# Source code for skrf.mathFunctions

"""
mathFunctions (:mod:skrf.mathFunctions)
=============================================

Provides commonly used mathematical functions.

Mathematical Constants
----------------------
Some convenient constants are defined in the :mod:skrf.constants module.

Complex Component Conversion
---------------------------------
.. autosummary::
:toctree: generated/

complex_2_reim
complex_2_magnitude
complex_2_db
complex_2_db10
complex_2_degree
complex_2_magnitude

Phase Unwrapping
--------------------------------
.. autosummary::
:toctree: generated/

sqrt_phase_unwrap

Unit Conversion
--------------------------------
.. autosummary::
:toctree: generated/

np_2_db
db_2_np

Scalar-Complex Conversion
---------------------------------
These conversions are useful for wrapping other functions that don't
support complex numbers.

.. autosummary::
:toctree: generated/

complex2Scalar
scalar2Complex

Special Functions
---------------------------------
.. autosummary::
:toctree: generated/

dirac_delta
neuman
null
cross_ratio

Various Utility Functions
--------------------------
.. autosummary::
:toctree: generated/

psd2TimeDomain
rational_interp
ifft
irfft
is_square
is_symmetric
is_Hermitian
is_positive_definite
is_positive_semidefinite
get_Hermitian_transpose

"""
from typing import Callable
import numpy as npy
from numpy import pi, angle, unwrap, real, imag, array
from scipy import signal
from scipy.interpolate import interp1d

from . constants import NumberLike, INF, ALMOST_ZERO, LOG_OF_NEG

# simple conversions
[docs]def complex_2_magnitude(z: NumberLike):
"""
Return the magnitude of the complex argument.

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
mag : ndarray or scalar

"""
return npy.abs(z)

[docs]def complex_2_db(z: NumberLike):
r"""
Return the magnitude in dB of a complex number (as :math:20\log_{10}(|z|))..

The magnitude in dB is defined as :math:20\log_{10}(|z|)
where :math:z is a complex number.

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
mag20dB : ndarray or scalar
"""
return magnitude_2_db(npy.abs(z))

[docs]def complex_2_db10(z: NumberLike):
r"""
Return the magnitude in dB of a complex number (as :math:10\log_{10}(|z|)).

The magnitude in dB is defined as :math:10\log_{10}(|z|)
where :math:z is a complex number.

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
mag10dB : ndarray or scalar
"""
return mag_2_db10(npy.abs(z))

"""
Return the angle complex argument in radian.

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
The counterclockwise angle from the positive real axis on the complex
plane in the range (-pi, pi], with dtype as numpy.float64.
"""
return npy.angle(z)

[docs]def complex_2_degree(z: NumberLike):
"""
Returns the angle complex argument in degree.

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
ang_deg : ndarray or scalar
"""
return npy.angle(z, deg=True)

r"""
Take a complex number and returns quadrature, which is (length, arc-length from real axis)

Arc-length is calculated as :math:|z| \arg(z).

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
mag : array like or scalar
magnitude (length)
arc_length : array like or scalar
arc-length from real axis: angle*magnitude
"""
return (npy.abs(z), npy.angle(z)*npy.abs(z))

[docs]def complex_2_reim(z: NumberLike):
"""
Return real and imaginary parts of a complex number.

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
real : array like or scalar
real part of input
imag : array like or scalar
imaginary part of input
"""
return (npy.real(z), npy.imag(z))

def complex_components(z: NumberLike):
"""
Break up a complex array into all possible scalar components.

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
c_real : array like or scalar
real part
c_imag : array like or scalar
imaginary part
c_angle : array like or scalar
angle in degrees
c_mag : array like or scalar
magnitude
c_arc : array like or scalar
arclength from real axis, angle*magnitude
"""

def magnitude_2_db(z: NumberLike, zero_nan: bool = True):
"""
Convert linear magnitude to dB.

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers
zero_nan : bool, optional
Replace NaN with zero. The default is True.

Returns
-------
z : number or array_like
Magnitude in dB given by 20*log10(|z|)
"""
out = 20 * npy.log10(z)
if zero_nan:
return npy.nan_to_num(out, copy=False, nan=LOG_OF_NEG, neginf=-npy.inf)
return out

mag_2_db = magnitude_2_db

def mag_2_db10(z: NumberLike, zero_nan:bool = True):
"""
Convert linear magnitude to dB.

Parameters
----------
z : array_like
A complex number or sequence of complex numbers
zero_nan : bool, optional
Replace NaN with zero. The default is True.

Returns
-------
z : array_like
Magnitude in dB given by 10*log10(|z|)
"""
out = 10 * npy.log10(z)
if zero_nan:
return npy.nan_to_num(out, copy=False, nan=LOG_OF_NEG, neginf=-npy.inf)
return out

def db_2_magnitude(z: NumberLike):
"""
Convert dB to linear magnitude.

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
z : number or array_like
10**((z)/20) where z is a complex number
"""
return 10**((z)/20.)

db_2_mag = db_2_magnitude

def db10_2_mag(z: NumberLike):
"""
Convert dB to linear magnitude.

Parameters
----------
z : array_like
A complex number or sequence of complex numbers

Returns
-------
z : array_like
10**((z)/10) where z is a complex number
"""
return 10**((z)/10.)

def magdeg_2_reim(mag: NumberLike, deg: NumberLike):
"""
Convert linear magnitude and phase (in deg) arrays into a complex array.

Parameters
----------
mag : number or array_like
A complex number or sequence of real numbers
deg : number or array_like
A complex number or sequence of real numbers

Returns
-------
z : array_like
A complex number or sequence of complex numbers

"""
return mag*npy.exp(1j*deg*pi/180.)

def dbdeg_2_reim(db: NumberLike, deg: NumberLike):
"""
Converts dB magnitude and phase (in deg) arrays into a complex array.

Parameters
----------
db : number or array_like
A realnumber or sequence of real numbers
deg : number or array_like
A real number or sequence of real numbers

Returns
-------
z : array_like
A complex number or sequence of complex numbers
"""
return magdeg_2_reim(db_2_magnitude(db), deg)

[docs]def db_2_np(db: NumberLike):
"""
Converts a value in decibel (dB) to neper (Np).

Parameters
----------
db : number or array_like
A real number or sequence of real numbers

Returns
-------
np : number or array_like
A real number of sequence of real numbers
"""
return (npy.log(10)/20) * db

[docs]def np_2_db(x: NumberLike):
"""
Converts a value in Nepers (Np) to decibel (dB).

Parameters
----------
np : number or array_like
A real number or sequence of real numbers

Returns
-------
db : number or array_like
A real number of sequence of real numbers
"""
return 20/npy.log(10) * x

"""
Convert angles from radians to degrees.

Parameters
----------

Returns
-------
deg : number or array_like
Angle in degree
"""

"""
Convert angles from degrees to radians.

Parameters
----------
deg : number or array_like

Returns
-------
Angle in degree
"""
return (deg)*pi/180.

def feet_2_meter(feet: NumberLike = 1):
"""
Convert length in feet to meter.

1 foot is equal to 0.3048 meters.

Parameters
----------
feet : number or array-like, optional
length in feet. Default is 1.

Returns
-------
meter: number or array-like
length in meter

--------
meter_2_feet
"""
return 0.3048*feet

def meter_2_feet(meter: NumberLike = 1):
"""
Convert length in meter to feet.

1 meter is equal to 0.3.28084 feet.

Parameters
----------
meter : number or array-like, optional
length in meter. Default is 1.

Returns
-------
feet : number or array-like
length in feet

--------
feet_2_meter
"""
return 3.28084*meter

def db_per_100feet_2_db_per_100meter(db_per_100feet: NumberLike = 1):
"""
Convert attenuation values given in dB/100ft to dB/100m.

db_per_100meter = db_per_100feet * rf.meter_2_feet()

Parameters
----------
db_per_100feet : number or array-like, optional
Attenuation in dB/ 100 ft. Default is 1.

Returns
-------
db_per_100meter : number or array-like
Attenuation in dB/ 100 m

--------
meter_2_feet
feet_2_meter
np_2_db
db_2_np
"""
return db_per_100feet * 100 / feet_2_meter(100)

"""
Unwraps a phase given in radians.

Parameters
----------
phi : number or array_like

Returns
-------
phi : number of array_like
"""
return unwrap(phi, axis=0)

def sqrt_known_sign(z_squared: NumberLike, z_approx: NumberLike):
"""
Return the square root of a complex number, with sign chosen to match z_approx.

Parameters
----------
z_squared : number or array-like
the complex to be square-rooted
z_approx : number or array-like
the approximate value of z. sign of z is chosen to match that of
z_approx

Returns
-------
z : number, array-like (same type as z_squared)
square root of z_squared.
"""
z = npy.sqrt(z_squared)
return npy.where(
npy.sign(npy.angle(z)) == npy.sign(npy.angle(z_approx)),
z, z.conj())

def find_correct_sign(z1: NumberLike, z2: NumberLike, z_approx: NumberLike):
r"""
Create new vector from z1, z2 choosing elements with sign matching z_approx.

This is used when you have to make a root choice on a complex number.
and you know the approximate value of the root.

.. math::

z1,z2 = \pm \sqrt(z^2)

Parameters
----------
z1 : array-like
root 1
z2 : array-like
root 2
z_approx : array-like

Returns
-------
z3 : npy.array
array built from z1 and z2 by
z1 where sign(z1) == sign(z_approx), z2 else

"""
return npy.where(
npy.sign(npy.angle(z1)) == npy.sign(npy.angle(z_approx)),z1, z2)

def find_closest(z1: NumberLike, z2: NumberLike, z_approx: NumberLike):
"""
Return z1 or z2  depending on which is  closer to z_approx.

Parameters
----------
z1 : array-like
root 1
z2 : array-like
root 2
z_approx : array-like

Returns
-------
z3 : npy.array
array built from z1 and z2

"""
z1_dist = abs(z1-z_approx)
z2_dist = abs(z2-z_approx)

return npy.where(z1_dist<z2_dist,z1, z2)

[docs]def sqrt_phase_unwrap(z: NumberLike):
r"""
Take the square root of a complex number with unwrapped phase.

This idea came from Lihan Chen.

.. math::

\sqrt{|z|} \exp( \arg_{unwrap}(z) / 2 )

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
z : number of array_like
A complex number or sequence of complex numbers
"""
return npy.sqrt(abs(z))*\

# mathematical functions
[docs]def dirac_delta(x: NumberLike):
r"""
Calculate Dirac function.

Dirac function :math:\delta(x) defined as :math:\delta(x)=1 if x=0,
0 otherwise.

Parameters
----------
x : number of array_like
A real number or sequence of real numbers

Returns
-------
delta : number of array_like
1 or 0

References
----------
https://en.wikipedia.org/wiki/Dirac_delta_function

"""
return (x==0)*1. + (x!=0)*0.

[docs]def neuman(x: NumberLike):
r"""
Calculate Neumans number.

It is defined as:

.. math::

2 - \delta(x)

where :math:\delta is the Dirac function.

Parameters
----------
x : number or array_like
A real number or sequence of real numbers

Returns
-------
y : number or array_like
A real number or sequence of real numbers

--------
dirac_delta
"""
return 2. - dirac_delta(x)

[docs]def null(A: npy.ndarray, eps: float = 1e-15):
"""
Calculate the null space of matrix A.

Parameters
----------
A : array_like
eps : float

Returns
-------
null_space : array_like

References
----------
https://stackoverflow.com/questions/5889142/python-numpy-scipy-finding-the-null-space-of-a-matrix
"""
u, s, vh = npy.linalg.svd(A)
null_space = npy.compress(s <= eps, vh, axis=0)
return null_space.T

def inf_to_num(x: NumberLike):
"""
Convert inf and -inf's to large numbers.

Parameters
------------
x : array-like or number
the input array or number

Returns
-------
x : Number of array_like
Input without with +/- inf replaced by large numbers
"""
x = npy.nan_to_num(x, copy=False, nan=npy.nan, posinf=INF, neginf=-1*INF)
return x

[docs]def cross_ratio(a: NumberLike, b: NumberLike, c: NumberLike, d:NumberLike):
r"""
Calculate the cross ratio of a quadruple of distinct points on the real line.

The cross ratio is defined as:

.. math::

r = \frac{ (a-b)(c-d) }{ (a-d)(c-b) }

Parameters
----------
a,b,c,d :  array-like or number

Returns
-------
r : array-like or number

References
----------
https://en.wikipedia.org/wiki/Cross-ratio

"""
return ((a-b)*(c-d))/((a-d)*(c-b))

def complexify(f: Callable, name: str = None):
"""
Make a function f(scalar) into f(complex).

If f(x) then it returns f_c(z) = f(real(z)) + 1j*f(imag(z))

If the real/imag arguments are not first, then you may specify the
name given to them as kwargs.

Parameters
----------
f : Callable
function of real variable
name : string, optional
name of the real/imag argument names if they are not first

Returns
-------
f_c : Callable
function of a complex variable

Examples
--------
>>> def f(x): return x
>>> f_c = rf.complexify(f)
>>> z = 0.2 -1j*0.3
>>> f_c(z)

"""
def f_c(z, *args, **kw):
if name is not None:
kw_re = {name: real(z)}
kw_im = {name: imag(z)}
kw_re.update(kw)
kw_im.update(kw)
return f(*args, **kw_re) + 1j*f(*args, **kw_im)
else:
return f(real(z), *args,**kw) + 1j*f(imag(z), *args, **kw)
return f_c

# old functions just for reference
[docs]def complex2Scalar(z: NumberLike):
"""
Serialize a list/array of complex numbers

Parameters
----------
z : number or array_like
A complex number or sequence of complex numbers

Returns
-------
re_im : array_like
produce the following array for an input z:
z[0].real, z[0].imag, z[1].real, z[1].imag, etc.

--------
scalar2Complex
"""
z = npy.array(z)
re_im = []
for k in z:
re_im.append(npy.real(k))
re_im.append(npy.imag(k))
return npy.array(re_im).flatten()

[docs]def scalar2Complex(s: NumberLike):
"""
Unserialize a list/array of real and imag numbers into a complex array.

Inverse of :func:complex2Scalar.

Parameters
----------
s : array_like
an array with real and imaginary parts ordered as:
z[0].real, z[0].imag, z[1].real, z[1].imag, etc.

Returns
-------
z : Number or array_like
A complex number or sequence of complex number

--------
complex2Scalar
"""
s = npy.array(s)
z = []

for k in range(0,len(s),2):
z.append(s[k] + 1j*s[k+1])
return npy.array(z).flatten()

def flatten_c_mat(s: NumberLike, order: str = 'F'):
"""
Take a 2D (mxn) complex matrix and serialize and flatten it.

by default (using order='F') this generates the following
from a  2x2

[s11,s12;s21,s22]->[s11re,s11im,s21re,s12im, ...]

Parameters
------------
s : ndarray
input 2D array
order : sting, optional
either 'F' or 'C', for the order of flattening
"""
return complex2Scalar(s.flatten(order='F'))

def rand_c(*args) -> npy.ndarray:
"""
Creates a complex random array of shape s.

The bounds on real and imaginary values are (-1,1)

Parameters
-----------
s : list-like
shape of array

Examples
---------
>>> x = rf.rand_c(2,2)
"""
return 1-2*npy.random.rand(*args) + \
1j-2j*npy.random.rand(*args)

[docs]def psd2TimeDomain(f: npy.ndarray, y: npy.ndarray, windowType: str = 'hamming'):
"""
Convert a one sided complex spectrum into a real time-signal.

Parameters
----------
f : list or npy.ndarray
frequency array
y : list of npy.ndarray
complex PSD array
windowType: string
windowing function, defaults to 'hamming''

Returns
-------
timeVector : array_like
inverse units of the input variable f,
signalVector : array_like

Note
----
If spectrum is not baseband then, timeSignal is modulated by exp(t*2*pi*f[0]).
So keep in mind units. Also, due to this, f must be increasing left to right.
"""

# apply window function
# make sure windowType exists in scipy.signal
if callable(getattr(signal, windowType)) and (windowType != 'rect' ):
window = getattr(signal, windowType)(len(f))
y = y * window

#create other half of spectrum
spectrum = (npy.hstack([npy.real(y[:0:-1]),npy.real(y)])) + \
1j*(npy.hstack([-npy.imag(y[:0:-1]),npy.imag(y)]))

# do the transform
df = abs(f[1]-f[0])
T = 1./df
timeVector = npy.linspace(-T/2.,T/2,2*len(f)-1)
signalVector = npy.fft.ifftshift(npy.fft.ifft(npy.fft.ifftshift(spectrum)))

#the imaginary part of this signal should be from fft errors only,
signalVector= npy.real(signalVector)
# the response of frequency shifting is
# exp(1j*2*pi*timeVector*f[0])
# but I would have to manually undo this for the inverse, which is just
# another variable to require. The reason you need this is because
# you can't transform to a bandpass signal, only a lowpass.
#
return timeVector, signalVector

[docs]def rational_interp(x: npy.ndarray, y: npy.ndarray, d: int = 4, epsilon: float = 1e-9, axis: int = 0, assume_sorted: bool = False) -> Callable:
"""
Interpolates function using rational polynomials of degree d.

Interpolating function is singular when xi is exactly one of the
original x points. If xi is closer than epsilon to one of the original points,
then the value at that points is returned instead.

Implementation is based on [#]_.

Parameters
----------
x : npy.ndarray
y : npy.ndarray
d : int, optional
order of the polynomial, by default 4
epsilon : float, optional
numerical tolerance, by default 1e-9
axis : int, optional
axis to operate on, by default 0
assume_sorted : bool, optional
If False, values of x can be in any order and they are sorted first.
If True, x has to be an array of monotonically increasing values.

Returns
-------
fx : Callable
Interpolate function

Raises
------
NotImplementedError
if axis != 0.

References
------------
.. [#] M. S. Floater and K. Hormann, "Barycentric rational interpolation with no poles and high rates of approximation," Numer. Math., vol. 107, no. 2, pp. 315-331, Aug. 2007
"""
if axis != 0:
raise NotImplementedError("Axis other than 0 is not implemented")

if not assume_sorted:
sort_indices = npy.argsort(x, axis=axis)
x = x[sort_indices]
y = y[sort_indices]

n = len(x)
if n <= d:
raise ValueError('Not enough x-axis points')

w = npy.zeros(n)
# Scaling to give close to 1 weights
hd = (x[n//2] - x[n//2-1])**d
for k in range(n):
for i in range(max(0,k-d), min(k+1, n-d)):
p = hd
for j in range(i,min(n,i+d+1)):
if j == k:
continue
p *= 1/(x[k] - x[j])
if i % 2 == 1:
w[k] -= p
else:
w[k] += p

# Add dimensions to match y shape
w_shape = [1]*len(y.shape)
w_shape[0] = -1
w = w.reshape(w_shape)

def fx(xi):
# The method will divide by zero if new x value is exactly existing x value.
# To avoid this we need to check for too close values and replace them with
# y value at that position.
idx = npy.searchsorted(x, xi)
idx[idx == len(x)] = len(x) - 1
nearest_idx = npy.where(npy.abs(x[idx] - xi) < epsilon)[0]
nearest_value = y[idx[nearest_idx]]

xi = xi.reshape(*w_shape)
with npy.errstate(divide='ignore', invalid='ignore'):
assert axis == 0
v = sum(y[i]*w[i]/(xi - x[i]) for i in range(n))\
/sum(w[i]/(xi - x[i]) for i in range(n))

# Fix divide by zero errors
for e, i in enumerate(nearest_idx):
v[i] = nearest_value[e]

return v

return fx

[docs]def ifft(x: npy.ndarray) -> npy.ndarray:
"""
Transforms S-parameters to time-domain bandpass.

Parameters
----------
x : array_like
S-parameters vs frequency array.

Returns
-------
X : array_like
Fourier transformed array

--------
irfft
"""
return npy.fft.fftshift(npy.fft.ifft(x, axis=0), axes=0)

[docs]def irfft(x: npy.ndarray, n:int = None) -> npy.ndarray:
"""
Transforms S-parameters to time-domain, assuming complex conjugates for
values corresponding to negative frequencies.

Parameters
----------
x : array_like
S-parameters vs frequency array.
n : int, optional
n parameter passed to :func:numpy.fft.irfft. Defaults to None.

Returns
-------
X : array_like
Fourier transformed array

--------
ifft
"""
return npy.fft.fftshift(npy.fft.irfft(x, axis=0, n=n), axes=0)

[docs]def is_square(mat: npy.ndarray) -> bool:
"""
Tests whether mat is a square matrix.

Parameters
----------
mat : npy.ndarray
Matrix to test for being square

Returns
-------
res : boolean

--------
is_unitary
is_symmetric
"""
return mat.shape[0] == mat.shape[1]

def is_unitary(mat: npy.ndarray, tol: float = ALMOST_ZERO) -> bool:
"""
Tests mat for unitariness.

Parameters
----------
mat : npy.ndarray
Matrix to test for unitariness
tol : float
Absolute tolerance. Defaults to :data:ALMOST_ZERO

Returns
-------
res : boolean or array of boolean

--------
is_square
is_symmetric

"""
if not is_square(mat):
return False
return npy.allclose(get_Hermitian_transpose(mat) @ mat,
npy.identity(mat.shape[0]), atol=tol)

[docs]def is_symmetric(mat: npy.ndarray, tol: int = ALMOST_ZERO) -> bool:
"""
Tests mat for symmetry.

Parameters
----------
mat : npy.ndarray
Matrix to test for symmetry
tol : float, optional
Absolute tolerance. Defaults to :data:ALMOST_ZERO

Returns
-------
res : boolean or array of boolean

--------
is_square
is_unitary
"""
if not is_square(mat):
return False
return npy.allclose(mat, mat.transpose(), atol=tol)

[docs]def get_Hermitian_transpose(mat: npy.ndarray) -> npy.ndarray:
"""
Returns the conjugate transpose of mat.

Parameters
----------
mat : npy.ndarray
Matrix to compute the conjugate transpose of

Returns
-------
mat : npy.ndarray

"""
return mat.transpose().conjugate()

[docs]def is_Hermitian(mat: npy.ndarray, tol: float = ALMOST_ZERO) -> bool:
"""
Tests whether mat is Hermitian.

Parameters
----------
mat : npy.ndarray
Matrix to test for being Hermitian
tol : float
Absolute tolerance

Returns
-------
res : boolean

--------
is_positive_definite
is_positive_semidefinite
"""
if not is_square(mat):
return False
return npy.allclose(mat, get_Hermitian_transpose(mat), atol=tol)

[docs]def is_positive_definite(mat: npy.ndarray, tol: float = ALMOST_ZERO) -> bool:
"""
Tests mat for positive definiteness.

Verifying that
(1) mat is symmetric
(2) it's possible to compute the Cholesky decomposition of mat.

Parameters
----------
mat : npy.ndarray
Matrix to test for positive definiteness
tol : float, optional
Absolute tolerance. Defaults to :data:ALMOST_ZERO

Returns
-------
res : bool or array of bool

--------
is_Hermitian
is_positive_semidefinite
"""
if not is_Hermitian(mat, tol=tol):
return False
try:
npy.linalg.cholesky(mat)
return True
except npy.linalg.LinAlgError:
return False

[docs]def is_positive_semidefinite(mat: npy.ndarray, tol: float = ALMOST_ZERO) -> bool:
"""
Tests mat for positive semidefiniteness.

Checking whether all eigenvalues of mat are nonnegative within a certain tolerance

Parameters
----------
mat : npy.ndarray
Matrix to test for positive semidefiniteness
tol : float, optional
Absolute tolerance in determining nonnegativity due to loss of precision
when computing the eigenvalues of mat. Defaults to :data:ALMOST_ZERO

Returns
-------
res : bool or array of bool

--------
is_Hermitian
is_positive_definite
"""
if not is_Hermitian(mat):
return False
try:
v = npy.linalg.eigvalsh(mat)
except npy.linalg.LinAlgError:
return False
return npy.all(v > -tol)

def rsolve(A: npy.ndarray, B: npy.ndarray) -> npy.ndarray:
r"""Solves x @ A = B.

Calls numpy.linalg.solve with transposed matrices.

Same as B @ npy.linalg.inv(A) but avoids calculating the inverse and
should be numerically slightly more accurate.

Input should have dimension of similar to (nfreqs, nports, nports).

Parameters
----------
A : npy.ndarray
B : npy.ndarray

Returns
-------
x : npy.ndarray
"""
return npy.transpose(npy.linalg.solve(npy.transpose(A, (0, 2, 1)).conj(),
npy.transpose(B, (0, 2, 1)).conj()), (0, 2, 1)).conj()

def nudge_eig(mat: npy.ndarray, cond: float = 1e-9, min_eig: float = 1e-12) -> npy.ndarray:
r"""Nudge eigenvalues with absolute value smaller than
max(cond * max(eigenvalue), min_eig) to that value.
Can be used to avoid singularities in solving matrix equations.

Input should have dimension of similar to (nfreqs, nports, nports).

Parameters
----------
mat : npy.ndarray
Matrices to nudge
cond : float, optional
Minimum eigenvalue ratio compared to the maximum eigenvalue
min_eig : float, optional
Minimum eigenvalue
Returns
-------
res : npy.ndarray
Nudged matrices
"""
# Eigenvalues and vectors
eigw, eigv = npy.linalg.eig(mat)
# Max eigenvalue for each frequency
max_eig = npy.amax(npy.abs(eigw), axis=1)
# Calculate mask for positions where problematic eigenvalues are
mask = npy.logical_or(npy.abs(eigw) < cond * max_eig[:, None], npy.abs(eigw) < min_eig)