By the way, my the second test function in interpolation/test.c uses randomly-generated data points, but actually serves to nicely illustrate the difference between major non-linear interpolation methods. See the linked graph for a comparison of the interpolation for those data using my implementation of steffen, and the existing GSL akima and cubic spline methods.
https://github.com/jfcaron3/gsl-steffen-devel/blob/steffen/interpolation/compare.pdf (I couldn’t send a pdf to the mailing list, and I don’t know how to view a pdf on github’s website, but I guess you can just get the image when you clone the repo.) While the cubic spline and akima methods preserve continuity of the second derivatives, they are not monotonic and can have oscillations that are often undesireable. The steffen method sacrifices continuity of the second derivative (but maintains it for the first) in order to maintain monoticity, which also eliminates weird oscillations. In Steffen’s paper, there is also an example graph where the akima method is unstable (a very small change in one data point makes a large change in the interpolated function), while the steffen method is stable by construction. Jean-François > > On Mar 27, 2014, at 01:10 , Patrick Alken <[email protected]> wrote: > >> The code is looking very good - I will try to find time in the next few days >> to do some tests and import it into GSL >> >> Thanks >> Patrick >> ________________________________________ >> From: [email protected] [[email protected]] On >> Behalf Of Jean-François Caron [[email protected]] >> Sent: Wednesday, March 26, 2014 7:10 PM >> To: [email protected] >> Subject: Re: Compiling & Testing New Interpolation Type >> >> I have now fixed the problems with the tests and added a more robust test >> with lots of data points. I am effectively ready to give a pull request >> from my github repo. Let me know what I need to do to facilitate this. >> >> Jean-François >> >> On Mar 25, 2014, at 15:51 , Jean-François Caron <[email protected]> wrote: >> >>> Git and Github weren’t as intimidating as I expected. I have a repo here >>> with the “steffen” branch including my changes: >>> >>> https://github.com/jfcaron3/gsl-steffen-devel >>> >>> The Savannah git repo didn’t include a configure script, and I got my >>> modified GSL+Steffen code to compile by directly modifying >>> interpolation/Makefile AFTER running ./configure, so I’m not sure how to >>> compile the files cloned from my github repo. At least it’s easier to see >>> the changes now. >>> >>> Jean-François >>> >>> On Mar 25, 2014, at 14:56 , Jean-François Caron <[email protected]> wrote: >>> >>>> I’ve improved my initial code greatly. You can find it here: >>>> >>>> http://bazaar.launchpad.net/~jfcaron/+junk/my_steffen/files >>>> >>>> You can compile it into GSL by adding in the interpolation/Makefile >>>> references to “steffen.c”, “steffen.lo”, and “steffen.Plo” exactly where >>>> there are currently references to “akima.*”. >>>> >>>> I’ve tried adding an “integ” method, but I’m afraid I don’t even >>>> understand the workings of the integ methods for the existing >>>> interpolation types. I couldn’t just copy from the akima.c integ method >>>> because they use a build-in spline calculation function (which I also >>>> don’t understand). Reading uncommented C code is pretty hard. My test >>>> program shows that the integration method isn’t obviously broken, but it >>>> fails the tests I wrote in interpolation/test.c The actual interpolation >>>> and derivatives seem to work and pass the tests. >>>> >>>> I’ve not used github before, so I guess my next move should be to learn >>>> the basics and start using that, since otherwise describing my additions & >>>> changes are hard to follow. In the meantime, is anyone able to explain >>>> how the heck the “integ” methods work? >>>> >>>> Jean-François >>>> >>>> On Mar 20, 2014, at 11:30 , Patrick Alken <[email protected]> >>>> wrote: >>>> >>>>> Yes that green curve is rather strange and doesn't seem much better than >>>>> the cubic spline. I like simplicity too so lets proceed with importing >>>>> the steffen code. >>>>> >>>>> On 03/20/2014 12:18 PM, Jean-François Caron wrote: >>>>>> Definitely an advantage of a) is that it is conceptually simple. b) is >>>>>> 44 pages while a) is only 7. Even if b) is somehow mathematically >>>>>> superior, I like the idea of understanding the tools that I am using >>>>>> (and being able to explain it to my academic supervisor/conference >>>>>> attendees). >>>>>> >>>>>> The MESA astrophysics library (C++ unfortunately) actually includes both >>>>>> types, and has a little graph to show differences: >>>>>> http://mesa.sourceforge.net/interp_1D.html >>>>>> >>>>>> Actually their graph is confusing, blue is supposed to be a), green b), >>>>>> but the green curve isn’t piece-wise monotonic between the data points. >>>>>> I’m starting to think maybe Stetten and Huynh mean different things when >>>>>> they say “monotonic”. I’ll try to read Huynh’s paper to see if they >>>>>> define what they are trying to do. Steffen is pretty clear about his >>>>>> technique being a for an interpolating function that is monotonic >>>>>> between data points - i.e. the interpolating function doesn’t change >>>>>> sign between data points, and extrema can only occur at said data points. >>>>>> >>>>>> Jean-François >>>>>> >>>>>> On Mar 20, 2014, at 11:03 , Patrick Alken <[email protected]> >>>>>> wrote: >>>>>> >>>>>>> I see question 1) is answered by section 4 of Steffen's paper - the >>>>>>> method works on all data sets, and preserves monotonicity in each >>>>>>> interval, which is nice. They also state that method (c) has some >>>>>>> serious drawbacks. >>>>>>> >>>>>>> Unfortunately paper (b) doesn't reference (a) and so its difficult to >>>>>>> tell whether (b) offers any advantage over (a) >>>>>>> >>>>>>> On 03/20/2014 11:52 AM, Patrick Alken wrote: >>>>>>>> Hi, I'm moving this discussion over to gsl-discuss which is more suited >>>>>>>> for development issues. >>>>>>>> >>>>>>>> I have 2 naive questions which you may be able to answer since you've >>>>>>>> been working on this code. >>>>>>>> >>>>>>>> 1) If the Steffen algorithm is applied to non-monotonic data, will it >>>>>>>> still provide a solution or does the method encounter an error? >>>>>>>> >>>>>>>> 2) Earlier on the GSL list it was mentioned that there are 3 different >>>>>>>> methods for interpolating monotonic data: >>>>>>>> >>>>>>>> (a) M.Steffen, "A simple method for monotonic interpolation in one >>>>>>>> dimension", Astron. Astrophys. 239, 443-450 (1990). >>>>>>>> >>>>>>>> (b) H.T.Huynh, "Accurate Monotone Cubic Interpolation", SIAM J. Numer. >>>>>>>> Anal. 30, 57-100 (1993). >>>>>>>> >>>>>>>> (c) Fritsch, F. N.; Carlson, R. E., "Monotone Piecewise Cubic >>>>>>>> Interpolation", SIAM J. Numer. Anal. 17 (2), 238–246 (1980). >>>>>>>> >>>>>>>> I haven't looked at (c) but it seems that (a) and (b) both use >>>>>>>> piecewise >>>>>>>> cubic polynomials and preserve monotonicity. Do you happen to know if >>>>>>>> one method is superior to the other? If one method is significantly >>>>>>>> better than the other two it would make more sense to include that one >>>>>>>> in GSL. >>>>>>>> >>>>>>>> Patrick >>>>>>>> >>>>>>>> On 03/20/2014 11:37 AM, Jean-François Caron wrote: >>>>>>>>> Yes, I didn’t bother doing the integration function at the time >>>>>>>>> because I was having trouble just compiling. I will add the >>>>>>>>> integration function, and re-write the eval and deriv/deriv2 >>>>>>>>> functions to use Horner’s scheme for the polynomials. I can generate >>>>>>>>> some comparison graphs using fake data like in Steffen’s paper, that >>>>>>>>> sounds easy enough. >>>>>>>>> >>>>>>>>> I’ll look at the interpolation/test.c file and see if I can come up >>>>>>>>> with similar tests. >>>>>>>>> >>>>>>>>> Thanks for offering to help with the integration into GSL itself. I >>>>>>>>> don’t know a lot of the procedures (or even politics sometimes!) >>>>>>>>> involved. >>>>>>>>> >>>>>>>>> Jean-François >>>>>>>>> >>>>>>>>> On Mar 20, 2014, at 10:22 , Patrick Alken >>>>>>>>> <[email protected]> wrote: >>>>>>>>> >>>>>>>>>> I did notice you talking about 1.6 in your earlier messages, but >>>>>>>>>> assumed it was a typo and you meant 1.16, oops. >>>>>>>>>> >>>>>>>>>> On 03/20/2014 11:11 AM, Jean-François Caron wrote: >>>>>>>>>>> My original problem was that I wanted to add an interpolation type >>>>>>>>>>> to GSL. Specifically I want monotonic cubic-splines following the >>>>>>>>>>> description in Steffen (1990): >>>>>>>>>>> http://adsabs.harvard.edu/full/1990A%26A...239..443S >>>>>>>>>> I took a quick look at your code earlier and it looks pretty nice. I >>>>>>>>>> noticed you commented out the _integ function - is this something >>>>>>>>>> you could add to make it feature complete with the other >>>>>>>>>> interpolation types? >>>>>>>>>> >>>>>>>>>> It is important to add automated tests for this. Can you look at >>>>>>>>>> interpolation/test.c and design similar tests for your new method? >>>>>>>>>> Also I think it would be nice to add a figure to the manual >>>>>>>>>> illustrating the differences between cubic, akima, and your new >>>>>>>>>> steffen method (similar to the figures in the Steffen paper). This >>>>>>>>>> would help users a lot when trying to decide what method to use. Do >>>>>>>>>> you happen to have a dataset which shows a nice contrast like Figs >>>>>>>>>> 1, 3 and 8 from that paper? >>>>>>>>>> >>>>>>>>>> When everything is ready I would be happy to add it to GSL, as we >>>>>>>>>> are already planning to update the interpolation module for the next >>>>>>>>>> release. When I find some time I want to import the 2D interpolation >>>>>>>>>> extension discussed previously, and also add Hermite interpolation. >>>>>>>>>> >>>>>>>>>> It would be easiest for us if you could clone the GSL git repository >>>>>>>>>> and make your changes there. You could make a new branch called >>>>>>>>>> 'steffen' or something and publish it to github, or just send a >>>>>>>>>> patch file to me, whichever is easiest. >>>>>>>>>> >>>>>>>>>> Patrick >>>>>>>>>> >>>>>>>>>> On Mar 19, 2014, at 18:40 , Dave Allured - NOAA Affiliate >>>>>>>>>> <[email protected]> wrote: >>>>>>>>>>>> More data. I tried the same plain build recipe, GSL 1.16 on our >>>>>>>>>>>> test >>>>>>>>>>>> machine which is at Mac OS 10.9.3. Got another perfect build, no >>>>>>>>>>>> make >>>>>>>>>>>> check errors, no PPC-related issues. Outputs on request, please be >>>>>>>>>>>> specific. >>>>>>>>>>>> >>>>>>>>>>>> CC=clang >>>>>>>>>>>> CFLAGS=-g >>>>>>>>>>>> ./configure --prefix /Users/dallured/Disk/3rd/gsl/1.16.os10.9 >>>>>>>>>>>> >>>>>>>>>>>> mac27:~/Disk/3rd/gsl/1.16.os10.9 57> sw_vers >>>>>>>>>>>> ProductName: Mac OS X >>>>>>>>>>>> ProductVersion: 10.9.3 >>>>>>>>>>>> BuildVersion: 13D17 >>>>>>>>>>>> >>>>>>>>>>>> mac27:~/Disk/3rd/gsl/1.16.os10.9/src 36> \ >>>>>>>>>>>> ? grep -i '# [a-z]' ../logfiles/make-check.0319a.log | sort | uniq >>>>>>>>>>>> -c >>>>>>>>>>>> 45 # ERROR: 0 >>>>>>>>>>>> 45 # FAIL: 0 >>>>>>>>>>>> 42 # PASS: 1 >>>>>>>>>>>> 3 # PASS: 2 >>>>>>>>>>>> 45 # SKIP: 0 >>>>>>>>>>>> 42 # TOTAL: 1 >>>>>>>>>>>> 3 # TOTAL: 2 >>>>>>>>>>>> 45 # XFAIL: 0 >>>>>>>>>>>> 45 # XPASS: 0 >>>>>>>>>>>> >>>>>>>>>>>> mac27:~/Disk/3rd/gsl/1.16.os10.9 62> \ >>>>>>>>>>>> ? grep -c -i ppc logfiles/*319a*log >>>>>>>>>>>> logfiles/configure.0319a.os10.9.log:0 >>>>>>>>>>>> logfiles/install.0319a.log:0 >>>>>>>>>>>> logfiles/make-check.0319a.log:0 >>>>>>>>>>>> logfiles/make.0319a.log:0 >>>>>>>>>>>> >>>>>>>>>>>> mac27:~/Disk/3rd/gsl/1.16.os10.9 65> \ >>>>>>>>>>>> ? grep -i ppc src/config.h src/config.log src/config.status >>>>>>>>>>>> src/config.h:/* #undef HAVE_GNUPPC_IEEE_INTERFACE */ >>>>>>>>>>>> src/config.log:HAVE_GNUPPC_IEEE_INTERFACE='' >>>>>>>>>>>> src/config.status:S["HAVE_GNUPPC_IEEE_INTERFACE"]="" >>>>>>>>>>>> >>>>>>>>>>>> --Dave >>>>>>>>>>>> >>>>>>>>>>>> On Wed, Mar 19, 2014 at 5:27 PM, Jean-Francois Caron >>>>>>>>>>>> <[email protected]> >>>>>>>>>>>> wrote: >>>>>>>>>>>>> Dave is correct, I am using an "i686" 64-bit x86 mac. For some >>>>>>>>>>>>> reason >>>>>>>>>>>>> it is still looking for the PPC mac header file. The ./configure >>>>>>>>>>>>> stage correctly identifies my system, so it's a bit strange. >>>>>>>>>>>>> Also GSL >>>>>>>>>>>>> installs without errors when I do it from MacPorts, and MacPorts >>>>>>>>>>>>> doesn't seem to do anything other than ./configure && make, from >>>>>>>>>>>>> my >>>>>>>>>>>>> reading of the portfile. >>>>>>>>>>>>> >>>>>>>>>>>>> When I get back to my Mac, I will look at the NOTES file to see if >>>>>>>>>>>>> anything needs to be done for 10.9. >>>>>>>>>>>>> >>>>>>>>>>>>> Jean-François >>>>> >>>> >>> >> >
