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Cant exit while loop on Simpson's Rule

I am trying to calculate an integral using Simpson's Rule formula.The catch is that the value of the integral is the one that satisfies the following condition:You find the Ih and Ih/2.If the absolute of (Ih-Ih/2)<error the loop is complete.Otherwise you repeat the process with half the h,which means it calculates the absolute of (Ih/2-Ih/4) and so on and so on.
while True:
###Ih part
h = (b - a) / N
y1 = np.linspace(a, b, N)
Ez11 = (np.sqrt(ra ** 2 + R ** 2 - 2 * ra * R * np.cos(y1 - fa))) / (
(aa ** 2 - 2 * ra * R * np.cos(y1 - fa)) ** (3 / 2))
I11 = (h/3) * (Ez11[0] + 2*sum(Ez11[:N-2:2]) \
+ 4*sum(Ez11[1:N-1:2]) + Ez11[N-1])
#####Ih/2 part
h = (b-a)/(2*N)
y2 = np.linspace(a, b, 2*N)
Ez22 = (np.sqrt(ra ** 2 + R ** 2 - 2 * ra * R * np.cos(y2 - fa))) / (
(aa ** 2 - 2 * ra * R * np.cos(y2 - fa)) ** (3 / 2))
print(Ez22)
I22 = (h/ 3) * (Ez22[0] + 2 * sum(Ez22[:N - 2:2]) \
+ 4 * sum(Ez22[1:N - 1:2]) + Ez22[N - 1])
# error condition I1=Ih I2=Ih/2
if np.abs(I11 - I22) < error:
break
else:
N = 2*N # h/2
print(np.abs(I11 - I22))
As far as I can tell,my approach should be correct.However the loop goes on and on,never to stop.
My code is as follows:
import numpy as np
from scipy.integrate import simps
import scipy.integrate as integrate
import scipy.special as special
# variables
a = 0
b = np.pi * 2
N = 100
ra = 0.1 # ρα
R = 0.05
fa = 35 * (np.pi / 180) # φα
za = 0.4
Q = 10 ** (-6)
k = 9 * 10 ** 9
aa = np.sqrt(ra ** 2 + R ** 2 + za ** 2)
error = 0.55 * 10 ** (-8)
h=(b-a)/N
I1 = np.nan
I11 = np.nan
#Simpsons section
############ Ez
#automated Simpson
while True:
###Ih part
y1 = np.linspace(a, b, N)
Ez1 = (np.sqrt(ra ** 2 + R ** 2 - 2 * ra * R * np.cos(y1 - fa))) / (
(aa ** 2 - 2 * ra * R * np.cos(y1 - fa)) ** (3 / 2))
print(len(Ez1))
I1 = simps(Ez1, y1)
#####Ih/2 part
y2 = np.linspace(a, b, 2*N)
Ez2 = (np.sqrt(ra ** 2 + R ** 2 - 2 * ra * R * np.cos(y2 - fa))) / (
(aa ** 2 - 2 * ra * R * np.cos(y2 - fa)) ** (3 / 2))
I2 = simps(Ez2, y2)
# error condition I1=Ih I2=Ih/2
if np.abs(I1 - I2) < error:
break
else:
N *= 2 # h/2
#custom-made Simpson
N = 100
while True:
###Ih part
h = (b - a) / N
y1 = np.linspace(a, b, N)
Ez11 = (np.sqrt(ra ** 2 + R ** 2 - 2 * ra * R * np.cos(y1 - fa))) / (
(aa ** 2 - 2 * ra * R * np.cos(y1 - fa)) ** (3 / 2))
I11 = (h/3) * (Ez11[0] + 2*sum(Ez11[:N-2:2]) \
+ 4*sum(Ez11[1:N-1:2]) + Ez11[N-1])
#####Ih/2 part
h = (b-a)/(2*N)
y2 = np.linspace(a, b, 2*N)
Ez22 = (np.sqrt(ra ** 2 + R ** 2 - 2 * ra * R * np.cos(y2 - fa))) / (
(aa ** 2 - 2 * ra * R * np.cos(y2 - fa)) ** (3 / 2))
print(Ez22)
I22 = (h/ 3) * (Ez22[0] + 2 * sum(Ez22[:N - 2:2]) \
+ 4 * sum(Ez22[1:N - 1:2]) + Ez22[N - 1])
# error condition I1=Ih I2=Ih/2
if np.abs(I11 - I22) < error:
break
else:
N = 2*N # h/2
print(np.abs(I11 - I22))
print(I1)
print(I11)
Simpson's Rule is as follows:
After a while it's stuck in this situation
The 5.23 part is the absolute diff of those 2 which shouldnt be that high.

Python - Converting list of coordinates to latitude and longitude

I recently started learning programing and Python.
Now I've been trying to convert a list of coordinates x,y to latitude and longitude. I searched and found a method in python in this post: How to convert from UTM to LatLng in python or Javascript
But when I try to apply the function to a list of floats from a dataframe I get an error:" cannot convert the series to <class 'float'> ".
What could I be doing wrong?
My code:
import math
def utmToLatLng(zone, easting, northing, northernHemisphere=True):
if not northernHemisphere:
northing = 10000000 - northing
a = 6378137
e = 0.081819191
e1sq = 0.006739497
k0 = 0.9996
arc = northing / k0
mu = arc / (a * (1 - math.pow(e, 2) / 4.0 - 3 * math.pow(e, 4) / 64.0 - 5 * math.pow(e, 6) / 256.0))
ei = (1 - math.pow((1 - e * e), (1 / 2.0))) / (1 + math.pow((1 - e * e), (1 / 2.0)))
ca = 3 * ei / 2 - 27 * math.pow(ei, 3) / 32.0
cb = 21 * math.pow(ei, 2) / 16 - 55 * math.pow(ei, 4) / 32
cc = 151 * math.pow(ei, 3) / 96
cd = 1097 * math.pow(ei, 4) / 512
phi1 = mu + ca * math.sin(2 * mu) + cb * math.sin(4 * mu) + cc * math.sin(6 * mu) + cd * math.sin(8 * mu)
n0 = a / math.pow((1 - math.pow((e * math.sin(phi1)), 2)), (1 / 2.0))
r0 = a * (1 - e * e) / math.pow((1 - math.pow((e * math.sin(phi1)), 2)), (3 / 2.0))
fact1 = n0 * math.tan(phi1) / r0
_a1 = 500000 - easting
dd0 = _a1 / (n0 * k0)
fact2 = dd0 * dd0 / 2
t0 = math.pow(math.tan(phi1), 2)
Q0 = e1sq * math.pow(math.cos(phi1), 2)
fact3 = (5 + 3 * t0 + 10 * Q0 - 4 * Q0 * Q0 - 9 * e1sq) * math.pow(dd0, 4) / 24
fact4 = (61 + 90 * t0 + 298 * Q0 + 45 * t0 * t0 - 252 * e1sq - 3 * Q0 * Q0) * math.pow(dd0, 6) / 720
lof1 = _a1 / (n0 * k0)
lof2 = (1 + 2 * t0 + Q0) * math.pow(dd0, 3) / 6.0
lof3 = (5 - 2 * Q0 + 28 * t0 - 3 * math.pow(Q0, 2) + 8 * e1sq + 24 * math.pow(t0, 2)) * math.pow(dd0, 5) / 120
_a2 = (lof1 - lof2 + lof3) / math.cos(phi1)
_a3 = _a2 * 180 / math.pi
latitude = 180 * (phi1 - fact1 * (fact2 + fact3 + fact4)) / math.pi
if not northernHemisphere:
latitude = -latitude
longitude = ((zone > 0) and (6 * zone - 183.0) or 3.0) - _a3
return (latitude, longitude)
import pandas as pd
df = pd.read_csv('Coord_rj.csv')
x = df['x']
y = df['y']
for i in range(len(df)):
lati,longi = utmToLatLng(23,x,y, False)
What my data looks like:
x y
529025.0 7422210.0
529114.0 7422343.0
545227.0 7435702.0
545582.0 7435741.0
The error:
TypeError Traceback (most recent call last)
~\AppData\Local\Temp/ipykernel_7368/1776418913.py in <module>
1 for i in range(len(df)):
----> 2 lati,longi = utmToLatLng(23,x,y, False)
3
~\AppData\Local\Temp/ipykernel_7368/3107957551.py in utmToLatLng(zone, easting, northing, northernHemisphere)
20 cc = 151 * math.pow(ei, 3) / 96
21 cd = 1097 * math.pow(ei, 4) / 512
---> 22 phi1 = mu + ca * math.sin(2 * mu) + cb * math.sin(4 * mu) + cc * math.sin(6 * mu) + cd * math.sin(8 * mu)
23
24 n0 = a / math.pow((1 - math.pow((e * math.sin(phi1)), 2)), (1 / 2.0))
~\anaconda3\lib\site-packages\pandas\core\series.py in wrapper(self)
183 if len(self) == 1:
184 return converter(self.iloc[0])
--> 185 raise TypeError(f"cannot convert the series to {converter}")
186
187 wrapper.__name__ = f"__{converter.__name__}__"
TypeError: cannot convert the series to <class 'float'>

When you're reading in your input file you're assigning all of the first column to a variable, and all of the second column to a variable:
>>> import pandas as pd
>>> df = pd.read_csv('Coord_rj.csv')
>>> df
x y
0 529025.0 7422210.0
1 529114.0 7422343.0
2 545227.0 7435702.0
3 545582.0 7435741.0
>>> df['x']
0 529025.0
1 529114.0
2 545227.0
3 545582.0
Name: x, dtype: float64
>>> df['y']
0 7422210.0
1 7422343.0
2 7435702.0
3 7435741.0
Name: y, dtype: float64
When you call your function you're passing the entire column to it for x and y, rather than just a row.
Try this instead:
for i in range(len(df)):
lati,longi = utmToLatLng(23, x[i], y[i], False)

'IndexError: index 4 is out of bounds for axis 1 with size 4'

I'm doing this code but I have a problem in the line 40, with a Index Error. I don't know if there is a problem with the nr-1, inside the for.
The code:
import numpy
Z = 1
R = 1
nr = 5
nz = 4
kn = 1
ks = 1
ke = 1
kw = 1
T = numpy.zeros((nr, nz), dtype=float)
vetorq = 1
for a in range(1, 100, 1):
for i in range(0, nz, 1):
for j in range(0, nr, 1):
an = (i + (1 / 2)) * (R ** 2)*nz*kn/ ((nr**2)* Z)
ae = (i + 2) * Z*ke / nz
aw = ((i+1)*kw * Z) / nz
asul = (i + (1 / 2)) * (R ** 2)*nz*ks / (Z * (nr**2))
ap = an + ae + aw + asul
aef = 0
if (i == 0 and j == 0):
T[i, j] = ae * ((T[i, j + 1]) / ap) + asul * (
(T[i + 1, j]) / ap) + 1846.35 * Z * (R ** 2) * (i + 1/2) / ((nr ** 2) * nz * ap) + aef / ap
elif(i == 0 and j != 0 and j != nr - 1):
T[i, j] = ae * ((T[i, j + 1]) / ap) + aw * ((T[i, j - 1]) / ap) + asul * (
(T[i + 1, j]) / ap) + 1846.35 * Z * (R ** 2) * (i + 1/2) / ((nr ** 2) * nz * ap) + aef / ap
elif(i == 0 and j == nr - 1):
aef = (i + 2) * R * Z * vetorq / (nr * nz)
T[i, j] = aw * ((T[i, j - 1]) / ap) + asul * (
(T[i + 1, j]) / ap) + 1846.35 * Z * (R ** 2) * (i + 1/2) / ((nr ** 2) * nz * ap) + aef / ap
elif(i != 0 and i != nz - 1 and j == nr - 1):
aef = (i + 2) * R * Z * vetorq / (nr * nz)
T[i, j] = aw * ((T[i, j - 1]) / ap) + an * ((T[i - 1, j]) / ap) + asul * (
(T[i + 1, j]) / ap) + 1846.35 * Z * (R ** 2) * (i + 1/2) / ((nr ** 2) * nz * ap) + aef / ap
elif(i == nz - 1 and j == nr - 1):
aef = (i + 2) * R * Z * vetorq / (nr * nz)
T[i, j] = aw * ((T[i, j - 1]) / ap) + an * ((T[i - 1, j]) / ap) + 1846.35 * Z * (R ** 2) * (i + 1/2) /((nr ** 2) * nz * ap) + aef / ap
elif(i == nz - 1 and j != 0 and j != nr - 1):
T[i, j] = ae * ((T[i, j + 1]) / ap) + aw * ((T[i, j - 1]) / ap) + an * ((T[i - 1, j]) / ap) + 1846.35 * Z * (R ** 2) * (i + 1/2) / ((nr ** 2) * nz * ap) + aef / ap
elif(i == nz - 1 and j == 0):
T[i, j] = ae * ((T[i, j + 1]) / ap) + an * ((T[i - 1, j]) / ap) + 1846.35 * Z * (R ** 2) * (i + 1/2) / ((nr ** 2) * nz * ap) + aef / ap
elif(i != 0 and i != nz - 1 and j == 0):
T[i, j] = ae * ((T[i, j + 1]) / ap) + an * ((T[i - 1, j]) / ap) + asul * (
(T[i + 1, j]) / ap) + 1846.35 * Z * (R ** 2) * (i + 1/2) / ((nr ** 2) * nz * ap) + aef / ap
elif(i != 0 and i != nz - 1 and j != 0 and j!= nr - 1):
T[i, j] = ae * ((T[i, j + 1]) / ap) + aw * ((T[i, j - 1]) / ap) + an * ((T[i - 1, j]) / ap) + asul * (
(T[i + 1, j]) / ap) + 1846.35 * Z * (R ** 2) * (i - 1/2) / ((nr ** 2) * nz * ap) + aef / ap
print(T)

The problem with your code is you have interchanged nr(rows) and nz(columns) with the iterators i and j; Your two loops must be as shown below and change all if, elif conditions on i with (nr-1) and on j with (nz-1). It works fine.
for i in range(0, nr, 1):
for j in range(0, nz, 1):

Python: My python plot is reflecting across y = 0.5 line

Why would my Python plot reflect across the y = 0.5 line? The same plot in Mathematica doesn't. I checked the equations 5-10 times and I don't see a difference. If I put a -1 in front of the python plot it will flip over and drop down 1 unit to y = -0.5.
Additionally, the definitions for alphag and betag are correct.
import numpy as np
import pylab
r1 = 1 # AU Earth
r2 = 1.524 # AU Mars
deltanu = 75 * np.pi / 180 # angle in radians
mu = 38.86984154054163
c = np.sqrt(r1 ** 2 + r2 ** 2 - 2 * r1 * r2 * np.cos(deltanu))
s = (r1 + r2 + c) / 2
am = s / 2
def g(a):
alphag = 2* np.pi - 2 * np.arcsin(np.sqrt(s / (2 * a)))
betag = -2 * np.arcsin(np.sqrt((s - c) / (2 * a)))
return (np.sqrt(a ** 3 / mu)
* (alphag - betag - (np.sin(alphag) - np.sin(betag)))
- dt)
a = np.linspace(am, 2, 500000)
dt = np.linspace(0, 2, 500000)
fig = pylab.figure()
ax = fig.add_subplot(111)
ax.plot(a, g(a), color = 'r')
pylab.xlim((0.9, 2))
pylab.ylim((0, 2))
pylab.show()
Python:
Edit 2:
There are actually 2 plots I am plotting and thanks to the comments, I noticed that there is something even stranger occurring.
The two plots I am plotting are:
dt = np.sqrt(a ** 3 / mu) * (alpha - beta - (sin(alpha) - sin(beta)))
where alpha is 2 * np.arcsin(np.sqrt(s / (2 * a))) or 2 * np.pi - 2 * np.arcsin(np.sqrt(s / (2 * a))) and beta is 2 * np.arcsin(np.sqrt((s - c) / (2 * a))) or the negative of the first.
In[13]:= r1 = 1;
r2 = 1.524;
dnu = 75 Degree;
mu = 38.86984154054163;
In[17]:= c = Sqrt[r1^2 + r2^2 - 2*r1*r2*Cos[dnu]]
Out[17]= 1.59176
In[18]:= s = (r1 + r2 + c)/2
Out[18]= 2.05788
In[19]:= alp = 2 \[Pi] - 2*ArcSin[Sqrt[s/(2*a)]];
bet = -2*ArcSin[Sqrt[(s - c)/(2*a)]];
In[22]:= Plot[
Sqrt[a^3/mu]*(alp - bet - (Sin[alp] - Sin[bet])), {a, 0, 2},
PlotRange -> {{.8, 2}, {0, 2}}]
This produces:
and
alp2 = 2*ArcSin[Sqrt[s/(2*a)]];
bet2 = 2*ArcSin[Sqrt[(s - c)/(2*a)]];
Plot[Sqrt[a^3/mu]*(alp2 - bet2 - (Sin[alp2] - Sin[bet2])), {a, 0, 2},
PlotRange -> {{.8, 2}, {0, 2}}]
So the Python code matches the first Mathematica code but the plots the second picture and my python code for the the second Mathematica codes produces the flipped image for the first Mathematica picture.

I think you simply have to remove the -dt from the Python code:
import numpy as np
import matplotlib.pyplot as plt
r1 = 1 # AU Earth
r2 = 1.524 # AU Mars
deltanu = 75 * np.pi / 180 # angle in radians
mu = 38.86984154054163
c = np.sqrt(r1 ** 2 + r2 ** 2 - 2 * r1 * r2 * np.cos(deltanu))
s = (r1 + r2 + c) / 2
am = s / 2
def g(a):
alphag = 2 * np.pi - 2 * np.arcsin(np.sqrt(s / (2 * a)))
betag = -2 * np.arcsin(np.sqrt((s - c) / (2 * a)))
return (np.sqrt(a ** 3 / mu)
* (alphag - betag - (np.sin(alphag) - np.sin(betag))))
def g2(a):
alphag = 2 * np.arcsin(np.sqrt(s / (2 * a)))
betag = 2 * np.arcsin(np.sqrt((s - c) / (2 * a)))
return (np.sqrt(a ** 3 / mu)
* (alphag - betag - (np.sin(alphag) - np.sin(betag))))
a = np.linspace(am, 2, 500000)
dt = np.linspace(0, 2, 500000)
fig, ax = plt.subplots(ncols=2)
ax[0].plot(a, g(a), color = 'r')
ax[1].plot(a, g2(a), color = 'r')
ax[0].set_xlim((0.9, 2))
ax[0].set_ylim((0, 2))
ax[1].set_xlim((0.9, 2))
ax[1].set_ylim((0, 2))
plt.show()
yields

Python: Filling in a gap between two plots

How can I connect two plots at a discontinues point? I have an equation for the point of discontinuity.
import numpy as np
import pylab
r1 = 1 # AU Earth
r2 = 1.524 # AU Mars
deltanu = 75 * np.pi / 180 # angle in radians
mu = 38.86984154054163
c = np.sqrt(r1 ** 2 + r2 ** 2 - 2 * r1 * r2 * np.cos(deltanu))
s = (r1 + r2 + c) / 2
am = s / 2
def f(a):
alpha = 2 * np.arcsin(np.sqrt(s / (2 * a)))
beta = 2 * np.arcsin(np.sqrt((s - c) / (2 * a)))
return (np.sqrt(a **3 / mu) * (alpha - beta - (np.sin(alpha)
- np.sin(beta))))
def g(a):
alphag = 2* np.pi - 2 * np.arcsin(np.sqrt(s / (2 * a)))
betag = -2 * np.arcsin(np.sqrt((s - c) / (2 * a)))
return (np.sqrt(a ** 3 / mu)
* (alphag - betag - (np.sin(alphag) - np.sin(betag))))
a = np.linspace(am, 2, 500000)
fig = pylab.figure()
ax = fig.add_subplot(111)
ax.plot(a, f(a), color = '#000000')
ax.plot(a, g(a), color = '#000000')
pylab.xlim((0.9, 2))
pylab.ylim((0, 2))
pylab.show()
The equation that reflects the point is: dt = np.sqrt(s ** 3 / 8) * (np.pi - betam + np.sin(betam)) where betam = 2 * np.arcsin(np.sqrt(1 - c / s)) so dt = 0.5 at a = s / 2. However, the gap between the plots looks bigger than a point.
I added: ax.plot([am, am], [.505, .55], color = '#000000') which fills in the gap but it feels out of place.

It seems as though perhaps you should only be using one value for betag:
import numpy as np
import matplotlib.pyplot as plt
r1 = 1 # AU Earth
r2 = 1.524 # AU Mars
deltanu = 75 * np.pi / 180 # angle in radians
mu = 38.86984154054163
c = np.sqrt(r1 ** 2 + r2 ** 2 - 2 * r1 * r2 * np.cos(deltanu))
s = (r1 + r2 + c) / 2
am = s / 2
def g(a, alphag, betag):
return (np.sqrt(a ** 3 / mu)
* (alphag - betag - (np.sin(alphag) - np.sin(betag))))
a = np.linspace(am, 2, 500000)
fig, ax = plt.subplots()
alphag = 2 * np.pi - 2 * np.arcsin(np.sqrt(s / (2 * a)))
betag = 2 * np.arcsin(np.sqrt((s - c) / (2 * a)))
ax.plot(a, g(a, alphag, betag), color = 'r')
alphag = 2 * np.arcsin(np.sqrt(s / (2 * a)))
ax.plot(a, g(a, alphag, betag), color = 'r')
plt.show()
yields
I really don't know what's going on here; I found this serendipitously.

ax.plot([am,am],[f(am),g(am)],color== '#000000')