Integration =========== :ref:`O2scl ` Integration contents -------------------- - :ref:`One-dimensional integration based on GSL, CERNLIB, and Boost` - :ref:`GSL-based integration details` - :ref:`GSL-based integration error messages` - :ref:`One-dimensional integration example` - :ref:`Gauss-Kronrod integration coefficients` - :ref:`Non-adaptive quadrature integration coefficients` .. This section is commented out for now .. - :ref:`Multi-dimensional integration routines` One-dimensional integration based on GSL, CERNLIB, and Boost ------------------------------------------------------------ Several classes integrate arbitrary one-dimensional functions: - Integration over a finite interval: * :ref:`inte_adapt_cern_tl ` * :ref:`inte_gauss_cern ` * :ref:`inte_gauss56_cern ` * :ref:`inte_kronrod_boost ` * :ref:`inte_double_exp_boost ` * :ref:`inte_qag_gsl ` * :ref:`inte_qng_gsl ` - Integration from :math:`a` to :math:`\infty`: * :ref:`inte_qagiu_gsl ` * :ref:`inte_double_exp_boost ` * :ref:`inte_transform ` - Integration from :math:`-\infty` to :math:`b`: * :ref:`inte_qagil_gsl ` * :ref:`inte_double_exp_boost ` * :ref:`inte_transform ` - Integration from :math:`-\infty` to :math:`\infty`: * :ref:`inte_qagi_gsl `, * :ref:`inte_double_exp_boost ` * :ref:`inte_transform `. - Integration over a finite interval for a function with singularities: * :ref:`inte_qags_gsl ` * :ref:`inte_qaws_gsl ` * :ref:`inte_double_exp_boost ` can handle singularities at either endpoint. - Cauchy principal value integration over a finite interval: * :ref:`inte_cauchy_cern ` * :ref:`inte_qawc_gsl ` - Integration over a function weighted by ``cos(x)`` or ``sin(x)``: * :ref:`inte_qawo_gsl_cos ` * :ref:`inte_qawo_gsl_sin ` - Fourier integrals: * :ref:`inte_qawf_gsl_cos ` * :ref:`inte_qawf_gsl_sin ` - Integration over a weight function .. math:: W(x)=(x-a)^{\alpha}(b-x)^{\beta}\log^{\mu}(x-a)\log^{\nu}(b-x) is performed by :ref:`inte_qaws_gsl `. Note that some of the integrators support multiprecision, see :ref:`Multiprecision Support`. There are two competing factors that can slow down an adaptive integration algorithm: (1) each evaluation of the integrand can be numerically expensive, depending on how the function is defined, and (2) the process of subdividing regions and recalculating values is almost always numerically expensive in its own right. For integrands that are very smooth (e.g., analytic functions), a high-order Gauss-Kronrod rule (e.g., 61-point) will achieve the desired error tolerance with relatively few subdivisions. For integrands with discontinuities or singular derivatives, a low-order rule (e.g., 15-point) is often more efficient. GSL-based integration details ----------------------------- For the GSL-based integration routines, the variables :cpp:var:`o2scl::inte::tol_abs` and :cpp:var:`o2scl::inte::tol_rel` have the same role as the quantities usually denoted in the GSL integration routines by ``epsabs`` and ``epsrel``. In particular, the integration classes attempt to ensure that .. math:: |\mathrm{result}-I| \leq \mathrm{Max}(\mathrm{tol\_abs}, \mathrm{tol\_rel}|I|) and returns an error to attempt to ensure that .. math:: |\mathrm{result}-I| \leq \mathrm{abserr} \leq \mathrm{Max}(\mathrm{tol\_abs},\mathrm{tol\_rel}|I|) where ``I`` is the integral to be evaluated. Even when the corresponding descendant of :cpp:func:`o2scl::inte::integ()` returns success, these inequalities may fail for sufficiently difficult functions. All of the GSL integration routines except for :ref:`inte_qng_gsl ` use a workspace given in :ref:`inte_workspace_gsl ` which holds the results of the various subdivisions of the original interval. The GSL routines were originally based on QUADPACK, which is available at http://www.netlib.org/quadpack . For adaptive GSL integration classes, the type of Gauss-Kronrod quadrature rule that is used to approximate the integral and estimate the error of a subinterval is set by :cpp:func:`o2scl::inte_kronrod_gsl::set_rule()`. The number of subdivisions of the integration region is limited by the size of the workspace, set in :cpp:func:`o2scl::inte_kronrod_gsl::set_limit()`. The number of subdivisions required for the most recent call to :cpp:func:`o2scl::inte::integ()` or :cpp:func:`o2scl::inte::integ_err()` is given in :cpp:var:`o2scl::inte::last_iter`. This number will always be less than or equal to the workspace size. .. note:: The GSL integration routines can sometimes lose precision if the integrand is everywhere much smaller than unity. Some rescaling may be required in these cases. GSL-based integration error messages ------------------------------------ The error messages given by the adaptive GSL integration routines tend to follow a standard form and are documented here. There are several error messages which indicate improper usage and cause the error handler to be called regardless of the value of :cpp:var:`o2scl::inte::err_nonconv`: - ``Iteration limit exceeds workspace in class::function().`` [ :cpp:enumerator:`exc_einval` ] - ``Could not integrate function in class::function() (it may have returned a non-finite result).`` [ :cpp:enumerator:`exc_efailed` ] This often occurs when the user-specified function returns ``inf`` or ``nan``. - ``Tolerance cannot be achieved with given value of tol_abs and tol_rel in class::function().`` [ :cpp:enumerator:`exc_ebadtol` ] This occurs if the user supplies unreasonable values for :cpp:var:`o2scl::inte::tol_abs` and :cpp:var:`o2scl::inte::tol_rel`. All positive values for :cpp:var:`o2scl::inte::tol_abs` are allowed. If zero-tolerance for :cpp:var:`o2scl::inte::tol_abs` is desired, then :cpp:var:`o2scl::inte::tol_rel` must be at least :math:`50 \cdot \epsilon_\mathrm{mach}` (:math:`\approx 1.11 \times 10^{-14}` ). - ``Cannot integrate with singularity on endpoint in inte_qawc_gsl::qawc().`` [ :cpp:enumerator:`exc_einval` ] The class \ref o2scl::inte_qawc_gsl cannot handle the case when a singularity is one of the endpoints of the integration. There are also convergence errors which will call the error handler unless :cpp:var:`o2scl::inte::err_nonconv` is false. See :ref:`What is an error?` for more discussion on convergence errors versus fatal errors: - ``Cannot reach tolerance because of roundoff error on first attempt in class::function().`` [ :cpp:enumerator:`exc_eround` ] Each integration attempt tests for round-off errors by comparing the computed integral with that of the integrand's absolute value (i.e., :math:`L^1` -norm). A highly oscillatory integrand may cause this error. - ``A maximum of 1 iteration was insufficient in class::function().`` [ :cpp:enumerator:`exc_emaxiter` ] This occurs if the workspace is allocated for one interval and a single Gauss-Kronrod integration does not yield the accuracy demanded by :cpp:var:`o2scl::inte::tol_abs` and :cpp:var:`o2scl::inte::tol_rel`. - ``Bad integrand behavior in class::function().`` [ :cpp:enumerator:`exc_esing` ] This occurs if the integrand is (effectively) singular in a region, causing the subdivided intervals to become too small for floating-point precision. - ``Maximum number of subdivisions 'value' reached in class::function().`` [ :cpp:enumerator:`exc_emaxiter` ] This occurs if the refinement algorithm runs out of allocated workspace. The number of iterations required for the most recent call to :cpp:func:`o2scl::inte::integ()` or :cpp:func:`o2scl::inte::integ_err()` is given in :cpp:var:`o2scl::inte::last_iter`. This number will always be less than or equal to the workspace size. - ``Roundoff error prevents tolerance from being achieved in class::function().`` [ :cpp:enumerator:`exc_eround` ] The refinement procedure counts round-off errors as they occur and terminates if too many such errors accumulate. - ``Roundoff error detected in extrapolation table in inte_singular_gsl::qags().`` [ :cpp:enumerator:`exc_eround` ] This occurs when error-terms from the :math:`\varepsilon` -algorithm are are monitored and compared with the error-terms from the refinement procedure. The algorithm terminates if these sequences differ by too many orders of magnitude. See :cpp:func:`o2scl::inte_singular_gsl::qelg()`. - ``Integral is divergent or slowly convergent in inte_singular_gsl::qags().`` [ :cpp:enumerator:`exc_ediverge` ] This occurs if the approximations produced by the refinement algorithm and the extrapolation algorithm differ by too many orders of magnitude. - ``Exceeded limit of trigonometric table in inte_qawo_gsl_sin()::qawo().`` [ :cpp:enumerator:`exc_etable` ] This occurs if the maximum level of the table of Chebyshev moments is reached. .. This section is commented out for now Multi-dimensional integration routines -------------------------------------- O₂scl reimplements the Cubature library for multi-dimensional integration. The h-adaptive and p-adaptive integration methods are implemented in :ref:`inte_hcubature ` and :ref:`inte_pcubature `. See also the Monte Carlo integration routines in :ref:`Monte Carlo Integration`. Multi-dimensional hypercubic integration is performed by children of :ref:`inte_multi . Currently in O₂scl, only the General multi-dimensional integration is performed by \ref o2scl::inte_gen_comp, the sole descendant of :ref:inte_gen. The user is allowed to specify a upper and lower limits which are functions of the variables for integrations which have not yet been performed, i.e. the n-dimensional integral \f[ \int_{x_0=a_0}^{x_0=b_0} f(x_0) \int_{x_1=a_1(x_0)}^{x_1=b_1(x_0)} f(x_0, x_1) ... \int_{x_{\mathrm{n}-1}=a_{\mathrm{n}-1}(x_0,x_1,..,x_{\mathrm{n}-2})}^ {x_{\mathrm{n}-1}=b_{\mathrm{n}-1}(x_0,x_1,..,x_{\mathrm{n}-2})} f(x_0,x_1,...,x_{\mathrm{n-1}})~d x_{\mathrm{n}-1}~...~d x_1~d x_0 \f] Again, one specifies a set of inte objects to apply to each variable to be integrated over. One-dimensional integration example ----------------------------------- This example computes the integral :math:`\int_{-\infty}^{\infty} e^{-x^2} ~dx` with :ref:`inte_qagi_gsl `, the integral :math:`\int_0^{\infty} e^{-x^2} ~dx` with :ref:`inte_qagiu_gsl `, the integral :math:`\int_{-\infty}^{0} e^{-x^2} ~dx` with :ref:`inte_qagil_gsl `, and the integral :math:`\int_0^1 \left[ \sin (2 x) + \frac{1}{2} \right]~dx` with both :ref:`inte_qag_gsl ` and :ref:`inte_adapt_cern_tl `, and compares the computed results with the exact results. .. literalinclude:: ../../../examples/ex_inte.cpp :language: c++ :start-after: sphinx-example-start .. Multi-dimensional integration example This example computes the integral :math:`\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \sqrt{x^3+y^3+z^3+x y^2 z}~dx~dy~dz` with \ref o2scl::inte_multi_comp . \dontinclude ex_minte.cpp \skip Example: \until End of example Gauss-Kronrod integration coefficients -------------------------------------- :ref:`Top ` .. doxygennamespace:: o2scl_inte_gk_coeffs Non-adaptive quadrature integration coefficients ------------------------------------------------ :ref:`Top ` .. doxygennamespace:: o2scl_inte_qng_coeffs