Interpolation ============= :ref:`O2scl ` Interpolation contents ---------------------- - :ref:`Interpolation introduction` - :ref:`Lookup and binary search` Interpolation introduction -------------------------- Basic interpolation of generic vector types is performed by :ref:`interp_vec ` and its children. The vector representing the independent variable must be monotonic, but need not be equally-spaced. The difference between the two classes is analogous to the difference between using ``gsl_interp_eval()`` and ``gsl_spline_eval()`` in GSL. You can create a :ref:`interp_vec ` object and use it to interpolate among any pair of chosen vectors. For example, cubic spline interpolation with natural boundary conditions:: boost::numeric::ublas::vector x(20), y(20); // fill x and y with data o2scl::interp<> oi(itp_cspline); double y_half=oi.eval(0.5,20,x,y); Alternatively, you can create a :ref:`interp_vec ` object which can be optimized for a pair of vectors that you specify in advance (now using linear interpolation instead):: boost::numeric::ublas::vector x(20), y(20); // fill x and y with data o2scl::interp_vec<> oi(20,x,y,itp_linear); double y_half=oi.eval(0.5); These interpolation classes require that the vector ``x`` is either monotonically increasing or monotonically decreasing, but these classes do not verify that this is the case. If the vectors or not monotonic, then the interpolation functions are not defined. These classes give identical results to the GSL interpolation routines when the vector is monotonically increasing. These interpolation classes will extrapolate to regions outside those defined by the user-specified vectors and will not warn you when they do this (this is not the same behavior as in GSL). One-dimensional Gaussian process interpolation (i.e. Kriging) is also provided in :ref:`interp_krige ` for a generic user-specified covariance function and :ref:`interp_krige_optim ` which allows one to optimize the parameters to fit the data. The different interpolation types are defined in ``src/base/interp.h`` .. doxygenenumvalue:: itp_linear .. doxygenenumvalue:: itp_cspline .. doxygenenumvalue:: itp_cspline_peri .. doxygenenumvalue:: itp_akima .. doxygenenumvalue:: itp_akima_peri .. doxygenenumvalue:: itp_monotonic .. doxygenenumvalue:: itp_steffen .. doxygenenumvalue:: itp_nearest_neigh .. doxygenenumvalue:: itp_gp_rbf_noise_loo_cv (experimental) .. doxygenenumvalue:: itp_gp_rbf_noise_max_lml (experimental) Integrals are always computed assuming that if the limits are ordered so that if the upper limit appears earlier in the array ``x`` in comparison to the lower limit, that the value of the integral has the opposite sign than if the upper limit appears later in the array ``x``. The class :ref:`interp_vec ` is based on the lower-level interpolation classes of type :ref:`interp_base `. Also, the interpolation classes based on :ref:`interp_base ` and also the class :ref:`interp_vec ` also have defined a function ``operator()`` which also returns the result of the interpolation. Lookup and binary search ------------------------ The classes :ref:`search_vec ` and :ref:`search_vec_ext ` contain searching functions for generic vector types which contain monotonic (either increasing or decreasing) data. It is :ref:`search_vec ` which is used internally by the interpolation classes to perform cached binary searching. These classes also allow one to to exhaustively search for the index of an element in a vector without specifying in advance if the vector is increasing or decreasing, e.g. :cpp:func:`o2scl::search_vec::ordered_lookup()`. Interpolation example --------------------- .. literalinclude:: ../../../examples/ex_interp.cpp :language: c++ :start-after: sphinx-example-start .. image ../../../examples/plot/ex_fptr_plot.png width: 60% alt: alt text .. todo:: Fix the interpolation plot for this example. Two and higher-dimensional interpolation ---------------------------------------- Support for multi-dimensional interpolation is documented in :ref:`Higher-dimensional Interpolation`. Inverse interpolation and related functions ------------------------------------------- The equivalent to "inverse" linear interpolation, which computes all the abcissae which have a fixed value of the ordinate, is implemented in the template function :cpp:func:`o2scl::vector_find_level()`. This function together with :cpp:func:`o2scl::vector_invert_enclosed_sum()` can be used to determine confidence limits surrounding the peak of a 1-dimensional data set using linear interpolation. To count level crossings in a function, use :cpp:func:`o2scl::vector_level_count()`. The function :cpp:func:`o2scl::vector_integ_interp()` uses interpolation to compute the integral defined by a set of vectors, and the function :cpp:func:`o2scl::vector_region_fracint()` finds the set of regions which gives a fraction of the integral reported by :cpp:func:`o2scl::vector_integ_interp()`. Derivatives and integrals on a fixed grid ----------------------------------------- If the indepedent variable is represented by a uniform (equally-spaced) then the functions in ``src/deriv/vector_derint.h`` can provide faster (and occasionally more accurate) results. See: - :cpp:func:`o2scl::vector_deriv_fivept()` - :cpp:func:`o2scl::vector_deriv_fivept_tap()` - :cpp:func:`o2scl::vector_deriv_interp()` - :cpp:func:`o2scl::vector_deriv_threept()` - :cpp:func:`o2scl::vector_deriv_threept_tap()` - :cpp:func:`o2scl::vector_integ_durand()` - :cpp:func:`o2scl::vector_integ_extended4()` - :cpp:func:`o2scl::vector_integ_extended8()` - :cpp:func:`o2scl::vector_integ_threept()` - :cpp:func:`o2scl::vector_integ_trap()`