# Polynomials, exact arithmetic and number theory


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``` python
import rustymath as rm
```

## Polynomials

`rm.polynomial` evaluates polynomials, finds roots, constructs
coefficients from roots, differentiates and fits data, under the names
of `numpy.polynomial.polynomial`. Coefficients start from the constant
term. Evaluation, construction and derivatives take real or complex
arrays. `polyvalfromroots(x, roots, m=1)` evaluates directly from roots
without forming coefficients. The roots of x^3 - 6x^2 + 11x - 6 are 1, 2
and 3:

``` python
rm.polynomial.polyroots([-6, 11, -6, 1])
```

    Array([(1+0j), (2-0j), (3+0j)], dtype=complex128)

## Exact arithmetic

`rm.exact` computes with integers and `fractions.Fraction`. It provides
rational gcd, lcm, remainder and binomial coefficients, matrix solves,
pseudoinverses, determinants, permanents and row reduction, and
polynomial construction, evaluation and derivatives. Arguments can be
integers, Fractions, nested lists of them, or integer arrays. Results
are Fractions, in nested lists for arrays. Matrix calls take one matrix.
Ordinary `rm.linalg` calls stay floating-point.

``` python
from fractions import Fraction
```

``` python
rm.exact.solve([[2, 1], [1, 2]], [Fraction(1, 2), 0]), rm.exact.binom(Fraction(3, 2), 2)
```

    ([Fraction(1, 3), Fraction(-1, 6)], Fraction(3, 8))

## Number theory

`rm.number_theory` tests primality, factorises, finds the next or
previous prime, gives Euler’s totient and exact factorials, and lists
and counts primes, on Python integers of any size. Ctrl-C stops a long
call:

``` python
nt = rm.number_theory
nt.factor(2**64 + 1), nt.prime_count(10**8), nt.next_prime(10**20)
```

    ([(274177, 1), (67280421310721, 1)], 5761455, 100000000000000000039)

`rm.integer.encode` and `decode` convert mixed-radix integers in
`int64`. A result that does not fit raises `OverflowError`.
`rm.polynomial.decode` and `rm.exact.decode` accept field-valued bases
and digits.