+thton5tjmrh7myfdktsoszdey9oiqezijfumnvkidyjo7/qusazglklqnagt817v9mabga+tkncvdfgqsskuekuwzqxhgjfoi8tuejrk0vumvbtqou1cnk5aze1ki/p3ulubukuwtlibkvzeufushhrdh0ha6f7khpqszwc7ysvdfgzcra0byzrocyvadzuyq1e9loi5am8ymdmjl4og1e3tsmdiwmr/im+z3w3zb6vxczxv94wc4dxytjdvqd3zpu/98aw8uaijs3bumgrmrvgsxcjfrfgyq18asvopdsg0m3a7su69tizosv8eixrlbsvutkofwkvozg7vcdw1d/k+bnls8n7vc1wh5lenikz6r9jmrcwxfjn5s+ucubv+vsyrsmdjh3+vs4vyidsptjtg6os67bcbbengq4sd6f9s8+7cw7cj4hl4ktianebwfb+a8gac8/bjksamxkym+rd+jh1eubirerbsrftzl3uz7n0=</lateit> <lateit sha1_base64="mvp7rx54gw5vmkbdg6r49g7lz0=">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</lateit> 9 >> format long >> es( double ) ans = e-16 >> ^(-5) ans = e-16 Machine recision Let be the smallest reresentable number that is (strictly) greater than 1 Then the machine recision is a real number defined as = 1 Here, 'double' means double recision, which is the standard reresentation in matlab With the definitions above, we eect to be close to 1/A 10 Absolute vs. relative Suose, c is a number of interest is the number stored by the comuter then, c c is the absolute error of is the relative error of c c Two eamles of things that can go wrong... Errors that originate in the machine recision are called "round-off" errors. Since is very small, erhas we don t have to worry about it (unless we care about the 16th decimal lace... ) <lateit sha1_base64="rikicke/ezvyfvchdoqjvoy9mdc=">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</lateit> <lateit sha1_base64="rikicke/ezvyfvchdoqjvoy9mdc=">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</lateit> <lateit sha1_base64="rikicke/ezvyfvchdoqjvoy9mdc=">aaac93icbvjnj9mwee3c1k+chy5whqri4rwyqobj7rslylonuvmqqaojpgqr9koy1vfme/4ya48np4c/wazjjdhogui1kavfdmn7jxhnmxzl8cqnr1/cvhvwo75z9979b4ohj86sqg3fcvvcmfmclhimceky43iudyliou7z1ajjs0lin5yw01zgusjsszbeeheb3shk6notls0tzzzsgtb+jbemcqnvukaljsngqbticrkpbm1lb77htas3netdvsn7url9fcglkssdftwlv7ijevymkgjct5sjgl/nbygvw/jonbvoszziciwfo1lbgoltzkomv0guakeyi0e15gh3ogoge8jvsnlxriifuuybesbylyoz1/e7wiwti6txzcrsdonw6cp8wlw5fswqb0lio1sztrbt6acyy9k+slwqgk1jizkcsbn5s9tjs555cclmv5ir3bovuncgu3ivfkzjh7meva/3gzvv5wtunyo6cwgsuadsdcscemuucxwdcghvm7elqbaer8n9ibuiffovjhilzqj0si1xk7ot4zq5tj+cde/f9zydbe+c8frkazvgtpgftaojgen+e6/bj+jbbrt+h79oncgov9zengl6kffwgfopit</lateit> <lateit sha1_base64="rikicke/ezvyfvchdoqjvoy9mdc=">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</lateit> 11 1

4 Eamle 1 -- Quadratic equation <lateit sha1_base64="jeothf0u4fybmj/ta/khztvmiw=">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</lateit> We know that if A + b = 0 then = A ± A Result: If we comute A b with an accuracy of 6 digits ( = 10 our estimate of already has errors in the nd digit (relative error 0.) b Suose we are given A, b and we want to comute. <lateit sha1_base64="pqem3jlw8asjcdb3447kv9mk348=">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</lateit> <lateit sha1_base64="dnjhohlauc4l0pnquozyio617g=">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</lateit>... we have to aroimate it by some reresentable number C Then we estimate the roots as 1 = A + C and = A C. <lateit sha1_base64="q+ibo4meqqfslwo8a9ubkmx84=">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</lateit> Suose that C has relative error. Then the relative errors on 1 and can be shown to be: <lateit sha1_base64="w3dekvpgmoa+b0o94hrclvga0=">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</lateit> A b A+ A b and A b A A b For = 10 7, A = 1000, b = 1, these relative errors are (aro) (on 1 ) and 0. (on (!)) We can estimate this as <lateit sha1_base64="p5ajepsgftz7cbdu5wli38y4lha=">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</lateit> 13 an+1 = an 1 an <lateit sha1_base64="mwpyzanowv5k3evrxcsdsco=">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</lateit> where, + 5 1!n Then = 1 and = 0 so an = 15 n ( 5 Difference equation for i = 1:n+1 eactseq(i) = ((sqrt(5)-1)/.0)^(i-1); end are constants that deend on a0 and a1 Suose initial conditions are a0 = 1 & a1 = 1 ( 5 <lateit sha1_base64="gmgdbckve0muip60nrwvqktm=">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</lateit> which reduces the relative error. for i = 3:n+1 numericalseq(i) = numericalseq(i-) - numericalseq(i-1); end Its eact general solution is an = b A+C, % comute and store n iterations of a(n) = a(n-) - a(n-1) n = 100; % the sequence is a matri of size 1 (n+1) numericalseq = zeros(1,n+1); % set the first two terms numericalseq(1) = 1; numericalseq() = 0.5*(sqrt(5)-1); Consider the difference equation!n <lateit sha1_base64="xv5yxhbak31cvywro+yv9sf4=">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 sbeltztt0+91b733ve++sn1zdjtrnzfehvtm/wazsz+</lateit> 14 Eamle -- Difference equation 5 1 The roblem usually arises if we have to subtract two similar numbers to get a much smaller number (the sacing between reresentable numbers is large when the numbers themselves are large) With good lanning it is sometimes ossible to rearrange equations to avoid this roblem, eg A + A b b = A A b= A A b = A + A b A + A b resectively ), This is a dangerous situation and shows that we must be careful! b is likely not reresentable A 7 <lateit sha1_base64="xffb3y1bvcqa1yteibi81lvdk=">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</lateit> <lateit sha1_base64="duab+neqjsy6x6m7ojdummo4=">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</lateit> The number Eamle 1 -- Quadratic equation 1)n % comute a vector with the (absolute) errors diffseq = numericalseq - eactseq; 1) % diff is the error, rel is the relative error dis('** First 5 terms') for i = 1:5 dis(['i=' numstr(i)... ' numeric ', numstr(numericalseq(i))... ' eact ' numstr(eactseq(i))... ' diff ' numstr(diffseq(i))... ' rel ' numstr(diffseq(i) / eactseq(i))]) end 16 Eamle: diffeq.m

5 <lateit sha1_base64="xosfzc+v39jfv9vibt67heevwq=">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</lateit> <lateit 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sha1_base64="jrbynwmpdkwtincs/d6bil39b8=">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</lateit> Difference equation Result of numerical eeriment: the difference between eact and comuted sequences starts small but grows eonentially <lateit 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sha1_base64="ffk+j7kdrovdzzwxrhs3cwa=">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</lateit> <lateit 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sha1_base64="keiytfn/35eqjwoyufcdzxihys=">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</lateit> <lateit sha1_base64="keiytfn/35eqjwoyufcdzxihys=">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</lateit> <lateit sha1_base64="keiytfn/35eqjwoyufcdzxihys=">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</lateit> <lateit sha1_base64="keiytfn/35eqjwoyufcdzxihys=">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</lateit> <lateit sha1_base64="keiytfn/35eqjwoyufcdzxihys=">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</lateit> Also: if we comute an eact sequence but for slightly different initial data, we get similar behaviour to the comuted solution <lateit sha1_base64="kc3v90mnvsiu+cw6eeigvzqsds=">aaaddhicbzlnjtmwemed8lwurwjhlhyvegdunsskeae0qbeoi0s3kzvvnxemibx+clbtbwxlcza4wztwq15b16ekzhthegucr///am5dntedwjuy/o/jk1wvxbzc7n6fefuvf79bydwn4bhhgmhzwkgfgvxohhcctytdylmbe6zs3hnt5dolnfqvvvxojdqkl5wbi5ii/7vtaat6zvpeizb30ttuydkyk17qcjzquelft0pg59ksfvdiqft++o53u3czz084mre0zr5/xrcakevmds+rshc4clnh3wir1euf/qcashtnw4kjqircupkcury/nkgy0qtncgwtevcxflevggqeuov9edqhb8+6yiu6anmyaaznsiild6io09rvfzvl1grrdadarx8wgo4qs8nys4zkf0cl/q/0lyzrobmtic1sruu7kh4zgtgezvlnbazqdewugvslrzv3molj4jsr4zddh8hv13melvrmywdhu170eve/3mzhuv556rbjrftokrmzmdq8egblkw3latedgkkvabmzhb+1qvasmcdfezpzguouyrmcnbwok9ewexc4ohq9q3zahhh5cljyatyrn6syzihlmloy/q5+hj/ir/g3+lv61tdvzkof/ompo+n98a==</lateit> <lateit sha1_base64="kc3v90mnvsiu+cw6eeigvzqsds=">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</lateit> <lateit sha1_base64="kc3v90mnvsiu+cw6eeigvzqsds=">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</lateit> <lateit sha1_base64="kc3v90mnvsiu+cw6eeigvzqsds=">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</lateit> Interretation: to obtain the eact (decreasing) solution with =0we would need to set initial condition a 1 =( 5 1)/, which is not reresentable... so in fact we get a different solution a n =! n shrinks with n 5 1! n absolute value grows with n... only one shrinking solution, large family of growing solutions 17 Round-off: main oints Finite recision of floating oint numbers can affect calculations, via "rounding errors" You need to bear in mind that these errors might roagate and affect your calculation Small numerical errors can be amlified (for eamle) when comuting small differences between large numbers, or if the solution to your equation is very sensitive to initial conditions 18 Truncation error - tye 1 Truncation error We now consider a different kind of error, that does not come from finite recision. 19 Suose we want a comutational estimate of e for some secific value of We know that e = P 1 n=0 n n! The comuter can t do the infinite summation but we can fi some N and estimate e as NX n C = n! Since we truncate the series after N terms, we can define 1X =e n C = n! as the truncation error n=0 n=n+1 (Comuters often do calculations in this way but they tyically take N large enough the truncation error is comarable to the round-off error) 0

6 Truncation error -- tye Truncation also occurs in aroimating derivatives, eg df d () = 1 [f( + h) h f()] + O(h) These kinds of aroimation are often used when solving ordinary differential equations (see net lecture) Making a Taylor eansion and rearranging, truncation error is... net lecture numerical solution of ODEs, and associated errors... likely to be relevant for a art IB core roject 1X n= 1 d n f n! d n ()hn 1 (assuming all derivatives eist at and that the sum converges) 1