=begin = Numerical Integration This section describes how to compute numerical integration of a function in one dimension. In Ruby/GSL, all the GSL routines for numerical integration is provided as methods of (()) objects. For example, a (({GSL::Function})) object which represents the sine function ((|sin(x)|)) can be expressed as f = GSL::Function.alloc { |x| sin(x) } To compute numerical integration of ((|sin(x)|)) over the range ((|(a, b)|)), one can use the methods ((|integrate_xxx|)) or simply ((|xxx|)), as * (({f.integrate_xxx([a, b])})), or (({f.xxx([a, b])})) * (({f.integrate_xxx(a, b)})), or (({f.xxx(a, b)})) == QNG non-adaptive Gauss-Kronrod integration --- GSL::Function#integration_qng([a, b], [epsabs = 0.0, epsrel = 1e-10]) --- GSL::Function#qng(...) --- GSL::Integration.qng(...) These methods apply the Gauss-Kronrod integration rules in succession until an estimate of the integral of the reciever function (a (({GSL::Function})) object) over ((|(a,b)|)) is achieved within the desired absolute and relative error limits, ((|epsabs|)) and ((|epsrel|)) (these are optional, the default values are 0,0 and 1e-10 respectively). These methods return an array of four elements (({[result, err, neval, status]})), those are the final approximation of the integration, an estimate of the absolute error, the number of function evaluation, and the status which is returned by the GSL (({integration_qng()})) function. * Ex: Integrate sin(x) over ((|x = 0 -- 2|)) with accuracies ((|epsabs = 0, epsrel = 1.0e-7|)). require 'gsl' f = GSL::Function.alloc { |x| sin(x) } ans = f.integration_qng([0, 2], [0, 1.0e-7]) # or shortly f.qng(...) p ans[0] <- result For all the methods described in this section, the arguments ((|[epsabs, epsrel]|)) are optional, and the default values are ((|[epsabs = 0.0, epsrel = 1e-10]|)). == QAG algorithm and the GSL::Integ::Workspace class The QAG algorithm is a simple adaptive integration procedure. The integration region is divided into subintervals, and on each iteration the subinterval with the largest estimated error is bisected. This reduces the overall error rapidly, as the subintervals become concentrated around local difficulties in the integrand. These subintervals are managed by a GSL::Integ::Workspace object, which handles the memory for the subinterval ranges, results and error estimates. --- GSL::Integ::Workspace.alloc(n = 1000) This creates a workspace sufficient to hold n double precision intervals, their integration results and error estimates. --- GSL::Integ::Workspace#limit --- GSL::Integ::Workspace#size === Algorithms which require the workspace The algorithms described below require (({gsl_integration_workspace})) struct in C. In Ruby/GSL, the corresponding methods require a (({GSL::Integ::Workspace})) object in their arguments. But it is also possible to use these methods without workspace arguments: if it is not given, a workspace is created/destroyed internally. Thus method calls are as f = GSL::Function.alloc { |x| Math::sin(x)/x } p f.qag([a, b]) or w = GSL::Integ::Workspace.alloc(limit) p f.qag([a, b], w) Explicit uses of a (({Workspace})) object reduce C function calls for memory allocations of workspace objects. == QAG adaptive integration --- GSL::Function#integration_qag([a, b], key = GSL::Integ::GAUSS61) --- GSL::Function#integration_qag([a, b], key, w) --- GSL::Function#integration_qag([a, b], w) --- GSL::Function#integration_qag([a, b], [epsabs, epsrel], key) --- GSL::Function#integration_qag([a, b], [epsabs, epsrel], key, w) --- GSL::Function#qag(...) --- GSL::Integration.qag(...) These methods apply an integration rule adaptively until an estimate of the integral of the reciever function over ((|(a,b)|)) is achieved within the desired absolute and relative error limits, ((|epsabs|)) and ((|epsrel|)). One can give a (({GSL::Integ::Workspace})) object ((|w|)) with the last argument (option: if not given, the workspace is internally allocated and freed). The method returns an array with four elements (({[result, err, neval, status]})). The integration rule is determined by the value of key, which should be chosen from the following symbolic names, GSL::Integ::GAUSS15 (key = 1) GSL::Integ::GAUSS21 (key = 2) GSL::Integ::GAUSS31 (key = 3) GSL::Integ::GAUSS41 (key = 4) GSL::Integ::GAUSS51 (key = 5) GSL::Integ::GAUSS61 (key = 6) corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod rules. The higher-order rules give better accuracy for smooth functions, while lower-order rules save time when the function contains local difficulties, such as discontinuities. == QAGS adaptive integration with singularities The presence of an integrable singularity in the integration region causes an adaptive routine to concentrate new subintervals around the singularity. As the subintervals decrease in size the successive approximations to the integral converge in a limiting fashion. This approach to the limit can be accelerated using an extrapolation procedure. The QAGS algorithm combines adaptive bisection with the Wynn epsilon-algorithm to speed up the integration of many types of integrable singularities. --- GSL::Function#integration_qags([a, b], [epsabs = 0.0, epsrel = 1e-10], limit) --- GSL::Function#integration_qags([a, b], [epsabs, epsrel], limit, w) --- GSL::Function#integration_qags([a, b], [epsabs, epsrel], w) --- GSL::Function#qags(...) --- GSL::Integration.qags(...) These methods apply the Gauss-Kronrod 21-point integration rule adaptively until an estimate of the integral over ((|(a,b)|)) is achieved within the desired absolute and relative error limits, ((|epsabs|)) and ((|epsrel|)). The results are extrapolated using the epsilon-algorithm, which accelerates the convergence of the integral in the presence of discontinuities and integrable singularities. The maximum number of subintervals is given by ((|limit|)). * ex: proc = Proc.new{ |x, alpha| # integrant log(alpha*x)/sqrt(x) } # create the function, with the parameter alpha = 1.0 f = GSL::Function.alloc(proc, 1.0) p f.integration_qags(0, 1) == QAGP adaptive integration with known singular points --- GSL::Function#integration_qagp(pts, [epsabs = 0.0, epsrel = 1e-10], limit = 1000, w) --- GSL::Function#qagp(...) --- GSL::Integration.qagp(...) These methods apply the adaptive integration algorithm QAGS taking account of the user-supplied locations of singular points. The array ((|pts|)) (a Ruby array or a GSL::Vector object) should contain the endpoints of the integration ranges defined by the integration region a nd locations of the singularities. For example, to integrate over the region ((|(a,b)|)) with break-points at x_1, x_2, x_3 (where a < x_1 < x_2 < x_3 < b) the following ((|pts|)) array should be used pts[0] = a pts[1] = x_1 pts[2] = x_2 pts[3] = x_3 pts[4] = b If you know the locations of the singular points in the integration region then this routine will be faster than QAGS. * ex: f454 = Function.alloc{ |x| x2 = x*x x3 = x2*x x3*log(((x2-1)*(x2-2)).abs) } pts = [0, 1, sqrt(2), 3] # range: [0, 3], singular points: [1, sqrt(2)] p f454.qagp(pts, 0.0, 1e-3) # <---- [52.7408061167272, 0.000175570384826074, 20, 0] # Expect: 61 log(2) + (77/4) log(7) - 27 = 52.7408061167272 == QAGI adaptive integration on infinite intervals --- GSL::Function#integration_qagi([epsabs = 0.0, epsrel = 1e-10], limit = 1000, w) --- GSL::Function#qagi(...) --- GSL::Integration.qagi(...) These methods compute the integral of the function over the infinite interval (-infty,+infty). The integral is mapped onto the interval (0,1] using the transformation x = (1-t)/t. It is then integrated using the QAGS algorithm. The normal 21-point Gauss-Kronrod rule of QAGS is replaced by a 15-point rule, because the transformation can generate an integrable singularity at the origin. In this case a lower-order rule is more efficient. * ex f = Function.alloc{ |x| Math::exp(-x*x) } exact = Math::sqrt(Math::PI) result, = f.qagi puts("exp(-x*x), x = -infty --- +infty") printf("exact = %.18f\n", exact) printf("result = %.18f\n\n", result) --- GSL::Function#integration_qagiu(a, epsabs = 0.0, epsrel = 1e-10, limit = 1000) --- GSL::Function#integration_qagiu(a, epsabs = 0.0, epsrel = 1e-10, w) --- GSL::Function#qagiu(...) --- GSL::Integration.qagiu(...) These methods compute the integral of the function over the semi-infinite interval (a,+infty). --- GSL::Function#integration_qagil(b, epsabs = 0.0, epsrel = 1e-10, limit = 1000) --- GSL::Function#integration_qagil(b, epsabs = 0.0, epsrel = 1e-10, w) --- GSL::Function#integration_qagil(b, [epsabs, epsrel], limit, w) --- GSL::Function#qagil(...) --- GSL::Integration.qagil(...) These methods compute the integral of the function over the semi-infinite interval (-infty,b). == QAWC adaptive integration for Cauchy principal values --- GSL::Function#integration_qawc([a, b], c, [epsabs = 0.0, epsrel = 1e-10], limit. 1000) --- GSL::Function#qawc(...) --- GSL::Function#qawc(...) These methods compute the Cauchy principal value of the integral over ((|(a,b)|)), with a singularity at ((|c|)). The adaptive bisection algorithm of QAG is used, with modifications to ensure that subdivisions do not occur at the singular point ((|x = c|)). When a subinterval contains the point ((|x = c|)) or is close to it then a special 25-point modified Clenshaw-Curtis rule is used to control the singularity. Further away from the singularity the algorithm uses an ordinary 15-point Gauss-Kronrod integration rule. * ex: require 'gsl' f459 = Function.alloc { |x| 1.0/(5.0*x*x*x + 6.0) } p f459.qawc([-1.0, 5.0], 0, [0.0, 1e-3]) # Expect: log(125/631)/18 == QAWS adaptive integration for singular functions The QAWS algorithm is designed for integrands with algebraic-logarithmic singularities at the end-points of an integration region. In order to work efficiently the algorithm requires a precomputed table of Chebyshev moments. --- GSL::Function#integration_qaws([a, b], table, [epsabs = 0.0, epsrel = 1e-10], limit = 1000) --- GSL::Function#integration_qaws(a, b, table, epsabs, epsrel, limit, w) --- GSL::Function#qaws(...) --- GSL::Integration.qaws(...) These methods compute the integral of the function over the interval ((|(a,b)|)) with the singular weight function (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x) The parameters (({[alpha, beta, mu, nu]})) is given by a Ruby array ((|table|)), or by a (({GSL::Integ::QAWS_Table})) object. The adaptive bisection algorithm of QAG is used. When a subinterval contains one of the endpoints then a special 25-point modified Clenshaw-Curtis rule is used to control the singularities. For subintervals which do not include the endpoints an ordinary 15-point Gauss-Kronrod integration rule is used. * ex1: f458 = Function.alloc { |x| if x.zero? val = 0.0 else u = log(x) v = 1.0 + u*u val = 1.0/(v*v) end val } table = [0.0, 0.0, 1, 0] p f458.qaws([0.0, 1.0], table, [0.0, 1e-10]) # Expect: -0.1892752 * ex2: table = Integ::QAWS_Table.alloc(0.0, 0.0, 1, 0) p f458.qaws([0.0, 1.0], table, [0.0, 1e-10]) == QAWO adaptive integration for oscillatory functions The QAWO algorithm is designed for integrands with an oscillatory factor, sin(omega x) or cos(omega x). In order to work efficiently the algorithm requires a table of Chebyshev moments. --- GSL::Function#integration_qawo(a, [epsabs = 0.0, epsrel = 1e-10], limit = 1000, w, table, ) --- GSL::Function#qawo(...) --- GSL::Integration.qawo(...) This method uses an adaptive algorithm to compute the integral over ((|[a,b]|)) with the weight function sin(omega x) or cos(omega x) defined by the table ((|table|)). * ex1: require("gsl") f456 = Function.alloc { |x| if x.zero? val = 0.0 else val = Math::log(x) end val } table = [10.0*Math::PI, 1.0, Integ::SINE, 1000] p f456.qawo(0.0, [0.0, 1e-10], table) * ex2: table = Integ::QAWO_Table.alloc(10.0*Math::PI, 1.0, Integ::SINE, 1000) p f456.qawo(0.0, [0.0, 1e-10], table) == QAWF adaptive integration for Fourier integrals --- GSL::Function#integration_qawf(a, epsabs = 1e-10, limit = 1000, w, wc, table) --- GSL::Function#integration_qawf(a, epsabs = 1e-10, limit = 1000, table) --- GSL::Function#integration_qawf(a, epsabs = 1e-10, table) --- GSL::Function#integration_qawf(a, table = 1000, table) --- GSL::Function#integration_qawf(a, table) --- GSL::Function#qawf(...) --- GSL::Integration.qawf(...) This method attempts to compute a Fourier integral of the function over the semi-infinite interval [a,+infty). I = \int_a^{+infty} dx f(x) sin(omega x) I = \int_a^{+infty} dx f(x) cos(omega x) The parameter omega is taken from the table ((|table|)) (the length ((|L||)) can take any value, since it is overridden by this function to a value appropriate for the fourier integration). * ex: f457 = Function.alloc { |x| if x.zero? val = 0.0 else val = 1.0/Math::sqrt(x) end val } table = [PI/2.0, 1.0, GSL::Integ::COSINE, 1000] p f457.qawf(0.0, 1e-10, table) # 0.999999999927979, Expect 1 In other style: limit = 1000 table = Integ::QAWO_Table.alloc(PI/2.0, 1.0, GSL::Integ::COSINE, 1000) w = Integ::Workspace.alloc # default n is 1000 wc = Integ::Workspace.alloc(limit) p f457.qawf(0.0, table) p f457.qawf(0.0, 1e-10, table) p f457.qawf(0.0, 1e-10, limit, table) p f457.qawf(0.0, limit, table) p f457.qawf(0.0, 1e-10, limit, w, wc, table) p f457.qawf(0.0, w, wc, table) p f457.qawf(0.0, limit, w, wc, table) p f457.qawf(0.0, limit, w, table) # Error p f457.qawf(0.0, limit, wc, table) # Error (()) (()) (()) (()) =end