API Reference¶

kronrod provides a Python interface to the Gauss-Kronrod quadrature C library (Piessens & Branders, 1974; C translation by Burkardt) for computing abscissas and weights of Gauss-Kronrod rules. The numerical core is implemented in C and compiled via f2py, providing near-native performance with a clean Python interface. See the Theory and Quickstart for mathematical background and usage examples.

Main Features¶

Gauss-Kronrod rule computation: kronrod
Interval adjustment: kronrod_adjust
Embedded Gauss points reused in Kronrod extension
Configurable accuracy parameter eps

Function Summary¶

Function	Category	Computes
`kronrod(n, eps)`	Rule computation	Abscissas and weights for 2N+1 point Gauss-Kronrod rule
`kronrod_adjust(a, b, n, x, w1, w2)`	Interval mapping	Adjust rule from \([-1, +1]\) to \([a, b]\)

`kronrod(n, eps=1e-6)`¶

Add N+1 Kronrod points to an N-point Gaussian rule.

Compute the abscissas and weights of the 2N+1 point Gauss-Kronrod quadrature formula for integration over [-1, +1]. Only the non-negative abscissas are returned (the rule is symmetric).

Parameter	Type	Description
`n`	`int`	Order of the Gauss rule
`eps`	`float`	Requested absolute accuracy of the abscissas (default 1e-6)
return	`tuple[ndarray, ndarray, ndarray]`	`(x, w1, w2)` — abscissas, Gauss-Kronrod weights, Gauss weights

import kronrod

x, w1, w2 = kronrod.kronrod(3)
print(x)  # [0.96049127 0.77459667 0.43424375 0.        ]
print(w1)  # [0.10465623 0.26848809 0.40139741 0.45091654]
print(w2)  # [0.         0.55555556 0.         0.88888889]

`kronrod_adjust(a, b, n, x, w1, w2)`¶

Adjust a Gauss-Kronrod rule from [-1, +1] to [a, b].

Linearly maps the abscissas and scales the weights so that the rule integrates over the interval [a, b] instead of [-1, +1].

Parameter	Type	Description
`a`	`float`	Left endpoint of the new interval
`b`	`float`	Right endpoint of the new interval
`n`	`int`	Order of the Gauss rule
`x`	`ndarray`	Abscissas from `kronrod()`
`w1`	`ndarray`	Gauss-Kronrod weights from `kronrod()`
`w2`	`ndarray`	Gauss weights from `kronrod()`
return	`tuple[ndarray, ndarray, ndarray]`	`(x, w1, w2)` — adjusted abscissas and weights

import kronrod

x, w1, w2 = kronrod.kronrod(3)
x_adj, w1_adj, w2_adj = kronrod.kronrod_adjust(0.0, 1.0, 3, x, w1, w2)
print(x_adj)  # adjusted abscissas in [0, 1]

List of C routines¶

Routine	Description
`abwe1`	Calculate a Kronrod abscissa and weight
`abwe2`	Calculate a Gaussian abscissa and two weights
`kronrod`	Add N+1 points to an N-point Gaussian rule
`r8_abs`	Return the absolute value of a double
`timestamp`	Print the current YMDHMS date as a time stamp