polexpr reference — polexpr 0.8.7a documentation

# polexpr reference

## Syntax overview via examples

The syntax to define a new polynomial is:

    \poldef polname(x):= expression in variable x;

The expression will be parsed by the services of xintexpr, with some polynomial aware functions added to its syntax; they are described in detail below. The parser accepts and will handle exactly arbitrarily big integers or fractions.

Note

xintexpr does not automatically reduce fractions to lowest terms, and, so far (but this may change in future) neither does \poldef. See rdcoeffs() and the macro \PolReduceCoeffs.

- In place of `x` an arbitrary dummy variable is authorized, i.e. per default one `a, .., z, A, .., Z` (more letters can be declared under Unicode engines).

- `polname` is a word (no space) built with letters, digits, and the `@`, `_` and `'` characters are allowed. The polynomial name must start with a letter.

  For guidelines regarding `_` and `@` see Technicalities.

- The colon before the equality sign is optional and its (reasonable) catcode does not matter.

- The semi-colon at the end of the expression is mandatory. It is not allowed to arise from expansion (despite the fact that the expression itself will be parsed using only expansion), it must be “visible” immediately.

There are some potential problems (refer to the Technicalities section at bottom of this page) with the semi-colon as expression terminator, so an alternative syntax is provided, which avoids it altogether:

    \PolDef[optional letter]{<polname>}{<expr. using letter as indeterminate>}

The `\PolDef` optional first argument defaults to `x` and must be used as the indeterminate in the expression.

Examples:

`\poldef f(x):= 1 - x + quo(x^5,1 - x + x^2);`

`\PolDef{f}{1 - x + quo(x^5,1 - x + x^2)}`

  Both parse the polynomial expression, and they create internally macros serving to incarnate the polynomial, its coefficients, and the associated polynomial function.

  The polynomial can then be used in further polynomial definitions, be served as argument to package macros, or appear as a variable in various functions described later.

  Warning

  Both the function `quo()` (as shown in the example above), and the infix operator `/` are mapped to the Euclidean quotient.

  This usage of `/` to stand for the Euclidean quotient is deprecated and reserved for a (somewhat improbable) possible extension of the package to handle rational functions as well.

  Attention

  Tacit multiplication rules let the parser when encountering `1/2 x^2` skip the space and thus handle it as `1/(2*x^2)`. But then it gives zero, because / stands for the Euclidean quotient operation here.

  Thus one must use `(1/2)x^2` or `1/2*x^2` or `(1/2)*x^2` for disambiguation: `x - 1/2*x^2 + 1/3*x^3...`. It is simpler to move the denominator to the right: `x - x^2/2 + x^3/3 - ...`.

  It is worth noting that `1/2(x-1)(x-2)` suffers the same issue: xintexpr‘s tacit multiplication always “ties more”, hence this gets interpreted as `1/(2*(x-1)*(x-2))` not as `(1/2)*(x-1)*(x-2)` and then gives zero by polynomial division. Thus, in such cases, use one of `(1/2)(x-1)(x-2)`, `1/2*(x-1)(x-2)` or `(x-1)(x-2)/2`.

  `\poldef P(x):=...;` defines `P` as a polynomial function, which can be used inside `\xinteval`, as:

      \xinteval{P(3 + 7 + 11)}

  or even as:

      \xinteval{P(Q1 + Q2 + Q3)}

  where `Q1`, `Q2`, `Q3` are polynomials. The evaluation result, if not a scalar, will then be printed as `pol([c0,c1,...])` which stands for a polynomial variable having the listed coefficients; see pol().

  Indeed, as seen above with `Q1`, the symbol `P` also stands for a variable of polynomial type, which serves as argument to polynomial specific functions such as deg() or polgcd(), or as argument to other polynomials (as above), or even simply stands for its own in algebraic expressions such as:

      \poldef Q(z):= P^2 + z^10;

  Notice that in the above, the `(z)` part is mandatory, as it informs `\poldef` of the letter used for the indeterminate. In the above `P(z)^2` would give the same as `P^2` but the latter is slightly more efficient.

  One needs to acquire a good understanding of when the symbol `P` will stand for a function and when it will stand for a variable.

  - If `P` and `Q` are both declared polynomials then:

        (P+Q)(3)%  <--- attention, does (P+Q)*3, not P(3)+Q(3)

    is currently evaluated as `(P+Q)*3`, because `P+Q` is not known as a function, but only as a variable of polynomial type. Note that evalp(P+Q,3) gives as expected the same as `P(3)+Q(3)`.

  - Also:

        (P)(3)%  <--- attention, does P*3, not P(3)

    will compute `P*3`, because one can not in current xintexpr syntax enclose a function name in parentheses: consequently it is the variable which is used here.

  There is a meager possibility that in future some internal changes to xintexpr would let `(P)(3)` actually compute `P(3)` and `(P+Q)(3)` compute `P(3) + Q(3)`, but note that `(P)(P)` will then do `P(P)` and not `P*P`, the latter, current interpretation, looking more intuitive. Anyway, do not rely too extensively on tacit `*` and use explicit `(P+Q)*(1+2)` if this is what is intended.

`\PolLet{g}={f}`

  saves a copy of `f` under name `g`. Also usable without `=`.

  Has exactly the same effect as `\poldef g(x):=f;` or `\poldef g(w):=f(w);`.

`\poldef f(z):= f^2;`

  redefines `f` in terms of itself. Prior to `0.8` one needed the right hand side to be `f(z)^2`. Also, now `sqr(f)` is possible (also `sqr(f(x))` but not `sqr(f)(x)`).

  It may look strange that an indeterminate variable is used on left-hand-side even though it may be absent of right-hand-side, as it seems to define `f` always as a polynomial function.

  This is a legacy of pre-`0.8` context.

  Important

  Note that `f^2(z)` or `sqr(f)(z)` will give a logical but perhaps unexpected result: first `f^2` is computed, then the opening parenthesis is seen which inserts a tacit multiplication `*`, so in the end it is as if the input had been `f^2 * z`. Although `f` is both a variable and a function, `f^2` is computed as a polynomial variable and ceases being a function.

`\poldef f(T):= f(f);`

  again modifies `f`. Here it is used both as variable and as a function. Prior to `0.8` it needed to be `f(f(T))`.

`\poldef k(z):= f-g(g^2)^2;`

  if everybody followed, this should now define the zero polynomial… And `f-sqr(g(sqr(g)))` computes the same thing.

  We can check this in a typeset document like this:

      \poldef f(x):= 1 - x + quo(x^5,1 - x + x^2);%
      \PolLet{g}={f}%
      \poldef f(z):= f^2;%
      \poldef f(T):= f(f);%
      \poldef k(w):= f-sqr(g(sqr(g)));%
      $$f(x) = \vcenter{\hsize10cm \PolTypeset{f}} $$
      $$g(z) = \PolTypeset{g} $$
      $$k(z) = \PolTypeset{k} $$
      \immediate\write128{f(x)=\PolToExpr{f}}% ah, here we see it also

`\poldef f'(x):= diff1(f);`

  (new at `0.8`)

`\PolDiff{f}{f'}`

  Both set `f'` (or any other chosen name) to the derivative of `f`.

  Important

  This is not done automatically. If some new definition needs to use the derivative of some available polynomial, that derivative polynomial must have been previously defined: something such as `f'(3)^2` will not work without a prior definition of `f'`.

  But one can now use `diff1(f)` for on-the-spot construction with no permanent declaration, so here `evalp(diff1(f),3)^2`. And `diff1(f)^2` is same as `f'^2`, assuming here `f'` was declared to be the derived polynomial.

  Notice that the name `diff1()` is experimental and may change. Use `\PolDiff{f}{f'}` as the stable interface.

`\PolTypeset{P}`

  Typesets (switching to math mode if in text mode):

      \poldef f(x):=(3+x)^5;%
      \PolDiff{f}{f'}\PolDiff{f'}{f''}\PolDiff{f''}{f'''}%
      $$f(z)   = \PolTypeset[z]{f}    $$
      $$f'(z)  = \PolTypeset[z]{f'}   $$
      $$f''(z) = \PolTypeset[z]{f''}  $$
      $$f'''(z)= \PolTypeset[z]{f'''} $$

  See its documentation for the configurability via macros.

  Since `0.8` \PolTypeset accepts directly an expression, it does not have to be a pre-declared polynomial name:

      \PolTypeset{mul(x-i,i=1..5)}

`\PolToExpr{P}`

  Expandably (contrarily to \PolTypeset) produces `c_n*x^n + ... + c_0` starting from the leading coefficient. The `+` signs are omitted if followed by negative coefficients.

  This is useful for console or file output. This syntax is Maple and PSTricks `\psplot[algebraic]` compatible; and also it is compatible with `\poldef` input syntax, of course. See \PolToExprCaret for configuration of the `^`, for example to use rather `**` for Python syntax compliance.

  Changed at `0.8`: the `^` in output is by default of catcode 12 so in a draft document one can use `\PolToExpr{P}` inside the typesetting flow (without requiring math mode, where the `*` would be funny and `^12` would only put the `1` as exponent anyhow; but arguably in text mode the `+` and `-` are not satisfactory for math, except sometimes in monospace typeface, and anyhow TeX is unable to break the expression across lines, barring special help).

  See \PolToExpr{<pol. expr.>} and related macros for customization.

  Extended at `0.8` to accept as argument not only the name of a polynomial variable but more generally any polynomial expression.

## Using defined polynomials in floating point context

Exact manipulations with fractional coefficients may quickly lead to very large denominators. For numerical evaluations, it is advisable to a use a floating point context. But for the polynomial to be usable as a function in floating point context, an extra step beyond `\poldef` is required: see \PolGenFloatVariant. Then the `\xintfloateval` macro from xintexpr will recognize the polynomial as a genuine function (with already float-rounded coefficients, and using a Horner scheme).

But \PolGenFloatVariant must be used each time the polynomial gets redefined or a new polynomial is created out of it. Functions such as for example deg() which handle the polynomial as an entity are only available within the `\poldef` and `\xinteval` (or `\xintexpr`) parsers. Inside `\xintfloateval` a polynomial can only serve as a numerical function (and only after declaration via \PolGenFloatVariant), and not as a variable.

In some cases one may wish to replace a polynomial having acquired very big fractional coefficients with a new one whose coefficients have been float-rounded. See \PolMapCoeffs which can be used for example with the `\xintFloat` macro from the xintfrac package to achieve this.

## The polexpr `0.8` extensions to the `\xintexpr` syntax

All the syntax elements described in this section can be used in the `\xintexpr/\xinteval` context (where polynomials can be obtained from the `pol([])` constructor, once polexpr is loaded): their usage is not limited to only `\poldef` context.

Note

If a variable `myPol` defined via `\xintdefvar` turns out to be a polynomial, the difference with those declared via `\poldef` will be:

1. `myPol` is not usable as function, but only as a variable. Attention that `f(x)` if `f` is only a variable (even a polynomial one) will actually compute `f * x`.

2. `myPol` is not known to the polexpr package, hence for example the macros to achieve localization of its roots are unavailable.

   In a parallel universe I perhaps have implemented this expandably which means it could then be accessible with syntax such as `rightmostroot(pol([42,1,34,2,-8,1]))` but…

### Warning about unstability of the new syntax

Warning

Consider the entirety of this section as UNSTABLE and EXPERIMENTAL (except perhaps regarding `+`, `-` and `*`).

And this applies even to items not explicitly flagged with one of unstable, Unstable, or UNSTABLE which only reflect that documentation was written over a period of time exceeding one minute, enough for the author mood changes to kick in.

It is hard to find good names at the start of a life-long extension program of functionalities, and perhaps in future it will be preferred to rename everything or give to some functions other meanings. Such quasi-complete renamings happened already a few times during the week devoted to development.

### Infix operators `+, -, *, /, **, ^`

> As has been explained in the Syntax overview via examples section these infix operators have been made polynomial aware, not only in the `\poldef` context, but generally in any `\xintexpr/\xinteval` context, inclusive of `\xintdeffunc`.

> Conversely functions declared via `\xintdeffunc` and making use of these operators will automatically be able to accept polynomials declared from `\poldef` as variables.

> Usage of `/` for euclidean division of polynomials is deprecated. Only in case of a scalar denominator is it to be considered stable. Please use rather `quo()`.

### Experimental infix operators `//, /:`

> Here is the tentative behaviour of `A//B` according to types:

> - `A` non scalar and `B` non scalar: euclidean quotient,

> - `A` scalar and `B` scalar: floored division,

> - `A` scalar and `B` non scalar: produces zero,

> - `A` non scalar and `B` scalar: coefficient per coefficient floored division.

> This is an experimental overloading of the `//` and `/:` from `\xintexpr`.

> The behaviour in the last case, but not only, is to be considerd unstable. The alternative would be for `A//B` with `B` scalar to act as `quo(A,B)`. But, we have currently chosen to let `//B` for a scalar `B` act coefficient-wise on the numerator. Beware that it thus means it can be employed with the idea of doing euclidean division only by checking that `B` is non-scalar.

> The `/:` operator provides the associated remainder so always `A` is reconstructed from `(A//B)*B + A/:B`.

> If `:` is active character use `/\string:` (it is safer to use `/\string :` if it is not known if `:` has catcode other, letter, or is active, but note that `/:` is fine and needs no precaution if `:` has catcode letter, it is only an active `:` which is problematic, like for all other characters possibly used in an expression).

> > UNSTABLE

> > As explained above, there are (among other things) hesitations about behaviour with `pol2` a scalar.

### Comparison operators `<, >, <=, >=, ==, !=`

> NOT YET IMPLEMENTED

> As the internal representation by xintfrac and xintexpr of fractions does not currently require them to be in reduced terms, such operations would be a bit costly as they could not benefit from the `\pdfstrcmp` engine primitive. In fact xintexpr does not use it yet anywhere, even for normalized pure integers, although it could speed up signifcantly certain aspects of core arithmetic.

> Equality of polynomials can currently be tested by computing the difference, which is a bit costly. And of course the `deg()` function allows comparing degrees. In this context note the following syntax:

>     (deg(Q)) ?? { zero } { non-zero scalar } { non-scalar }

> for branching.

### `pol(<nutple expression>)`

> This converts a nutple `[c0,c1,...,cN]` into the polynomial variable having these coefficients. Attention that the square brackets are mandatory, except of course if the argument is actually an expression producing such a “nutple”.

> > Currently, this process will not normalize the coefficients (such as reducing to lowest terms), it only trims out the leading zero coefficients.

> Inside `\xintexpr`, this is the only (allowed) way to create ex nihilo a polynomial variable; inside `\poldef` it is an alternative input syntax which is more efficient than the input `c0 + c1 * x + c2 * x^2 + ...`.

Important

Whenever an expression with polynomials collapses to a constant, it becomes a scalar. There is currently no distinction during the parsing of expressions by `\poldef` or `\xintexpr` between constant polynomial variables and scalar variables.

Naturally, `\poldef` can be used to declare a constant polynomial `P`, then `P` can also be used as function having a value independent of argument, but as a variable, it is non-distinguishable from a scalar (of course functions such as `deg()` tacitly consider scalars to be constant polynomials).

Notice that we tend to use the vocable “variable” to refer to arbitrary expressions used as function arguments, without implying that we are actually referring to pre-declared variables in the sense of `\xintdefvar`.

### `lpol(<nutple expression>)`

> This converts a nutple `[cN,...,c1,c0]` into the polynomial variable having these coefficients, with leading coefficients coming first in the input. Attention that the square brackets are mandatory, except of course if the argument is actually an expression producing such a “nutple”.

> > Currently, this process will not normalize the coefficients (such as reducing to lowest terms), it only trims out the leading zero coefficients.

> > NAME UNSTABLE

> It can be used in `\poldef` as an alternative input syntax, which is more efficient than using the algebraic notation with monomials.

> (new with `0.8.1`, an empty nutple will cause breakage)

### `\xinteval{<pol. expr.>}`

> This is documented here for lack of a better place: it evaluates the polynomial expression then outputs the “string” `pol([c0, c1, ..., cN])` if the degree `N` is at least one (and the usual scalar output else).

> The “pol” word uses letter catcodes, which is actually mandatory for this output to be usable as input, but it does not make sense to use this inside `\poldef` or `\xintexpr` at it means basically executing `pol(coeffs(..expression..))` which is but a convoluted way to obtain the same result as `(..expression..)` (the parentheses delimiting the polynomial expression).

> For example, `\xinteval{(1+pol([0,1]))^10}` expands (in two steps) to:

>     pol([1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1])

> You do need loading polexpr for this, else of course `pol([])` remains unknown to `\xinteval{}` as well as the polynomial algebra ! This example can also be done as `\xinteval{subs((1+x)^10,x=pol([0,1]))}`.

> I hesitated using as output the polynomial notation as produced by \PolToExpr{}, but finally opted for this.

### `evalp(<pol. expr.>, <pol. expr>)`

> Evaluates the first argument as a polynomial function of the second. Usually the second argument will be scalar, but this is not required:

>     \poldef K(x):= evalp(-3x^3-5x+1,-27x^4+5x-2);

> If the first argument is an already declared polynomial `P`, use rather the functional form `P()` (which can accept a numerical as well as polynomial argument) as it is more efficient.

> One can also use `subs()` syntax [1] (see xintexpr documentation):

>     \poldef K(x):= subs(-3y^3-5y+1, y = -27x^4+5x-2);

> but the `evalp()` will use a Horner evaluation scheme which is usually more efficient.

> [1]

> by the way Maple uses the opposite, hence wrong, order `subs(x=..., P)` but was written before computer science reached the xintexpr heights. However it makes validating Maple results by polexpr sometimes cumbersome, but perhaps they will update it at some point.

> > name unstable

> > `poleval`? `evalpol`? `peval`? `evalp`? `value`? `eval`? `evalat`? `eval1at2`? `evalat2nd`?

> > Life is so complicated when one asks questions. Not everybody does, though, as is amply demonstrated these days.

> > syntax unstable

> > I am hesitating about permuting the order of the arguments.

### `deg(<pol. expr.>)`

> Computes the degree.

Important

As `\xintexpr` does not yet support infinities, the degree of the zero polynomial is `-1`. Beware that this breaks additivity of degrees, but `deg(P)<0` correctly detects the zero polynomial, and `deg(P)<=0` detects scalars.

### `coeffs(<pol. expr.>)`

> Produces the nutple `[c0,c1,...,cN]` of coefficients. The highest degree coefficient is always non zero (except for the zero polynomial…).

> > name unstable

> > I am considering in particular using `polcoeffs()` to avoid having to overload `coeffs()` in future when matrix type will be added to xintexpr.

### `lcoeffs(<pol. expr.>)`

> Produces the nutple `[cN,....,c1,c0]` of coefficients, starting with the highest degree coefficient.

> (new with `0.8.1`)

### `coeff(<pol. expr.>, <num. expr.>)`

> As expected. Produces zero if the numerical index is negative or higher than the degree.

> > name, syntax and output unstable

> > I am hesitating with `coeff(n,pol)` syntax and also perhaps using `polcoeff()` in order to avoid having to overload `coeff()` when matrix type will be added to xintexpr.

> > The current behaviour is at odds with legacy \PolNthCoeff{<polname>}{<index>} regarding negative indices. Accessing leading or sub-leading coefficients can be done with other syntax, see lc(<pol. expr.>), and in some contexts it is useful to be able to rely on the fact that coefficients with negative indices do vanish, so I am for time being maintaining this.

### `lc(<pol. expr.>)`

> The leading coefficient. The same result can be obtained from `coeffs(pol)[-1]`, which shows also how to generalize to access sub-leading coefficients. See the xintexpr documentation for Python-like indexing syntax.

### `monicpart(<pol. expr.>)`

> Divides by the leading coefficient, except that `monicpart(0)==0`.

> > unstable

> > Currently the coefficients are reduced to lowest terms (contrarily to legacy behaviour of \PolMakeMonic), and additionally the xintfrac `\xintREZ` macro is applied which extracts powers of ten from numerator or denominator and stores them internally separately. This is generally beneficial to efficiency of multiplication.

### `cont(<pol. expr.>)`

> The (fractional) greatest common divisor of the polynomial coefficients. It is always produced as an irreducible (non-negative) fraction. According to Gauss theorem the content of a product is the product of the contents.

> > name and syntax unstable

> > At `0.8` it was created as `icontent()` to match the legacy macro \PolIContent, whose name in 2018 was chosen in relation to Maple’s function `icontent()`, possibly because at that time I had not seen that Maple also had a `content()` function. Name changed at `0.8.1`.

> > It will change syntax if in future multivariate polynomials are supported, and `icontent()` will then make a come-back.

### `primpart(<pol. expr.>)`

> The quotient (except for the zero polynomial) by `cont(<pol. expr.>)`. This is thus a polynomial with integer coefficients having `1` as greatest common divisor. The sign of the leading coefficient is the same as in the original.

> And `primpart(0)==0`.

> The trailing zeros of the integer coefficients are extracted into a power of ten exponent part, in the internal representation.

### `quorem(<pol. expr.>, <pol. expr.>)`

> Produces a nutple `[Q,R]` with `Q` the euclidean quotient and `R` the remainder.

> > name unstable

> > `poldiv()`?

### `quo(<pol. expr.>, <pol. expr.>)`

> The euclidean quotient.

> The deprecated `pol1/pol2` syntax computes the same polynomial.

### `rem(<pol. expr.>, <pol. expr.>)`

> The euclidean remainder. If `pol2` is a (non-zero) scalar, this is zero.

> There is no infix operator associated to this, for lack of evident notation. Please advise.

> `/:` can be used if one is certain that `pol2` is of degree at least one. But read the warning about it being unstable even in that case.

### `prem(<pol. expr. 1>, <pol. expr. 2>)`

> Produces a nutple `[m, spR]` where `spR` is the (special) pseudo Euclidean remainder. Its description is:

> - the standard euclidean remainder `R` is `spR/m`

> - `m = b^f` with `b` equal to the absolute value of the leading coefficient of `pol2`,

> - `f` is the number of non-zero coefficients in the euclidean quotient, if `deg(pol2)>0` (even if the remainder vanishes).

>   If `pol2` is a scalar however, the function outputs `[1,0]`.

> With these definitions one can show that if both `pol1` and `pol2` have integer coefficients, then this is also the case of `spR`, which makes its interest (and also `m*Q` has integer coefficients, with `Q` the euclidean quotient, if `deg(pol2)>0`). Also, `prem()` is computed faster than `rem()` for such integer coefficients polynomials.

> Hint

> If you want the euclidean quotient `R` evaluated via `spR/m` (which may be faster, even with non integer coefficients) use `subs(last(x)/first(x),x=prem(P,Q))` syntax as it avoids computing `prem(P,Q)` twice. This does the trick both in `\poldef` or in `\xintdefvar`.

> However, as is explained in the xintexpr documentation, using such syntax in an `\xintdeffunc` is (a.t.t.o.w) illusory, due to technicalities of how `subs()` gets converted into nested expandable macros. One needs an auxiliary function like this:

>     \xintdeffunc lastoverfirst(x):=last(x)/first(x);
>     \xintdeffunc myR(x)=lastoverfirst(prem(x));

> Then, `myR(pol1,pol2)` will evaluate `prem(pol1,pol2)` only once and compute a polynomial identical to the euclidean remainder (internal representations of coefficients may differ).

> In this case of integer coefficients polynomials, the polexpr internal representation of the integer coefficients in the pseudo remainder will be with unit denominators only if that was already the case for those of `pol1` and `pol2` (no automatic reduction to lowest terms is made prior or after computation).

> Pay attention here that `b` is the absolute value of the leading coefficient of `pol2`. Thus the coefficients of the pseudo-remainder have the same signs as those of the standard remainder. This diverges from Maple’s function with the same name.

### `divmod(<pol. expr. 1>, <pol. expr. 2>)`

> Overloads the scalar `divmod()` and associates it with the experimental `//` and `/:` as extended to the polynomial type.

> In particular when both `pol1` and `pol2` are scalars, this is the usual `divmod()` (as in Python) and for `pol1` and `pol2` non constant polynomials, this is the same as `quorem()`.

> > Highly unstable overloading of `\xinteval`‘s `divmod()`.

### `mod(<pol. expr. 1>, <pol. expr. 2>)`

> The `R` of the `divmod()` output. Same as `R` of `quorem()` when the second argument `pol2` is of degree at least one.

> > Highly unstable overloading of `\xinteval`‘s `mod()`.

### `polgcd(<pol. expr. 1>, <pol. expr. 2>, ...)`

> Evaluates to the greatest common polynomial divisor of all the polynomial inputs. The output is a primitive (in particular, with integer coefficients) polynomial. It is zero if and only if all inputs vanish.

> Attention, there must be either at least two polynomial variables, or alternatively, only one argument which then must be a bracketed list or some expression or variable evaluating to such a “nutple” whose items are polynomials (see the documentation of the scalar `gcd()` in xintexpr).

> > The two variable case could (and was, during development) have been defined at user level like this:

> >     \xintdeffunc polgcd_(P,Q):=
> >       (deg(Q))??{P}{1}{polgcd_(Q,primpart(last(prem(P,Q))))};
> >     \xintdeffunc polgcd(P,Q):=polgcd_(primpart(P),primpart(Q));%

> > This is basically what is done internally for two polynomials, up to some internal optimizations.

> UNSTABLE

> I hesitate between returning a primitive or a monic polynomial. Maple returns a primitive polynomial if all inputs [2] have integer coefficients, else it returns a monic polynomial, but this is complicated technically for us to add such a check and would add serious overhead.

> Internally, computations are done using primitive integer-coefficients polynomials (as can be seen in the function template above). So I decided finally to output a primitive polynomial, as one can always apply `monicpart()` to it.

> Attention that this is at odds with behaviour of the legacy \PolGCD (non expandable) macro.

> [2]

> actually, only two polynomial arguments are allowed by Maple’s `gcd()` as far as I know.

### `resultant(<pol. expr. 1>, <pol. expr. 2>)`

> The resultant.

> > NOT YET IMPLEMENTED

### `disc(<pol. expr.>)`

> The discriminant.

> > NOT YET IMPLEMENTED

### `polpowmod(<pol. expr. 1>, <num. expr.>, <pol. expr. 2>)`

> Modular exponentiation: `mod(pol1^N, pol2)` in a more efficient manner than first computing `pol1^N` then reducing modulo `pol2`.

> Attention that this is using the `mod()` operation, whose current experimental status is as follows:

> - if `deg(pol2)>0`, the euclidean remainder operation,

> - if `pol2` is a scalar, coefficient-wise reduction modulo `pol2`.

> UNSTABLE

> > This is currently implemented at high level via `\xintdeffunc` and recursive definitions, which were copied over from a scalar example in the xintexpr manual:

> >     \xintdeffunc polpowmod_(P, m, Q) :=
> >            isone(m)?
> >                % m=1: return P modulo Q
> >                {   mod(P,Q)  }
> >                % m > 1: test if odd or even and do recursive call
> >                {   odd(m)? {  mod(P*sqr(polpowmod_(P, m//2, Q)), Q) }
> >                            {  mod(  sqr(polpowmod_(P, m//2, Q)), Q) }
> >                 }
> >              ;%
> >     \xintdeffunc polpowmod(P, m, Q) := (m)?{polpowmod_(P, m, Q)}{1};%

> > Negative exponents are not currently implemented.

> > For example:

> >     \xinteval{subs(polpowmod(1+x,100,x^7),x=pol([0,1]))}
> >     \xinteval{subs(polpowmod(1+x,20,10), x=pol([0,1]))}

> > produce respectively:

> >     pol([1, 100, 4950, 161700, 3921225, 75287520, 1192052400])
> >     pol([1, 0, 0, 0, 5, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 4, 5, 0, 0, 0, 1])

### `rdcoeffs(<pol. expr.>)`

> This operates on the internal representation of the coefficients, reducing them to lowest terms.

> > name HIGHLY undecided

### `rdzcoeffs(<pol. expr.>)`

> This operates on the internal representation of the coefficients, reducing them to lowest terms then extracting from numerator or denominator the maximal power of ten to store as a decimal exponent.

> This is sometimes favourable to more efficient polynomial algebra computations.

> > name HIGHLY undecided

### `diff1(<pol. expr.>)`

> The first derivative.

> > name UNSTABLE

> > This name may be used in future to be the partial derivative with respect to a first variable.

### `diff2(<pol. expr.>)`

> The second derivative.

> > name UNSTABLE

> > This name may be used in future to be the partial derivative with respect to a second variable.

### `diffn(<pol. expr. P>, <num. expr. n>)`

> The `n`th derivative of `P`. For `n<0` computes iterated primitives vanishing at the origin.

> The coefficients are not reduced to lowest terms.

> > name and syntax UNSTABLE

> > I am also considering reversing the order of the arguments.

### `antider(<pol. expr. P>)`

> The primitive of `P` with no constant term. Same as `diffn(P,-1)`.

### `intfrom(<pol. expr. P>, <pol. expr. c>)`

> The primitive of `P` vanishing at `c`, i.e. `\int_c^x P(t)dt`.

> Also `c` can be a polynomial… so if `c` is monomial `x` this will give zero!

> > UNSTABLE

> > Allowing general polynomial variable for `c` adds a bit of overhead to the case of a pure scalar. So I am hesitating maintaining this feature whose interest appears dubious.

> Attention

> As the two arguments are both allowed to be polynomials, if by inadvertance one exchanges the two, there is no error but the meaning of `intfrom(c,P)` is completely otherwise, as it produces `c*(x - P)` if `c` is a scalar:

>     >>> &pol
>     pol mode (i.e. function definitions use \poldef)
>     >>> P(x):=1+x^2;
>     P = x^2+1
>         --> &GenFloat(P) lets P become usable as function in fp mode
>         --> &ROOTS(P) (resp. &ROOTS(P,N)) finds all rational roots exactly and
>             all irrational roots with at least 10 (resp. N) fractional digits
>     >>> intfrom(P,1);
>     @_1     pol([-4/3, 1, 0, 1/3])
>     >>> intfrom(1,P);
>     @_2     pol([-1, 1, -1])
>     >>> &bye

### `integral(<pol. expr. P>, [<pol. expr. a>, <pol. expr. b>])`

> `\int_a^b P(t)dt`.

> Warning

> The brackets here are not denoting an optional argument but a mandatory nutple argument `[a, b]` with two items. No real recoverable-from error check is done on the input syntax. The input can be an xintexpr variable which happens to be a nutple with two items, or any expression which evaluates to such a nutple.

> `a` and `b` are not restricted to be scalars, they are allowed to be themselves polynomial variables or even polynomial expressions.

> > To compute `\int_{x-1}^x P(t)dt` it is more efficient to use `intfrom(x-1)`.

> > Similary to compute `\int_x^{x+1} P(t)dt`, use `-intfrom(x+1)`.

> > UNSTABLE

> > Am I right to allow general polynomials `a` and `b` hence add overhead to the pure scalar case ?

## Non-expandable macros

Note

At `0.8` `polexpr` is usable with Plain TeX and not only with LaTeX. Some examples given in this section may be using LaTeX syntax such as `\renewcommand`.

### `\poldef polname(letter):= expression using the letter as indeterminate;`

> This evaluates the polynomial expression and stores the coefficients in a private structure accessible later via other package macros, used with argument `polname`. Of course the expression can make use of previously defined polynomials.

> Polynomial names must start with a letter and are constituted of letters, digits, underscores, the `@` (see Technicalities) and the right tick `'`.

> The whole xintexpr syntax is authorized, as long as the final result is of polynomial type:

>     \poldef polname(z) := add((-1)^i z^(2i+1)/(2i+1)!, i = 0..10);

> With fractional coefficients, beware the tacit multiplication issue.

> Furthermore:

> - a variable `polname` is defined which can be used in `\poldef` as well as in `\xinteval` for algebraic computations or as argument to polynomial aware functions,

> - a function `polname()` is defined which can be used in `\poldef` as well as in `\xinteval`. It accepts there as argument scalars and also other polynomials (via their names, thanks to previous item).

> Any function defined via `\xintdeffunc` and only algebraic operations, as well as ople indexing or slicing operations, should work fine in `\xintexpr/\xinteval` with such polynomial names as argument.

> In the case of a constant polynomial, the xintexpr variable (not the internal data structure on which the package macros operate) associated to it is indistinguishable from a scalar, it is actually a scalar and has lost all traces from its origins as a polynomial (so for example can be used as argument to the `cos()` function). The function on the other hand remains a one-argument function, which simply has a constant value.

> Attention

> The function `polname()` is defined only for `\xintexpr/\xinteval` context. It will be unknown to `\xintfloateval`.

> Worse, a previously existing floating point function of the same name will be made undefined again, to avoid hard to debug mismatches between exact and floating point polynomials. This also applies when the polynomial is produced not via `\poldef` or `\PolDef` but as result of usage of the other package macros.

> See \PolGenFloatVariant{<polname>} to generate a function usable in `\xintfloateval`.

> Attention

> Using the variable `mypol` inside `\xintfloateval` will generate low-level errors because the infix operators there are not polynomial-aware, and the polynomial specific functions such as `deg()` are only defined for usage inside `\xintexpr`.

> In short, currently polynomials defined via `polexpr` can be used in floating point context only for numerical evaluations, via functions obtained from \PolGenFloatVariant{<polname>} usage.

> Changes to the original polynomial via package macros are not automatically mapped to the numerical floating point evaluator which must be manually updated as necessary when the original rational coefficient polynomial is modified.

> The original expression is lost after parsing, and in particular the package provides no way to typeset it (of course the package provides macros to typeset the computed polynomial). Typesetting the original expression has to be done manually, if needed.

### `\PolDef[<letter>]{<polname>}{<expr. using the letter as indeterminate>}`

> Does the same as \poldef in an undelimited macro format, the main interest is to avoid potential problems with the catcode of the semi-colon in presence of some packages. In absence of a `[<letter>]` optional argument, the variable is assumed to be `x`.

### `\PolGenFloatVariant{}`

Syntax: `\PolGenFloatVariant{<polname>}`

> Makes the polynomial also usable in the `\xintfloatexpr/\xintfloateval` parser. It will therein evaluates via an Horner scheme using polynomial coefficients already pre-rounded to the float precision.

> See also \PolToFloatExpr{<pol. expr.>}.

> Attention

> Any operation, for example generating the derivative polynomial, or dividing two polynomials or using the `\PolLet`, must be followed by explicit usage of `\PolGenFloatVariant{<polname>}` if the new polynomial is to be used in `\xintfloateval`.

### `\PolTypeset[]{}`

Syntax: `\PolTypeset[<letter>]{<pol. expr.>}`

> Typesets in descending powers, switching to math mode if in text mode, after evaluating the polynomial expression:

>     \PolTypeset{mul(x-i,i=1..5)}% possible since polexpr 0.8

> The letter used in the input is by default assumed to be `x`, but can be modified by a redefinition of \PolToExprInVar.

> The letter used in the output is also by default `x`. This one can be changed on-the-fly via the optional `<letter>`:

>     \PolTypeset[z]{polname or polynomial expression}

> By default zero coefficients are skipped (use `\poltypesetalltrue` to get all of them in output).

> The following macros (whose meanings will be found in the package code) can be re-defined for customization. Their default definitions are expandable, but this is not a requirement.

#### `\PolTypesetCmd{}`

Syntax: `\PolTypesetCmd{<raw_coeff>}`

> Its package definition checks if the coefficient is `1` or `-1` and then skips printing the `1`, except for the coefficient of degree zero. Also it sets the conditional deciding behaviour of \PolIfCoeffIsPlusOrMinusOne{T}{F}.

> The actual printing of the coefficients, when not equal to plus or minus one, is handled by \PolTypesetOne{<raw_coeff>}.

##### `\PolIfCoeffIsPlusOrMinusOne{}{}`

Syntax: `\PolIfCoeffIsPlusOrMinusOne{T}{F}`

> This macro is a priori undefined.

> It is defined via the default \PolTypesetCmd{<raw_coeff>} to be used if needed in the execution of \PolTypesetMonomialCmd, e.g. to insert a `\cdot` in front of `\PolVar^{\PolIndex}` if the coefficient is not plus or minus one.

> The macro will execute `T` if the coefficient has been found to be plus or minus one, and `F` if not. It chooses expandably between `T` and `F`.

#### `\PolTypesetOne{}`

Syntax: `\PolTypesetOne{<raw_coeff>}`

> Defaults to `\xintTeXsignedFrac` (LaTeX) or `\xintTeXsignedOver` (else). But these xintfrac old legacy macros are a bit annoying as they insist in exhibiting a power of ten rather than using simpler decimal notation.

> As alternative, one can do definitions such as:

>     \def\PolTypesetOne#1{\xintDecToString{\xintREZ{#1}}}
>     % or with LaTeX+siunitx for example
>     \renewcommand\PolTypesetOne[1]{\num{\xintPFloat[5]{#1}}}
>     % (as \num of siunitx understands floating point notation)
>     \renewcommand\PolTypesetOne[1]{\num{\xintRound{4}{#1}}}

#### `\PolTypesetMonomialCmd`

> This decides how a monomial (in variable `\PolVar` and with exponent `\PolIndex`) is to be printed. The default does nothing for the constant term, `\PolVar` for the first degree and `\PolVar^{\PolIndex}` for higher degrees monomials. Beware that `\PolIndex` expands to digit tokens and needs termination in `\ifnum` tests.

#### `\PolTypesetCmdPrefix{}`

Syntax: `\PolTypesetCmdPrefix{<raw_coeff>}`

> Expands to a `+` if the `raw_coeff` is zero or positive, and to nothing if `raw_coeff` is negative, as in latter case the `\xintTeXsignedFrac` (or `\xintTeXsignedOver`) used by \PolTypesetCmd{<raw_coeff>} will put the `-` sign in front of the fraction (if it is a fraction) and this will thus serve as separator in the typeset formula. Not used for the first term.

### `\PolTypeset*[]{}`

Syntax: `\PolTypeset*[<letter>]{<pol. expr.>}`

> Typesets in ascending powers. The `<letter>` optional argument (after the `*`) declares the letter to use in the output. As for \PolTypeset, it defaults to `x`.

> To modify the expected `x` in the input, see \PolToExprInVar.

> Extended at `0.8` to accept general expressions and not only polynomial names.

### `\PolLet{}={}`

Syntax: `\PolLet{<polname_2>}={<polname_1>}`

> Makes a copy of the already defined polynomial `polname_1` to a new one `polname_2`. This has the same effect as `\PolDef{<polname_2>}{<polname_1>(x)}` or (better) `\PolDef{<polname_2>}{<polname_1>}` but with less overhead. The `=` is optional.

### `\PolGlobalLet{}={}`

Syntax: `\PolGlobalLet{<polname_2>}={<polname_1>}`

> Acts globally.

### `\PolAssign{}\toarray{}`

Syntax: `\PolAssign{<polname>}\toarray{<\macro>}`

> Defines a one-argument expandable macro `\macro{#1}` which expands to the (raw) #1th polynomial coefficient.

> - Attention, coefficients here are indexed starting at 1. This is an unfortunate legacy situation related to the original indexing convention in xinttools arrays.

> - With #1=-1, -2, …, `\macro{#1}` returns leading coefficients.

> - With #1=0, returns the number of coefficients, i.e. `1 + deg f` for non-zero polynomials.

> - Out-of-range #1’s return `0/1[0]`.

> See also \PolNthCoeff{<polname>}{<index>}.

### `\PolGet{}\fromarray{}`

Syntax: `\PolGet{<polname>}\fromarray{<\macro>}`

> Does the converse operation to `\PolAssign{<polname>}\toarray\macro`. Each individual `\macro{<value>}` gets expanded in an `\edef` and then normalized via xintfrac‘s macro `\xintRaw`.

> The leading zeros are removed from the polynomial.

> (contrived) Example:

>     \xintAssignArray{1}{-2}{5}{-3}\to\foo
>     \PolGet{f}\fromarray\foo

> This will define `f` as would have `\poldef f(x):=1-2x+5x^2-3x^3;`.

### `\PolFromCSV{}{}`

Syntax: `\PolFromCSV{<polname>}{<csv>}`

> Defines a polynomial directly from the comma separated list of values (or a macro expanding to such a list) of its coefficients, the first item gives the constant term, the last item gives the leading coefficient, except if zero, then it is dropped (iteratively). List items are each expanded in an `\edef` and then put into normalized form via xintfrac‘s macro `\xintRaw`.

> As leading zero coefficients are removed:

>     \PolFromCSV{f}{0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

> defines the zero polynomial, which holds only one coefficient.

> See also expandable macro \PolToCSV{<polname>}.

### `\PolMapCoeffs{}{}`

Syntax: `\PolMapCoeffs{\macro}{<polname>}`

> It modifies (‘in-place’: original coefficients get lost) each coefficient of the defined polynomial via the expandable macro `\macro`. The degree is adjusted as necessary if some leading coefficients vanish after the operation.

> In the replacement text of `\macro`, `\index` expands to the coefficient index (starting at zero for the constant term).

> Notice that `\macro` will have to handle inputs in the xintfrac internal format. This means that it probably will have to be expressed in terms of macros from the xintfrac package.

> Example:

>     \def\foo#1{\xintMul{#1}{\the\numexpr\index*\index\relax}}

> (or with `\xintSqr{\index}`) to replace `n`-th coefficient `f_n` by `f_n*n^2`.

### `\PolReduceCoeffs{}`

Syntax: `\PolReduceCoeffs{<polname>}`

> Reduces the internal representations of the coefficients to their lowest terms.

### `\PolReduceCoeffs*{}`

Syntax: `\PolReduceCoeffs*{<polname>}`

> Reduces the internal representations of the coefficients to their lowest terms, but ignoring a possible separated “power of ten part”.

> For example, xintfrac stores an `30e2/50` input as `30/50` with a separate `10^2` part. This will thus get replaced by `3e^2/5` (or rather whatever xintfrac uses for internal representation), and not by `60` as would result from complete reduction.

> Evaluations with polynomials treated by this can be much faster than with those handled by the non-starred variant \PolReduceCoeffs{<polname>}: as the numerators and denominators remain generally smaller.

### `\PolMakeMonic{}`

Syntax: `\PolMakeMonic{<polname>}`

> Divides by the leading coefficient. It is recommended to execute \PolReduceCoeffs*{<polname>} immediately afterwards. This is not done automatically, in case the original polynomial had integer coefficients and the user wants to keep the leading one as common denominator for typesetting purposes.

### `\PolMakePrimitive{}`

Syntax: `\PolMakePrimitive{<polname>}`

> Divides by the integer content see (\PolIContent). This thus produces a polynomial with integer coefficients having no common factor. The sign of the leading coefficient is not modified.

### `\PolDiff{}{}`

Syntax: `\PolDiff{<polname_1>}{<polname_2>}`

> This sets `polname_2` to the first derivative of `polname_1`. It is allowed to issue `\PolDiff{f}{f}`, effectively replacing `f` by `f'`.

> Coefficients of the result `polname_2` are irreducible fractions (see Technicalities for the whole story.)

### `\PolDiff[]{}{}`

Syntax: `\PolDiff[N]{<polname_1>}{<polname_2>}`

> This sets `polname_2` to the `N`-th derivative of `polname_1`. Identical arguments is allowed. With `N=0`, same effect as `\PolLet{<polname_2>}={<polname_1>}`. With negative `N`, switches to using `\PolAntiDiff`.

### `\PolAntiDiff{}{}`

Syntax: `\PolAntiDiff{<polname_1>}{<polname_2>}`

> This sets `polname_2` to the primitive of `polname_1` vanishing at zero.

> Coefficients of the result `polname_2` are irreducible fractions (see Technicalities for the whole story.)

### `\PolAntiDiff[]{}{}`

Syntax: `\PolAntiDiff[N]{<polname_1>}{<polname_2>}`

> This sets `polname_2` to the result of `N` successive integrations on `polname_1`. With negative `N`, it switches to using `\PolDiff`.

### `\PolDivide{}{}{}{}`

Syntax: `\PolDivide{<polname_1>}{<polname_2>}{<polname_Q>}{<polname_R>}`

> This sets `polname_Q` and `polname_R` to be the quotient and remainder in the Euclidean division of `polname_1` by `polname_2`.

### `\PolQuo{}{}{}`

Syntax: `\PolQuo{<polname_1>}{<polname_2>}{<polname_Q>}`

> This sets `polname_Q` to be the quotient in the Euclidean division of `polname_1` by `polname_2`.

### `\PolRem{}{}{}`

Syntax: `\PolRem{<polname_1>}{<polname_2>}{<polname_R>}`

> This sets `polname_R` to be the remainder in the Euclidean division of `polname_1` by `polname_2`.

### `\PolGCD{}{}{}`

Syntax: `\PolGCD{<polname_1>}{<polname_2>}{<polname_GCD>}`

> This sets `polname_GCD` to be the (monic) GCD of `polname_1` and `polname_2`. It is a unitary polynomial except if both `polname_1` and `polname_2` vanish, then `polname_GCD` is the zero polynomial.

### Root localization routines via the Sturm Theorem

As \PolToSturm{<polname>}{<sturmname>} and \PolSturmIsolateZeros{<sturmname>} and variants declare additional polynomial or scalar variables with names based on `<sturmname>` as prefix, it is advisable to keep the `<sturmname>` namespace separate from the one applying to `\xintexpr` variables generally, or to polynomials.

#### `\PolToSturm{}{}`

Syntax: `\PolToSturm{<polname>}{<sturmname>}`

> With `<polname>` being for example `P`, and `<sturmname>` being for example `S`, the macro starts by computing the derivative `P'`, then computes the opposite of the remainder in the euclidean division of `P` by `P'`, then the opposite of the remainder in the euclidean division of `P'` by the first obtained polynomial, etc… Up to signs following the `--++--++...` pattern, these are the same remainders as in the Euclide algorithm applied to the computation of the GCD of `P` and `P'`.

> The precise process differs from the above description: the algorithm first sets `S_0_` to be the primitive part of `P` and `S_1_` to be the primitive part of `P'` (see \PolIContent{<polname>}), then at each step the remainder is made primitive and stored for internal reference as `S_k_`, so only integer-coefficients polynomials are manipulated.

> Warning

> This exact procedure will perhaps in future be replaced by a sub-resultant algorithm, which may bring some speed gain in obtaining a pseudo-Sturm sequence, but some experimenting is needed, in the context of realistically realizable computations by the package; primitive polynomials although a bit costly have the smallest coefficients hence are the best for the kind of computations done for root localization, after having computed a Sturm sequence.

> The last non-zero primitivized remainder `S_N_` is, up to sign, the primitive part of the GCD of `P` and `P'`. Its roots (real and complex) are the multiple roots of the original `P`. The original `P` was “square-free” (i.e. did not have multiple real or complex roots) if and only if `S_N_` is a constant, which is then `+1` or `-1` (its value before primitivization is lost).

> The macro then divides each `S_k_` by `S_N_` and declares the quotients `S_k` as user polynomials for future use. By Gauss theorem about the contents of integer-coefficients polynomials, these `S_k` also are primitive integer-coefficients polynomials.

> This step will be referred to as normalization, and in this documentation the obtained polynomials are said to constitute the “Sturm chain” (or “Sturm sequence”), i.e. by convention the “Sturm chain polynomials” are square-free and primitive. The possibly non-square-free ones are referred to as non-normalized.

> As an exception to the rule, if the original `P` was “square-free” (i.e. did not have multiple real or complex roots) then normalization is skipped (in that case `S_N_` is either `+1` or `-1`), so `S_0_` is exactly the primitive part of starting polynomial `P`, in the “square-free” case.

> The next logical step is to execute \PolSturmIsolateZeros{S} or one of its variants. Be careful not to use the names `sturmname_0`, `sturmname_1`, etc… for defining other polynomials after having done `\PolToSturm{<polname>}{<sturmname>}` and before executing `\PolSturmIsolateZeros{<sturmname>}` or its variants else the latter will behave erroneously.

> Note

> The declaration of the `S_k`‘s will overwrite with no warning previously declared polynomials with identical names `S_k`, i.e. `<sturmname>_k`. This is why the macro was designed to expect two names: `<polname>` and `<sturmname>`.

> It is allowed to use the polynomial name `P` as Sturm chain name `S`: `\PolToSturm{P}{P}`, but this is considered bad practice for the reason mentioned in the previous paragraph.

> Furthermore, \PolSturmIsolateZeros creates xintexpr variables whose names start with `<sturmname>L_`, `<sturmname>R_`, and `<sturmname>Z_`, also `<sturmname>M_` for holding the multiplicities, and this may overwrite pre-existing user-defined xintexpr variables.

> Warning

> The reason why the `S_k`‘s are declared as polynomials is that the associated polynomial functions are needed to compute the sign changes in the Sturm sequence evaluated at a given location, as this is the basis mechanism of \PolSturmIsolateZeros (on the basis of the Sturm theorem).

> It is possible that in future the package will only internally construct such polynomial functions and only the starred variant will make the normalized (i.e. square-free) Sturm sequence public.

> The integer `N` giving the length of the Sturm chain `S_0`, `S_1`, …, `S_N` is available as \PolSturmChainLength{<sturmname>}. If all roots of original `P` are real, then `N` is both the number of distinct real roots and the degree of `S_0`. In the case of existence of complex roots, the number of distinct real roots is at most `N` and `N` is at most the degree of `S_0`.

#### `\PolToSturm*{}{}`

Syntax: `\PolToSturm*{<polname>}{<sturmname>}`

> Does the same as un-starred version and additionally it keeps for user usage the memory of the un-normalized (but still made primitive) Sturm chain polynomials `sturmname_k_`, `k=0,1, ..., N`, with `N` being \PolSturmChainLength{<sturmname>}.

#### `\PolSturmIsolateZeros{}`

Syntax: `\PolSturmIsolateZeros{<sturmname>}`

> The macro locates, using the Sturm Theorem, as many disjoint intervals as there are distinct real roots.

> Important

> The Sturm chain must have been produced by an earlier \PolToSturm{<polname>}{<sturmname>}.

> After its execution they are two types of such intervals (stored in memory and accessible via macros or xintexpr variables, see below):

> - singleton `{a}`: then `a` is a root, (necessarily a decimal number, but not all such decimal numbers are exactly identified yet).

> - open intervals `(a,b)`: then there is exactly one root `z` such that `a < z < b`, and the end points are guaranteed to not be roots.

> The interval boundaries are decimal numbers, originating in iterated decimal subdivision from initial intervals `(-10^E, 0)` and `(0, 10^E)` with `E` chosen initially large enough so that all roots are enclosed; if zero is a root it is always identified as such. The non-singleton intervals are of the type `(a/10^f, (a+1)/10^f)` with `a` an integer, which is neither `0` nor `-1`. Hence either `a` and `a+1` are both positive or they are both negative.

> One does not a priori know what will be the lengths of these intervals (except that they are always powers of ten), they vary depending on how many digits two successive roots have in common in their respective decimal expansions.

> Important

> If some two consecutive intervals share an end-point, no information is yet gained about the separation between the two roots which could at this stage be arbitrarily small.

> See \PolRefineInterval*{<sturmname>}{<index>} which addresses this issue.

> Let us suppose `<sturmname>` is `S`.

> The interval boundaries (and exactly found roots) are made available for future computations in `\xintexpr/xinteval` or `\poldef` as variables `SL_1`, `SL_2`, etc…, for the left end-points and `SR_1`, `SR_2`, …, for the right end-points.

> Additionally, xintexpr variable `SZ_1_isknown` will have value `1` if the root in the first interval is known, and `0` otherwise. And similarly for the other intervals.

> Important

> The variable declarations are done with no check of existence of previously existing variables with identical names.

> Also, macros \PolSturmIsolatedZeroLeft{<sturmname>}{<index>} and \PolSturmIsolatedZeroRight{<sturmname>}{<index>} are provided which expand to these same values, written in decimal notation (i.e. pre-processed by \PolDecToString.) And there is also \PolSturmIfZeroExactlyKnown{<sturmname>}{<index>}{T}{F}.

> Important

> Trailing zeroes in the stored decimal numbers accessible via the macros are significant: they are also present in the decimal expansion of the exact root, so as to be able for example to print out bounds of real roots with as many digits as is significant, even if the digits are zeros.

> The start of the decimal expansion of the `<index>`-th root is given by \PolSturmIsolatedZeroLeft{<sturmname>}{<index>} if the root is positive, and by PolSturmIsolatedZeroRight{<sturmname>}{<index>} if the root is neagtive. These two decimal numbers are either both zero or both of the same sign.

> The number of distinct roots is obtainable expandably as \PolSturmNbOfIsolatedZeros{<sturmname>}.

> Furthermore \PolSturmNbOfRootsOf{<sturmname>}\LessThanOrEqualTo{<value>} and \PolSturmNbOfRootsOf{<sturmname>}\LessThanOrEqualToExpr{<num. expr.>}. will expandably compute respectively the number of real roots at most equal to `value` or `expression`, and the same but with multiplicities.

> These variables and macros are automatically updated in case of subsequent usage of \PolRefineInterval*{<sturmname>}{<index>} or other localization improving macros.

> Note

> The current polexpr implementation defines the xintexpr variables and xinttools arrays as described above with global scope. On the other hand the Sturm sequence polynomials obey the current scope.

> This is perhaps a bit inconsistent and may change in future.

> Note

> The results are exact bounds for the mathematically exact real roots.

> Future releases will perhaps also provide macros based on Newton or Regula Falsi methods. Exact computations with such methods lead however quickly to very big fractions, and this forces usage of some rounding scheme for the abscissas if computation times are to remain reasonable. This raises issues of its own, which are studied in numerical mathematics.

#### `\PolSturmIsolateZeros*{}`

Syntax: `\PolSturmIsolateZeros*{<sturmname>}`

> The macro does the same as \PolSturmIsolateZeros{<sturmname>} and then in addition it does the extra work to determine all multiplicities of the real roots.

> After execution, \PolSturmIsolatedZeroMultiplicity{<sturmname>}{<index>} expands to the multiplicity of the root located in the `index`-th interval (intervals are enumerated from left to right, with index starting at `1`).

> Furthermore, if for example the `<sturmname>` is `S`, xintexpr variables `SM_1`, `SM_2`… hold the multiplicities thus computed.

> Note

> Somewhat counter-intuitively, it is not necessary to have executed the \PolToSturm* starred variant: during its execution, \PolToSturm, even though it does not declare the non-square-free Sturm chain polynomials as user-level genuine polynomials, stores their data in private macros.

> See `The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots` example in `polexpr-examples.pdf`.

##### `\PolSturmIsolateZerosAndGetMultiplicities{}`

Syntax: `\PolSturmIsolateZerosAndGetMultiplicities{<sturmname>}`

> This is another name for \PolSturmIsolateZeros*{<sturmname>}.

#### `\PolSturmIsolateZeros**{}`

Syntax: `\PolSturmIsolateZeros**{<sturmname>}`

> The macro does the same as \PolSturmIsolateZeros*{<sturmname>} and in addition it does the extra work to determine all the rational roots.

> Note

> After execution of this macro, a root is “known” if and only if it is rational.

> Furthermore, primitive polynomial `sturmname_sqf_norr` is created to match the (square-free) `sturmname_0` from which all rational roots have been removed. The number of distinct rational roots is thus the difference between the degrees of these two polynomials (see also \PolSturmNbOfRationalRoots{<sturmname>}).

> And `sturmname_norr` is `sturmname_0_` from which all rational roots have been removed, i.e. it contains the irrational roots of the original polynomial, with the same multiplicities.

> See `A degree five polynomial with three rational roots` in `polexpr-examples.pdf`.

##### `\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots`

Syntax: `\PolSturmIsolateZerosGetMultiplicitiesAndRationalRoots`

> This is another name for \PolSturmIsolateZeros**{<sturmname>}.

#### `\PolSturmIsolateZerosAndFindRationalRoots{}`

Syntax: `\PolSturmIsolateZerosAndFindRationalRoots{<sturmname>}`

> This works exactly like \PolSturmIsolateZeros**{<sturmname>} (inclusive of declaring the polynomials `sturmname_sqf_norr` and `sturmname_norr` with no rational roots) except that it does not compute the multiplicities of the non-rational roots.

> Note

> There is no macro to find the rational roots but not compute their multiplicities at the same time.

> Attention

> This macro does not define xintexpr variables `sturmnameM_1`, `sturmnameM_2`, … holding the multiplicities and it leaves the multiplicity array (whose accessor is \PolSturmIsolatedZeroMultiplicity{<sturmname>}{<index>}) into a broken state, as all non-rational roots will supposedly have multiplicity one. This means that the output of \PolPrintIntervals* will be erroneous regarding the multiplicities of irrational roots.

> I decided to document it because finding multiplicities of the non rational roots is somewhat costly, and one may be interested only into finding the rational roots (of course random polynomials with integer coefficients will not have any rational root anyhow).

#### `\PolRefineInterval*{}{}`

Syntax: `\PolRefineInterval*{<sturmname>}{<index>}`

> The `index`-th interval (starting indexing at one) is further subdivided as many times as is necessary in order for the newer interval to have both its end-points distinct from the end-points of the original interval. As a consequence, the `k`th root is then strictly separated from the other roots.

#### `\PolRefineInterval[]{}{}`

Syntax: `\PolRefineInterval[N]{<sturmname>}{<index>}`

> The `index`-th interval (starting count at one) is further subdivided once, reducing its length by a factor of 10. This is done `N` times if the optional argument `[N]` is present.

#### `\PolEnsureIntervalLength{}{}{}`

Syntax: `\PolEnsureIntervalLength{<sturmname>}{<index>}{<exponent>}`

> The `index`-th interval is subdivided until its length becomes at most `10^E`. This means (for `E<0`) that the first `-E` digits after decimal mark of the `k`th root will then be known exactly.

#### `\PolEnsureIntervalLengths{}{}`

Syntax: `\PolEnsureIntervalLengths{<sturmname>}{<exponent>}`

> The intervals as obtained from `\PolSturmIsolateZeros` are (if necessary) subdivided further by (base 10) dichotomy in order for each of them to have length at most `10^E`.

> This means that decimal expansions of all roots will be known with `-E` digits (for `E<0`) after decimal mark.

#### `\PolSetToSturmChainSignChangesAt{}{}{}`

Syntax: `\PolSetToSturmChainSignChangesAt{\foo}{<sturmname>}{<value>}`

> Sets macro `\foo` to store the number of sign changes in the already computed normalized Sturm chain with name prefix `<sturmname>`, at location `<value>` (which must be in format as acceptable by the xintfrac macros.)

> The definition is made with global scope. For local scope, use `[\empty]` as extra optional argument.

> One can use this immediately after creation of the Sturm chain.

#### `\PolSetToNbOfZerosWithin{}{}{}{}`

Syntax: `\PolSetToNbOfZerosWithin{\foo}{<sturmname>}{<value_left>}{<value_right>}`

> Sets, assuming the normalized Sturm chain has been already computed, macro `\foo` to store the number of roots of `sturmname_0` in the interval `(value_left, value_right]`. The macro first re-orders end-points if necessary for `value_left <= value_right` to hold.

> In accordance to Sturm Theorem this is computed as the difference between the number of sign changes of the Sturm chain at `value_right` and the one at `value_left`.

> The definition is made with global scope. For local scope, use `[\empty]` as extra optional argument.

> One can use this immediately after creation of a Sturm chain.

> See also the expandable \PolSturmNbOfRootsOf{<sturmname>}\LessThanOrEqualTo{value}, which however requires prior execution of \PolSturmIsolateZeros.

> See also the expandable \PolSturmNbWithMultOfRootsOf{<sturmname>}\LessThanOrEqualTo{value} which requires prior execution of \PolSturmIsolateZeros*.

### Displaying the found roots: `\PolPrintIntervals[<varname>]{}`

Syntax: `\PolPrintIntervals[<varname>]{<sturmname>}`

> This is a convenience macro which prints the bounds for the roots `Z_1`, `Z_2`, … (the optional argument `varname` allows to specify a replacement for the default `Z`). This will be done (by default) in a math mode `array`, one interval per row, and pattern `rcccl`, where the second and fourth column hold the `<` sign, except when the interval reduces to a singleton, which means the root is known exactly.

> Note

> The explanations here and in this section are for LaTeX. With other TeX macro formats, the LaTeX syntax such as for example `\begin{array}{rcccl}` which appears in the documentation here is actually replaced with quasi-equivalent direct use of TeX primitives.

> The next macros which govern its output.

#### `\PolPrintIntervalsNoRealRoots`

> Executed in place of an `array` environment, when there are no real roots. Default definition:

>     \newcommand\PolPrintIntervalsNoRealRoots{}

#### `\PolPrintIntervalsBeginEnv`

> Default definition (given here for LaTeX, Plain has a variant):

>     \newcommand\PolPrintIntervalsBeginEnv{\[\begin{array}{rcccl}}

> A simpler `center` environment provides a straightforward way to obtain a display allowing pagebreaks. Of course redefinitions must at any rate be kept in sync with \PolPrintIntervalsKnownRoot and \PolPrintIntervalsUnknownRoot.

> Prior to `0.8.6` it was not possible to use here for example `\begin{align}` due to the latter executing twice in contents.

#### `\PolPrintIntervalsEndEnv`

> Default definition:

>     \newcommand\PolPrintIntervalsEndEnv{\end{array}\]}

#### `\PolPrintIntervalsRowSeparator`

> Expands by default to `\\` with LaTeX and to `\cr` with Plain

> Added at `0.8.6`.

#### `\PolPrintIntervalsKnownRoot`

> Default definition:

>     \newcommand\PolPrintIntervalsKnownRoot{%
>       &&\PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}%
>       &=&\PolPrintIntervalsPrintExactZero
>     }

#### `\PolPrintIntervalsUnknownRoot`

> Default definition:

>     \newcommand\PolPrintIntervalsUnknownRoot{%
>       \PolPrintIntervalsPrintLeftEndPoint&<&%
>       \PolPrintIntervalsTheVar_{\PolPrintIntervalsTheIndex}&<&%
>       \PolPrintIntervalsPrintRightEndPoint
>     }

#### `\PolPrintIntervalsPrintExactZero`

> Default definition:

>     \newcommand\PolPrintIntervalsPrintExactZero{\PolPrintIntervalsTheLeftEndPoint}

#### `\PolPrintIntervalsPrintLeftEndPoint`

> Default definition:

>     \newcommand\PolPrintIntervalsPrintLeftEndPoint{\PolPrintIntervalsTheLeftEndPoint}

#### `\PolPrintIntervalsPrintRightEndPoint`

> Default definition is:

>     \newcommand\PolPrintIntervalsPrintRightEndPoint{\PolPrintIntervalsTheRightEndPoint}

#### `\PolPrintIntervals*[<varname>]{}`

Syntax: `\PolPrintIntervals*[<varname>]{<sturmname>}`

> This starred variant produces an alternative output (which displays the root multiplicity), and is provided as an example of customization.

> As replacement for \PolPrintIntervalsKnownRoot, \PolPrintIntervalsPrintExactZero, \PolPrintIntervalsUnknownRoot it uses its own `\POL@@PrintIntervals...` macros. We only reproduce here one definition:

>     \newcommand\POL@@PrintIntervalsPrintExactZero{%
>        \displaystyle
>        \xintTeXsignedFrac{\PolPrintIntervalsTheLeftEndPoint}%
>     }%

> Multiplicities are printed using this auxiliary macro:

##### `\PolPrintIntervalsPrintMultiplicity`

> whose default definition is:

>     \newcommand\PolPrintIntervalsPrintMultiplicity{(\mbox{mult. }\PolPrintIntervalsTheMultiplicity)}

## Expandable macros

Note

At `0.8` `polexpr` is usable with Plain TeX and not only with LaTeX. Some examples given in this section may be using LaTeX syntax such as `\renewcommand`. Convert to TeX primitives as appropriate if testing with a non LaTeX macro format.

These macros expand completely in two steps except `\PolToExpr` and `\PolToFloatExpr` which need a `\write`, `\edef` or a `\csname...\endcsname` context.

### `\PolToExpr{}`

Syntax: `\PolToExpr{<pol. expr.>}`

> Produces expandably [3] the string `coeff_N*x^N+...`, i.e. the polynomial is using descending powers.

> [3]

> requires exhaustive expansion, for example as triggered by `\write` or `\edef`.

> Since `0.8` the input is not restricted to be a polynomial name but is allowed to be an arbitrary expression. Then `x` is expected as indeterminate but this can be customized via \PolToExprInVar.

> The output uses the letter `x` by default, this is customizable via \PolToExprVar. The default output is compatible both with

> - the Maple’s input format,

> - and the PSTricks `\psplot[algebraic]` input format.

> Attention that it is not compatible with Python, see further \PolToExprCaret in this context.

> The following applies:

> - vanishing coefficients are skipped (issue `\poltoexpralltrue` to override this and produce output such as `x^3+0*x^2+0*x^1+0`),

> - negative coefficients are not prefixed by a `+` sign (else, Maple would not be happy),

> - coefficients numerically equal to `1` (or `-1`) are present only via their sign,

> - the letter `x` is used and the degree one monomial is output as `x`, not as `x^1`.

> - (`0.8`) the caret `^` is of catcode 12. This means that one can for convenience typeset in regular text mode, for example using `\texttt` (in LaTeX). But TeX will not know how to break the expression across end-of-lines anyhow. Formerly `^` was suitable for math mode but as the exponent is not braced this worked only for polynomials of degrees at most 9. Anyhow this is not supposed to be a typesetting macro.

> Complete customization is possible, see the next macros. Any user redefinition must maintain the expandability property.

#### `\PolToExprVar`

> Defaults to `x`. The letter used in the macro output.

#### `\PolToExprInVar`

> Defaults to `x`: the letter used as the polynomial indeterminate in the macro input:

>     \def\PolToExprInVar{x}% (default)

> Recall that declared polynomials are more efficiently used in algebraic expressions without the `(x)`, i.e. `P*Q` is better than `P(x)*Q(x)`. Thus the input, even if an expression, does not have to contain any `x`.

> (new with `0.8`)

#### `\PolToExprTimes`

> Defaults to `*`.

#### `\PolToExprCaret`

> Defaults to `^` of catcode 12. Set it to expand to `**` for Python compatible output.

> (new with `0.8`)

#### `\PolToExprCmd{}`

Syntax: `\PolToExprCmd{<raw_coeff>}`

> Defaults to `\xintPRaw{\xintRawWithZeros{#1}}`.

> This means that the coefficient value is printed-out as a fraction `a/b`, skipping the `/b` part if `b` turns out to be one.

> Configure it to be `\xintPRaw{\xintIrr{#1}}` if the fractions must be in irreducible terms.

> An alternative is `\xintDecToString{\xintREZ{#1}}` which uses integer or decimal fixed point format such as `23.0071` if the internal representation of the number only has a power of ten as denominator (the effect of `\xintREZ` here is to remove trailing decimal zeros). The behaviour of `\xintDecToString` is not yet stable for other cases, and for example at time of writing no attempt is made to identify inputs having a finite decimal expansion so for example `23.007/2` or `23.007/25` can appear in output and not their finite decimal expansion with no denominator.

#### `\PolToExprOneTerm{}{}`

Syntax: `\PolToExprOneTerm{<raw_coeff>}{<exponent>}`

> This is the macro which from the coefficient and the exponent produces the corresponding term in output, such as `2/3*x^7`.

> For its default definition, see the source code. It uses \PolToExprCmd, \PolToExprTimes, \PolToExprVar and \PolToExprCaret.

#### `\PolToExprOneTermStyleA{}{}`

Syntax: `\PolToExprOneTermStyleA{<raw_coeff>}{<exponent>}`

> This holds the default package meaning of `\PolToExprOneTerm`.

#### `\PolToExprOneTermStyleB{}{}`

Syntax: `\PolToExprOneTermStyleB{<raw_coeff>}{<exponent>}`

> This holds an alternative meaning, which puts the fractional part of a coefficient after the monomial, i.e. like this:

>     2*x^11/3+3*x^8/7-x^5-x^4/4-x^3-x^2/2-2*x+1

> \PolToExprCmd isn’t used at all in this style. But \PolToExprTimes, \PolToExprVar and \PolToExprCaret are obeyed.

> To activate it use `\let\PolToExprOneTerm\PolToExprOneTermStyleB`. To revert to the package default behaviour, issue `\let\PolToExprOneTerm\PolToExprOneTermStyleA`.

#### `\PolToExprTermPrefix{}`

Syntax: `\PolToExprTermPrefix{<raw_coeff>}`

> It receives as argument the coefficient. Its default behaviour is to produce a `+` if the coefficient is positive, which will thus serve to separate the monomials in the output. This is to match the default for \PolToExprCmd{<raw_coeff>} which in case of a positive coefficient does not output an explicit `+` prefix.

### `\PolToFloatExpr{}`

Syntax: `\PolToFloatExpr{<pol. expr.>}`

> Similar to \PolToExpr{<pol. expr.>} but using \PolToFloatExprCmd{<raw_coeff>} which by default rounds and converts the coefficients to floating point format.

> Note

> This is unrelated to \PolGenFloatVariant{<polname>}: \PolToFloatExprCmd{<raw_coeff>} operates on the exact coefficients anew (and may thus produce something else than the coefficients of the polynomial function acting in `\xintfloateval` if the floating point precision was changed in between).

> Extended at `0.8` to accept general expressions as input.

#### `\PolToFloatExprOneTerm{}{}`

Syntax: `\PolToFloatExprOneTerm{<raw_coeff>}{<exponent>}`

> Similar to \PolToExprOneTerm{<raw_coeff>}{<exponent>}. But does not treat especially coefficients equal to plus or minus one.

#### `\PolToFloatExprCmd{}`

Syntax: `\PolToFloatExprCmd{<raw_coeff>}`

> The one-argument macro used by `\PolToFloatExprOneTerm`. It defaults to `\xintPFloat{#1}`, which trims trailing zeroes.

> changed at 0.8.2 Formerly it was using `\xintFloat`.

### `\PolToExpr*{}`

Syntax: `\PolToExpr*{<pol. expr.>}`

> Ascending powers: `coeff_0+coeff_1*x+coeff_2*x^2+...`.

> Extended at `0.8` to accept general expressions as input.

> Customizable with the same macros as for \PolToExpr{<pol. expr.>}.

### `\PolToFloatExpr*{}`

Syntax: `\PolToFloatExpr*{<pol. expr.>}`

> Ascending powers.

> Extended at `0.8` to accept general expressions as input.

### `\PolNthCoeff{}{}`

Syntax: `\PolNthCoeff{<polname>}{<index>}`

> It expands to the raw `N`-th coefficient (`N=0` corresponds to the constant coefficient). If `N` is out of range, zero (in its default xintfrac format `0/1[0]`) is returned.

> Negative indices `N=-1`, `-2`, … return the leading coefficient, sub-leading coefficient, …, and finally `0/1[0]` for `N<-1-degree`.

### `\PolLeadingCoeff{}`

Syntax: `\PolLeadingCoeff{<polname>}`

> Expands to the leading coefficient.

### `\PolDegree{}`

Syntax: `\PolDegree{<polname>}`

> It expands to the degree. This is `-1` if zero polynomial but this may change in future. Should it then expand to `-\infty` ?

### `\PolIContent{}`

Syntax: `\PolIContent{<polname>}`

> It expands to the contents of the polynomial, i.e. to the positive fraction such that dividing by this fraction produces a polynomial with integer coefficients having no common prime divisor.

> See \PolMakePrimitive.

### `\PolToList{}`

Syntax: `\PolToList{<polname>}`

> Expands to `{coeff_0}{coeff_1}...{coeff_N}` with `N` = degree, and `coeff_N` the leading coefficient (the zero polynomial does give `{0/1[0]}` and not an empty output.)

### `\PolToCSV{}`

Syntax: `\PolToCSV{<polname>}`

> Expands to `coeff_0, coeff_1, coeff_2, ....., coeff_N`, starting with constant term and ending with leading coefficient. Converse to \PolFromCSV{<polname>}{<csv>}.

### `\PolEval{}\AtExpr{}`

Syntax: `\PolEval{<polname>}\AtExpr{<num. expr.>}`

> Same output as `\xinteval{polname(numerical expression)}`.

### `\PolEval{}\At{}`

Syntax: `\PolEval{<polname>}\At{<value>}`

> Evaluates the polynomial at the given value which must be in (or expand to) a format acceptable to the xintfrac macros.

### `\PolEvalReduced{}\AtExpr{}`

Syntax: `\PolEvalReduced{<polname>}\AtExpr{<num. expr.>}`

> Same output as `\xinteval{reduce(polname(numerical expression))}`.

### `\PolEvalReduced{}\At{}`

Syntax: `\PolEvalReduced{<polname>}\At{<value>}`

> Evaluates the polynomial at the value which must be in (or expand to) a format acceptable to the xintfrac macros, and outputs an irreducible fraction.

### `\PolFloatEval{}\AtExpr{}`

Syntax: `\PolFloatEval{<polname>}\AtExpr{<num. expr.>}`

> Same output as `\xintfloateval{polname(numerical expression)}`.

> Attention

> \PolGenFloatVariant must have been issued before.

> To use the exact coefficients with exactly executed additions and multiplications and do the rounding only as the final last step, the following syntax can be used: [4]

>     \xintfloateval{3.27*\xintexpr f(2.53)\relax^2}

> [4]

> Cf. xintexpr documentation about nested expressions.

### `\PolFloatEval{}\At{}`

Syntax: `\PolFloatEval{<polname>}\At{<value>}`

> Evaluates the polynomial at the value which must be in (or expand to) a format acceptable to the xintfrac macros.

### Expandable macros in relation to root localization via Sturm Theorem

#### `\PolSturmChainLength{}`

Syntax: `\PolSturmChainLength{<sturmname>}`

> Returns the integer `N` such that `sturmname_N` is the last one in the Sturm chain `sturmname_0`, `sturmname_1`, …

> See \PolToSturm{<polname>}{<sturmname>}.

#### `\PolSturmIfZeroExactlyKnown{}{}{}{}`

Syntax: `\PolSturmIfZeroExactlyKnown{<sturmname>}{<index>}{T}{F}`

> Executes `T` if the `index`-th interval reduces to a singleton, i.e. the root is known exactly, else `F`.

#### `\PolSturmIsolatedZeroLeft{}{}`

Syntax: `\PolSturmIsolatedZeroLeft{<sturmname>}{<index>}`

> Expands to the left end-point for the `index`-th interval, as computed by some earlier \PolSturmIsolateZeros{<sturmname>}.

> Note

> Execution of this macro after some \PolRefineInterval{<sturmname>}{<index>} will take into account the now known tighter bounds.

> The value is pre-formatted using \PolDecTostring.

#### `\PolSturmIsolatedZeroRight{}{}`

Syntax: `\PolSturmIsolatedZeroRight{<sturmname>}{<index>}`

> Expands to the right end-point for the `index`-th interval as computed by some earlier \PolSturmIsolateZeros{<sturmname>} and possibly refined afterwards.

> The value is pre-formatted using \PolDecTostring.

#### `\PolSturmIsolatedZeroMultiplicity{}{}`

Syntax: `\PolSturmIsolatedZeroMultiplicity{<sturmname>}{<index>}`

> Expands to the multiplicity of the unique root contained in the `index`-th interval.

> Attention

> A prior execution of \PolSturmIsolateZeros*{<sturmname>} is mandatory.

> See `The degree nine polynomial with 0.99, 0.999, 0.9999 as triple roots` in `polexpr-examples.pdf`.

#### `\PolSturmNbOfIsolatedZeros{}`

Syntax: `\PolSturmNbOfIsolatedZeros{<sturmname>}`

> Expands to the number of real roots of the polynomial `<sturmname>_0`, i.e. the number of distinct real roots of the polynomial originally used to create the Sturm chain via \PolToSturm{<polname>}{<sturmname>}.

Warning

The next few macros counting roots, with or without multiplicities, less than or equal to some value, are under evaluation and may be removed from the package if their utility is judged to be not high enough. They can be re-coded at user level on the basis of the other documented package macros anyway.

#### `\PolSturmNbOfRootsOf{}\LessThanOrEqualTo{}`

Syntax: `\PolSturmNbOfRootsOf{<sturmname>}\LessThanOrEqualTo{<value>}`

> Expands to the number of distinct roots (of the polynomial used to create the Sturm chain) less than or equal to the `value` (i.e. a number of fraction recognizable by the xintfrac macros).

> Attention

> \PolSturmIsolateZeros{<sturmname>} must have been executed beforehand.

> And the argument is a `<sturmname>`, not a `<polname>` (this is why the macro contains Sturm in its name), simply to be reminded of the above constraint.

#### `\PolSturmNbOfRootsOf{}\LessThanOrEqualToExpr{}`

Syntax: `\PolSturmNbOfRootsOf{<sturmname>}\LessThanOrEqualToExpr{<num. expr.>}`

> Expands to the number of distinct roots (of the polynomial used to create the Sturm chain) which are less than or equal to the given numerical expression.

> Attention

> \PolSturmIsolateZeros{<sturmname>} must have been executed beforehand.

#### `\PolSturmNbWithMultOfRootsOf{}\LessThanOrEqualTo{}`

Syntax: `\PolSturmNbWithMultOfRootsOf{<sturmname>}\LessThanOrEqualTo{<value>}`

> Expands to the number counted with multiplicities of the roots (of the polynomial used to create the Sturm chain) which are less than or equal to the given `value`.

> Attention

> \PolSturmIsolateZeros*{<sturmname>} (or the double starred variant) must have been executed beforehand.

#### `\PolSturmNbWithMultOfRootsOf{}\LessThanOrEqualToExpr{}`

Syntax: `\PolSturmNbWithMultOfRootsOf{<sturmname>}\LessThanOrEqualToExpr{<num. expr.>}`

> Expands to the total number of roots (counted with multiplicities) which are less than or equal to the given `expression`.

> Attention

> \PolSturmIsolateZeros*{<sturmname>} (or the double starred variant) must have been executed beforehand.

#### `\PolSturmNbOfRationalRoots{}`

Syntax: `\PolSturmNbOfRationalRoots{<sturmname>}`

> Expands to the number of rational roots (without multiplicities).

> Attention

> \PolSturmIsolateZeros**{<sturmname>} must have been executed beforehand.

#### `\PolSturmNbOfRationalRootsWithMultiplicities{}`

Syntax: `\PolSturmNbOfRationalRootsWithMultiplicities{<sturmname>}`

> Expands to the number of rational roots (counted with multiplicities).

> Attention

> \PolSturmIsolateZeros**{<sturmname>} must have been executed beforehand.

#### `\PolSturmRationalRoot{}{}`

Syntax: `\PolSturmRationalRoot{<sturmname>}{<k>}`

> Expands to the k-th rational root. They are enumerated from left to right starting at index value `1`.

> Attention

> \PolSturmIsolateZeros**{<sturmname>} must have been executed beforehand.

#### `\PolSturmRationalRootIndex{}{}`

Syntax: `\PolSturmRationalRootIndex{<sturmname>}{<k>}`

> Expands to the index of the `k`th rational root as part of the ordered real roots (counted without multiplicities). So \PolSturmRationalRoot{<sturmname>}{<k>} is equivalent to this nested call:

>     \PolSturmIsolatedZeroLeft{<sturmname>}{\PolSturmRationalRootIndex{<sturmname>}{<k>}}

> Attention

> \PolSturmIsolateZeros**{<sturmname>} must have been executed beforehand.

#### `\PolSturmRationalRootMultiplicity{}{}`

Syntax: `\PolSturmRationalRootMultiplicity{<sturmname>}{<k>}`

> Expands to the multiplicity of the `k`th rational root.

> Attention

> \PolSturmIsolateZeros**{<sturmname>} must have been executed beforehand.

#### `\PolIntervalWidth{}{}`

Syntax: `\PolIntervalWidth{<sturmname>}{<index>}`

> The `10^E` width of the current `index`-th root localization interval. Output is in xintfrac raw `1/1[E]` format (if not zero).

### Expandable macros for use within execution of `\PolPrintIntervals`

These macros are for usage within custom user redefinitions of \PolPrintIntervalsKnownRoot, \PolPrintIntervalsUnknownRoot, or in redefinitions of PolPrintIntervalsPrintExactZero (used in the default for the former) and of \PolPrintIntervalsPrintLeftEndPoint, \PolPrintIntervalsPrintRightEndPoint (used in the default for the latter).

#### `\PolPrintIntervalsTheVar`

> Expands to the name (default `Z`) used for representing the roots, which was passed as optional argument `varname` to \PolPrintIntervals[varname]{<sturmname>}.

#### `\PolPrintIntervalsTheIndex`

> Expands to the index of the considered interval (indexing starting at 1 for the leftmost interval).

#### `\PolPrintIntervalsTheSturmName`

> Expands to the argument which was passed as `<sturmname>` to \PolPrintIntervals[varname]{<sturmname>}.

#### `\PolPrintIntervalsTheLeftEndPoint`

> The left end point of the interval, as would be produced by \PolSturmIsolatedZeroLeft if it was used with arguments the Sturm chain name and interval index returned by \PolPrintIntervalsTheSturmName and \PolPrintIntervalsTheIndex.

#### `\PolPrintIntervalsTheRightEndPoint`

> The right end point of the interval, as would be produced by \PolSturmIsolatedZeroRight for this Sturm chain name and index.

#### `\PolPrintIntervalsTheMultiplicity`

> The multiplicity of the unique root within the interval of index \PolPrintIntervalsTheIndex. Makes sense only if the starred (or double-starred) variant of \PolSturmIsolateZeros was used earlier.

## TeX Booleans (with names enacting their defaults)

### `\xintverbosefalse`

> This is actually an xintexpr configuration. Setting it to `true` triggers the writing of information to the log when new polynomial or scalar variables are defined.

> Caution

> The macro and variable meanings as written to the log are to be considered unstable and undocumented internal structures.

### `\polnewpolverbosefalse`

> When `\poldef` is used, both a variable and a function are defined. The default `\polnewpolverbosefalse` setting suppresses the print-out to the log and terminal of the function macro meaning, as it only duplicates the information contained in the variable which is already printed out to the log and terminal.

> However \PolGenFloatVariant{<polname>} does still print out the information relative to the polynomial function it defines for use in `\xintfloateval{}` as there is no float polynomial variable, only the

> function, and it is the only way to see its rounded coefficients (`\xintverbosefalse` suppresses also that info).

> If set to `true`, it overrides in both cases `\xintverbosefalse`. The setting only affects polynomial declarations. Scalar variables such as those holding information on roots obey only the `\xintverbose...` setting.

> (new with `0.8`)

### `\poltypesetallfalse`

> If `true`, \PolTypeset will also typeset the vanishing coefficients.

### `\poltoexprallfalse`

> If `true`, \PolToExpr{<pol. expr.>} and \PolToFloatExpr{<pol. expr.>} will also include the vanishing coefficients in their outputs.

## Utilities

### `\PolDecToString{}`

Syntax: `\PolDecToString{decimal number}`

> This is a utility macro to print decimal numbers. It is an alias for `\xintDecToString`.

> For example `\PolDecToString{123.456e-8}` will expand to `0.00000123456` and `\PolDecToString{123.450e-8}` to `0.00000123450` which illustrates that trailing zeros are not trimmed.

> To trim trailing zeroes, one can use `\PolDecToString{\xintREZ{#1}}`.

> Attention that a.t.t.o.w. if the argument is for example `1/5`, the macro does not identify that this is in fact a number with a finite decimal expansion and it outputs `1/5`. See current xintfrac documentation.

### `\polexprsetup{key=val,...}`

> Serves to customize the package. Currently only two keys are recognized:

> - `norr`: the postfix that \PolSturmIsolateZeros**{<sturmname>} should append to `<sturmname>` to declare the primitive polynomial obtained from original one after removal of all rational roots. The default value is `_norr` (standing for “no rational roots”).

> - `sqfnorr`: the postfix that \PolSturmIsolateZeros**{<sturmname>} should append to `<sturmname>` to declare the primitive polynomial obtained from original one after removal of all rational roots and suppression of all multiplicities. The default value is `_sqf_norr` (standing for “square-free with no rational roots”).

> The package executes `\polexprsetup{norr=_norr, sqfnorr=_sqf_norr}` as default.

## Technicalities

- Do not use the underscore `_` as the first character of a polynomial name, even if of catcode letter. This may cause an infinite loop.

- The `@` is allowed in the names of polynomials, independently of whether it is of catcode letter or other. In defining macros which will use `\poldef` to create polynomials it seems reasonable to adopt the convention that `@` as first character in polynomial names is to be reserved to temporary auxiliary polynomials.

  Attention

  Do not use `@_` at start of polynomial names. This is reserved for internal usage by the package.

- Catcodes are set temporarily by \poldef macro to safe values prior to grab the polynomial expression up to the terminator `;`, and also by \PolDef prior to grab the brace-enclosed polynomial expression. This gives a layer of protection in case some package (for example the `babel-french` module) has made some characters active. It will fail though if the whole thing is located inside some definition of a macro done at a time the characters are active.

- Attention

  Contrarily to `\xintdefvar` and `\xintdeffunc` from xintexpr, `\poldef` uses a naive delimited macro to fetch up to the expression terminator `";"`, hence it will be fooled if some `;` is used inside the expression (which is possible as it appears in some xintexpr constructs). Work-around is to use curly braces around the inner semi-colons, or simpler to use `\PolDef`.

- As a consequence of xintfrac addition and subtraction always using least common multiples for the denominators, user-chosen common denominators (currently) survive additions and multiplications. For example, this:

      \poldef P(x):= 1/2 + 2/2*x + 3/2*x^3 + 4/2*x^4;
      \poldef Q(x):= 1/3 + (2/3)x + (3/3)x^3 + (4/3)x^4;
      \poldef PQ(x):= P*Q;

  gives internally the polynomial:

      1/6+4/6*x^1+4/6*x^2+6/6*x^3+20/6*x^4+16/6*x^5+9/6*x^6+24/6*x^7+16/6*x^8

  where all coefficients have the same denominator 6. Notice though that `\PolToExpr{PQ}` outputs the `6/6*x^3` as `x^3` because (by default) it recognizes and filters out coefficients equal to one or minus one. One can use for example `\PolToCSV{PQ}` to see the internally stored coefficients.

- \PolDiff{<polname_1>}{<polname_2>} always applies `\xintPIrr` to the resulting coefficients, which means that fractions are reduced to lowest terms but ignoring an already separated power of ten part `[N]` present in the internal representation. This is tentative and may change.

  Same remark for \PolAntiDiff{<polname_1>}{<polname_2>}.

- Currently, the package stores all coefficients from index `0` to index equal to the polynomial degree inside a single macro, as a list. This data structure is obviously very inefficient for polynomials of high degree and few coefficients (as an example with `\poldef f(x):=x^1000 + x^500;` the subsequent definition `\poldef g(x):= f(x)^2;` will do of the order of 1,000,000 multiplications and additions involvings only zeroes… which does take time). This may change in the future.

- As is to be expected internal structures of the package are barely documented and unstable. Don’t use them.