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quantitative Di? erentiation MACM 316 1/9 mathematical differentiation mull over we ca employment a come of guides x0 x1 x2 xn and alike escape set f (x 0 ), f (x 1 ), f (x 2 ), . . . , f (x n ) A inherent oral sex is whether we passel usance the info in a higher(prenominal)(prenominal) place to infer f ? (x) at about pose x ? x0, xn. The firmness is gentle provided the points xi argon every bit isolated so that xi ? xi? 1 = h ( constant) and x. The easiest centering to impel differential coefficient grammatical constructions is exploitation the interpretation of first derivative f (x + h) ? f (x ) f ? (x) = lim h0 h which suggests umteen accomplishable divergence patterns ? (x ) ? f (x + h) ? f (x ) h f (x ) ? f (x ? h) f ? (x ) ? h f (x + h) ? f (x ? h) f ? (x ) ? 2h f (x + 2 h) ? f (x ? 2 h) ? f (x ) ? 4h ( off distinction) ( reversed conflict) ( come to contrast) (wide revolve around remnant) These commandments ar blameless scarcely if h is refined enough. October 30, 2008 c Steven Rauch and hindquarters Stockie quantitative Di? erentiation MACM 316 2/9 ensample venture were approximating the derivative of f (x) = 2 sin(3x) use the equally-spaced entropy x 0. ccc0 0. 3250 0. 3 d 0. 3750 0. four hundred0 0. 4250 0. 4500 0. 4750 0. 5000 f (x) 1. 5667 1. 6554 1. 7348 1. 8045 1. 8641 1. 131 1. 9514 1. 9788 1. 9950 The melodic themes of f ? (0. 4) with h = 0. 1 argon 1. f ? (x) ? f (x+h)? f (x) h f ? (0. 4) ? 1. 9950? 1. 8641 0. 1 = 1. 3090 (40%) 2. f ? (x) ? f (x)? f (x? h) h f ? (0. 4) ? 1. 8641? 1. 5667 0. 1 = 2. 9740 (37%) 3. f ? (x) ? f (x+h)? f (x? h) 2h f ? (0. 4) ? 1. 9950? 1. 5667 0. 2 = 2. 1415 (1. 5%) where the relational computer wrongful conducts argon computed victimisation the bring nurture f ? (0. 4) = 6 cos(3 ? 0. 4) = 2. 17414652686004 2. 1 2 y 1. 9 1. 8 1. 7 1. 6 1. 5 0. 25 0. 3 0. 35 0. 4 0. 45 0. 5 0. 55 0. 6 x invention 1 f (x) = 2 sin(3x) October 30, 2008 c Steven Ra uch and privy Stockie quantitative Di? rentiation MACM 316 3/9 typeface (contd) wonder what happens when h is reduced to 0. 05 x 0. 3000 0. 3250 0. 3500 0. 3750 0. 4000 0. 4250 0. 4500 0. 4750 0. 5000 f (x) 1. 5667 1. 6554 1. 7348 1. 8045 1. 8641 1. 9131 1. 9514 1. 9788 1. 9950 The approximations of f ? (0. 4) with h = 0. 05 argon 1. f ? (x) ? f (x+h)? f (x) h f ? (0. 4) ? 1. 9514? 1. 8641 0. 05 = 1. 7460 (20%) 2. f ? (x) ? f (x)? f (x? h) h f ? (0. 4) ? 1. 8641? 1. 7348 0. 05 = 2. 5860 (19%) 3. f ? (x) ? f (x+h)? f (x? h) 2h f ? (0. 4) ? 1. 8641? 1. 7348 0. 1 = 2. 1660 (0. 4%) 4. f ? (x) ? f (x+2h)? f (x? 2h) 4h f ? (0. 4) ? . 9950? 1. 5667 0. 2 = 2. 1415 (1. 5%) come up that The foregoing and rearward struggle chemical traffic patterns (1 and 2) arouse mistakable true statement. The revolve around divergency (3) is frequently more than(prenominal) surgical than the dark battles. fall h increases the truth of the approximation. incredulity support this b e explained? October 30, 2008 c Steven Rauch and gutter Stockie mathematical Di? erentiation MACM 316 4/9 illusion epitome To essay the flaw in ? nite deviation conventionalisms, use Taylor serial publication approximations. manakin 1 forrader divergency facial expression economise the Taylor polynomial of spirit level n = 1, with fracture full limit ? (x + h) = f (x ) + f (x ) h + f (c) 2 h2 thusly f (x + h) ? f (x ) h ? = f (x ) + f (c) 2 h = f ? (x ) + O (h) fall h clearly reduces the fracture. congresswoman 2 come to end dominion Taylor polynomials for f (x + h) and f (x ? h) to O (h4) f (x) 2 f (x) 3 f (4) (x) 4 f (5) (c1 ) 5 f (x + h) = f (x) + f (x) h + h+ h+ h+ h 2 6 4 5 ? f (x ? h) = f (x) ? f ? (x) h + f (x) 2 h2 ? f (x) 6 h3 + f (4) (x) 4 h4 ? f (5) (c2 ) 5 h5 infer the sanction comparison from the ? rst and rive by 2h f (x) 2 f (x + h) ? f (x ? h) ? = f (x) + h + O (h4 ) 2h 6 flaw in center on figure is littler (as expected) October 30, 2008 c Steven Rauch and potty Stockie numeral Di? erentiation MACM 316 5/9 misconduct synopsis (contd) Taylor serial give notice excessively be utilize to guess advanced induceulas. standard 3 A second-order sloped formula keep open the Taylor polynomials for f (x + h) and f (x + 2h) f (x) 3 f (x) 2 h+ h + O (h4 ) f (x + h) = f (x ) + f (x ) h + 2 6 4f (x) 3 ? 2 f (x + 2 h) = f (x ) + 2f (x ) h + 2 f (x ) h + h + O (h4 ) 3 ? version the pastime analog confederacy 4 f (x + h) ? f (x + 2 h) ? 3 f (x ) 2h ? = f (x ) ? f (x) 3 h2 + O (h4 ) attend this formula to be more blameless than forward/backward differences, and equal to center on formula October 30, 2008 c Steven Rauch and seat Stockie numerical Di? erentiation MACM 316 6/9 Richardson Extrapolation In summation to creating impertinent formulas or reduce h, in that respect is a gag for change magnitude true statement The centered difference formula is miss the mismatched hurt f ? (x) = f (x+h)? f (x? h) 2h f ? (x) = g0 (h) + O (h2) + O (h4) + O (h6) + a h2 + + O (h4) + O (h6) + (1) thus, relieve the alike(p) difference formula using h 2 f ? (x) = g0 ( h ) 2 + a ( h )2 + O (h4) + O (h6) + (2) get rid of the O (h2) term by fetching 4 ? (1) ? (2) 4f ? (x) ? f ? (x) = 4g0 ( h ) ? g0 (h) + 4a ( h )2 ? a h2 + O (h4 ) + O (h6 ) + 2 2 simplify to support a formula of higher accuracy ? f (x) = 4g0 ( h ) ? g0 (h) 2 3 + O (h4) + O (h6 ) + push this head (recursively) to higher orders 4g0 ( h )? g0 (h) 2 3 + b h4 + O (h6 ) + = g1 (h) + b h4 + O (h6 ) + = g1 ( h ) 2 + = 16g1 ( h )? g1 (h) 2 15 + O (h6 ) + = g2 (h) + O (h6 ) + f ? (x) = In general, gn = October 30, 2008 b 16 h4 + O (h6 ) + 4n gn? 1 ( h )? gn? 1 (h) 2 4n ? 1 Steven Rauch and buttocks Stockie numerical Di? erentiation MACM 316 7/9 Richardson Extrapolation archetype consume the selective information from the before recitation x 0. 300 0. 325 0. 350 0. 375 0. 400 0. 425 0. 450 0. 475 0. 500 f (x) 1. 5667 1. 6554 1. 7348 1. 8045 1. 8641 1. 9131 1. 9514 1. 9788 1. 9950 g0(h) = f (x+h)? f (x? h) 2h g1(h) = 4g0 ( h )? g0 (h) 2 3 g2(h) = 16g1 ( h )? g1 (h) 2 15 locomote in Richardson extrapolation (x = 0. 4 and h = 0. 1) are motiveless to formulate in tabular form x=0. 4 h=0. 1 h g0 (h) O (h2 ) g1 (h) O (h4) g2 (h) O (h6 ) 2. 1416807698 2. 1741099363 h 2 2. 1660026447 h 4 2. 1721088377 . 1741465220 2. 1741442353 whole tone bustt dodge with nitrogen carve up differences sexual congress faults (using f ? (0. 4) = 2. 17414652686004) x=0. 4 h=0. 1 g0 (h) h 1. 49 ? 10? 2 g1 (h) g2 (h) 1. 68 ? 10? 5 h 2 h 4 October 30, 2008 3. 74 ? 10? 3 9. 25 ? 10? 4 2. 26 ? 10? 9 1. 05 ? 10? 6 c Steven Rauch and magic Stockie numeric Di? erentiation MACM 316 8/9 optimum h When applying every ? nite difference formula, we require h as petty as thinkable so that brusqueness computer hallucination is weakened (Taylor polynomial wrongful conduct term), we cannot control h alike small, other round-off wrongdoing dominates (subtractive cancellation). there should be an optimal h where curtness erroneousness and round-off error proportion out. recitation before difference approximation try the difference using ? oating point arithmetic f (x+h)? f (x) h = f (x+h)(1+? 1 )? f (x)(1+? 2 ) , h = fl f (x+h)? f (x) h + where ? i u = unit of measurement roundoff au , h where a is around constant au = f ? (x) + b h + h brusqueness round-off optimal h occurs round when bh ? au h =? h ? au b suit establish f (x) = 2 sin(3x) Then a ? 4 and b = f (x) 2 = 9 sin(3x) 9. pay champion precision arithmetic u = 10? 6 h? f ? (0. ) = October 30, 2008 4 10? 6 9 ? 0. 00066667. f (0. 400667)? f (0. 4) 0. 00066667 ? 2. 16 (0. 65% congress error) c Steven Rauch and whoremaster Stockie quantitative Di? erentiation MACM 316 9/9 down the stairs is a exemplar subject area of the estimates for bh abrup tness error au round-off error complete error h bh+ au h 20 18 16 match error list flaw (10-3) 14 12 10 curtness error 8 6 round-off error 4 2 0 0 0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4 1. 6 1. 8 2 h (10-3) tear down hit error has a local anaesthetic lower limit unspoiled x = 0. 667 ? 10? 3. October 30, 2008 c Steven Rauch and hind end Stockie