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Numerical Integration

Integration functions
val trapz : ?n:int -> ?eps:float -> (float -> float) -> float -> float -> float

``trapz f a b`` computes the integral of ``f`` on the interval ``a,b`` using the trapezoidal rule, i.e. :math:`\int_a^b f(x) dx`.

Parameters: * ``f``: function to be integrated. * ``n``: the maximum allowed number of steps. The default value is ``20``. * ``eps``: the desired fractional accuracy. The default value is ``1e-6``. * ``a``: lower bound of the integrated interval. * ``b``: upper bound of the integrated interval.

Returns: * ``y``: the integral of ``f`` on ``a, b``.

val simpson : ?n:int -> ?eps:float -> (float -> float) -> float -> float -> float

``simpson f a b`` computes the integral of ``f`` on the interval ``a,b`` using the Simpson's rule, i.e. :math:`\int_a^b f(x) dx`.

Parameters: * ``f``: function to be integrated. * ``n``: the maximum allowed number of steps. The default value is ``20``. * ``eps``: the desired fractional accuracy. The default value is ``1e-6``. * ``a``: lower bound of the integrated interval. * ``b``: upper bound of the integrated interval.

Returns: * ``y``: the integral of ``f`` on ``a, b``.

val romberg : ?n:int -> ?eps:float -> (float -> float) -> float -> float -> float

``romberg f a b`` computes the integral of ``f`` on the interval ``a,b`` using the Romberg method, i.e. :math:`\int_a^b f(x) dx`. Note that this algorithm is much faster than ``trapz`` and ``simpson``.

Parameters: * ``f``: function to be integrated. * ``n``: the maximum allowed number of steps. The default value is ``20``. * ``eps``: the desired fractional accuracy. The default value is ``1e-6``. * ``a``: lower bound of the integrated interval. * ``b``: upper bound of the integrated interval.

Returns: * ``y``: the integral of ``f`` on ``a, b``.

val gaussian_fixed : ?n:int -> (float -> float) -> float -> float -> float

``gaussian_fixed f a b`` computes the integral of ``f`` on the interval ``a,b`` using the Gaussian quadrature of fixed order. Note that this algorithm is much faster than others due to cached weights.

Parameters: * ``f``: function to be integrated. * ``n``: the order of polynomial. The default value is ``10``. * ``a``: lower bound of the integrated interval. * ``b``: upper bound of the integrated interval.

Returns: * ``y``: the integral of ``f`` on ``a, b``.

val gaussian : ?n:int -> ?eps:float -> (float -> float) -> float -> float -> float

``gaussian f a b`` computes the integral of ``f`` on the interval ``a,b`` using adaptive Gaussian quadrature of fixed tolerance.

Parameters: * ``f``: function to be integrated. * ``n``: the maximum order. The default value is ``50``. * ``eps``: the desired fractional accuracy. The default value is ``1e-6``. * ``a``: lower bound of the integrated interval. * ``b``: upper bound of the integrated interval.

Returns: * ``y``: the integral of ``f`` on ``a, b``.

Helper functions
val trapzd : (float -> float) -> float -> float -> int -> float

The function computes the nth stage of refinement of an extended trapezoidal rule. It is the workhorse of several integration functions including ``trapz``, ``simpson``, and ``romberg``.

Parameters: * ``f``: function to be integrated. * ``a``: lower bound of the integrated interval. * ``b``: upper bound of the integrated interval. * ``n``: the nth stage.

Returns: * ``y``: the integral of ``f`` on ``a, b``.

val gauss_legendre : ?eps:float -> ?a:float -> ?b:float -> int -> float array * float array

Given the lower and upper limits of integration ``a`` and ``b``, and order ``n``, the function computes the abscissas and weights of the Gauss-Legendre n-point quadrature formula.

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