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# Numerical Integration

Contents:

1. Introduction

2. QNG non-adaptive Gauss-Kronrod integration

3. QAG adaptive integration

   1. GSL::Integration::Workspace class

   2. Methods

4. QAGS adaptive integration with singularities

5. QAGP adaptive integration with known singular points

6. QAGI adaptive integration on infinite intervals

7. QAWC adaptive integration for Cauchy principal values

8. QAWS adaptive integration for singular functions

9. QAWO adaptive integration for oscillatory functions

10. QAWF adaptive integration for Fourier integrals

## Introduction

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 GSL::Function 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 adaptive integration

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::Integration::Workspace object, which handles the memory for the subinterval ranges, results and error estimates.

### Workspace class

- GSL::Integration::Workspace.alloc(n = 1000)

  This creates a workspace sufficient to hold n double precision intervals, their integration results and error estimates.

- GSL::Integration::Workspace#limit

- GSL::Integration::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::Integration::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::Integration::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.

### Methods

- GSL::Function#integration_qag([a, b], key = GSL::Integration::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::Integration::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::Integration::GAUSS15  (key = 1)
      GSL::Integration::GAUSS21  (key = 2)
      GSL::Integration::GAUSS31  (key = 3)
      GSL::Integration::GAUSS41  (key = 4)
      GSL::Integration::GAUSS51  (key = 5)
      GSL::Integration::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::Integration::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 = Integration::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, Integration::SINE, 1000]
        p f456.qawo(0.0, [0.0, 1e-10], table)

  - ex2:

        table = Integration::QAWO_Table.alloc(10.0*Math::PI, 1.0, Integration::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::Integration::COSINE, 1000]
        p f457.qawf(0.0, 1e-10, table)     #  0.999999999927979, Expect 1

  In other style:

      limit = 1000
      table = Integration::QAWO_Table.alloc(PI/2.0, 1.0, GSL::Integration::COSINE, 1000)
      w = Integration::Workspace.alloc          # default n is 1000
      wc = Integration::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

## Gauss-Legendre integration

(GSL-1.14) The fixed-order Gauss-Legendre integration routines are provided for fast integration of smooth functions with known polynomial order. The n-point Gauss-Legendre rule is exact for polynomials of order 2*n-1 or less. For example, these rules are useful when integrating basis functions to form mass matrices for the Galerkin method. Unlike other numerical integration routines within the library, these routines do not accept absolute or relative error bounds.

- GSL::Integration::Glfixed_table.alloc(n)

  Determines the Gauss-Legendre abscissae and weights necessary for an `n`-point fixed order integration scheme. If possible, high precision precomputed coefficients are used. If precomputed weights are not available, lower precision coefficients are computed on the fly.

- GSL::Function#glfixed(a, b, t)

  Applies the Gauss-Legendre integration rule contained in table `t` and returns the result.

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