def phi3(x0, *,
dps): # (epx(x)-1-x-0.5*x*x)/(x*x*x) , |x| > 1e-32 -> dps > 100
with mp.workdps(dps):
y = mp.matrix([
mp.fdiv(
mp.fsub(mp.fsub(mp.expm1(x), x), mp.fmul('0.5', mp.fmul(
x, x))), mp.power(x, '3')) if x != 0.0 else mpf(1) / mpf(6)
for x in x0
])
return np.array(y.tolist(), dtype=x0.dtype)[:, 0]
def FreeFermions(eigvec, subsystem, FermiVector): r=range(FermiVector) Cij=mp.matrix([[mp.fsum([eigvec[i,k]*eigvec[j,k] for k in r]) for i in subsystem] for j in subsystem]) C_eigval=mp.eigsy(Cij, eigvals_only=True) EH_eigval=mp.matrix([mp.log(mp.fdiv(mp.fsub(mp.mpf(1.0),x),x)) for x in C_eigval]) S=mp.re(mp.fsum([mp.log(mp.mpf(1.0)+mp.exp(-x))+mp.fdiv(x,mp.exp(x)+mp.mpf(1.0)) for x in EH_eigval])) return(S)
def phi2(x0, *, dps): # (exp(x) - 1 - x) / (x * x), |x| > 1e-32 -> dps > 40
with mp.workdps(dps):
y = mp.matrix([
mp.fdiv(mp.fsub(mp.expm1(x), x), mp.fmul(x, x))
if x != 0.0 else mpf(1) / mpf(2) for x in x0
])
return np.array(y.tolist(), dtype=x0.dtype)[:, 0]
def diff(EP: float, AP: float):
'''
Purpose: to calculate the difference between the exact and approx value
Return: returns the error
'''
return mp.fsub(EP, AP)
def FreeFermions(subsystem, C):
C = mp.matrix([[C[x, y] for x in subsystem] for y in subsystem])
C_eigval = mp.eigh(C, eigvals_only=True)
EH_eigval = mp.matrix(
[mp.log(mp.fdiv(mp.fsub(mp.mpf(1.0), x), x)) for x in C_eigval])
S = mp.re(
mp.fsum([
mp.log(mp.mpf(1.0) + mp.exp(-x)) + mp.fdiv(x,
mp.exp(x) + mp.mpf(1.0))
for x in EH_eigval
]))
return (S)
def FreeFermions(subsystem,
C_t): #implements free fermion technique by peschel
C = mp.matrix([[C_t[x, y] for x in subsystem] for y in subsystem])
C_eigval = mp.eigh(C, eigvals_only=True)
EH_eigval = mp.matrix(
[mp.log(mp.fdiv(mp.fsub(mp.mpf(1.0), x), x)) for x in C_eigval])
S = mp.re(
mp.fsum([
mp.log(mp.mpf(1.0) + mp.exp(-x)) + mp.fdiv(x,
mp.exp(x) + mp.mpf(1.0))
for x in EH_eigval
]))
return (S)
def rate_process_RMSE(rate_log_dic, correct_rate, size):
l = []
dic = {}
correct_rate_mpf = mp.mpf(correct_rate)
for i in rate_log_dic:
if rate_log_dic[i] == None:
continue
else:
element = mp.fsub(rate_log_dic[i], correct_rate_mpf)
l.append(element)
square = mp.fsum(l, squared=True)
mean = mp.fdiv(square, size)
root = mp.sqrt(mean)
dic['rate_RMSE'] = root
return dic
def Sub(a, b):
return mp.fsub(a, b, exact=True)
def Sub(a, b):
return mp.fsub(a, b, exact=True)
def dipole(r, theta, charge, a):
'''
Purpose : To calculate Electric Potential due to Dipole considering very small values
Formula Used :
(q*const_k)*(1/r_1 - 1/r_2)
where,
r_1 = distance between negative charge and point of observation
= (r^2 + a^2 - 2*a*Cos(theta))**0.5
r_2 = distance between positive charge and point of observation
= (r^2 + a^2 + 2*a*Cos(theta))**0.5
const_k = 4*pi*epsilon_not
= 8.9875518 x 10^9
Parameters :
a) r - distance between center of the dipole and point of observation
b) theta - Angle between positive charge and point of observation
c) charge - either charge irrespective of sign
d) a - distance between either charge and center of dipole
Return: returns exact value of electric field calculated
'''
# Editing value of r as per the unit
if r[1].lower() in ('meters', 'm'):
r = r[0]
elif r[1].lower() in ('centimeters', 'cm'):
r = mp.fmul(r[0], 10**(-2))
elif r[1].lower() in ('millimeters', 'mm'):
r = mp.fmul(r[0], 10**(-3))
elif r[1].lower() in ('angstroms', 'a', 'A'):
r = mp.fmul(r[0], 10**(-10))
# Editing value of a as per the unit
if a[1].lower() in ('meters', 'm'):
a = a[0]
elif a[1].lower() in ('centimeters', 'cm'):
a = mp.fmul(a[0], 10**(-2))
elif a[1].lower() in ('millimeters', 'mm'):
a = mp.fmul(a[0], 10**(-3))
elif a[1].lower() in ('angstroms', 'a', 'A'):
a = mp.fmul(a[0], 10**(-10))
# Calculating Value of Cos(theta)
if theta[1].lower() == 'radians':
cos_theta = round(mp.cos(theta[0]), 5)
elif theta[1].lower() == 'degrees':
cos_theta = round(mp.cos(mp.radians(theta[0])), 5)
# Editing Value of charge as per the unit
if charge[1] in ('Coulomb', 'C'):
charge = charge[0]
elif charge[1] in ('microCoulomb', 'uC'):
charge = mp.fmul(charge[0], 10**(-6))
elif charge[1] in ('milliCoulomb', 'mC'):
charge = mp.fmul(charge[0], 10**(-3))
elif charge[1] in ('electronCharge', 'eC'):
charge = mp.fmul(charge[0], mp.fmul(1.60217646,
mp.fmul(10,
-19))) # 1.60217646⋅10-19
elif charge[1] in ('nanoCoulomb', 'nC'):
charge = mp.fmul(charge[0], mp.power(10, -10))
elif charge[1] in ('picoCharge', 'pC'):
charge = mp.fmul(charge[0], mp.power(10, -12))
# Calculating value of r_1 and r_2
r_1 = mp.sqrt(
mp.fsub(mp.fadd(mp.power(r, 2), mp.power(a, 2)),
mp.fmul(2, mp.fmul(a, mp.fmul(r, cos_theta)))))
r_2 = mp.sqrt(
mp.fadd(mp.fadd(mp.power(r, 2), mp.power(a, 2)),
mp.fmul(2, mp.fmul(a, mp.fmul(r, cos_theta)))))
# Calculating final result
result = mp.fmul(mp.fmul(charge, const_k),
mp.fsub(mp.fdiv(1, r_1), mp.fdiv(1, r_2)))
# returning final result
return result
# -*- coding: utf-8 -*-
"""
Created on Thu Jun 6 13:44:19 2019
arbitrary prescision
@author: Chris
"""
from mpmath import mp
mp.dps = 100 #set prescision (number of digits)
pi = mp.mpf(mp.pi) #mpmath has a good built in pi
pi_string = '3.1415926535897932384626433832795028841971693993751058209749445923078164062'
print(len(pi_string), ' digits of pi')
#mpf input should be a string, entering a float is imprecise
pi2 = mp.mpf(pi_string)
difference = mp.fsub(pi2, pi)
digits_to_print = mp.dps - len(pi_string) #number of sig. figs. to print
print('difference: ', end='')
mp.nprint(difference, n=digits_to_print) #need nprint for mpf truncation
import decimal as de
de.getcontext().prec = 50 #set prescision for decimal module
pi_de = de.Decimal(pi_string)
### uncomment below
#print('decimal pi', pi_de)
#print('1/pi', 1/pi_de)