homelab/home-bro-brain
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

switching to high quality piper tts and added label translations

Browse Source
This commit is contained in:
Matthias Hinrichs
2026-01-29 23:48:19 +01:00
commit d80c619df9
3934 changed files with 1451600 additions and 0 deletions
Download Patch File Download Diff File Expand all files Collapse all files
venv_piper/lib/python3.12/site-packages/mpmath/calculus/polynomials.py
+213
View File
@@ -0,0 +1,213 @@
from ..libmp.backend import xrange
from .calculus import defun
#----------------------------------------------------------------------------#
# Polynomials #
#----------------------------------------------------------------------------#
# XXX: extra precision
@defun
def polyval(ctx, coeffs, x, derivative=False):
r"""
Given coefficients `[c_n, \ldots, c_2, c_1, c_0]` and a number `x`,
:func:`~mpmath.polyval` evaluates the polynomial
.. math ::
P(x) = c_n x^n + \ldots + c_2 x^2 + c_1 x + c_0.
If *derivative=True* is set, :func:`~mpmath.polyval` simultaneously
evaluates `P(x)` with the derivative, `P'(x)`, and returns the
tuple `(P(x), P'(x))`.
>>> from mpmath import *
>>> mp.pretty = True
>>> polyval([3, 0, 2], 0.5)
2.75
>>> polyval([3, 0, 2], 0.5, derivative=True)
(2.75, 3.0)
The coefficients and the evaluation point may be any combination
of real or complex numbers.
"""
if not coeffs:
return ctx.zero
p = ctx.convert(coeffs[0])
q = ctx.zero
for c in coeffs[1:]:
if derivative:
q = p + x*q
p = c + x*p
if derivative:
return p, q
else:
return p
@defun
def polyroots(ctx, coeffs, maxsteps=50, cleanup=True, extraprec=10,
error=False, roots_init=None):
"""
Computes all roots (real or complex) of a given polynomial.
The roots are returned as a sorted list, where real roots appear first
followed by complex conjugate roots as adjacent elements. The polynomial
should be given as a list of coefficients, in the format used by
:func:`~mpmath.polyval`. The leading coefficient must be nonzero.
With *error=True*, :func:`~mpmath.polyroots` returns a tuple *(roots, err)*
where *err* is an estimate of the maximum error among the computed roots.
**Examples**
Finding the three real roots of `x^3 - x^2 - 14x + 24`::
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> nprint(polyroots([1,-1,-14,24]), 4)
[-4.0, 2.0, 3.0]
Finding the two complex conjugate roots of `4x^2 + 3x + 2`, with an
error estimate::
>>> roots, err = polyroots([4,3,2], error=True)
>>> for r in roots:
... print(r)
...
(-0.375 + 0.59947894041409j)
(-0.375 - 0.59947894041409j)
>>>
>>> err
2.22044604925031e-16
>>>
>>> polyval([4,3,2], roots[0])
(2.22044604925031e-16 + 0.0j)
>>> polyval([4,3,2], roots[1])
(2.22044604925031e-16 + 0.0j)
The following example computes all the 5th roots of unity; that is,
the roots of `x^5 - 1`::
>>> mp.dps = 20
>>> for r in polyroots([1, 0, 0, 0, 0, -1]):
... print(r)
...
1.0
(-0.8090169943749474241 + 0.58778525229247312917j)
(-0.8090169943749474241 - 0.58778525229247312917j)
(0.3090169943749474241 + 0.95105651629515357212j)
(0.3090169943749474241 - 0.95105651629515357212j)
**Precision and conditioning**
The roots are computed to the current working precision accuracy. If this
accuracy cannot be achieved in ``maxsteps`` steps, then a
``NoConvergence`` exception is raised. The algorithm internally is using
the current working precision extended by ``extraprec``. If
``NoConvergence`` was raised, that is caused either by not having enough
extra precision to achieve convergence (in which case increasing
``extraprec`` should fix the problem) or too low ``maxsteps`` (in which
case increasing ``maxsteps`` should fix the problem), or a combination of
both.
The user should always do a convergence study with regards to
``extraprec`` to ensure accurate results. It is possible to get
convergence to a wrong answer with too low ``extraprec``.
Provided there are no repeated roots, :func:`~mpmath.polyroots` can
typically compute all roots of an arbitrary polynomial to high precision::
>>> mp.dps = 60
>>> for r in polyroots([1, 0, -10, 0, 1]):
... print(r)
...
-3.14626436994197234232913506571557044551247712918732870123249
-0.317837245195782244725757617296174288373133378433432554879127
0.317837245195782244725757617296174288373133378433432554879127
3.14626436994197234232913506571557044551247712918732870123249
>>>
>>> sqrt(3) + sqrt(2)
3.14626436994197234232913506571557044551247712918732870123249
>>> sqrt(3) - sqrt(2)
0.317837245195782244725757617296174288373133378433432554879127
**Algorithm**
:func:`~mpmath.polyroots` implements the Durand-Kerner method [1], which
uses complex arithmetic to locate all roots simultaneously.
The Durand-Kerner method can be viewed as approximately performing
simultaneous Newton iteration for all the roots. In particular,
the convergence to simple roots is quadratic, just like Newton's
method.
Although all roots are internally calculated using complex arithmetic, any
root found to have an imaginary part smaller than the estimated numerical
error is truncated to a real number (small real parts are also chopped).
Real roots are placed first in the returned list, sorted by value. The
remaining complex roots are sorted by their real parts so that conjugate
roots end up next to each other.
**References**
1. http://en.wikipedia.org/wiki/Durand-Kerner_method
"""
if len(coeffs) <= 1:
if not coeffs or not coeffs[0]:
raise ValueError("Input to polyroots must not be the zero polynomial")
# Constant polynomial with no roots
return []
orig = ctx.prec
tol = +ctx.eps
with ctx.extraprec(extraprec):
deg = len(coeffs) - 1
# Must be monic
lead = ctx.convert(coeffs[0])
if lead == 1:
coeffs = [ctx.convert(c) for c in coeffs]
else:
coeffs = [c/lead for c in coeffs]
f = lambda x: ctx.polyval(coeffs, x)
if roots_init is None:
roots = [ctx.mpc((0.4+0.9j)**n) for n in xrange(deg)]
else:
roots = [None]*deg;
deg_init = min(deg, len(roots_init))
roots[:deg_init] = list(roots_init[:deg_init])
roots[deg_init:] = [ctx.mpc((0.4+0.9j)**n) for n
in xrange(deg_init,deg)]
err = [ctx.one for n in xrange(deg)]
# Durand-Kerner iteration until convergence
for step in xrange(maxsteps):
if abs(max(err)) < tol:
break
for i in xrange(deg):
p = roots[i]
x = f(p)
for j in range(deg):
if i != j:
try:
x /= (p-roots[j])
except ZeroDivisionError:
continue
roots[i] = p - x
err[i] = abs(x)
if abs(max(err)) >= tol:
raise ctx.NoConvergence("Didn't converge in maxsteps=%d steps." \
% maxsteps)
# Remove small real or imaginary parts
if cleanup:
for i in xrange(deg):
if abs(roots[i]) < tol:
roots[i] = ctx.zero
elif abs(ctx._im(roots[i])) < tol:
roots[i] = roots[i].real
elif abs(ctx._re(roots[i])) < tol:
roots[i] = roots[i].imag * 1j
roots.sort(key=lambda x: (abs(ctx._im(x)), ctx._re(x)))
if error:
err = max(err)
err = max(err, ctx.ldexp(1, -orig+1))
return [+r for r in roots], +err
else:
return [+r for r in roots]