Package PyDSTool :: Module common :: Class KroghInterpolator
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Class KroghInterpolator

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object --+
         |
        KroghInterpolator

The interpolating polynomial for a set of points

Constructs a polynomial that passes through a given set of points, optionally with specified derivatives at those points. Allows evaluation of the polynomial and all its derivatives. For reasons of numerical stability, this function does not compute the coefficients of the polynomial, although they can be obtained by evaluating all the derivatives.

Be aware that the algorithms implemented here are not necessarily the most numerically stable known. Moreover, even in a world of exact computation, unless the x coordinates are chosen very carefully - Chebyshev zeros (e.g. cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon. In general, even with well-chosen x values, degrees higher than about thirty cause problems with numerical instability in this code.

Based on Krogh 1970, "Efficient Algorithms for Polynomial Interpolation and Numerical Differentiation"

Instance Methods [hide private]
 
__init__(self, xi, yi)
Construct an interpolator passing through the specified points
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__call__(self, x)
Evaluate the polynomial at the point x
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derivatives(self, x, der=None)
Evaluate many derivatives of the polynomial at the point x
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derivative(self, x, der)
Evaluate one derivative of the polynomial at the point x
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Inherited from object: __delattr__, __getattribute__, __hash__, __new__, __reduce__, __reduce_ex__, __repr__, __setattr__, __str__

Properties [hide private]

Inherited from object: __class__

Method Details [hide private]

__init__(self, xi, yi)
(Constructor)

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Construct an interpolator passing through the specified points

The polynomial passes through all the pairs (xi,yi). One may additionally
specify a number of derivatives at each point xi; this is done by
repeating the value xi and specifying the derivatives as successive
yi values.

Parameters
----------
xi : array-like, length N
    known x-coordinates
yi : array-like, N by R
    known y-coordinates, interpreted as vectors of length R,
    or scalars if R=1

Example
-------
To produce a polynomial that is zero at 0 and 1 and has
derivative 2 at 0, call

>>> KroghInterpolator([0,0,1],[0,2,0])

Overrides: object.__init__

__call__(self, x)
(Call operator)

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Evaluate the polynomial at the point x

Parameters
----------
x : scalar or array-like of length N

Returns
-------
y : scalar, array of length R, array of length N, or array of length N by R
    If x is a scalar, returns either a vector or a scalar depending on
    whether the interpolator is vector-valued or scalar-valued.
    If x is a vector, returns a vector of values.

derivatives(self, x, der=None)

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Evaluate many derivatives of the polynomial at the point x

Produce an array of all derivative values at the point x.

Parameters
----------
x : scalar or array-like of length N
    Point or points at which to evaluate the derivatives
der : None or integer
    How many derivatives to extract; None for all potentially
    nonzero derivatives (that is a number equal to the number
    of points). This number includes the function value as 0th
    derivative.
Returns
-------
d : array
    If the interpolator's values are R-dimensional then the
    returned array will be der by N by R. If x is a scalar,
    the middle dimension will be dropped; if R is 1 then the
    last dimension will be dropped.

Example
-------
>>> KroghInterpolator([0,0,0],[1,2,3]).derivatives(0)
array([1.0,2.0,3.0])
>>> KroghInterpolator([0,0,0],[1,2,3]).derivatives([0,0])
array([[1.0,1.0],
       [2.0,2.0],
       [3.0,3.0]])

derivative(self, x, der)

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Evaluate one derivative of the polynomial at the point x

Parameters
----------
x : scalar or array-like of length N
    Point or points at which to evaluate the derivatives
der : None or integer
    Which derivative to extract. This number includes the
    function value as 0th derivative.
Returns
-------
d : array
    If the interpolator's values are R-dimensional then the
    returned array will be N by R. If x is a scalar,
    the middle dimension will be dropped; if R is 1 then the
    last dimension will be dropped.

Notes
-----
This is computed by evaluating all derivatives up to the desired
one and then discarding the rest.