Class inte_qawf_gsl_sin (o2scl)

O2scl : Class List

template<class func_t>
class inte_qawf_gsl_sin : public o2scl::inte_qawo_gsl_sin<func_t>

Adaptive integration for oscillatory integrals (GSL)

The Fourier integral

\[ \int_a^{\infty} f(x) \sin(\omega x)~dx \]

is computed for some frequency parameter \( \omega \), stored in inte_qawo_gsl_sin::omega .

The integral is computed using the same method as inte_qawo_gsl_sin and inte_qawo_gsl_cos over each of the subintervals,

\[\begin{split}\begin{eqnarray*} C_1 &=& [a, a+c] \\ C_2 &=& [a+c, a+2c] \\ &\vdots & \\ C_k &=& [a +(k-1)c,\, a+kc], \end{eqnarray*}\end{split}\]

where \( c = (2\mathrm{floor}(|\omega|)+1)\pi/|\omega|\). This width is chosen to cover an odd number of periods so that the contributions from the intervals alternate in sign and are monotonically decreasing when \( f \) is positive and monotonically decreasing. The sum of this sequence of contributions is accelerated using the \( \varepsilon \) algorithm.

The algorithm uses zero for the relative tolerance inte::tol_rel and attempts to compute the integral to an overall absolute tolerance set by inte::tol_abs. The following strategy is used: on each interval \( C_k\), the algorithm tries to achieve the tolerance

\[ \mathrm{TOL}_k = u_k\cdot \epsilon_{\mathrm{abs}} \]

where \( u_k = (1-p)p^{k-1} \) and \( p = 0.9\). The sum of the geometric series of contributions from each interval gives an overall tolerance of \( \epsilon_{\mathrm{abs}}\). If the integration of a subinterval leads to difficulties then the accu racy requirement for subsequent intervals is relaxed,

\[ \mathrm{TOL}_k = u_k\cdot \max\{\epsilon_{\mathrm{abs}}, E_1, \ldots, E_{k-1} \} \]

where \( E_k\) is the estimated error on the interval \( C_k\).

See GSL-based integration details in the User’s guide for general information about the GSL integration classes.

When verbose output is enabled, this class outputs information from both the subintegrations performed by inte_qawo_gsl_sin and the overall integration progress in this class.

Todo

Class inte_qawf_gsl_sin: More documentation and examples for the qawf, qawo and qawc integrators.

Subclassed by o2scl::inte_qawf_gsl_cos< func_t >

Public Functions

inline inte_qawf_gsl_sin()

inline virtual ~inte_qawf_gsl_sin()

inline virtual int integ_err(func_t &func, double a, double b, double &res, double &err)

Integrate function func from a to b and place the result in res and the error in err.

Protected Functions

inline int qawf(func_t &func, const double a, const double epsabs, double *result, double *abserr)

The full GSL integration routine called by integ_err()

inline virtual double transform(double t, func_t &func)

Add the oscillating part to the integrand.

inline virtual const char *type()

Return string denoting type (“inte_qawf_gsl_sin”)

Protected Attributes

inte_workspace_gsl *cyclew

The integration workspace.