Welcome to baryrat’s documentation!

A Python package for barycentric rational approximation.

class baryrat.BarycentricRational(z, f, w)

A class representing a rational function in barycentric representation.

Parameters:
  • z (array) – the interpolation nodes
  • f (array) – the values at the interpolation nodes
  • w (array) – the weights

The rational function has the interpolation property r(z_j) = f_j at all nodes where w_j != 0.

__call__(x)

Evaluate rational function at all points of x.

degree(tol=1e-12)

Compute the pair (m,n) of true degrees of the numerator and denominator.

degree_denom(tol=1e-12)

Compute the true degree of the denominator polynomial.

Uses a result from [Berrut, Mittelmann 1997].

degree_numer(tol=1e-12)

Compute the true degree of the numerator polynomial.

Uses a result from [Berrut, Mittelmann 1997].

denominator()

Return a new BarycentricRational which represents the denominator polynomial.

gain()

The gain in a poles-zeros-gain representation of the rational function, or equivalently, the value at infinity.

numerator()

Return a new BarycentricRational which represents the numerator polynomial.

order

The order of the barycentric rational function, that is, the maximum degree that its numerator and denominator may have, or the number of interpolation nodes minus one.

poles(use_mp=False)

Return the poles of the rational function.

If use_mp is True, uses the mpmath package to compute the result using 100-digit precision arithmetic. If an integer is passed, uses that number of digits to compute the result.

polres(use_mp=False)

Return the poles and residues of the rational function.

The use_mp argument has the same meaning as for poles() and is only used during computation of the poles.

reciprocal()

Return a new BarycentricRational which is the reciprocal of this one.

reduce_order()

Return a new BarycentricRational which represents the same rational function as this one, but with minimal possible order.

See (Ionita 2013), PhD thesis.

zeros(use_mp=False)

Return the zeros of the rational function.

If use_mp is True, uses the mpmath package to compute the result using 100-digit precision arithmetic. If an integer is passed, uses that number of digits to compute the result.

baryrat.aaa(Z, F, tol=1e-13, mmax=100, return_errors=False)

Compute a rational approximation of F over the points Z using the AAA algorithm.

Parameters:
  • Z (array) – the sampling points of the function. Unlike for interpolation algorithms, where a small number of nodes is preferred, since the AAA algorithm chooses its support points adaptively, it is better to provide a finer mesh over the support.
  • F – the function to be approximated; can be given as a function or as an array of function values over Z.
  • tol – the approximation tolerance
  • mmax – the maximum number of iterations/degree of the resulting approximant
  • return_errors – if True, also return the history of the errors over all iterations
Returns:

an object which can be called to evaluate the rational function, and can be queried for the poles, residues, and zeros of the function.
Return type:

BarycentricRational

For more information, see the paper

The AAA Algorithm for Rational Approximation
Yuji Nakatsukasa, Olivier Sete, and Lloyd N. Trefethen
SIAM Journal on Scientific Computing 2018 40:3, A1494-A1522
https://doi.org/10.1137/16M1106122

as well as the Chebfun package <http://www.chebfun.org>. This code is an almost direct port of the Chebfun implementation of aaa to Python.

baryrat.brasil(f, interval, deg, tol=0.0001, maxiter=1000, max_step_size=0.1, step_factor=0.1, npi=100, init_steps=100, info=False)

Best Rational Approximation by Successive Interval Length adjustment.

Computes best rational or polynomial approximations in the maximum norm by the BRASIL algorithm (see reference below).

References

https://doi.org/10.1007/s11075-020-01042-0

Parameters:
  • f – the scalar function to be approximated
  • interval – the bounds (a, b) of the approximation interval
  • deg – the degree of the numerator m and denominator n of the rational approximation; either an integer (m=n) or a pair (m, n). If n = 0, a polynomial best approximation is computed.
  • tol – the maximum allowed deviation from equioscillation
  • maxiter – the maximum number of iterations
  • max_step_size – the maximum allowed step size
  • step_factor – factor for adaptive step size choice
  • npi – points per interval for error calculation. If npi < 0, golden section search with -npi iterations is used instead of sampling. For high-accuracy results, npi=-30 is typically a good choice.
  • init_steps – how many steps of the initialization iteration to run
  • info – whether to return an additional object with details
Returns:

the computed rational approximation. If info is True, instead returns a pair containing the approximation and an object with additional information (see below).
Return type:

BarycentricRational

The info object returned along with the approximation if info=True has the following members:

  • converged (bool): whether the method converged to the desired tolerance tol
  • error (float): the maximum error of the approximation
  • deviation (float): the relative error between the smallest and the largest equioscillation peak. The convergence criterion is deviation <= tol.
  • nodes (array): the abscissae of the interpolation nodes (2*deg + 1)
  • iterations (int): the number of iterations used, including the initialization phase
  • errors (array): the history of the maximum error over all iterations
  • deviations (array): the history of the deviation over all iterations
  • stepsizes (array): the history of the adaptive step size over all iterations

Additional information about the resulting rational function, such as poles, residues and zeroes, can be queried from the BarycentricRational object itself.

baryrat.floater_hormann(nodes, values, blending)

Compute the Floater-Hormann rational interpolant for the given nodes and values. See (Floater, Hormann 2007), DOI 10.1007/s00211-007-0093-y.

The blending parameter (usually called d in the literature) is an integer between 0 and n (inclusive), where n+1 is the number of interpolation nodes. For functions with higher smoothness, the blending parameter may be chosen higher. For d=n, the result is the polynomial interpolant.

Returns an instance of BarycentricRational.

baryrat.interpolate_poly(nodes, values)

Compute the interpolating polynomial for the given nodes and values in barycentric form.

baryrat.interpolate_rat(nodes, values)

Compute a rational function which interpolates the given nodes/values.

Parameters:
  • nodes (array) – the interpolation nodes; must have odd length and be passed in strictly increasing or decreasing order
  • values (array) – the values at the interpolation nodes
Returns:

the rational interpolant. If there are 2n + 1 nodes, both the numerator and denominator have degree at most n.
Return type:

BarycentricRational

References

https://doi.org/10.1109/LSP.2007.913583

baryrat.interpolate_with_degree(nodes, values, deg)

Compute a rational function which interpolates the given nodes/values with given degree m of the numerator and n of the denominator.

Parameters:
  • nodes (array) – the interpolation nodes
  • values (array) – the values at the interpolation nodes
  • deg – a pair (m, n) of the degrees of the interpolating rational function. The number of interpolation nodes must be m + n + 1.
Returns:

the rational interpolant
Return type:

BarycentricRational

References

https://doi.org/10.1016/S0377-0427(96)00163-X

baryrat.interpolate_with_poles(nodes, values, poles)

Compute a rational function which interpolates the given values at the given nodes and which has the given poles.

The arrays nodes and values should have length n, and poles should have length n - 1.

Indices and tables