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Github: {Signature} {Implementation}

## 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. $$\int_a^b f(x) dx$$.

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].

source code

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. $$\int_a^b f(x) dx$$.

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].

source code

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. $$\int_a^b f(x) dx$$. 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].

source code

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].

source code

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].

source code

## 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].

source code

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.

source code