python Python.Math.Algebra
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# When parsing positive / unsigned integers you can use:
>>> a = "03523"
>>> a.isdigit()
True
>>> b = "963spam"
>>> b.isdigit()
False
######################################################
# benchmark solutions
def is_number_tryexcept(s):
""" Returns True is string is a number. """
try:
float(s)
return True
except ValueError:
return False
import re
def is_number_regex(s):
""" Returns True is string is a number. """
if re.match("^\d+?\.\d+?$", s) is None:
return s.isdigit()
return True
def is_number_repl_isdigit(s):
""" Returns True is string is a number. """
return s.replace('.','',1).isdigit()
# Then iterate over the function list.
funcs = [
is_number_tryexcept,
is_number_regex,
is_number_repl_isdigit
]
a_float = '.1234'
print('Float notation ".1234" is not supported by:')
for f in funcs:
if not f(a_float):
print('\t -', f.__name__)
#############################################################
from re import match as re_match
from re import compile as re_compile
def is_number_tryexcept(s):
""" Returns True is string is a number. """
try:
float(s)
return True
except ValueError:
return False
def is_number_regex(s):
""" Returns True is string is a number. """
if re_match("^\d+?\.\d+?$", s) is None:
return s.isdigit()
return True
comp = re_compile("^\d+?\.\d+?$")
def compiled_regex(s):
""" Returns True is string is a number. """
if comp.match(s) is None:
return s.isdigit()
return True
def is_number_repl_isdigit(s):
""" Returns True is string is a number. """
return s.replace('.','',1).isdigit()
#######################
# Check if complex or float
def is_number(s):
try:
float(s) # for int, long and float
except ValueError:
try:
complex(s) # for complex
except ValueError:
return False
#######################################
# Check for various types
# x.isdigit() works well for checking if x is an integer.
# x.replace('-','').isdigit() works well for checking if x is a negative.(Check - in first position)
# x.replace('.','').isdigit() works well for checking if x is a decimal.
# x.replace(':','').isdigit() works well for checking if x is a ratio.
# x.replace('/','',1).isdigit() works well for checking if x is a fraction
Contents:
1. factors = Calculates factorilization of number
2. gcd = Calculates greatest common divisor (Iterative/Recursive)
3. lcm = Calculates least common multiple
4. square_root = Calculates square root
5. full_euclid = Calculates euclid relation
6. is_prime = Calculates whether a number is prime or not
7. binomial_coefficient = Calculates Binomial Coefficient of two numbers
8. find_last_index = Finds index of element so that List[element] <= bound
9. prime factorization = Finds all prime factors of a number and appends to list
10. fibonacci_numbers = Prints / Returns all fibonacci numbers up to n
11.fibo_sum_series = Calculate the sum of the first n elements of fibonacci
12.biggest_number = Returns the biggest number out of all provided
13.smallest_number = Returns the smallest number out of all provided
14.distancefromzero = Returns the absolute value out of all provided
15.normalizenumbers = Normalize all numbers in list
16.geometric_series = Fill an array with geometric series up to n
17.geo_series_sum = Calculate the sum of first n elements of geometric list
18.harmonic_series = Fill an array with harmonic series up to n
19.harm_series_sum = Calculate the sum of first n elements of harmonic series
20.prime_generate = Generate primes in various ways
21.apply_to_array = Apply calculation to all array elements or Boolean
22.mean_value = Calculates the mean value of list elements
23.stdev = Calculates the standard deviation of list elements
24.quotientremainder.py = Calculates a tuple of quotient / remainder for two numbers
25.Compoundinterest.py = Calculates compound interest
26.Permutations.py = Many forms of combinatronics and permutations
27.Checkifdigitfloat.py = Check if a string is a float or digit# -*- coding: utf-8 -*-
"""
Created on Fri Mar 31 09:49:49 2017
@author: P.Doulgeridis
"""
import itertools
def permutations(iterable, r=None):
# permutations('ABCD', 2) --> AB AC AD BA BC BD CA CB CD DA DB DC
# permutations(range(3)) --> 012 021 102 120 201 210
pool = tuple(iterable)
n = len(pool)
r = n if r is None else r
if r > n:
print("Combination length required longer than populace")
return
indices = list(range(n))
cycles = list(range(n, n-r, -1))
yield tuple(pool[i] for i in indices[:r])
while n:
for i in reversed(range(r)):
cycles[i] -= 1
if cycles[i] == 0:
indices[i:] = indices[i+1:] + indices[i:i+1]
cycles[i] = n - i
else:
j = cycles[i]
indices[i], indices[-j] = indices[-j], indices[i]
yield tuple(pool[i] for i in indices[:r])
break
else:
return
def permutations2(iterable, r=None):
pool = tuple(iterable)
n = len(pool)
r = n if r is None else r
for indices in product(range(n), repeat=r):
if len(set(indices)) == r:
yield tuple(pool[i] for i in indices)
a = [ 1, 2, 3, 7, "the" ]
c = [ 5, 6, 8, 9, "ifthen" ]
b = '4567'
# print(list(permutations(b, 3)))
# print(list(permutations(a, 5)))
# print(list(permutations(a, 4)))
# Permutation (Order matters)
#print (list(itertools.permutations(a, 3)))
# Combination (Order does not matter)
#print (list(itertools.combinations(a, 3)))
# Cartesian product (with several iterables - combine each with each one by one)
#print (list(itertools.product(a, c)))
# Cartesian product (with one iterable by itself, repeat factor)
#print (list(itertools.product(a, repeat=4)))
def permutation_generate(input, slots=2):
try:
import itertools
except ImportError:
return([],"ImportError: Itertools")
output_list = []
count = 0
calc = list(itertools.permutations(input, slots))
count = len(calc)
return [count, calc]
def combination_generate(input, slots=2):
try:
import itertools
except ImportError:
return([],"ImportError: Itertools")
output_list = []
count = 0
calc = list(itertools.combinations(input, slots))
count = len(calc)
return [count, calc]
def combi_repl_generate(input, slots=2):
checks = True
try:
import itertools
except ImportError:
return([],"ImportError: Itertools")
output_list = []
count = 0
calc = list(itertools.combinations_with_replacement(input, slots))
count = len(calc)
return [count, calc]
def cartesian_prod_generate(input1, input2):
try:
import itertools
except ImportError:
return([],"ImportError: Itertools")
output_list = []
count = 0
calc = list(itertools.product(a, c))
count = len(calc)
return [count, calc]
def cartesian_prod_repeat(input1, integer):
try:
import itertools
except ImportError:
return([],"ImportError: Itertools")
output_list = []
count = 0
calc = list(itertools.product(a, repeat=integer))
count = len(calc)
return [count, calc]
#print (permutation_generate(a, 4))
# print (combination_generate(a, 3))
# print (combi_repl_generate(a, 3))
#print (cartesian_prod_generate(a, c))
#print (cartesian_prod_repeat(c, 2))#One important formula in finance and accounting is the formula for continually
#compounding interest. This formula takes as input an amount of money invested,
#an interest rate, and an amount of time (in years), and returns the value of
#the investment after that period of time.
#
#The formula, shown above, is Amount = Principal * e ^ (Rate * Time).
#
#In the code below, we have created a function called CalculateAmount. This
#function has access to three variables: Principal, Rate, and Time. Note that
#e is a constant; you may access it with math.e. Complete this function by
#completing the marked lines so that the function returns the value of an
#investment given a certain Principal, Rate, and Time.
import math
def CalculateAmount(Principal, Rate, Time):
#If you want, you may add additional code here. Note that in the code you
#write here, you have access to three variables: Principal, Rate, and Time.
amount = Principal * math.e**(Rate*Time)
#We'll round the answer to the nearest dollar to avoid long decimals.
return round(amount)
#You may modify the variables below to test your code above.
testPrincipal = 5000
testRate = 0.05
testTime = 5
testAmount = CalculateAmount(testPrincipal, testRate, testTime)
print("After", testTime, "years invested with a", testRate, "interest rate, a", testPrincipal, "dollar investment will be worth", testAmount, "dollars.")def _ss(data):
"""Return sum of square deviations of sequence data."""
c = mean(data)
ss = sum((int(x)-c)**2 for x in data)
return ss
def pstdev(data):
"""Calculates the population standard deviation."""
n = len(data)
if n < 2:
raise ValueError('variance requires at least two data points')
ss = _ss(data)
pvar = ss/n # the population variance
return pvar**0.5
#####################################################################
# One function form
def pstdev(data):
def _ss(data2):
"""Return sum of square deviations of sequence data."""
c = mean(data2)
ss = sum((int(x)-c)**2 for x in data2)
return ss
"""Calculates the population standard deviation."""
n = len(data)
if n < 2:
raise ValueError('variance requires at least two data points')
ss = _ss(data)
pvar = ss/n # the population variance
return pvar**0.5def square_root( n ):
"""
Function: square_root
Description: Calculates sq.root
Input : Integer
Output: Sqrt of integer
Usage: sqrt( n )
Is infinite-precision long-integer square-root evaluator
Uses Newton's method, not floating-point
"""
if n < 1000000:
return int(math.sqrt(n))
digits = len( repr(n) ) - 1
if digits <= 0: digits = 1
x = 10 ** (digits/2) ## smaller than actual sqrt
decr = (x - n/x)/2
while decr >= 1 or decr <= -1:
x = x - decr
decr = (x - n/x)/2
return x
print (sqrt(34300003))def smallest_number(*args):
'''
Function: smallest_number
Description: Finds the smallest number
Input: numeric arguments
Output: smallest number
Usage: smallest_number(-10, -5, 5, 10)
Notes: Convert to biggest by changing min with max
'''
print min(args)
return min(args)
smallest_number(-10, -5, 5, 10)
def quotientAndRemainder(dividend, divisor):
quotient = dividend // divisor
#Complete this line such that quotient receives the
#quotient of dividend and divisor
remainder = dividend % divisor
#Complete this line such that remainder receives the
#remainder of dividend and divisor
return (quotient, remainder)
#You may modify these two variables to test your code
testDividend = 7
testDivisor = 3
#Don't modify these lines; they're used to test your code above!
(quo, rem) = quotientAndRemainder(testDividend, testDivisor)
print("The quotient is", quo)
print("The remainder is", rem)def prime_factors(n):
'''
Function: prime_factors
Description: Calculates prime factors of an integer
Input: Integer
Output: Prime factors of integer
Usage: prime_factors(n)
'''
i = 2
factors = []
while i * i <= n:
if n % i:
i += 1
else:
n //= i
factors.append(i)
if n > 1:
factors.append(n)
return factors
print (prime_factors(78))def normalize(numbers):
'''
Function: normalize
Description: Normalizes a set of numbers
Input: array of numbers
Output: normalized array
Usage: norm = normalize(a)
Notes: Used to morph raw data
'''
max_number = max(numbers)
for i in range(len(numbers)):
numbers[i] /= float(max_number)
return numbers
a = [ 13, 11, 40, 10, 9 ]
print (normalize(a))def mean(list):
mean = 0
for j in list:
mean += int(j)
return mean / len(list)
# First way - With repeated modulo
def lcm(p,q):
'''
Function : lcm
Description : Calculates lcm of two numbers
Input : p, q
Output : lcm
Usage : a = lcm(3,4)
Notes : Using repeated modulo
'''
p, q = abs(p), abs(q)
m = p * q
if not m: return 0
while True:
p %= q
if not p: return m // q
q %= p
if not q: return m // p
# For more than one value
def lcm(*values):
'''
Function: lcm
Description: Calculates lcm (for more than 2 values)
Input: a, b, c, d
Output: lcm
Usage: print (lcm(3,4,8))
Notes: Can take any arguments
'''
values = set([abs(int(v)) for v in values])
if values and 0 not in values:
n = n0 = max(values)
values.remove(n)
while any( n % m for m in values ):
n += n0
return n
return 0
print (lcm(3,4,8))def is_prime( n ):
'''
Function: is_prime
Description : Checks if a number is prime
Input: Integer
Output: Boolean (T/F)
Usage: is_prime(10)
Usage: is is_prime(11):
Notes: Contains call to "Factors" function(Embedded)
'''
import math
def factors( m, verbose = 0 ):
lfactors = []
sq = math.sqrt(m)
while m % 2 == 0:
lfactors.append( 2 )
m = m/2
sq = math.sqrt(m)
if verbose:
print (2)
d = 3
while d <= sq:
if m % d == 0:
lfactors.append( d )
m = m/d
sq = math.sqrt(m)
if verbose:
print (d)
else:
d = d+2
if m > 1:
lfactors.append( m )
if verbose:
print (m)
return lfactors
if (factors(n) == [n]):
return True
else:
return False
print (is_prime(18))
print (is_prime(17))def harmonic(n):
'''
Function: harmonic
Description: Evaluate harmonic series up to n into array
Input : n > 0
Output : Array with n first elements of harmonic series
Usage: b = harmonic(50)
Notes: Only integer arguments
'''
harmonic_series = [1]
for i in range(2,n,1):
temp = 1.0/i
print (i)
print (temp)
harmonic_series.append(temp)
return harmonic_series
n = input("Print n first numbers of Harmonic: ")
print (harmonic(n))def harmonicsum(n):
'''
Function: harmonicsum
Description: Compute partial sum of the harmonic series:
1/1 + 1/2 + 1/3 + 1/4 + ... + 1/n.
Input: n must be an integer > 0.
Output: Returns the harmonic sum, as a float.
Usage: a = harmonicsum(100)
Notes: Harmonic sum has no upper limit
'''
sum = 0.0
for i in range(int(n)):
sum += 1.0 / (i+1)
return sum
# The rest of this is a testbed program;
# the exam did not require this.
# n = input("Compute harmonic sum up to (n>0) n=") # Py3: int(input(...
# print "The sum is: %.10f" % harmonic(n) # Py3: print(...
def geometric(n):
'''
Function: geometric
Description: Evaluate geometric series up to n into array
Input: n > 0 (Integer)
Output: Array with n first elements of geometric series
Usage: llista = geometric(5)
Notes: Only integer arguments
'''
geometric_series = [0]
for i in range(1,n,1):
temp = 1.0/(2*i)
# print (i)
# print (temp)
geometric_series.append(temp)
return geometric_series
n = input("Print n first numbers of geometric: ")
print (geometric(n))def geometricsum(n):
'''
Function: geometricsum
Description: Evaluate geometric series sum up to n
Input : n > 0
Output : Sum of n first elements of geometric series
Usage: a = geometricsum(5)
Notes: Only integer arguments
'''
geometric_series = [0]
for j in range(1,n,1):
temp = 1.0/(2*j)
# print (i)
# print (temp)
geometric_series.append(temp)
total = 0
for i in geometric_series:
total += i
return total
print (geometricsum(n)) def genPrimesFn(amount):
'''
Function: genPrimesFn
Description: Generates a set amount of primes
Input: Amount of primes
Output: List with primes
Usage: genPrimesFn(5)
Notes: Only integer input
Function to return 1000000 prime numbers
'''
primes = [] # primes generated so far
last = 1 # last number tried
while len(primes) < amount:
last += 1
for p in primes:
if last % p == 0:
break
else:
primes.append(last)
return primes
def genPrimesinterval(amount, interval):
'''
Function: genPrimesinterval
Description: Function to print every 10th prime
number, until you've printed 20 of them.
Input: Amount(Integer) - amount of primes to be stored
Interval - amount of primes to be skipped
Output: List
Usage: a = genPrimesinterval(5,10)
Notes: Arguments can only be integers
'''
primes = [] # primes generated so far
intervalprimes = [] # interval primes to be stored
last = 1 # last number tried
counter = 1
while True:
last += 1
for p in primes:
if last % p == 0:
break
else:
primes.append(last)
counter += 1
if counter % interval == 0:
# Print every 10th prime
intervalprimes.append(last)
if counter % (amount*interval) == 0:
# Quit when we've printed the 10th prime 20 times (ie we've
# printed the 200th prime)
return intervalprimes
c = genPrimesFn2(10, 5)
print (c)
def genPrimesmanual():
'''
Function: genPrimesmanual
Description: Function to keep printing the prime number until the user stops the program.
This way uses user input; you can also just run an infinite loop (while True)
that the user can quit out of by hitting control-c
Input: None
Output: STDOUT, print
Usage: genPrimesmanual()
Notes: No arguments
'''
primes = [] # primes generated so far
last = 1 # last number tried
uinp = 'y' # Assume we want to at least print the first prime...
while uinp != 'n':
last += 1
for p in primes:
if last % p == 0:
break
else:
primes.append(last)
print(last)
uinp = input("Print the next prime? [y/n] ")
while uinp != 'y' and uinp != 'n':
while uinp:
print("Sorry, I did not understand your input. Please enter 'y' for yes, or 'n' for no.")
uinp = input("Print the next prime? [y/n] ")
if uinp == 'n':
breakdef gcd(x,y):
"""
Function: gcd
Description : Finds greatest common divisor
Input : 2 integers
Output: Greatest common divisor
Usage: gcd(10,2)
"""
while ( y>0 ):
x = x % y
(x,y) = (y,x)
return xdef full_euclid(x,y):
"""
Function: full_euclid(x,y)
Description: Calculates euclid relation
Input: 2 integers
Output: List of integers [a,b,g]
Usage: full_euclid( x, y )
Returns [a,b,g] so that a.x + b.n = g = gcd(x,n)
"""
def sgn( n ):
if n > 0:
return 1
elif n < 0:
return -1
else:
return 0
a,b,c,d = 1,0,0,1
sgnx = sgn(x)
sgny = sgn(y)
x = abs(x)
y = abs(y)
while y != 0:
[q,r] = divmod(x,y)
x,y = y,r
a,b,c,d = c,d,a-c*q,b-q*d
# now a.x_abs + b.y_abs = gcd
return a*sgnx, b*sgny, x # now x is the gcd
print (full_euclid(30,10))def find_last_index( bound, List ):
import math
"""
Function: find_last_index( bound, List )
Description : finds highest index i so that List[i] <= bound
Input: Two integers
Output: Tuple of (Index, List Element)
Usage: find_last_index( bound, List )
Notes: Uses divide-and-conquer (binary expansion) search
"""
lg = int( math.log( len(List)-1 )/ math.log( 2 ) )
now = pow(2, lg) - 1
while ( lg > 0 and (List[now] > bound or
( now+1 < len(List) and List[now+1] <= bound ))):
lg = lg - 1
inc = pow(2,lg)
if List[now] > bound: # guess is too high
now = now - inc
else: # too low
if now + inc < len(List):
now = now + inc
else:
pass
return (now, List[now])
a = [ 1, 2, 3, 8, 2, 6, 7, 8 ]
print (find_last_index(5, a))# Fibonacci numbers module
def fib(n): # write Fibonacci series up to n
'''
Function: fib(n)
Description: prints the first n fibonacci numbers
Input: n (integer)
Output: STDOUT, screen
Usage: fib(n)
Notes: Prints on screen
'''
a, b = 0, 1
while b < n:
print b,
a, b = b, a+b
def fib2(n): # return Fibonacci series up to n
'''
Function: fib2(n)
Description: Stores the first n fibonacci numbers in list
Input: n (integer)
Output: List
Usage: fib2(n)
Notes: Stores output in list
'''
result = []
a, b = 0, 1
while b < n:
result.append(b)
a, b = b, a+b
return result
###################################
def fibonacci(n):
'''
Function: fibonacci(n)
Description: Calculates the sum of first n fibonacci
Input: Integer > 09FA
Output: Prints first n numbers of fibonacci
Usage: print (fibonacci(n))
Notes: Compute partial sum of fibonacci series : (0 + 1 + 1 + 2 + 3 + 5)
'''
fibonacci_numbers = [0, 1]
for i in range(2,n):
fibonacci_numbers.append(fibonacci_numbers[i-1]+fibonacci_numbers[i-2])
return fibonacci_numbers
n = input("Print n first numbers of Fibonacci: ")
print (fibonacci(n))def fibonaccisum(n):
'''
Function: fibonaccisum(n)
Description: Returns the sum of first n fibonacci numbers
Input: Target number
Output: Sum of fibonacci up to target number
Usage: b = fibonaccisum(n)
Notes: n = integer
'''
fibonacci_numbers = [0, 1]
for i in range(2,n):
fibonacci_numbers.append(fibonacci_numbers[i-1]+fibonacci_numbers[i-2])
for j in fibonacci_numbers:
j += j
return j
n = input("Print the sum of the first n Fibos: ")
print (fibonaccisum(n))def factors( m, verbose = 0 ):
import math
"""
Function: factors
Description: Finds a factor combination of a number
Input: Integer
Output: List of factors
Usage: factors( m, verbose = 0 )
returns [list_of_factors]
"""
list_of_factors = []
sq = math.sqrt(m)
while m % 2 == 0:
list_of_factors.append( 2 )
m = m/2
sq = math.sqrt(m)
if verbose:
print (2)
d = 3
while d <= sq:
if m % d == 0:
list_of_factors.append( d )
m = m/d
sq = math.sqrt(m)
if verbose:
print (d)
else:
d = d+2
if m > 1:
list_of_factors.append( m )
if verbose:
print (m)
return list_of_factors
print (factors(1485674))
def distance_from_zero(arg):
'''
Function: distance_from_zero
Description: returns the absolute value
Input: Number
Output: Absolute of number
Usage: distance_from_zero(number)
Notes: Works on any kind of number
'''
print abs(arg)
return abs(arg)
distance_from_zero(-10)def checkiftriangleexists(side1, side2, side3):
'''
Name: checkiftriangleexists
Function: check if a triangle exists by comparing sides
Description: Check if a triangle exists
Input: 3 integers
Output: Boolean
Usage: print (checkiftriangleexists(1,2,4))
'''
return (side1 + side2) > side3
print (checkiftriangleexists(1,2,4))def binomial_coefficient( n , k ):
'''
Function: binomial_coefficient(n.k)
Description: Calculates binom coefficient of n by k
Input : Integer (n,k)
Output: Integer (Binomial Coefficient)
Usage: binomial_coefficient(n,k)
Usage: print (binomial_coefficient(n,k))
'''
out = n + 0
for i in range(1,k):
out = out * (n-i)
for i in range(1,k+1):
out = out/i
return out
print (binomial_coefficient(20, 2))
#### BETTER VERSION
def binomial(x, y):
from math import factorial as fac
try:
binom = fac(x) // fac(y) // fac(x - y)
except ValueError:
binom = 0
return binom
print (binomial(45,5))def biggest_number(*args):
'''
Function: biggest_number
Description: Returns the greatest number provided
Input: Any number of numbers
Output: Largest number
Usage: biggest_number(5, 6, -10, -1)
Notes: Works for any kind of number
'''
print max(args)
return max(args)
biggest_number(-10, -5, 5, 10)
######################################################
# FUNCTION - CALC_ARRAY
# APPLIES CALCULATION OF FUNC TO ARRAY ELEMENTS
# ELEMENT BY ELEMENT
# NOTES : THE ARRAYS NEED TO BE OF THE SAME SIDE
# THE ARRAYS NEED TO BE OF THE SAME TYPE, OTHERWISE :
# MIXED NUMBERS WILL BE CONVERTED TO FLOATS
# MIXED ARRAYS WITH STRINGS WILL YIELD STRING
def func(x, y):
'''
Token function defined for calc purposes
'''
return x * y
def func2(x, y):
'''
Token function defined for calc purposes
'''
return y - x ** 2
def func3(x, y):
'''
Token boolean function used for calc purposes
'''
return x > y
def calc_array(list1, list2, f):
'''
Function: calc_array
Description: Element-wise calculations in lists
Input: list1, list2, function
Output: f(list1, list2)
Usage: print (calc_array(height, weight, func))
Notes: Lists must be of the same size
'''
import numpy as np
flist1 = np.array(list1)
flist2 = np.array(list2)
return f(flist1, flist2)
print (calc_array(height, weight, func))
print (calc_array(height, weight, func2))
print (calc_array(height, weight, func3))
print (calc_array(height, alt, func))
b = (calc_array(height, alt, func))
print (type(b[1]))
_# Float will either be given as XXX,YY
# or XXXX.YY
#xxx.yy - straight conversion
float_in = float(XXX.YY)
#XXX,YY - replace with ., then convert
string_in = "XXX,YY"
string_fixed = string_in.replace(",", ".")
float_in = float(string_fixed)
# Example with comparison
try:
value_fl = value_in.replace(',', '.')
except Exception as e:
print("cant convert to float with exception: " + str(e))
else:
value_fl_fixed = float(value_fl)
finally:
#print(key_in, value_in, value_fl, value_fl_fixed)
pass
if value_fl_fixed >= 150:
outfile_first.write(line)
else:
outfile_firstr.write(line)
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