# Newton’s Method¶

PyEphem comes with a simple implementation of Newton’s Method, named `newton`.

```>>> from ephem import newton
```

Given a continuous function and two x coordinates near which the function crosses zero, it returns the x coordinate of the actual zero crossing. For example, if asked to find the zero-crossing of the `sin()` function in the vicinity of the number three, it returns a quite good appoximation of π (“pi”):

```>>> import math
>>> n = newton(math.sin, 3.0, 3.1)
>>> print('%.11f' % n)
3.14159265359
```

Here is a real-world example of using Newton’s method: trying to figure out when the longitude of the Sun on Mars crosses one of the values 0°, 90°, 180°, or 270° that mark the solstices and equinoxes (inspired by a Stack Overflow question).

```# The angle that we call "the longitude of the Sun, as
# seen from Mars" should grow at the same rate as the
# "longitude of Mars as seen from the Sun" (since the
# two are the same line but viewed in opposite
# directions).
#
# The only problem is knowing what point to name "zero",
# so we have to learn what .hlon was when the first
# Martian year started:

from ephem import Mars, Date, degrees, newton
m = Mars()
m.compute('1955/4/11 23:00')
Ls0 = m.hlon

def Ls(date):
m.compute(date)
return degrees(m.hlon - Ls0).norm

# There!  Ls() should give Martian solar latitude.
# So the first round of seasons listed at the page
# http://www.planetary.org/explore/space-topics/mars/mars-calendar.html
# should give 90 degrees, 180 degrees, and 270 degrees:

for date in '1955/10/27', '1956/4/27', '1956/9/21':
print Ls(date)

# The output is close to what we would expect:
#
# 90:11:58.3
# 179:57:32.2
# 270:13:22.6
#
# Great!  So what if we want to know, say, the date
# of the upcoming Spring Equinox or Summer Solstice?
# We need functions that are smooth, well-behaved,
# and cross zero at those two times, so that we can
# unleash Newton's Method upon them:

def spring_equinox(date):
return Ls(date).znorm

def summer_solstice(date):
return Ls(date) - degrees('90:00:00')

def find_spring_equinox(start_date):
start_date = Date(start_date)
y0 = Ls(start_date)
y1 = Ls(start_date + 1)
rate = y1 - y0
angle_to_go = degrees(0.0 - y0).norm
closer_date = start_date + angle_to_go / rate
d = newton(spring_equinox, closer_date, closer_date + 1)
return Date(d)

def find_summer_solstice(start_date):
start_date = Date(start_date)
y0 = Ls(start_date)
y1 = Ls(start_date + 1)
rate = y1 - y0
angle_to_go = degrees(degrees('90:00:00') - y0).norm
closer_date = start_date + angle_to_go / rate
d = newton(summer_solstice, closer_date, closer_date + 1)
return Date(d)

d = find_spring_equinox('2015/1/22')
print d, Ls(d)

d = find_summer_solstice('2015/1/22')
print d, Ls(d)

# Output:
# 2015/6/16 15:03:15 0:00:00.0
# 2015/12/31 21:12:07 90:00:00.0
```