1、CSE 541-DifferentiationRoger CrawfisAugust 17,2023OSU/CIS 5412Numerical Differentiation The mathematical definition:Can also be thought of as the tangent line.0()()()limhf xhf xfxhxx+hAugust 17,2023OSU/CIS 5413Numerical Differentiation We can not calculate the limit as h goes to zero,so we need to a
2、pproximate it.Apply directly for a non-zero h leads to the slope of the secant curve.xx+hAugust 17,2023OSU/CIS 5414Numerical Differentiation This is called Forward Differences and can be derived using Taylors Series:22()()()()2!()()()()2!()()()()2!()()()0hf xhf xfx hfhf xhf xfx hff xhf xhfxfhf xhf x
3、fx as hhTheoretically speakingAugust 17,2023OSU/CIS 5415Truncation Errors Let f(x)=a+e,and f(x+h)=a+f.Then,as h approaches zero,ea and fa.With limited precision on our computer,our representation of f(x)a f(x+h).We can easily get a random round-off bit as the most significant digit in the subtractio
4、n.Dividing by h,leads to a very wrong answer for f(x).August 17,2023OSU/CIS 5416Error TradeoffUsing a smaller step size reduces truncation error.However,it increases the round-off error.Trade off/diminishing returns occurs:Always think and test!Log errorLog step sizeTruncation errorRound off errorTo
5、tal errorPoint of diminishingreturnsAugust 17,2023OSU/CIS 5417Numerical Differentiation This formula favors(or biases towards)the right-hand side of the curve.Why not use the left?xx+hx-hAugust 17,2023OSU/CIS 5418Numerical Differentiation This leads to the Backward Differences formula.2()()()()2!()(
6、)()()2!()()()0hf xhf xfx hff xf xhhfxfhf xf xhfx as hhAugust 17,2023OSU/CIS 5419Numerical Differentiation Can we do better?Lets average the two:This is called the Central Difference formula.1()()()()()()()22f xhf xf xf xhf xhf xhfxhhhForward difference Backward differenceAugust 17,2023OSU/CIS 54110C
7、entral Differences This formula does not seem very good.It does not follow the calculus formula.It takes the slope of the secant with width 2h.The actual point we are interested in is not even evaluated.xx+hx-hAugust 17,2023OSU/CIS 54111Numerical Differentiation Is this any better?Lets use Taylors S
8、eries to examine the error:232333()()()()()23!()()()()()23!()()2()()()3!3!hhf xhf xfx hfxfhhf xhf xfx hfxfsubtractinghhf xhf xhfx hffAugust 17,2023OSU/CIS 54112Central Differences The central differences formula has much better convergence.Approaches the derivative as h2 goes to zero!2()()1()(),26f
9、xhf xhfxfhxh xhh2()()()2f xhf xhfxO hhAugust 17,2023OSU/CIS 54113Warning Still have truncation error problem.Consider the case of:Build a table withsmaller values of h.What about largevalues of h for thisfunction?()100100100()21,0.000333,60.01000330.0099966()0.0100500.000666666Relative error:0.01-0.
10、0100500.5%0.01xf xxhxhfxhat xhwith significantdigitsfxAugust 17,2023OSU/CIS 54114Richardson Extrapolation Can we do better?Is my choice of h a good one?Lets subtract the two Taylor Series expansions again:2345452345455533()()()()()()()23!4!5!()()()()()()()23!4!5!()()()()2()222()3!3!5!hhhhf xhf xfx h
11、fxfxfxfxhhhhf xhf xfx hfxffxfxsubtractingfxfxhf xhf xhfx hhhfxAugust 17,2023OSU/CIS 54115Richardson Extrapolation Assuming the higher derivatives exist,we can hold x fixed(which also fixes the values of f(x),to obtain the following formula.Richardson Extrapolation examines the operator below as a fu
12、nction of h.2462461()()()2fxf xhf xha ha ha hh1()()()2hf xhf xhhAugust 17,2023OSU/CIS 54116Richardson Extrapolation This function approximates f(x)to O(h2)as we saw earlier.Lets look at the operator as h goes to zero.246246246246()()()()2222hfxa ha ha hhhhhfxaaaSame leading constantsAugust 17,2023OS
13、U/CIS 54117Richardson Extrapolation Using these two formulas,we can come up with another estimate for the derivative that cancels out the h2 terms.46464315()4()3()24161()()()()232hhfxa ha horhhfxhO h new estimatedifference between old and new estimatesExtrapolates by assuming the new estimate unders
14、hot.August 17,2023OSU/CIS 54118Richardson Extrapolation If h is small(h1),then h4 goes to zero much faster than h2.Cool!Can we cancel out the h6 term?Yes,by using h/4 to estimate the derivative.August 17,2023OSU/CIS 54119Richardson Extrapolation Consider the following property:where L is unknown,as
15、are the coefficients,a2k.221221()()kkkkkkhfxa hLa h 0lim()hLhfxAugust 17,2023OSU/CIS 54120Richardson Extrapolation Do not forget the formal definition is simply the central-differences formula:New symbology(is this a word?):1()()()2hf xhf xhh21,02,02nknkhD nhLA kFrom previous slideAugust 17,2023OSU/
16、CIS 54121Richardson Extrapolation D(n,0)is just the central differences operator for different values of h.Okay,so we proceed by computing D(n,0)for several values of n.Recalling our cancellation of the h2 term.441()()()()2321(1,0)(1,0)(0,0)4 1hhfxhO hDDDO hAugust 17,2023OSU/CIS 54122Richardson Extr
17、apolation If we let hh/2,then in general,we can write:Lets denote this operator as:41()(,0)(,0)(1,0)4 12nhfxD nD nD nO11,1(,0),01,041D nD nD nD nAugust 17,2023OSU/CIS 54123Richardson Extrapolation Now,we can formally define Richardsons extrapolation operator as:or41,(,1)1,1,14141mmmD n mD n mD nmmnn
18、ew estimateold estimate1,(,1),11,141mD n mD n mD n mD nmAugust 17,2023OSU/CIS 54124Richardson Extrapolation Now,we can formally define Richardsons extrapolation operator as:Memorize me!1,(,1),11,141mD n mD n mD n mD nmAugust 17,2023OSU/CIS 54125Richardson Extrapolation Theorem These terms approach f
19、(x)very quickly.21,2knk mhD n mLA k mOrder starts much higher!August 17,2023OSU/CIS 54126Richardson Extrapolation Since m n,this leads to a two-dimensional triangular array of values as follows:We must pick an initial value of h and a max iteration value N.0,01,01,12,02,12,2,0,1,2,DDDDDDD ND ND ND N
20、 NAugust 17,2023OSU/CIS 54127Example523cos(100()11.3,1280,016.6963861,040.5833932,0109.3225283,0135.0317474,0142.0686155,0143.866937xf xxxhDDDDDD10.002444096h August 17,2023OSU/CIS 54128Example0,016.6963861,040.5833931,148.5833932,0109.3225282,1132.2355743,0135.0317473,1143.6014874,0142.0686154,1144
21、.4142385,0143.8669375,1144.466377DDDDDDDDDDD13extrapolateAugust 17,2023OSU/CIS 54129Example0,016.6963861,040.5833931,148.5833932,0109.3225282,1132.2355742,2137.8148973,0135.0317473,1143.6014873,2144.3592144,0142.0686154,1144.4142384,2144.4684215,0143.8669375,1144DDDDDDDDDDDDDD.4663775,2144.469853D11
22、5extrapolateAugust 17,2023OSU/CIS 54130Example Which converges up to eight decimal places.Is it accurate?16.69638640.58339348.583393109.322528132.235574137.814897135.031747143.601487144.359214144.463092142.068615144.414238144.468421144.470154144.470182143.866937144.466377144.469853144.469876144.4698
23、755,D5144.46987511023extrapolate1255extrapolate163extrapolate115extrapolate13extrapolateAugust 17,2023OSU/CIS 54131Example We can look at the(theoretical)error term on this example.Taking the derivative:255 1122575,5,5211.3(6,5),540962kkkkhDLA khfAA k 2-144(1.3)144.469874253f Round-off errorAugust 1
24、7,2023OSU/CIS 54132Second Derivatives What if we need the second derivative?Any guesses?234545234545()()()()()()()23!4!5!()()()()()()()23!4!5!hhhhf xhf xfx hfxfxfxfxhhhhf xhf xfx hfxffxfxAugust 17,2023OSU/CIS 54133Second Derivatives Lets cancel out the odd derivatives and double up the even ones:Imp
25、lies adding the terms together.244()()2()2()2()24!hhf xhf xhf xfxfxAugust 17,2023OSU/CIS 54134Second Derivatives Isolating the second derivative term yields:With an error term of:2()2()()()f xhf xf xhfxh 42112Eh f August 17,2023OSU/CIS 54135Partial Derivatives Remember:Nothing special about partial derivatives:,2,2f xh yf xh yfx yxhf x yhf x yhfx yyhAugust 17,2023OSU/CIS 54136Calculating the Gradient For lab 2,you need to calculate the gradient.Just use central differences for each partial derivative.Remember to normalize it(divide by its length).谢谢你的阅读v知识就是财富v丰富你的人生 Thank you
