Numerical di erentiation. Quentin Louveaux (ULg - Institut Montefiore) Numerical analysis / 12
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1 Numerical di erentiation Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
2 Numerical di erentiation In general, symbolic di erentiation is very e Sometimes : no analytical form available Ex : Implicit functions, measures, Simple formulas but uge roundo errors cient In most of te cases we can do Symbolic differentiation, Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
3 so fax Naive metod (order 1 f : IR 742 f ' C x = im fcxt I Just apply te definition Write a table for going to 0 f (x + F := f (x Proposition Let f be a function twice continuously di erentiable, for wic we want to compute te first derivative in x Wedefineteerroras E( := f (x+ f (x f 0 (x For all, tereexists 2 [x, x + ] andc > 0suc tat E( apple C 2 Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
4 f a Proof : Taylor expansion Reminder : f C x = f Taylor around a Ca + ft la Cx tf + fmla H;a + ft' " " (a 4 ( B Cx 4 Formula of around te derivative ; f C x t T Cx x Taylor fait = fix t f ' Ix t t f " ag tf f C = x fix ' ' fhtf = f ily t z f to e proportional order CD (2 les wit } C Cxixt ] Atl of convergence Oc
5 Practical example Let us compute te derivative of f (x =x 4 in x =1 Teoretical result : f 0 (x =4x 3 and f 0 (1 = 4 In practice : Best approximation : 8 correct digits is unstable! Te formula Because subtract 2 close numbers we Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
6 o O I = to w ( a t 4=1 for te computer because > epsilon macine tl = O 10 = no 4t 14 = ( 1 I 1 o oooo oooo a " 0 o o o o o o o 4 y an r 0 O o 4? 4, O o o o e 6 digits are correct no digits lo are in correct 6 correct
7 ( Central di erence formula Symmetric around x Tabulate, for going down to 0 F := f (x + f (x 2 Proposition Let f be a function tree times continuously di erentiable Te sequence f (x+ f (x 2 converges to f 0 (x wit an order of convergence of 2 Practical example f (x =x 4 Up to 11 correct digits ( It " Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12 I 4 if e Em it =/ more accurate approximation but = I obtained wit larger e
8 f t f f' fait = f Cx t f f I x = fix ' ( x tf ' txt tf " " ( x t + f ' " (9 HI I x t " ( I f f Cxtl G = 2 f ' l x t f f ' " 19+1 t f' " l 9 HI flxtlzftxlfi I x 1 I f ' " Mean value teorem 4 flxtyfxtl = Hatt l 9+1 tf " ' CE I Ab K ( Corollary of intermediate S f (a fi 41 t f ' " les t values teorem ofdefugae GUY
9 Onesided formulas Goal : Exceptionnally we sometimes only ave values on one side of te interval Metodology To find a onesided formula and its order of convergence : Write te Lagrange formula tat yields te interpolation polynomial Compute te derivative of te formula Use te Taylor expansion to find te order of convergence Examples Onesided formulas of order 2 Twosided formula for te second derivative An unsolved issue : we do not get rid of roundo errors! Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
10 Tx is unavailable for x so µ Is it possible to approximate a devalue only on one side? : YI f Cx f = Cx fait = f Cx t f ' I x t f f Cxtg = f I Ht 2 f ' HI t 2L f " l x " ( x tat f ' " l x afl ayy + If 3 f ' " ( x a, 13 I ai tartan fix t f ' k tey ( az tan f O = o f " = O = O = 2 ' 'll
11 2 x t us allows Lagrange : Pff = fix filt! 4t It 2 to write explicitly quadratic polynomial interpolating te 3 points It t C It ( 24 t fait "t ( I te tfcxtz pl H gives Does not tell you you te formula you are looking for te order of convergence of te formula
12 Optimal coice of te step If increases! te teoretical error increases and te roundo errors decrease If decreases! te teoretical error decreases and te roundo errors increase How to find te rigt balance? Proposition For a centered formula, te total error (teoretical+roundo is approximately given by E( 2C M + D 6 2 Roundo error Teoretical error Optimal coice of te step For te centered formula, = 3 r 12C M 2D Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
13 extrapolate p te end te of sequence #annot trust How to obtain te best possible precision and get rid of roundo errors Ricardson extrapolation Centered di erences f (x+ f (x Step 2 E( We do see tat te sequence converges to 4 until someting wrong appens Idea Knowing tat te sequence converges quadratically, we can extrapolate from te beginnning of te sequence wat is te end of te sequence Tis way, we get rid of te roundo errors Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
14 Ricardson extrapolation We assume tat we ave Tus g converges linearly to g ie g = g + c 1 + c c lim g = g!0 Teorem Te sequence 2g 2 g converges quadratically to g Alternatively, te sequence 10g g converges quadratically to g 10 We can also go on and obtain a sequence tat converges cubically Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
15 2 27 gas gu go, gtcntcr # # ga gigtcntca ' ( I 942 gtciztcry guy, 't g 12 _ gtcnztczqt 822 2% g, (a 2gy= 2gzsk Is _ 0121 g 2g%g=g tc 2 I ' 2
16 d I df I gpyz g Converges quadratically d = g I 't 't I t, I Y t d 42 g t Ez a ' + 5 HI t 5, t 4 da = 3g to 5 HI I sift 4 du 3 " du = g ' Eg tt order of convergence 6 C 3
17 Application to te numerical computation of te derivative f Te centered di erence formula g converges already quadratically to g : Proposition Te sequence g 2 Examples f = g fht 2 g = g + c c c converges to g wit an order of convergence of 4 Computations of f 0 (1 for f (x =x 4 : too easy for Ricardson extrapolation! Computation of 0 (0 for (x =sinx, only even terms in te convergence formula Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
18 Array representation of te Ricardson extrapolation General formula We denote by G i,j te approximations of g were we got rid of te term of order j (order j + 1 approximation and were te approximation uses te values g 2 i j,g 2 i TevaluesG i,j from te Ricardson extrapolation are given by Table G i,j = G i,j 1 we could also 1 G 2 j i 1,j 1 use %94 2 j wo formula Koo 9 t ooo : Te depends te ratio between G 0,0 & on 2 G 1,0! G 1,1 2 consecutive & & te and on 4 G 2,0! G 2,1! G 2,2 order tat & & & 8 G 3,0! G 3,1! G 3,2! G You want 3,3 rid to & & & & got off, 16 G 4,0! G 4,1! G 4,2! G 4,3! G 4,4 Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
19 Table for te computation of te derivative Example : computation of te derivative of e x in 0 G i,0 G i,1 G i,2 G i, Quentin Louveaux (ULg Institut Montefiore Numerical analysis / 12
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