emmy.polynomial.richardson — org.mentat/emmy 0.31.0

Richardson interpolation is a special case of polynomial interpolation; knowing the ratios of successive x coordinates in the point sequence allows a more efficient calculation.

richardson-column clj/s

(richardson-column xs col t)

(richardson-column xs col t p-seq)

(richardson-column xs col t p q)

Function with an identical interface to richardson-sequence, except for an additional second argument col.

richardson-column will return that column offset the interpolation tableau instead of the first row. This will give you a sequence of nth-order Richardson accelerations taken between point i and the next n points.

As a reminder, this is the shape of the Richardson tableau:

p0 p01 p012 p0123 p01234
p1 p12 p123 p1234 .
p2 p23 p234 .     .
p3 p34 .    .     .
p4 .   .    .     .

So supplying a column of 1 gives a single acceleration by combining points from column 0; 2 kills two terms from the error sequence, etc.

NOTE Given a better interface for richardson-sequence this function could be merged with that function.

richardson-fold clj/s

(richardson-fold t)

(richardson-fold t initial-p next-p-fn)

Returns a fold expected to process the outputs of some function A for inputs of the form:

$$A(h), A(h/t), A(h/t^2) \ldots$$

and generate (when present is called) successively tighter estimates of A(0) using the algorithm described in richardson-sequence.

Takes as a required argument:

• t: the ratio between the successive inputs that generated the data to be processed by this fold (see above)

If initial-p and next-p-fn are not supplied, it's assumed that the order of the error terms in the taylor series expansion of A start at 1 and increase by 1 with each new term.

You can tune this by supplying:

• initial-p: The order of the first error term
• next-p-fn: a function that will generate the next term given the previous term

For the geometrically increasing error series [2, 4, 6, 8], for example, try

(richardson-fold <t> 2 #(+ % 2))

richardson-scan clj/s

(richardson-scan t & opts)

Returns a function that consumes an entire sequence xs of points of the form A(h), A(h/t), A(h/t^2),... (where t is the t argument supplied here) and returns a lazy sequence of successive approximations A(0) using the algorithm described in richardson-sequence.

Equivalent to ([[richardson-sequence]] t).

See richardson-fold for all supported arities; all arguments are passed through to richardson-fold.

richardson-sequence clj/s

(richardson-sequence xs t)

(richardson-sequence xs t p-sequence)

(richardson-sequence xs t p q)

Takes:

• xs: a (potentially lazy) sequence of points representing function values generated by inputs continually decreasing by a factor of t. For example: [f(x), f(x/t), f(x/t^2), ...]
• t: the ratio between successive inputs that generated xs.

And returns a new (lazy) sequence of 'accelerated' using Richardson extrapolation to cancel out error terms in the taylor series expansion of f(x) around the value the series to which the series is trying to converge.

Each term in the returned sequence cancels one of the error terms through a linear combination of neighboring terms in the sequence.

Custom P Sequence

The three-arity version takes one more argument:

• p-sequence: the orders of the error terms in the taylor series expansion of the function that xs is estimating. For example, if xs is generated from some f(x) trying to approximate A, then [p_1, p_2...] etc are the correction terms:

$$f(x) = A + B x^{p_1} + C x^{p_2}...$$

The two-arity version uses a default p-sequence of [1, 2, 3, ...]

Arithmetic Progression

The FOUR arity version takes xs and t as before, but instead of p-sequence makes the assumption that p-sequence is an arithmetic progression of the form p + iq, customized by:

• p: the exponent on the highest-order error term
• q: the step size on the error term exponent for each new seq element

Notes

Richardson extrapolation is a special case of polynomial extrapolation, implemented in polynomial.cljc.

Instead of a sequence of xs, if you generate an explicit series of points of the form [x (f x)] with successively smaller x values and polynomial-extrapolate it forward to x == 0 (with, say, (polynomial/modified-neville xs 0)) you'll get the exact same result.

Richardson extrapolation is more efficient since it can make assumptions about the spacing between points and pre-calculate a few quantities. See the namespace for more discussion.

References:

• Wikipedia, "Richardson Extrapolation"
• GJS, 'Abstraction in Numerical Methods'

richardson-sum clj/s

(richardson-sum t & opts)

Returns a function that consumes an entire sequence xs of points of the form A(h), A(h/t), A(h/t^2),... (where t is the t argument supplied here) and returns the best approximation of A(0) using the algorithm described in richardson-sequence.

Equivalent to (last ([[richardson-sequence]] t))

See richardson-fold for all supported arities; all arguments are passed through to richardson-fold.