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emmy.polynomial


->expressionclj/s

(->expression p vars)

Accepts a [[Polynomial]] p and a sequence of symbols for each indeterminate, and emits the canonical form of the symbolic expression that represents [[Polynomial]] p.

A similar result could be achieved by calling (apply p vars); but [[Polynomial]] application uses Horner's rule, and form of the returned result will be different.

NOTE: this is the output stage of Flat Polynomial canonical form simplification. The input stage is handled by expression->.

NOTE See analyzer for an instance usable by emmy.expression.analyze/make-analyzer.

Accepts a [[Polynomial]] `p` and a sequence of symbols for each indeterminate,
and emits the canonical form of the symbolic expression that
represents [[Polynomial]] `p`.

A similar result could be achieved by calling `(apply p vars)`;
but [[Polynomial]] application uses [Horner's
rule](https://en.wikipedia.org/wiki/Horner%27s_method), and form of the
returned result will be different.

NOTE: this is the output stage of Flat Polynomial canonical form
simplification. The input stage is handled by [[expression->]].

NOTE See [[analyzer]] for an instance usable
by [[emmy.expression.analyze/make-analyzer]].
sourceraw docstring

->power-seriesclj/s

(->power-series p)

Given a univariate polynomial p, returns a [[series/PowerSeries]] representation of the supplied [[Polynomial]].

Given a [[series/PowerSeries]], acts as identity.

Non-[[Polynomial]] coefficients return [[series/PowerSeries]] instances via series/constant; any multivariate [[Polynomial]] throws an exception.

NOTE use lower-arity to generate a univariate polynomial in the first indeterminate, given a multivariate polynomial.

Given a univariate polynomial `p`, returns a [[series/PowerSeries]]
representation of the supplied [[Polynomial]].

Given a [[series/PowerSeries]], acts as identity.

Non-[[Polynomial]] coefficients return [[series/PowerSeries]] instances
via [[series/constant]]; any multivariate [[Polynomial]] throws an exception.

NOTE use [[lower-arity]] to generate a univariate polynomial in the first
indeterminate, given a multivariate polynomial.
sourceraw docstring

->strclj/s

(->str p)
(->str p n)

Returns a string representation of the supplied [[Polynomial]] instance p.

The optional argument n specifies how many terms to include in the returned string before an ellipsis cuts them off.

Returns a string representation of the supplied [[Polynomial]] instance `p`.

The optional argument `n` specifies how many terms to include in the returned
string before an ellipsis cuts them off.
sourceraw docstring

->termsclj/s

(->terms p)

Given some [[Polynomial]], returns the terms entry of the type. Handles other types as well:

  • Acts as identity on vectors, interpreting them as vectors of terms
  • any zero-valued p returns []
  • any other coefficient returns a vector of a single constant term.
Given some [[Polynomial]], returns the `terms` entry of the type. Handles other types as well:

- Acts as identity on vectors, interpreting them as vectors of terms
- any zero-valued `p` returns `[]`
- any other coefficient returns a vector of a single constant term.
sourceraw docstring

absclj/s

(abs p)

If the leading-coefficient of p is negative, returns (negate p), else acts as identity.

If the [[leading-coefficient]] of `p` is negative, returns `(negate p)`, else
acts as identity.
sourceraw docstring

addclj/s

(add p q)

Returns the sum of polynomials p and q, with appropriate handling for non-[[Polynomial]] coefficient inputs on either or both sides.

Returns the sum of polynomials `p` and `q`, with appropriate handling for
non-[[Polynomial]] coefficient inputs on either or both sides.
sourceraw docstring

analyzerclj/s

Singleton [[emmy.expression.analyze/ICanonicalize]] instance.
sourceraw docstring

arg-scaleclj/s

(arg-scale p factors)

Given some [[Polynomial]] p, returns a new [[Polynomial]] generated by substituting each indeterminate x_i for f_i * x_i, where f_i is a factor supplied in the factors sequence.

When p is a multivariate [[Polynomial]], each factor must be either a non-[[Polynomial]] or a [[Polynomial]] with the same arity as p.

Given some [[Polynomial]] `p`, returns a new [[Polynomial]] generated by
substituting each indeterminate `x_i` for `f_i * x_i`, where `f_i` is a factor
supplied in the `factors` sequence.

When `p` is a multivariate [[Polynomial]], each factor must be either a
non-[[Polynomial]] or a [[Polynomial]] with the same [[arity]] as `p`.
sourceraw docstring

arg-shiftclj/s

(arg-shift p shifts)

Given some [[Polynomial]] p, returns a new [[Polynomial]] generated by substituting each indeterminate x_i for s_i + x_i, where s_i is a shift supplied in the shifts sequence.

When p is a multivariate [[Polynomial]], each shift must be either a non-[[Polynomial]] or a [[Polynomial]] with the same arity as p.

Given some [[Polynomial]] `p`, returns a new [[Polynomial]] generated by
substituting each indeterminate `x_i` for `s_i + x_i`, where `s_i` is a shift
supplied in the `shifts` sequence.

When `p` is a multivariate [[Polynomial]], each shift must be either a
non-[[Polynomial]] or a [[Polynomial]] with the same [[arity]] as `p`.
sourceraw docstring

arityclj/s

(arity p)

Returns the declared arity of the supplied [[Polynomial]], or 0 for non-polynomial arguments.

Returns the declared arity of the supplied [[Polynomial]], or `0` for
non-polynomial arguments.
sourceraw docstring

c*xnclj/s

(c*xn arity c n)

Given some arity, a coefficient c and an exponent n, returns a monomial representing $c{x_0}^n$. The first indeterminate is always exponentiated.

Similar to make, this function attempts to drop down to scalar-land if possible:

  • If c is [[emmy.value/zero?]], returns c
  • if n is zero?, returns (constant arity c)

NOTE that negative exponents are not allowed.

Given some `arity`, a coefficient `c` and an exponent `n`, returns a monomial
representing $c{x_0}^n$. The first indeterminate is always exponentiated.

Similar to [[make]], this function attempts to drop down to scalar-land if
possible:

- If `c` is [[emmy.value/zero?]], returns `c`
- if `n` is `zero?`, returns `(constant arity c)`

NOTE that negative exponents are not allowed.
sourceraw docstring

coeff?clj/s

(coeff? x)

Returns true if the input x is explicitly not an instance of [[Polynomial]], false otherwise.

Equivalent to (not (polynomial? x)).

Returns true if the input `x` is explicitly _not_ an instance
of [[Polynomial]], false otherwise.

Equivalent to `(not (polynomial? x))`.
sourceraw docstring

coefficientsclj/s

(coefficients p)

Returns a sequence of the coefficients of the supplied polynomial p. A coefficient is treated here as a monomial, and returns a sequence of itself.

If p is zero, returns an empty list.

Returns a sequence of the coefficients of the supplied polynomial `p`. A
coefficient is treated here as a monomial, and returns a sequence of itself.

If `p` is zero, returns an empty list.
sourceraw docstring

constantclj/s

(constant c)
(constant arity c)

Given some coefficient c, returns a [[Polynomial]] instance with a single constant term referencing c.

arity defaults to 1; supply it to set the arity of the returned [[Polynomial]].

Given some coefficient `c`, returns a [[Polynomial]] instance with a single
constant term referencing `c`.

`arity` defaults to 1; supply it to set the arity of the
returned [[Polynomial]].
sourceraw docstring

contractclj/s

(contract p n)

If p is contractible? at index n, returns a new [[Polynomial]] instance of arity 1 less than p with all variable indices > n decremented.

For non-[[Polynomial]] inputs, acts as identity. Throws if p is not explicitly contractible?.

For example:

(= (make 2 {[1 2] 3 [3 4] 5})
   (contract
     (make 3 {[0 1 2] 3 [0 3 4] 5}) 0))
If `p` is [[contractible?]] at index `n`, returns a new [[Polynomial]] instance
of [[arity]] `1` less than `p` with all variable indices > `n` decremented.

For non-[[Polynomial]] inputs, acts as identity. Throws if `p` is not
explicitly [[contractible?]].

For example:

```clojure
(= (make 2 {[1 2] 3 [3 4] 5})
   (contract
     (make 3 {[0 1 2] 3 [0 3 4] 5}) 0))
```
sourceraw docstring

contractible?clj/s

(contractible? p n)

Returns true if n is a valid variable index for the [[Polynomial]] p, and the variable with that index has no powers greater than 0 in p, false otherwise.

Returns true if `n` is a valid variable index for the [[Polynomial]] `p`, and
the variable with that index has no powers greater than `0` in `p`, false
otherwise.
sourceraw docstring

cubeclj/s

(cube p)

Returns the cube of polynomial p. Equivalent to (mul p (mul p p)).

Returns the cube of polynomial `p`. Equivalent to `(mul p (mul p p))`.
sourceraw docstring

degreeclj/s

(degree p)
(degree p i)

Returns the degree of the supplied polynomial.

the degree of a polynomial is the highest of the degrees of the polynomial's individual terms with non-zero coefficients. The degree of an individual term is the sum of all exponents in the term.

Optionally, degree takes an indeterminate index i; in this case, degree returns the maximum power found for the ith indeterminate across all terms.

NOTE when passed either a 0 or a zero-polynomial, degree returns -1. See Wikipedia's 'degree of the zero polynomial' for color on why this is the case.

Returns the [degree](https://en.wikipedia.org/wiki/Degree_of_a_polynomial) of
the supplied polynomial.

the degree of a polynomial is the highest of the degrees of the polynomial's
individual terms with non-zero coefficients. The degree of an individual term
is the sum of all exponents in the term.

Optionally, [[degree]] takes an indeterminate index `i`; in this
case, [[degree]] returns the maximum power found for the `i`th indeterminate
across all terms.

NOTE when passed either a `0` or a zero-polynomial, [[degree]] returns -1. See
Wikipedia's ['degree of the zero
polynomial'](https://en.wikipedia.org/wiki/Degree_of_a_polynomial#Degree_of_the_zero_polynomial)
for color on why this is the case.
sourceraw docstring

divideclj/s

(divide u v)

Given two polynomials u and v, returns a pair of polynomials of the form [quotient, remainder] using polynomial long division.

The contract satisfied by this returned pair is that

u == (add (mul quotient v) remainder)
Given two polynomials `u` and `v`, returns a pair of polynomials of the form
`[quotient, remainder]` using [polynomial long
division](https://en.wikipedia.org/wiki/Polynomial_long_division).

The contract satisfied by this returned pair is that

```
u == (add (mul quotient v) remainder)
```
sourceraw docstring

divisible?clj/s

(divisible? n d)

Returns true if the numerator n is evenly divisible by d (i.e., leaves no remainder), false otherwise.

NOTE that this performs a full division with divide. If you're planning on doing this, you may as well call divide and check that the remainder satisfies [[emmy.value/zero?]].

Returns true if the numerator `n` is evenly divisible by `d` (i.e., leaves no
remainder), false otherwise.

NOTE that this performs a full division with [[divide]]. If you're planning on
doing this, you may as well call [[divide]] and check that the remainder
satisfies [[emmy.value/zero?]].
sourceraw docstring

drop-leading-termclj/s

(drop-leading-term p)

Given some [[Polynomial]] p, returns p without its leading-term. non-[[Polynomial]] p inputs are treated at constant polynomials and return 0.

NOTE that drop-leading-term will return a non-[[Polynomial]] if the result of the mapping has only a constant term.

Given some [[Polynomial]] `p`, returns `p` without its [[leading-term]].
non-[[Polynomial]] `p` inputs are treated at constant polynomials and return
`0`.

NOTE that [[drop-leading-term]] will return a non-[[Polynomial]] if the result
of the mapping has only a constant term.
sourceraw docstring

eqclj/s

(eq this that)

Returns true if the [[Polynomial]] this is equal to that. If that is a [[Polynomial]], this and that are equal if they have equal terms and equal arity. Coefficients are compared using emmy.value/=.

If that is non-[[Polynomial]], eq only returns true if this is a monomial and its coefficient is equal to that (again using emmy.value/=).

Returns true if the [[Polynomial]] this is equal to `that`. If `that` is
a [[Polynomial]], `this` and `that` are equal if they have equal terms and
equal arity. Coefficients are compared using [[emmy.value/=]].

If `that` is non-[[Polynomial]], `eq` only returns true if `this` is a
monomial and its coefficient is equal to `that` (again
using [[emmy.value/=]]).
sourceraw docstring

evaluateclj/s

(evaluate p xs)

Returns the result of evaluating a multivariate [[Polynomial]] p at the values in the sequence xs using Horner's rule.

If p is a non-[[Polynomial]] coefficient, acts as identity.

Supplying too many arguments in xs (i.e., a greater number than the arity of p) will throw an exception. Too few arguments will result in a partial evaluation of p, leaving the remaining indeterminates with their variable indices shifted down.

For example:

(= (make [0 0 '(* 3 (expt x 2) y) 0 '(* 5 (expt x 2) (expt y 3))])
   (simplify
     (evaluate
       (make 3 {[2 1 2] 3 [2 3 4] 5}) ['x 'y])))
Returns the result of evaluating a multivariate [[Polynomial]] `p` at the
values in the sequence `xs` using [Horner's
rule](https://en.wikipedia.org/wiki/Horner%27s_method).

If `p` is a non-[[Polynomial]] coefficient, acts as identity.

Supplying too many arguments in `xs` (i.e., a greater number than the [[arity]]
of `p`) will throw an exception. Too few arguments will result in a partial
evaluation of `p`, leaving the remaining indeterminates with their variable
indices shifted down.

For example:

```clojure
(= (make [0 0 '(* 3 (expt x 2) y) 0 '(* 5 (expt x 2) (expt y 3))])
   (simplify
     (evaluate
       (make 3 {[2 1 2] 3 [2 3 4] 5}) ['x 'y])))
```
sourceraw docstring

evenly-divideclj/s

(evenly-divide u v)

Returns the result of dividing the polynomial u by v (non-[[Polynomial]] instances are allowed).

Throws an exception if the division leaves a remainder. Else, returns the quotient.

Returns the result of dividing the polynomial `u` by `v` (non-[[Polynomial]]
instances are allowed).

Throws an exception if the division leaves a remainder. Else, returns the
quotient.
sourceraw docstring

expression->clj/s

(expression-> expr)
(expression-> expr cont)
(expression-> expr cont v-compare)

Converts the supplied symbolic expression expr into Flat Polynomial canonical form (i.e., a [[Polynomial]] instance). expr should be a bare, unwrapped expression built out of Clojure data structures.

Returns the result of calling continuation cont with the [[Polynomial]] and the list of variables corresponding to each indeterminate in the [[Polynomial]]. (contdefaults to vector).

The second optional argument v-compare allows you to provide a Comparator between variables. Sorting indeterminates by v-compare will determine the order of the indeterminates in the generated [[Polynomial]]. The list of variables passed to cont will be sorted using v-compare.

Absorbing an expression with expression-> and emitting it again with ->expression will generate the canonical form of an expression, with respect to the operations in the [[operators-known]] set.

This kind of simplification proceeds purely symbolically over the known Flat Polynomial operations; other operations outside the arithmetic available in polynomials over commutative rings should be factored out by an expression analyzer (see emmy.expression.analyze/make-analyzer) before calling expression->.

NOTE See analyzer for an instance usable by emmy.expression.analyze/make-analyzer.

Converts the supplied symbolic expression `expr` into Flat Polynomial canonical
form (i.e., a [[Polynomial]] instance). `expr` should be a bare, unwrapped
expression built out of Clojure data structures.

Returns the result of calling continuation `cont` with the [[Polynomial]] and
the list of variables corresponding to each indeterminate in
the [[Polynomial]]. (`cont `defaults to `vector`).

The second optional argument `v-compare` allows you to provide a Comparator
between variables. Sorting indeterminates by `v-compare` will determine the
order of the indeterminates in the generated [[Polynomial]]. The list of
variables passed to `cont` will be sorted using `v-compare`.

Absorbing an expression with [[expression->]] and emitting it again
with [[->expression]] will generate the canonical form of an expression, with
respect to the operations in the [[operators-known]] set.

This kind of simplification proceeds purely symbolically over the known Flat
Polynomial operations; other operations outside the arithmetic available in
polynomials over commutative rings should be factored out by an expression
analyzer (see [[emmy.expression.analyze/make-analyzer]]) before
calling [[expression->]].

NOTE See [[analyzer]] for an instance usable
by [[emmy.expression.analyze/make-analyzer]].
sourceraw docstring

exptclj/s

(expt p n)

Returns a polynomial generated by raising the input polynomial p to the (integer) power n.

Negative exponents are not supported. For negative polynomial exponentation, see rational-function/expt.

Returns a polynomial generated by raising the input polynomial `p` to
the (integer) power `n`.

Negative exponents are not supported. For negative polynomial exponentation,
see [[rational-function/expt]].
sourceraw docstring

extendclj/s

(extend p n)

Interpolates a new variable into the supplied [[Polynomial]] p at index n by incrementing any existing variable index >= n.

Returns a new [[Polynomial]] of arity 1 greater than the arity of p, or equal to (inc n) if n is greater than the arity of p.

For non-[[Polynomial]] inputs (or negative n), acts as identity.

Interpolates a new variable into the supplied [[Polynomial]] `p` at index `n`
by incrementing any existing variable index >= `n`.

Returns a new [[Polynomial]] of [[arity]] 1 greater than the [[arity]] of `p`,
or equal to `(inc n)` if `n` is greater than the [[arity]] of `p`.

For non-[[Polynomial]] inputs (or negative `n`), acts as identity.
sourceraw docstring

from-pointsclj/s

(from-points xs)

Given a sequence of points of the form [x, f(x)], returns a univariate polynomial that passes through each input point.

The degree of the returned polynomial is equal to (dec (count xs)).

Given a sequence of points of the form `[x, f(x)]`, returns a univariate
polynomial that passes through each input point.

The degree of the returned polynomial is equal to `(dec (count xs))`.
sourceraw docstring

from-power-seriesclj/s

(from-power-series s n-terms)

Returns a univariate polynomial of all terms in the supplied [[series/PowerSeries]] instance, up to (and including) order n-terms.

(g/simplify
  ((from-power-series series/exp-series 3) 'x))
;; => (+ (* 1/6 (expt x 3)) (* 1/2 (expt x 2)) x 1)
Returns a univariate polynomial of all terms in the
supplied [[series/PowerSeries]] instance, up to (and including) order
`n-terms`.

```clojure
(g/simplify
  ((from-power-series series/exp-series 3) 'x))
;; => (+ (* 1/6 (expt x 3)) (* 1/2 (expt x 2)) x 1)
```
sourceraw docstring

horner-with-errorclj/s

(horner-with-error a z)
(horner-with-error a z cont)

Takes a univariate polynomial a, an argument z and a continuation cont (vector by default) and calls the continuation with (SEE BELOW).

This Horner's rule evaluator is restricted to numerical coefficients and univariate polynomials. It returns by calling cont with 4 arguments:

  • the computed value
  • the values of the first two derivatives
  • an estimate of the roundoff error incurred in computing the value

The recurrences used are from Kahan's 18 Nov 1986 paper 'Roundoff in Polynomial Evaluation', generalized for sparse representations and another derivative by GJS.

For $p = A(z)$, $q = A'(z)$, $r = A''(z)$, and $e$ = error in $A(x)$,

$$p_{j+n} = z^n p_j + a_{j+n}$$

$$e_{j+n} = |z|^n ( e_j + (n-1) p_j ) + |p_{j+n}|$$

$$q_{j+n} = z^n q_j + n z^{n-1} p_j$$

$$r_{j+n} = z^n r_j + n z^{n-1} q_j + 1/2 n (n-1) z^{n-2} p_j$$

Takes a univariate polynomial `a`, an argument `z` and a continuation
`cont` (`vector` by default) and calls the continuation with (SEE BELOW).

This Horner's rule evaluator is restricted to numerical coefficients and
univariate polynomials. It returns by calling `cont` with 4 arguments:

- the computed value
- the values of the first two derivatives
- an estimate of the roundoff error incurred in computing the value

The recurrences used are from Kahan's 18 Nov 1986 paper ['Roundoff in
Polynomial
Evaluation'](https://people.eecs.berkeley.edu/~wkahan/Math128/Poly.pdf),
generalized for sparse representations and another derivative by GJS.

For $p = A(z)$, $q = A'(z)$, $r = A''(z)$, and $e$ = error in $A(x)$,

$$p_{j+n} = z^n p_j + a_{j+n}$$

$$e_{j+n} = |z|^n ( e_j + (n-1) p_j ) + |p_{j+n}|$$

$$q_{j+n} = z^n q_j + n z^{n-1} p_j$$

$$r_{j+n} = z^n r_j + n z^{n-1} q_j + 1/2 n (n-1) z^{n-2} p_j$$
sourceraw docstring

identityclj/s

(identity)
(identity arity)
(identity arity i)

Generates a [[Polynomial]] instance representing a single indeterminate with constant 1.

When called with no arguments, returns a monomial of arity 1 that acts as identity in the first indeterminate.

The one-argument version takes an explicit arity, but still sets the identity to the first indeterminate.

The two-argument version takes an explicit i and returns a monomial of arity arity with an exponent of 1 in the ith indeterminate.

Generates a [[Polynomial]] instance representing a single indeterminate with
constant 1.

When called with no arguments, returns a monomial of arity 1 that acts as
identity in the first indeterminate.

The one-argument version takes an explicit `arity`, but still sets the
identity to the first indeterminate.

The two-argument version takes an explicit `i` and returns a monomial of arity
`arity` with an exponent of 1 in the `i`th indeterminate.
sourceraw docstring

leading-base-coefficientclj/s

(leading-base-coefficient p)

Similar to leading-coefficient, but of the coefficient itself is a [[Polynomial]], recurses down until it reaches a non-[[Polynomial]] lead coefficient.

If p is a non-[[Polynomial]] coefficient, acts as identity.

Similar to [[leading-coefficient]], but of the coefficient itself is
a [[Polynomial]], recurses down until it reaches a non-[[Polynomial]] lead
coefficient.

If `p` is a non-[[Polynomial]] coefficient, acts as identity.
sourceraw docstring

leading-coefficientclj/s

(leading-coefficient p)

Returns the coefficient of the leading (highest degree) term of the [[Polynomial]] p.

If p is a non-[[Polynomial]] coefficient, acts as identity.

Returns the coefficient of the leading (highest degree) term of
the [[Polynomial]] `p`.

If `p` is a non-[[Polynomial]] coefficient, acts as identity.
sourceraw docstring

leading-exponentsclj/s

(leading-exponents p)

Returns the exponents of the leading (highest degree) term of the [[Polynomial]] p.

If p is a non-[[Polynomial]] coefficient, returns [[exponent/empty]].

Returns the exponents of the leading (highest degree) term of
the [[Polynomial]] `p`.

If `p` is a non-[[Polynomial]] coefficient, returns [[exponent/empty]].
sourceraw docstring

leading-termclj/s

(leading-term p)

Returns the leading (highest degree) term of the [[Polynomial]] p.

If p is a non-[[Polynomial]] coefficient, returns a term with zero exponents and p as its coefficient.

Returns the leading (highest degree) term of the [[Polynomial]] `p`.

If `p` is a non-[[Polynomial]] coefficient, returns a term with zero exponents
and `p` as its coefficient.
sourceraw docstring

linearclj/s

(linear arity i root)

Given some arity, an indeterminate index i and some constant root, returns a polynomial of the form x_i - root. The returned polynomial represents a linear equation in the ith indeterminate.

If root is 0, linear is equivalent to the two-argument version of identity.

Given some `arity`, an indeterminate index `i` and some constant `root`,
returns a polynomial of the form `x_i - root`. The returned polynomial
represents a linear equation in the `i`th indeterminate.

If `root` is 0, [[linear]] is equivalent to the two-argument version
of [[identity]].
sourceraw docstring

lower-arityclj/s

(lower-arity p)

Given a multivariate [[Polynomial]] p, returns an equivalent univariate [[Polynomial]] whose coefficients are polynomials of arity equal to one less than the arity of p.

Use raise-arity to undo this transformation. See with-lower-arity for a function that packages these two transformations.

NOTE that lower-arity will drop a coefficient down to a non-[[Polynomial]] if the result of extracting the first variable leaves a constant term.

Given a multivariate [[Polynomial]] `p`, returns an equivalent
univariate [[Polynomial]] whose coefficients are polynomials of [[arity]]
equal to one less than the [[arity]] of `p`.

Use [[raise-arity]] to undo this transformation. See [[with-lower-arity]] for
a function that packages these two transformations.

NOTE that [[lower-arity]] will drop a coefficient down to a non-[[Polynomial]]
if the result of extracting the first variable leaves a constant term.
sourceraw docstring

lowest-degreeclj/s

(lowest-degree p)

Returns the lowest degree found across any term in the supplied [[Polynomial]]. If a non-[[Polynomial]] is supplied, returns either 0 or -1 if the input is itself a 0.

See degree for a discussion of this -1 case.

Returns the lowest degree found across any term in the supplied [[Polynomial]].
If a non-[[Polynomial]] is supplied, returns either `0` or `-1` if the input
is itself a `0`.

See [[degree]] for a discussion of this `-1` case.
sourceraw docstring

makeclj/s

(make dense-coefficients)
(make arity expts->coef)

Generates a [[Polynomial]] instance (or a bare coefficient!) from either:

  • a sequence of dense coefficients of a univariate polynomial (in ascending order)
  • an explicit arity, and a sparse mapping (or sequence of pairs) of exponent => coefficient

In the first case, the sequence is interpreted as a dense sequence of coefficients of an arity-1 (univariate) polynomial. The coefficients begin with the constant term and proceed to each higher power of the indeterminate. For example, x^2 - 1 can be constructed by (make [-1 0 1]).

In the 2-arity case,

  • arity is the number of indeterminates
  • expts->coef is a map of an exponent representation to a coefficient.

The exponent portion of the mapping can be any of:

  • a proper exponent entry created by emmy.polynomial.exponent
  • a map of the form {variable-index, power}
  • a dense vector of variable powers, like [3 0 1] for $x^3z$. The length of each vector should be equal to arity, in this case.

For example, any of the following would generate $4x^2y + 5xy^2$:

(make 2 [[[2 1] 4] [[1 2] 5]])
(make 2 {[2 1] 4, [1 2] 5})
(make 2 {{0 2, 1 1} 4, {0 1, 1 2} 5})

NOTE: make will try and return a bare coefficient if possible. For example, the following form will return a constant, since there are no explicit indeterminates with powers > 0:

(make 10 {{} 1 {} 2})
;;=> 3

See constant if you need an explicit [[Polynomial]] instance wrapping a constant.

Generates a [[Polynomial]] instance (or a bare coefficient!) from either:

- a sequence of dense coefficients of a univariate polynomial (in ascending
order)
- an explicit `arity`, and a sparse mapping (or sequence of pairs) of exponent
=> coefficient

In the first case, the sequence is interpreted as a dense sequence of
coefficients of an arity-1 (univariate) polynomial. The coefficients begin
with the constant term and proceed to each higher power of the indeterminate.
For example, x^2 - 1 can be constructed by (make [-1 0 1]).

In the 2-arity case,

- `arity` is the number of indeterminates
- `expts->coef` is a map of an exponent representation to a coefficient.

The `exponent` portion of the mapping can be any of:

- a proper exponent entry created by `emmy.polynomial.exponent`
- a map of the form `{variable-index, power}`
- a dense vector of variable powers, like `[3 0 1]` for $x^3z$. The length of
  each vector should be equal to `arity`, in this case.

For example, any of the following would generate $4x^2y + 5xy^2$:

```clojure
(make 2 [[[2 1] 4] [[1 2] 5]])
(make 2 {[2 1] 4, [1 2] 5})
(make 2 {{0 2, 1 1} 4, {0 1, 1 2} 5})
```

NOTE: [[make]] will try and return a bare coefficient if possible. For
example, the following form will return a constant, since there are no
explicit indeterminates with powers > 0:

```clojure
(make 10 {{} 1 {} 2})
;;=> 3
```

See [[constant]] if you need an explicit [[Polynomial]] instance wrapping a
constant.
sourceraw docstring

map-coefficientsclj/s

(map-coefficients f p)

Given a [[Polynomial]], returns a new [[Polynomial]] instance generated by applying f to the coefficient of each term in p and filtering out all resulting zeros.

Given a non-[[Polynomial]] coefficient, returns (f p).

NOTE that map-coefficients will return a non-[[Polynomial]] if the result of the mapping has only a constant term.

Given a [[Polynomial]], returns a new [[Polynomial]] instance generated by
applying `f` to the coefficient of each term in `p` and filtering out all
resulting zeros.

Given a non-[[Polynomial]] coefficient, returns `(f p)`.

NOTE that [[map-coefficients]] will return a non-[[Polynomial]] if the result
of the mapping has only a constant term.
sourceraw docstring

map-exponentsclj/s

(map-exponents f p)
(map-exponents f p new-arity)

Given a [[Polynomial]], returns a new [[Polynomial]] instance generated by applying f to the exponents of each term in p and filtering out all resulting zeros. The resulting [[Polynomial]] will have either the same arity as p, or the explicit, optional new-arity argument. (This is because f might increase or decrease the total arity.)

Given a non-[[Polynomial]] coefficient, if (f empty-exponents) produces a non-zero result, errors without an explicit new-arity argument..

NOTE that map-exponents will return a non-[[Polynomial]] if the result of the mapping has only a constant term.

Given a [[Polynomial]], returns a new [[Polynomial]] instance generated by
applying `f` to the exponents of each term in `p` and filtering out all
resulting zeros. The resulting [[Polynomial]] will have either the
same [[arity]] as `p`, or the explicit, optional `new-arity` argument. (This
is because `f` might increase or decrease the total arity.)

Given a non-[[Polynomial]] coefficient, if `(f empty-exponents)` produces a
non-zero result, errors without an explicit `new-arity` argument..

NOTE that [[map-exponents]] will return a non-[[Polynomial]] if the result
of the mapping has only a constant term.
sourceraw docstring

monic?clj/s

(monic? p)

Returns true if p is a monic polynomial, false otherwise.

A monic polynomial is a univariate polynomial with a leading coefficient that responds true to [[emmy.value/one?]]. This means that any coefficient that responds true to [[emmy.value/one?]] also qualifies as a monic polynomial.

Returns true if `p` is a [monic
polynomial](https://en.wikipedia.org/wiki/Monic_polynomial), false otherwise.

A monic polynomial is a univariate polynomial with a leading coefficient that
responds `true` to [[emmy.value/one?]]. This means that any coefficient
that responds `true` to [[emmy.value/one?]] also qualifies as a monic
polynomial.
sourceraw docstring

monomial?clj/s

(monomial? p)

Returns true if p is either:

  • a [[Polynomial]] instance with a single term, or
  • a non-[[Polynomial]] coefficient,

false otherwise.

Returns true if `p` is either:

- a [[Polynomial]] instance with a single term, or
- a non-[[Polynomial]] coefficient,

false otherwise.
sourceraw docstring

mulclj/s

(mul p q)

Returns the product of polynomials p and q, with appropriate handling for non-[[Polynomial]] coefficient inputs on either or both sides.

Returns the product of polynomials `p` and `q`, with appropriate handling for
non-[[Polynomial]] coefficient inputs on either or both sides.
sourceraw docstring

multivariate?clj/s

(multivariate? p)

Returns true if p is a [[Polynomial]] of arity > 1, false otherwise.

Returns true if `p` is a [[Polynomial]] of arity > 1, false otherwise.
sourceraw docstring

negateclj/s

(negate p)

Returns the negation of polynomial p, i.e., a polynomial with all coefficients negated.

Returns the negation of polynomial `p`, i.e., a polynomial with all coefficients
negated.
sourceraw docstring

negative?clj/s

(negative? p)

Returns true if the leading-base-coefficient of p is generic/negative?, false otherwise.

Returns true if the [[leading-base-coefficient]] of `p`
is [[generic/negative?]], false otherwise.
sourceraw docstring

new-variablesclj/s

(new-variables n)

Returns a sequence of n monomials of arity n, each with an exponent of 1 for the ith indeterminate (where i matches the position in the returned sequence).

Returns a sequence of `n` monomials of arity `n`, each with an exponent of `1`
for the `i`th indeterminate (where `i` matches the position in the returned
sequence).
sourceraw docstring

normalizeclj/s

(normalize p)
(normalize p c)

Given a polynomial p, returns a normalized polynomial generated by dividing through either the leading-coefficient of p or an optional, explicitly supplied scaling factor c.

For example:

(let [p (make [5 3 2 2 10])]
  (univariate->dense (normalize p)))
;;=> [1/2 3/10 1/5 1/5 1]
Given a polynomial `p`, returns a normalized polynomial generated by dividing
through either the [[leading-coefficient]] of `p` or an optional, explicitly
supplied scaling factor `c`.

For example:

```clojure
(let [p (make [5 3 2 2 10])]
  (univariate->dense (normalize p)))
;;=> [1/2 3/10 1/5 1/5 1]
```
sourceraw docstring

partial-derivativeclj/s

(partial-derivative p i)

Given some [[Polynomial]] p, returns the partial derivative of p with respect to the ith indeterminate. Throws if i is an invalid indeterminate index for p.

For non-[[Polynomial]] inputs, returns 0.

Given some [[Polynomial]] `p`, returns the partial derivative of `p` with
respect to the `i`th indeterminate. Throws if `i` is an invalid indeterminate
index for `p`.

For non-[[Polynomial]] inputs, returns `0`.
sourceraw docstring

partial-derivativesclj/s

(partial-derivatives p)

Returns the sequence of partial derivatives of [[Polynomial]] p with respect to each indeterminate. The returned sequence has length equal to the arity of p.

For non-[[Polynomial]] inputs, returns an empty sequence.

Returns the sequence of partial derivatives of [[Polynomial]] `p` with respect
to each indeterminate. The returned sequence has length equal to the [[arity]]
of `p`.

For non-[[Polynomial]] inputs, returns an empty sequence.
sourceraw docstring

polynomial?clj/s

(polynomial? x)

Returns true if the supplied argument is an instance of [[Polynomial]], false otherwise.

Returns true if the supplied argument is an instance of [[Polynomial]], false
otherwise.
sourceraw docstring

pseudo-remainderclj/s

(pseudo-remainder u v)

Returns the pseudo-remainder of univariate polynomials u and v.

NOTE: Fractions won't appear in the result; instead the divisor is multiplied by the leading coefficient of the dividend before quotient terms are generated so that division will not result in fractions.

Returns a pair of

  • the remainder
  • the integerizing factor needed to make this happen.

Similar in spirit to Knuth's algorithm 4.6.1R, except we don't multiply the remainder through during gaps in the remainder. Since you don't know up front how many times the integerizing multiplication will be done, we also return the number d for which d * u = q * v + r.

Returns the pseudo-remainder of univariate polynomials `u` and `v`.

NOTE: Fractions won't appear in the result; instead the divisor is multiplied
by the leading coefficient of the dividend before quotient terms are generated
so that division will not result in fractions.

Returns a pair of

- the remainder
- the integerizing factor needed to make this happen.

Similar in spirit to Knuth's algorithm 4.6.1R, except we don't multiply the
remainder through during gaps in the remainder. Since you don't know up front
how many times the integerizing multiplication will be done, we also return
the number d for which d * u = q * v + r.
sourceraw docstring

raise-arityclj/s

(raise-arity p a)

Given either a non-[[Polynomial]] coefficient or a univariate [[Polynomial]] with possibly-[[Polynomial]] coefficients, returns a new [[Polynomial]] of arity a generated by attaching the polynomial coefficients back as variables starting with 1.

raise-arity undoes the transformation of lower-arity. See with-lower-arity for a function that packages these two transformations.

Given either a non-[[Polynomial]] coefficient or a univariate [[Polynomial]]
with possibly-[[Polynomial]] coefficients, returns a new [[Polynomial]] of
arity `a` generated by attaching the polynomial coefficients back as variables
starting with `1`.

[[raise-arity]] undoes the transformation of [[lower-arity]].
See [[with-lower-arity]] for a function that packages these two
transformations.
sourceraw docstring

reciprocalclj/s

(reciprocal p)
(reciprocal p i)

Given a polynomial p, returns the reciprocal polynomial with respect to the ith indeterminate. i defaults to 0.

The reciprocal polynomial of p with respect to i is generated by

  • treating the polynomial as univariate with respect to i and pushing all other terms into the coefficients of the polynomial
  • reversing the order of these coefficients
  • flattening the polynomial out again

For example, note that the entries for the first indeterminate are reversed:

(= (make 3 {[3 0 0] 5 [2 0 1] 2 [0 2 1] 3})
   (reciprocal
     (make 3 {[0 0 0] 5 [1 0 1] 2 [3 2 1] 3})))
Given a polynomial `p`, returns the [reciprocal
polynomial](https://en.wikipedia.org/wiki/Reciprocal_polynomial) with respect
to the `i`th indeterminate. `i` defaults to 0.

The reciprocal polynomial of `p` with respect to `i` is generated by

- treating the polynomial as univariate with respect to `i` and pushing all
  other terms into the coefficients of the polynomial
- reversing the order of these coefficients
- flattening the polynomial out again

For example, note that the entries for the first indeterminate are reversed:

```clojure
(= (make 3 {[3 0 0] 5 [2 0 1] 2 [0 2 1] 3})
   (reciprocal
     (make 3 {[0 0 0] 5 [1 0 1] 2 [3 2 1] 3})))
```
sourceraw docstring

scaleclj/s

(scale p c)

Given some polynomial p and a coefficient c, returns a new [[Polynomial]] generated by multiplying each coefficient of p by c (on the right).

See scale-l if left multiplication is important.

NOTE that scale will return a non-[[Polynomial]] if the result of the mapping has only a constant term.

Given some polynomial `p` and a coefficient `c`, returns a new [[Polynomial]]
generated by multiplying each coefficient of `p` by `c` (on the right).

See [[scale-l]] if left multiplication is important.

NOTE that [[scale]] will return a non-[[Polynomial]] if the result of the
mapping has only a constant term.
sourceraw docstring

scale-lclj/s

(scale-l c p)

Given some polynomial p and a coefficient c, returns a new [[Polynomial]] generated by multiplying each coefficient of p by c (on the left).

See scale if right multiplication is important.

NOTE that scale-l will return a non-[[Polynomial]] if the result of the mapping has only a constant term.

Given some polynomial `p` and a coefficient `c`, returns a new [[Polynomial]]
generated by multiplying each coefficient of `p` by `c` (on the left).

See [[scale]] if right multiplication is important.

NOTE that [[scale-l]] will return a non-[[Polynomial]] if the result of the
mapping has only a constant term.
sourceraw docstring

squareclj/s

(square p)

Returns the square of polynomial p. Equivalent to (mul p p).

Returns the square of polynomial `p`. Equivalent to `(mul p p)`.
sourceraw docstring

subclj/s

(sub p q)

Returns the difference of polynomials p and q, with appropriate handling for non-[[Polynomial]] coefficient inputs on either or both sides.

Returns the difference of polynomials `p` and `q`, with appropriate handling
for non-[[Polynomial]] coefficient inputs on either or both sides.
sourceraw docstring

touchardclj/s

(touchard n)

Returns the nth Touchard polynomial.

These are also called Bell polynomials (in Mathematica, implemented as BellB) or /exponential polynomials/.

Returns the nth [Touchard
polynomial](https://en.wikipedia.org/wiki/Touchard_polynomials).

These are also called [Bell
polynomials](https://mathworld.wolfram.com/BellPolynomial.html) (in
Mathematica, implemented as `BellB`) or /exponential polynomials/.
sourceraw docstring

trailing-coefficientclj/s

(trailing-coefficient p)

Returns the coefficient of the trailing (lowest degree) term of the [[Polynomial]] p.

If p is a non-[[Polynomial]] coefficient, acts as identity.

Returns the coefficient of the trailing (lowest degree) term of
the [[Polynomial]] `p`.

If `p` is a non-[[Polynomial]] coefficient, acts as identity.
sourceraw docstring

univariate->denseclj/s

(univariate->dense x)
(univariate->dense x x-degree)

Given a univariate [[Polynomial]] (see univariate?) returns a dense vector of the coefficients of each term in ascending order.

For example:

(univariate->dense (make [1 0 0 2 3 4]))
;;=> [1 0 0 2 3 4]

Supplying the second argument x-degree will pad the right side of the returning coefficient vector to be the max of x-degree and (degree x).

NOTE use lower-arity to generate a univariate polynomial in the first indeterminate, given a multivariate polynomial.

Given a univariate [[Polynomial]] (see [[univariate?]]) returns a dense vector
of the coefficients of each term in ascending order.

For example:

```clojure
(univariate->dense (make [1 0 0 2 3 4]))
;;=> [1 0 0 2 3 4]
```

Supplying the second argument `x-degree` will pad the right side of the
returning coefficient vector to be the max of `x-degree` and `(degree x)`.

NOTE use [[lower-arity]] to generate a univariate polynomial in the first
indeterminate, given a multivariate polynomial.
sourceraw docstring

univariate?clj/s

(univariate? p)

Returns true if p is a [[Polynomial]] of arity 1, false otherwise.

Returns true if `p` is a [[Polynomial]] of arity 1, false otherwise.
sourceraw docstring

valid-arity?clj/s

(valid-arity? p i)

Given some input p and an indeterminate index i, returns true if 0 <= i < (arity p), false otherwise.

Given some input `p` and an indeterminate index `i`, returns true if `0 <= i
< (arity p)`, false otherwise.
sourceraw docstring

with-lower-arityclj/s

(with-lower-arity u v continue)

Given:

  • multivariate [[Polynomial]]s u and v
  • a continue function that accepts two univariate [[Polynomial]]s with possibly-[[Polynomial]] coefficients,

Returns the result of calling lower-arity on u and v, passing the results to continue and using raise-arity to raise the result back to the original shared arity of u and v.

The exception is that if continue returns a non-[[Polynomial]], with-lower-arity will not attempt to re-package it as a [[Polynomial]].

Given:

- multivariate [[Polynomial]]s `u` and `v`
- a `continue` function that accepts two univariate [[Polynomial]]s with
  possibly-[[Polynomial]] coefficients,

Returns the result of calling [[lower-arity]] on `u` and `v`, passing the
results to `continue` and using [[raise-arity]] to raise the result back to
the original shared [[arity]] of `u` and `v`.

The exception is that if `continue` returns a
non-[[Polynomial]], [[with-lower-arity]] will not attempt to re-package it as
a [[Polynomial]].
sourceraw docstring

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