sicmutils.series — sicmutils 0.19.0

This namespace contains an implementation of two data types:

• Series, which represents a generic infinite series of arbitrary values, and
• PowerSeries, a series that represents a power series in a single variable; in other words, a series where the nth entry is interpreted as the coefficient of $x^n$:

$$[a b c d ...] == $a + bx + cx^2 + dx^3 + ...$$

Many of the functions below draw on the [[sicmutils.series.impl]] namespace, which implements many of these operations on bare Clojure sequences.

The implementation follows Doug McIlroy's beautiful paper, "Power Series, Power Serious".

Doug also has a 10-line version in Haskell on his website.

->function clj/s

(->function s)

Accepts a Series or PowerSeries and coerces the input to a PowerSeries without any application. Returns the coerced PowerSeries instance.

Supplying a non-series will throw.

acos-series clj/s

acot-series clj/s

arg-scale clj/s

(arg-scale s factors)

Given a univariate PowerSeries and a singleton sequence of factors, returns a new PowerSeries that scales its argument by (first factor) on application.

Given a Series, recursively applies arg-scale to each element, making this ONLY appropriate in its current form for a Series of PowerSeries instances.

arg-shift clj/s

(arg-shift s shifts)

Given a univariate PowerSeries and a singleton sequence of shifts, returns a function that, when applied, returns a value equivalent to calling the original s with its argument shifted by (first shifts).

NOTE: arg-shift can't return a PowerSeries instance because the implementation of compose does not currently allow a constant element in the right-hand series.

Given a Series, recursively applies arg-shift to each element, making this ONLY appropriate in its current form for a Series of PowerSeries instances. Returns a Series of functions.

asin-series clj/s

asinh-series clj/s

atan-series clj/s

atanh-series clj/s

binomial-series clj/s

(binomial-series alpha)

Returns a PowerSeries instance representing a Binomial series, ie, the taylor series of the function $f$ given by

$$f(x) = (1 + x)^\alpha$$

catalan-series clj/s

compose clj/s

(compose s t)

Returns a new PowerSeries $U$ that represents the composition of the two input power series $S$ and $T$, where $U$ evaluates like:

$$U(x) = S(T(x))$$

constant clj/s

(constant c)

(constant c kind)

Returns a PowerSeries representing the supplied constant term.

Optionally, pass kind of either ::series or ::power-series to specify the type of series returned.

cos-series clj/s

cosh-series clj/s

exp-series clj/s

fib-series clj/s

fmap clj/s

(fmap f s)

Returns a new series generated by applying the supplied f to each element in the input series s. The returned series will be the same type as the input series, either Series or PowerSeries.

NOTE scmutils calls this series:elementwise.

generate clj/s

(generate f)

(generate f kind)

Returns a PowerSeries generated by (f i) for i in 0, 1, ...

Optionally, pass kind of either ::series or ::power-series to specify the type of series returned.

identity clj/s

PowerSeries instance representing the identity function.

inflate clj/s

(inflate s n)

Accepts an input series s and an exponent n, and expands the series in the nth power of its argument. Every term i maps to position i*n, with zeros padded in the new missing slots.

For example:

(inflate identity 3)
;; => (series 0 0 0 1)

(take 6 (inflate (generate inc) 3))
;; => (1 0 2 0 3 0)

NOTE this operation makes sense as described for a PowerSeries, where each entry represents the coefficient of some power of x; functionally it still works with Series objects.

integral clj/s

(integral s)

(integral s constant)

Returns a PowerSeries $U$ that represents the definite integral of the input power series $S$ with constant term $c$:

$$U = c + \int_0^{\infty} S$$

log1+x-series clj/s

log1-x-series clj/s

one clj/s

PowerSeries instance representing the constant 1.

partial-sums clj/s

(partial-sums s)

Returns a series (of the same type as the input) of partial sums of the terms in the supplied series s.

power-series clj/s

(power-series & prefix)

Return a PowerSeries starting with the supplied values. The remainder of the series will be filled with the zero-value corresponding to the first of the given values.

If you have a sequence already, prefer power-series*.

power-series* clj/s

(power-series* prefix)

Given a sequence, returns a new PowerSeries object that wraps that sequence (potentially padding its tail with zeros if it's finite).

power-series? clj/s

(power-series? s)

Returns true if s is specifically a PowerSeries, false otherwise.

PowerSeries cljs

revert clj/s

(revert s)

Returns a new PowerSeries $U$ that represents the compositional inverse (the 'reversion') of the input power series $S$, satisfying:

$$S(U(x)) = x$$

sec-series clj/s

series clj/s

(series & prefix)

Return a Series starting with the supplied values. The remainder of the series will be filled with the zero-value corresponding to the first of the given values.

If you have a sequence already, prefer series*.

Series cljs

series* clj/s

(series* prefix)

Given a sequence, returns a new Series object that wraps that sequence (potentially padding its tail with zeros if it's finite).

series? clj/s

(series? s)

Returns true if s is either a Series or a PowerSeries, false otherwise.

sin-series clj/s

sinh-series clj/s

sum clj/s

(sum s n)

Returns the sum of all elements in the input series s up to order n (inclusive). For example:

(sum (series 1 1 1 1 1 1 1) 3)
;; => 4

NOTE that sum sums the first n + 1 terms, since a series starts with an order 0 term.

tan-series clj/s

tanh-series clj/s

value clj/s

(value s xs)

Returns the value of the supplied Series or PowerSeries applied to xs.

If a PowerSeries is supplied, xs (despite its name) must be a single value. Returns a Series generated by multiplying each ith term in s by $x^i$, where $x$ is the xs argument.

If a Series s is supplied:

Assumes that s is a series of applicables of arity equal to the count of xs. If, in fact, s is a series of series-valued applicables, then the result will be a sort of layered sum of the values.

Concretely, suppose that s has the form:

[x => [A1 A2 A3...], x => [B1 B2 B3...], x => [C1 C2 C3...], ...]

Then, this series applied to x will yield the new series:

[A1 (+ A2 B1) (+ A3 B2 C1) ...]

The way to think about this is, that if a power series has some other series as the coefficient of the $x^n$ term, the series must shift by $n$ positions before being added into the final total.

xpow clj/s

(xpow n)

Returns a PowerSeries instance representing $x^n$.

zero clj/s

PowerSeries instance representing the constant 0.