Module interpolate

source code

Functions

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cubic_int(a, b, fa, fb, ga, gb)
Cubic interpolation using f(a), f(b), g(a), and g(b).

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cubic_ext(a, b, fa, fb, ga, gb, full_output=0)
Cubic Extrapolation using f(a), f(b), g(a), and g(b).

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quadratic_fafbga(a, b, fa, fb, ga)
Quadratic interpolation using f(a), f(b), and g(a).

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quadratic_gagb(a, b, ga, gb)
Quadratic interpolation using g(a) and g(b).

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Imports: sqrt

Function Details

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cubic_int(a, b, fa, fb, ga, gb)

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Cubic interpolation using f(a), f(b), g(a), and g(b).

Equations
~~~~~~~~~

f(a) = a'a**3 + b'a**2 + c'a + d'
f(b) = a'b**3 + b'b**2 + c'b + d'
g(a) = 3a'a**2 + 2b'a + c'
g(b) = 3a'b**2 + 2b'b + c'


Interpolation
~~~~~~~~~~~~~

The extrema are the roots of the quadratic equation:

    3a'*alpha**2 + 2b'*alpha + c' = 0

The cubic interpolant is given by the formula:

                       g(b) + beta2 - beta1
    ac = b - (b - a) . ---------------------
                       g(b) - g(a) + 2*beta2

where:
                              f(a) - f(b)
    beta1 = g(a) + g(b) - 3 . -----------
                                 a - b

    if a < b:
        beta2 = sqrt(beta1**2 - g(a).g(b))
    else:
        beta2 = -sqrt(beta1**2 - g(a).g(b))

cubic_ext(a, b, fa, fb, ga, gb, full_output=0)

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Cubic Extrapolation using f(a), f(b), g(a), and g(b).

Extrapolation
~~~~~~~~~~~~~

The extrema are the roots of the quadratic equation:

    3a'*alpha**2 + 2b'*alpha + c' = 0

The cubic extrapolant is given by the formula:

                       g(b) + beta2 - beta1
    ac = b - (b - a) . ---------------------
                       g(b) - g(a) + 2*beta2

where:

                              f(a) - f(b)
    beta1 = g(a) + g(b) - 3 . -----------
                                 a - b

    if a < b:
        beta2 = sqrt(max(0.0, beta1**2 - g(a).g(b)))
    else:
        beta2 = -sqrt(max(0.0, beta1**2 - g(a).g(b)))

quadratic_fafbga(a, b, fa, fb, ga)

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Quadratic interpolation using f(a), f(b), and g(a).

The extremum of the quadratic is given by:

                 1             g(a)
    aq  =  a  +  - . -------------------------
                 2   f(a) - f(b) - (a - b)g(a)

quadratic_gagb(a, b, ga, gb)

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Quadratic interpolation using g(a) and g(b).

The extremum of the quadratic is given by:

           bg(a) - ag(b)
    aq  =  -------------
            g(a) - g(b)