# [math] Numerical derivatives in Commons Math

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## [math] Numerical derivatives in Commons Math

 Hello, I have a proposal for a numerical derivatives framework for Commons Math. I'd like to add the ability to take any UnivariateRealFunction and produce another function that represents it's derivative for an arbitrary order. Basically, I'm saying add a factory-like interface that looks something like the following: public interface UniverateNumericalDeriver {  public UnivariateRealFunction derive(UnivariateRealFunction f, int derivOrder); } For an initial implementation of this interface, I propose using finite differences - either central, forward, or backward. Computing the finite difference coefficients, for any derivative order and any error order, is a relatively trivial linear algebra problem. The user will simply choose an error order and difference type when setting up an FD univariate deriver - everything else will happen automagically. You can compute the FD coefficients once the user invokes the function in the interface above (might be expensive), and determine an appropriate stencil width when they call evaluate(double) on the function returned by the aformentioned method - for example, if the user has asked for the nth derivative, we simply use the nth root of the machine epsilon/double ulp for the stencil width. It would also be pretty easy to let the user control this (which might be desirable in some cases). Wikipedia has decent article on FDs of all flavors: http://en.wikipedia.org/wiki/Finite_differenceThere are, of course, many other univariate numerical derivative schemes that could be added in the future - using Fourier transforms, Barak's adaptive degree polynomial method, etc. These could be added later. We could also add the ability to numerically differentiate at single point using an arbitrary or user-defined grid (rather than an automatically generated one, like above). Barak's method and Fornberg finite difference coefficients could be used in this case: http://pubs.acs.org/doi/abs/10.1021/ac00113a006http://amath.colorado.edu/faculty/fornberg/Docs/MathComp_88_FD_formulas.pdfIt would also make sense to add vectorial and matrix-flavored versions of interface above. These interfaces would be slightly more complex, but nothing too crazy. Again, the initial implementation would be finite differences. This would also be really easy to implement, since multivariate FD coefficients are nothing more than an outer product of their univariate cousins. The Wikipedia article also has some good introductory material on multivariate FDs. Cheers, Fran. --------------------------------------------------------------------- To unsubscribe, e-mail: [hidden email] For additional commands, e-mail: [hidden email]
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## Re: [math] Numerical derivatives in Commons Math

 Le 11/08/2011 23:27, Fran Lattanzio a écrit : > Hello, Hi Fran, > > I have a proposal for a numerical derivatives framework for Commons > Math. I'd like to add the ability to take any UnivariateRealFunction > and produce another function that represents it's derivative for an > arbitrary order. Basically, I'm saying add a factory-like interface > that looks something like the following: > > public interface UniverateNumericalDeriver { >   public UnivariateRealFunction derive(UnivariateRealFunction f, int derivOrder); > } This sound interesting. did you have a look at Commons Nabla UnivariateDifferentiator interface and its implementations ? Luc > > For an initial implementation of this interface, I propose using > finite differences - either central, forward, or backward. Computing > the finite difference coefficients, for any derivative order and any > error order, is a relatively trivial linear algebra problem. The user > will simply choose an error order and difference type when setting up > an FD univariate deriver - everything else will happen automagically. > You can compute the FD coefficients once the user invokes the function > in the interface above (might be expensive), and determine an > appropriate stencil width when they call evaluate(double) on the > function returned by the aformentioned method - for example, if the > user has asked for the nth derivative, we simply use the nth root of > the machine epsilon/double ulp for the stencil width. It would also be > pretty easy to let the user control this (which might be desirable in > some cases). Wikipedia has decent article on FDs of all flavors: > http://en.wikipedia.org/wiki/Finite_difference> > There are, of course, many other univariate numerical derivative > schemes that could be added in the future - using Fourier transforms, > Barak's adaptive degree polynomial method, etc. These could be added > later. We could also add the ability to numerically differentiate at > single point using an arbitrary or user-defined grid (rather than an > automatically generated one, like above). Barak's method and Fornberg > finite difference coefficients could be used in this case: > http://pubs.acs.org/doi/abs/10.1021/ac00113a006> http://amath.colorado.edu/faculty/fornberg/Docs/MathComp_88_FD_formulas.pdf> > It would also make sense to add vectorial and matrix-flavored versions > of interface above. These interfaces would be slightly more complex, > but nothing too crazy. Again, the initial implementation would be > finite differences. This would also be really easy to implement, since > multivariate FD coefficients are nothing more than an outer product of > their univariate cousins. The Wikipedia article also has some good > introductory material on multivariate FDs. > > Cheers, > Fran. > > --------------------------------------------------------------------- > To unsubscribe, e-mail: [hidden email] > For additional commands, e-mail: [hidden email] > > --------------------------------------------------------------------- To unsubscribe, e-mail: [hidden email] For additional commands, e-mail: [hidden email]
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## Re: [math] Numerical derivatives in Commons Math

 I like the idea of adding this feature. What about an abstract class that implements DifferentiableMultivariateRealFunction and provides the method for partialDerivative (). People could then override the partialDerivative method if they have an analytic derivative. Here's some code that I'm happy to contribute to Commons-math. It computes the derivative by the central difference meathod and the Hessian by finite difference. I can add this to JIRA when it's there. /**       * Numerically compute gradient by the central difference method. Override this method       * when the analytic gradient is available.       *       *       * @param x       * @return       */      public double[] derivativeAt(double[] x){          int n = x.length;          double[] grd = new double[n];          double[] u = Arrays.copyOfRange(x, 0, x.length);          double f1 = 0.0;          double f2 = 0.0;          double stepSize = 0.0001;          for(int i=0;i Le 11/08/2011 23:27, Fran Lattanzio a écrit : >> Hello, > > Hi Fran, > >> >> I have a proposal for a numerical derivatives framework for Commons >> Math. I'd like to add the ability to take any UnivariateRealFunction >> and produce another function that represents it's derivative for an >> arbitrary order. Basically, I'm saying add a factory-like interface >> that looks something like the following: >> >> public interface UniverateNumericalDeriver { >>   public UnivariateRealFunction derive(UnivariateRealFunction f, int >> derivOrder); >> } > > This sound interesting. did you have a look at Commons Nabla > UnivariateDifferentiator interface and its implementations ? > > Luc > >> >> For an initial implementation of this interface, I propose using >> finite differences - either central, forward, or backward. Computing >> the finite difference coefficients, for any derivative order and any >> error order, is a relatively trivial linear algebra problem. The user >> will simply choose an error order and difference type when setting up >> an FD univariate deriver - everything else will happen automagically. >> You can compute the FD coefficients once the user invokes the function >> in the interface above (might be expensive), and determine an >> appropriate stencil width when they call evaluate(double) on the >> function returned by the aformentioned method - for example, if the >> user has asked for the nth derivative, we simply use the nth root of >> the machine epsilon/double ulp for the stencil width. It would also be >> pretty easy to let the user control this (which might be desirable in >> some cases). Wikipedia has decent article on FDs of all flavors: >> http://en.wikipedia.org/wiki/Finite_difference>> >> There are, of course, many other univariate numerical derivative >> schemes that could be added in the future - using Fourier transforms, >> Barak's adaptive degree polynomial method, etc. These could be added >> later. We could also add the ability to numerically differentiate at >> single point using an arbitrary or user-defined grid (rather than an >> automatically generated one, like above). Barak's method and Fornberg >> finite difference coefficients could be used in this case: >> http://pubs.acs.org/doi/abs/10.1021/ac00113a006>> http://amath.colorado.edu/faculty/fornberg/Docs/MathComp_88_FD_formulas.pdf  >> >> >> It would also make sense to add vectorial and matrix-flavored versions >> of interface above. These interfaces would be slightly more complex, >> but nothing too crazy. Again, the initial implementation would be >> finite differences. This would also be really easy to implement, since >> multivariate FD coefficients are nothing more than an outer product of >> their univariate cousins. The Wikipedia article also has some good >> introductory material on multivariate FDs. >> >> Cheers, >> Fran. >> >> --------------------------------------------------------------------- >> To unsubscribe, e-mail: [hidden email] >> For additional commands, e-mail: [hidden email] >> >> > > > --------------------------------------------------------------------- > To unsubscribe, e-mail: [hidden email] > For additional commands, e-mail: [hidden email] > --------------------------------------------------------------------- To unsubscribe, e-mail: [hidden email] For additional commands, e-mail: [hidden email]
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## Re: [math] Numerical derivatives in Commons Math

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## Re: [math] Numerical derivatives in Commons Math

 As Patrick suggested, this approach should really be extended to multivariate functions. To cite but one example, I recently attended a conf where Pr. Prevost (Princeton) talked about non-linear finite elements calcs. The long standing approach had always been to implement the analytical expressions tangent stiffness (which is nothing but a jacobian matrix). He argued strongly against it for at least two reasons   - it is error-prone,   - most of the time, the expressions are so complex that their evaluation is just as time-consuming as a numerical derivative. So, having some robust algorithms for multidimensional functions already implemented in CM would in my view be invaluable. Sébastien --------------------------------------------------------------------- To unsubscribe, e-mail: [hidden email] For additional commands, e-mail: [hidden email]
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## Re: [math] Numerical derivatives in Commons Math

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## Re: [math] Numerical derivatives in Commons Math

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## Re: [math] Numerical derivatives in Commons Math

 In reply to this post by Sébastien Brisard Hi Sébastien, Le 12/08/2011 07:50, Sébastien Brisard a écrit : > As Patrick suggested, this approach should really be extended to > multivariate functions. To cite but one example, I recently attended a > conf where Pr. Prevost (Princeton) talked about non-linear finite > elements calcs. The long standing approach had always been to > implement the analytical expressions tangent stiffness (which is > nothing but a jacobian matrix). He argued strongly against it for at > least two reasons >    - it is error-prone, >    - most of the time, the expressions are so complex that their > evaluation is just as time-consuming as a numerical derivative. I don't fully agree with this. Both numerical and analytical approach are useful and have advantages and drawbacks. The fact analytical approach is error-prone is true only when analytical differentiation is done manually. Using automatic differentiation completely removes this problem (take a look at Nabla). The fact expression are has time consuming as numerical derivatives is simply false when speaking about multivariate functions. This result is known as the "cheap gradient" property. The relative computing effort for gradients or Jacobians using finite differences for an n variables function with respect to the basic function evaluation is roughly 2n. Using the automatic differentiation technique known as "reverse mode" (which is not implemented in Nabla but should be in the unknown future), this cost is about 4 and is *independent of n*, which is really an amazing result. > So, having some robust algorithms for multidimensional functions > already implemented in CM would in my view be invaluable. I agree with that. Luc > Sébastien > > --------------------------------------------------------------------- > To unsubscribe, e-mail: [hidden email] > For additional commands, e-mail: [hidden email] > > --------------------------------------------------------------------- To unsubscribe, e-mail: [hidden email] For additional commands, e-mail: [hidden email]
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## Re: [math] Numerical derivatives in Commons Math

 Hi Luc, > > I don't fully agree with this. Both numerical and analytical approach are > useful and have advantages and drawbacks. > Wowww! I certainly did not want to start a debate on this topic. I'm just reporting on a conference I heard which lead me think that CM users might find such a feature usefull. I think we can safely assume Prevost knows what he is talking about, he has been around long enough in the FE community. Having said that, I'm confident you also know what you are talking about... So maybe you both are talking about slightly different things. Prevost is differentiating a material constitutive law: multivariate, yes, but not too many variables. So maybe the time overhead is not so great at low dimensionality, you tell me. As for automatic differentiation: that topic was not raised in his talk, and I'm sure he is using an external CAS software and copy/pasting into his huge FORTRAN (ughhh) software. Sébastien 2011/8/12 Luc Maisonobe <[hidden email]>: > Hi Sébastien, > > Le 12/08/2011 07:50, Sébastien Brisard a écrit : >> >> As Patrick suggested, this approach should really be extended to >> multivariate functions. To cite but one example, I recently attended a >> conf where Pr. Prevost (Princeton) talked about non-linear finite >> elements calcs. The long standing approach had always been to >> implement the analytical expressions tangent stiffness (which is >> nothing but a jacobian matrix). He argued strongly against it for at >> least two reasons >>   - it is error-prone, >>   - most of the time, the expressions are so complex that their >> evaluation is just as time-consuming as a numerical derivative. > The fact analytical approach is error-prone is true only when analytical > differentiation is done manually. Using automatic differentiation completely > removes this problem (take a look at Nabla). > > The fact expression are has time consuming as numerical derivatives is > simply false when speaking about multivariate functions. This result is > known as the "cheap gradient" property. The relative computing effort for > gradients or Jacobians using finite differences for an n variables function > with respect to the basic function evaluation is roughly 2n. Using the > automatic differentiation technique known as "reverse mode" (which is not > implemented in Nabla but should be in the unknown future), this cost is > about 4 and is *independent of n*, which is really an amazing result. > >> So, having some robust algorithms for multidimensional functions >> already implemented in CM would in my view be invaluable. > Luc > >> Sébastien >> >> --------------------------------------------------------------------- >> To unsubscribe, e-mail: [hidden email] >> For additional commands, e-mail: [hidden email] >> >> > > > --------------------------------------------------------------------- > To unsubscribe, e-mail: [hidden email] > For additional commands, e-mail: [hidden email] > > --------------------------------------------------------------------- To unsubscribe, e-mail: [hidden email] For additional commands, e-mail: [hidden email]
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