# spectral-norm Python 3 #8 program

## source code

```# The Computer Language Benchmarks Game
# https://salsa.debian.org/benchmarksgame-team/benchmarksgame/
#
# Contributed by Sebastien Loisel
# Fixed by Isaac Gouy
# Sped up by Josh Goldfoot
# Dirtily sped up by Simon Descarpentries
# Used list comprehension by Vadim Zelenin
# Concurrency by Jason Stitt
# Concurrency simplified by Matt Vollrath
# Optimized math by Adam Beckmeyer

from itertools import repeat
from math import sqrt
from multiprocessing import Pool
from sys import argv

def eval_A(i, j):
ij = i + j
return ij * (ij + 1) // 2 + i + 1

def A_sum(u, i):
return sum(u_j / eval_A(i, j) for j, u_j in enumerate(u))

def At_sum(u, i):
return sum(u_j / eval_A(j, i) for j, u_j in enumerate(u))

def multiply_AtAv(u):
r = range(len(u))

tmp = pool.starmap(
A_sum,
zip(repeat(u), r)
)
return pool.starmap(
At_sum,
zip(repeat(tmp), r)
)

def main():
n = int(argv)
u =  * n

for _ in range(10):
v = multiply_AtAv(u)
u = multiply_AtAv(v)

vBv = vv = 0

for ue, ve in zip(u, v):
vBv += ue * ve
vv  += ve * ve

result = sqrt(vBv/vv)
print("{0:.9f}".format(result))

if __name__ == '__main__':
with Pool(processes=4) as pool:
main()
```

## notes, command-line, and program output

```NOTES:
Python 3.9.0

Wed, 07 Oct 2020 00:21:50 GMT

MAKE:
mv spectralnorm.python3-8.python3 spectralnorm.python3-8.py

0.07s to complete and log all make actions

COMMAND LINE:
/opt/src/Python-3.9.0/bin/python3 -OO spectralnorm.python3-8.py 5500

PROGRAM OUTPUT:
1.274224153
```