2
$\begingroup$

I have an interpolating function (which comes from an integral that cannot be done analytically), it looks nice and smooth:

Interpolating function

However, since what I need is the derivative of this function, it is not anywhere as nice, and creates even more problems because I need to take another derivative:

Derivative

What can I do to fix this? Would a finer interpolation (more points) help me get a nice and smooth derivative?

Edit: here is the Data:

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{2.17`, 1.18062102709483626743450116027108782721`10.150319754850342}, \
{2.18`, 1.18060406529779559723444717474235253855`10.150319735351882}, \
{2.19`, 1.18058665164381804732895291445055512447`10.150319820029056}, \
{2.2`, 1.18056617223435425175155706629953711257`10.150319880511699}, \
{2.21`, 1.18054689405580620013440462400070321051`10.150319861008644}, \
{2.22`, 1.18053066355023167469223980968670030235`10.150319904811823}, \
{2.23`, 1.18051433663530613279316577665188132458`10.150319943291501}, \
{2.24`, 1.18050561948186035190070601306403514893`10.150320013339043}, \
{2.25`, 1.18048735790823636898188577242636659913`10.150319936232428}, \
{2.2600000000000002`, 
    1.18047358256870323144781205364905150799`10.1503200024423}, \
{2.27`, 1.18045284425836003610183819602310982111`10.15032006948038}, \
{2.2800000000000002`, 
    1.18043318423569678280432921534772495023`10.150320097854435}, \
{2.29`, 1.18042205490595340640316049868951433544`10.150320139600492}, \
{2.3000000000000003`, 
    1.1804080664330128916841858733953962064`10.150320174993}, {2.31`, 
    1.18038941529283831613907223168965142054`10.150320177542772}, \
{2.32`, 1.18038455410165951359586395404854568313`10.150320276480821}, \
{2.33`, 1.18036718935573902329672139575135827985`10.150320223681874}, \
{2.34`, 1.18035775053568044946464396483048302296`10.150320335258167}, \
{2.35`, 1.18034230197134929817981075767585169461`10.150320348473715}, \
{2.36`, 1.18033056970535840931401072963026847378`10.150320387547922}, \
{2.37`, 1.18031710662004174760569617689021824006`10.150320413705433}, \
{2.38`, 1.18029850231687033272750676174596324433`10.150320399454788}, \
{2.39`, 1.18029315613612538342372210272335426953`10.150320616336812}, \
{2.4`, 1.18027136016388886609024108419546052058`10.15032049953037}, \
{2.41`, 1.1802683757398227804152332116438480561`10.150320532384328}, \
{2.42`, 1.18024818233949735052398592060266069481`10.150320539818848}, \
{2.43`, 1.18024066185043397345256423825981681447`10.15032057611617}, \
{2.44`, 1.18022432267000804084233260810718129599`10.150320559046884}, \
{2.45`, 1.18021116327552406361346016665879265821`10.150320563812805}, \
{2.46`, 1.18019936285550823302632808088898016066`10.150320637244697}, \
{2.47`, 1.18019101588451309001209519859425720985`10.150320637647223}, \
{2.48`, 1.18018786096338499449853791905465739753`10.150320636263197}, \
{2.49`, 1.18016459448109308818327022400910753323`10.150320604856626}, \
{2.5`, 1.18015519313440986415250313960183004747`10.15032074122922}, \
{2.5100000000000002`, 
    1.1801395093394907109725119477046775641`10.150320673420158}, \
{2.52`, 1.1801521189103215413303663227645363953`10.150320879595276}, \
{2.5300000000000002`, 
    1.1801307972602899695356580794903075229`10.150320703866537}, \
{2.54`, 1.1801219032572064046637548898437084353`10.15032077822856}, \
{2.5500000000000003`, 
    1.18010778328630934425504963985643981838`10.150320780679827}, \
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{2.59`, 1.18007131998070713493031563275406521632`10.150320850568097}, \
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{2.65`, 1.18001928474006261575689832562465017511`10.150320951961788}, \
{2.66`, 1.18001302942814922306819489909187068756`10.150320969148144}, \
{2.67`, 1.18000585845323368595610623797510024537`10.150320986058905}, \
{2.68`, 1.17999830319398077633955012371853220692`10.150320961943892}, \
{2.69`, 1.17998837347173506474908235560120900907`10.150321004684509}, \
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{2.77`, 1.17993882505107265878314852070597855445`10.150321086933197}, \
{2.7800000000000002`, 
    1.17993452640470845116188014746164546198`10.150321116360066}, \
{2.79`, 1.17992909870801670694847287322627336658`10.150321131596627}, \
{2.8000000000000003`, 
    1.17992439123620572181080232739291171754`10.150321147110823}, \
{2.81`, 1.17991912467737054169571074322045806068`10.15032114225105}, \
{2.82`, 1.17991480940324422553226187393945756974`10.150321154100451}, \
{2.83`, 1.17990958874666267274774447524562975543`10.150321181093055}, \
{2.84`, 1.17990564194203696094776943735162938735`10.150321172747566}, \
{2.85`, 1.17989994927618164916452938552678725266`10.15032120730178}, \
{2.86`, 1.17989755501309123951355728961865913188`10.150321198361539}, \
{2.87`, 1.17989232754353003171510898983896799535`10.150321194894682}, \
{2.88`, 1.17988845568961326992564992642119012053`10.150321197127615}, \
{2.89`, 1.17988589742719803584973730650738260995`10.150321212434863}, \
{2.9`, 1.17988176581893780962421803300552151788`10.15032119417042}, \
{2.91`, 1.17987875417937959585613984639853561569`10.150321197195288}, \
{2.92`, 1.17987965938957754449600847664661010376`10.150321205153638}, \
{2.93`, 1.17987402613409458459515189806653082172`10.150321201971883}, \
{2.94`, 1.17987222533192294311177015368629973929`10.150321206144982}, \
{2.95`, 1.17986957347002501590911694223837354359`10.150321212486556}, \
{2.96`, 1.17986677535605053823104454953534171341`10.15032121854462}, \
{2.97`, 1.17986323710674197821728392192023793707`10.150321215599114}, \
{2.98`, 1.17985949640129442747227095859703509117`10.150321207294247}, \
{2.99`, 1.17985551713664815217415823611544887084`10.15032118932838}, \
{3.`, 1.17985149480779907471182091433276932591`10.150321160381248}, \
{3.0100000000000002`, 
    1.17984770136888243517551916981756053725`10.150321118990874}, \
{3.02`, 1.17984424326886577914136687162659965423`10.150321065700378}, \
{3.0300000000000002`, 
    1.17984067863493941511370583225672568141`10.150321025113692}, \
{3.04`, 1.17983972280714366167920448006408726134`10.150320931432793}, \
{3.0500000000000003`, 
    1.17983713099698523369055172117943560253`10.150320881950467}, \
{3.06`, 1.17983805672222088042509478174748495443`10.150320777479246}, \
{3.0700000000000003`, 
    1.17983859248359339244678035006009651812`10.150320698649798}, \
{3.08`, 1.17984010278916593097573438815866905194`10.15032062546959}, \
{3.09`, 1.17984090117462542124244791687790076279`10.150320559809153}, 
   {3.1`, 
    1.17982997239924707255835635158905814981`10.150320733070526}, \
{3.11`, 1.17984393058236338402998580037742437307`10.150320451743152}, \
{3.12`, 1.1798458854886468657719264206187240623`10.150320429716823}, \
{3.13`, 1.17984624533975615219754358686757348567`10.150320394862455}, \
{3.14`, 1.17984664133333547726439513516606186835`10.150320383670744}};

Sorry that this became so long, it was the quickest way to share the data with you.

$\endgroup$
  • 5
    $\begingroup$ Could you provide the code that generated the interpolating function? $\endgroup$ – Chris K Jun 27 '18 at 16:56
  • $\begingroup$ I simply used Func = Interpolation[Data]; $\endgroup$ – Gordon Jun 27 '18 at 17:26
  • 5
    $\begingroup$ But what is Data? $\endgroup$ – Chip Hurst Jun 27 '18 at 17:27
  • $\begingroup$ Most likely your problem is sampling too fine. That causes the derivative to expose round off error in the samples. $\endgroup$ – John Doty Jun 27 '18 at 17:34
  • 2
    $\begingroup$ @Gordon Hard to say without the Data. $\endgroup$ – MarcoB Jun 27 '18 at 18:04
2
$\begingroup$

There are probably better ways to do it, but for lack of a better answer, a low pass filter will do it.

Interpolation[Data]
Interpolation[Transpose[{Most@#[[1]], LowpassFilter[Differences@#[[2]], .1]} &@ Transpose[ata]]]
Plot[%%[x], {x, 0, Pi}]
Plot[%%[x], {x, 0, Pi}]

enter image description here

Play around with the cutoff for better results.

... or use a moving average:

Interpolation[MovingAverage[Data, 15]]
Plot[%[x], {x, 0, \[Pi]}]
Plot[%%'[x], {x, 0, \[Pi]}]

enter image description here

$\endgroup$
  • $\begingroup$ Thank you! Could you perhaps mention what better methods you have in mind, so I could investigate? $\endgroup$ – Gordon Jun 27 '18 at 21:14
  • $\begingroup$ Note: I'm lazy and didn't divide by $\Delta x$; this would only change the scale of the graph, but not its overall shape. $\endgroup$ – AccidentalFourierTransform Jun 27 '18 at 21:15
  • $\begingroup$ @Gordon Off the top of my head, I cannot think of a better method; that's why I went with this one ;-) $\endgroup$ – AccidentalFourierTransform Jun 27 '18 at 21:15
  • $\begingroup$ Better? Maybe a fit to a Chebyshev series. Details left as an exercise. There are whole books on the art of function approximation. $\endgroup$ – John Doty Jun 27 '18 at 21:18
  • $\begingroup$ @AccidentalFourierTransform thanks. Out of curiousity, where would you divide by (\Delta x), as I need the actual values of the derivative, not just the shape. $\endgroup$ – Gordon Jun 27 '18 at 21:36

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