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---- Steffen's method guarantees the monotonicity of the interpolating function between the given data points. Therefore, minima and maxima can only occur exactly at the data points, and there can never be spurious oscillations between data points. The interpolated function is piecewise cubic in each interval. The resulting curve and its first derivative are guaranteed to be continuous, but the second derivative may be discontinuous. ---- Does this look ok? I added your name to test.c Patrick On 04/03/2014 01:58 PM, Jean-François Caron wrote:
Hi Patrick, yes feel free to change the example dataset. I used it because it’s the same as I put into the test.c code, and other interpolation methods used randomly-generated data. For the description in the docs, I might recommend a different wording: @deffn {Interpolation Type} gsl_interp_steffen Steffen’s method guarantees the monoticity of the interpolating function between the given data points. Thus minima and maxima can only occur exactly at the data points, and there can never be spurious oscillations between data points. The interpolated function and its first derivative are guaranteed to be continuous, but the second derivative may be discontinuous. @end deffn Thanks for supporting my work! I’m very excited to be officially contributing to an open-source project. Could you check the copyright & attribution parts of the code files that I modified? I’m not sure what is correct, but I see author’s names and dates. I added mine to the steffen.c, but should I add it also to test.c and the others? Jean-François On Mar 31, 2014, at 15:24 , Patrick Alken <patrick.alken@Colorado.EDU> wrote:I couldn't reproduce the figure in Steffen's paper, so I found another dataset which nicely illustrates oscillation issues with Akima: J. M. Hyman, Accurate Monotonicity preserving cubic interpolation, SIAM J. Sci. Stat. Comput. 4, 4, 1983. The dataset is simpler than your randomly generated plot and I think its a little easier to compare the different methods. I added an example program and a figure to the manual (in the steffen branch). I am hoping to finish everything up and merge into master by the end of the week. Thanks again, Patrick On 03/31/2014 02:37 PM, Patrick Alken wrote:Ok I made a new branch 'steffen' in the GSL repository with your latest changes, thanks for all your work on this. I still want to update the docs a little and do some more testing on my own before merging it into master. I made a blurb about gsl_interp_steffen in the docs: ---- @deffn {Interpolation Type} gsl_interp_steffen Steffen's method for monotonic interpolation (not allowing minima or maxima to occur between adjacent data points). The resulting curve is piecewise cubic on each interval with the slope at each grid point chosen to ensure monotonicity and prevent undesired oscillations. The first-order derivative is everywhere continuous. @end deffn ---- Can you read this and make sure I haven't said anything inaccurate? Or let me know any suggestions you think its important to add for the users benefit to understand what this method does. Thanks, Patrick On 03/27/2014 11:17 AM, Jean-François Caron wrote: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-FrançoisOn Mar 27, 2014, at 01:10 , Patrick Alken <patrick.alken@Colorado.EDU> 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: gsl-discuss-owner@sourceware.org [gsl-discuss-owner@sourceware.org] On Behalf Of Jean-François Caron [jfcaron@phas.ubc.ca] Sent: Wednesday, March 26, 2014 7:10 PM To: gsl-discuss@sourceware.org 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 <jfcaron@phas.ubc.ca> 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 <jfcaron@phas.ubc.ca> 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 <patrick.alken@Colorado.EDU> 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 <patrick.alken@colorado.edu> 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 <patrick.alken@Colorado.EDU> 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..443SI 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 <dave.allured@noaa.gov> 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 <jfcaron@phas.ubc.ca> 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
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