I'm calculating the matrix value with Python, but I want to distinguish the value of equtaion, is there a way?
x - y - 2z = 4
2x - y - z = 2
2x +y +4z = 16
I want to make the expression above like this when I print out the matrix from the function I created
1 -1 -2 | 4
2 -1 -1 | 2
2 1 4 | 16
Same as the rref result of this
1 0 0 | 24
0 1 0 | 72
0 0 1 | -26
def showMatrix():
print("\n")
for i in sd:
for j in i:
print(j, end="\t")
print("\n")
def getone(pp):
for i in range(len(sd[0])):
if sd[pp][pp] != 1:
q00 = sd[pp][pp]
for j in range(len(sd[0])):
sd[pp][j] = sd[pp][j] / q00
def getzero(r, c):
for i in range(len(sd[0])):
if sd[r][c] != 0:
q04 = sd[r][c]
for j in range(len(sd[0])):
sd[r][j] = sd[r][j] - ((q04) * sd[c][j])
sd = [
[1, 1, 2, 9],
[2, 4, -3, 1],
[3, 6, -5, 0]
]
showMatrix()
for i in range(len(sd)):
getone(i)
for j in range(len(sd)):
if i != j:
getzero(j, i)
showMatrix()
print("FiNAL result")
showMatrix()
Here is a function which takes a list of 4 numbers and returns a string representing an equation in x,y,z. It handles coefficients which are negative, zero, or +/-1 appropriately:
def make_equation(nums):
coefficients = nums[:3]
variables = 'xyz'
terms = []
for c,v in zip(coefficients,variables):
if c == 0:
continue
elif c == 1:
coef = ''
elif c == -1:
coef = '-'
else:
coef = str(c)
terms.append(coef + v)
s = ' + '.join(terms)
s = s.replace('+ -','- ')
return s + ' = ' + str(nums[3])
Typical example:
make_equation([2,-3,1,6])
With output:
'2x - 3y + z = 6'
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I tried to write code to solve the standard Integer Partition problem (Wikipedia). The code I wrote was a mess. I need an elegant solution to solve the problem, because I want to improve my coding style. This is not a homework question.
A smaller and faster than Nolen's function:
def partitions(n, I=1):
yield (n,)
for i in range(I, n//2 + 1):
for p in partitions(n-i, i):
yield (i,) + p
Let's compare them:
In [10]: %timeit -n 10 r0 = nolen(20)
1.37 s ± 28.7 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
In [11]: %timeit -n 10 r1 = list(partitions(20))
979 µs ± 82.9 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
In [13]: sorted(map(sorted, r0)) == sorted(map(sorted, r1))
Out[14]: True
Looks like it's 1370 times faster for n = 20.
Anyway, it's still far from accel_asc:
def accel_asc(n):
a = [0 for i in range(n + 1)]
k = 1
y = n - 1
while k != 0:
x = a[k - 1] + 1
k -= 1
while 2 * x <= y:
a[k] = x
y -= x
k += 1
l = k + 1
while x <= y:
a[k] = x
a[l] = y
yield a[:k + 2]
x += 1
y -= 1
a[k] = x + y
y = x + y - 1
yield a[:k + 1]
It's not only slower, but requires much more memory (but apparently is much easier to remember):
In [18]: %timeit -n 5 r2 = list(accel_asc(50))
114 ms ± 1.04 ms per loop (mean ± std. dev. of 7 runs, 5 loops each)
In [19]: %timeit -n 5 r3 = list(partitions(50))
527 ms ± 8.86 ms per loop (mean ± std. dev. of 7 runs, 5 loops each)
In [24]: sorted(map(sorted, r2)) == sorted(map(sorted, r3))
Out[24]: True
You can find other versions on ActiveState: Generator For Integer Partitions (Python Recipe).
I use Python 3.6.1 and IPython 6.0.0.
While this answer is fine, I'd recommend skovorodkin's answer.
>>> def partition(number):
... answer = set()
... answer.add((number, ))
... for x in range(1, number):
... for y in partition(number - x):
... answer.add(tuple(sorted((x, ) + y)))
... return answer
...
>>> partition(4)
set([(1, 3), (2, 2), (1, 1, 2), (1, 1, 1, 1), (4,)])
If you want all permutations(ie (1, 3) and (3, 1)) change answer.add(tuple(sorted((x, ) + y)) to answer.add((x, ) + y)
I've compared the solution with perfplot (a little project of mine for such purposes) and found that Nolen's top-voted answer is also the slowest.
Both answers supplied by skovorodkin are much faster. (Note the log-scale.)
To to generate the plot:
import perfplot
import collections
def nolen(number):
answer = set()
answer.add((number,))
for x in range(1, number):
for y in nolen(number - x):
answer.add(tuple(sorted((x,) + y)))
return answer
def skovorodkin(n):
return set(skovorodkin_yield(n))
def skovorodkin_yield(n, I=1):
yield (n,)
for i in range(I, n // 2 + 1):
for p in skovorodkin_yield(n - i, i):
yield (i,) + p
def accel_asc(n):
return set(accel_asc_yield(n))
def accel_asc_yield(n):
a = [0 for i in range(n + 1)]
k = 1
y = n - 1
while k != 0:
x = a[k - 1] + 1
k -= 1
while 2 * x <= y:
a[k] = x
y -= x
k += 1
l = k + 1
while x <= y:
a[k] = x
a[l] = y
yield tuple(a[: k + 2])
x += 1
y -= 1
a[k] = x + y
y = x + y - 1
yield tuple(a[: k + 1])
def mct(n):
partitions_of = []
partitions_of.append([()])
partitions_of.append([(1,)])
for num in range(2, n + 1):
ptitions = set()
for i in range(num):
for partition in partitions_of[i]:
ptitions.add(tuple(sorted((num - i,) + partition)))
partitions_of.append(list(ptitions))
return partitions_of[n]
perfplot.show(
setup=lambda n: n,
kernels=[nolen, mct, skovorodkin, accel_asc],
n_range=range(1, 17),
logy=True,
# https://stackoverflow.com/a/7829388/353337
equality_check=lambda a, b: collections.Counter(set(a))
== collections.Counter(set(b)),
xlabel="n",
)
I needed to solve a similar problem, namely the partition of an integer n into d nonnegative parts, with permutations. For this, there's a simple recursive solution (see here):
def partition(n, d, depth=0):
if d == depth:
return [[]]
return [
item + [i]
for i in range(n+1)
for item in partition(n-i, d, depth=depth+1)
]
# extend with n-sum(entries)
n = 5
d = 3
lst = [[n-sum(p)] + p for p in partition(n, d-1)]
print(lst)
Output:
[
[5, 0, 0], [4, 1, 0], [3, 2, 0], [2, 3, 0], [1, 4, 0],
[0, 5, 0], [4, 0, 1], [3, 1, 1], [2, 2, 1], [1, 3, 1],
[0, 4, 1], [3, 0, 2], [2, 1, 2], [1, 2, 2], [0, 3, 2],
[2, 0, 3], [1, 1, 3], [0, 2, 3], [1, 0, 4], [0, 1, 4],
[0, 0, 5]
]
I'm a bit late to the game, but I can offer a contribution which might qualify as more elegant in a few senses:
def partitions(n, m = None):
"""Partition n with a maximum part size of m. Yield non-increasing
lists in decreasing lexicographic order. The default for m is
effectively n, so the second argument is not needed to create the
generator unless you do want to limit part sizes.
"""
if m is None or m >= n: yield [n]
for f in range(n-1 if (m is None or m >= n) else m, 0, -1):
for p in partitions(n-f, f): yield [f] + p
Only 3 lines of code. Yields them in lexicographic order. Optionally allows imposition of a maximum part size.
I also have a variation on the above for partitions with a given number of parts:
def sized_partitions(n, k, m = None):
"""Partition n into k parts with a max part of m.
Yield non-increasing lists. m not needed to create generator.
"""
if k == 1:
yield [n]
return
for f in range(n-k+1 if (m is None or m > n-k+1) else m, (n-1)//k, -1):
for p in sized_partitions(n-f, k-1, f): yield [f] + p
After composing the above, I ran across a solution I had created almost 5 years ago, but which I had forgotten about. Besides a maximum part size, this one offers the additional feature that you can impose a maximum length (as opposed to a specific length). FWIW:
def partitions(sum, max_val=100000, max_len=100000):
""" generator of partitions of sum with limits on values and length """
# Yields lists in decreasing lexicographical order.
# To get any length, omit 3rd arg.
# To get all partitions, omit 2nd and 3rd args.
if sum <= max_val: # Can start with a singleton.
yield [sum]
# Must have first*max_len >= sum; i.e. first >= sum/max_len.
for first in range(min(sum-1, max_val), max(0, (sum-1)//max_len), -1):
for p in partitions(sum-first, first, max_len-1):
yield [first]+p
Much quicker than the accepted response and not bad looking, either. The accepted response does lots of the same work multiple times because it calculates the partitions for lower integers multiple times. For example, when n=22 the difference is 12.7 seconds against 0.0467 seconds.
def partitions_dp(n):
partitions_of = []
partitions_of.append([()])
partitions_of.append([(1,)])
for num in range(2, n+1):
ptitions = set()
for i in range(num):
for partition in partitions_of[i]:
ptitions.add(tuple(sorted((num - i, ) + partition)))
partitions_of.append(list(ptitions))
return partitions_of[n]
The code is essentially the same except we save the partitions of smaller integers so we don't have to calculate them again and again.
Here is a recursive function, which uses a stack in which we store the numbers of the partitions in increasing order.
It is fast enough and very intuitive.
# get the partitions of an integer
Stack = []
def Partitions(remainder, start_number = 1):
if remainder == 0:
print(" + ".join(Stack))
else:
for nb_to_add in range(start_number, remainder+1):
Stack.append(str(nb_to_add))
Partitions(remainder - nb_to_add, nb_to_add)
Stack.pop()
When the stack is full (the sum of the elements of the stack then corresponds to the number we want the partitions), we print it,
remove its last value and test the next possible value to be stored in the stack. When all the next values have been tested, we pop the last value of the stack again and we go back to the last calling function.
Here is an example of the output (with 8):
Partitions(8)
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1 + 2
1 + 1 + 1 + 1 + 1 + 3
1 + 1 + 1 + 1 + 2 + 2
1 + 1 + 1 + 1 + 4
1 + 1 + 1 + 2 + 3
1 + 1 + 1 + 5
1 + 1 + 2 + 2 + 2
1 + 1 + 2 + 4
1 + 1 + 3 + 3
1 + 1 + 6
1 + 2 + 2 + 3
1 + 2 + 5
1 + 3 + 4
1 + 7
2 + 2 + 2 + 2
2 + 2 + 4
2 + 3 + 3
2 + 6
3 + 5
4 + 4
8
The structure of the recursive function is easy to understand and is illustrated below (for the integer 31):
remainder corresponds to the value of the remaining number we want a partition (31 and 21 in the example above).
start_number corresponds to the first number of the partition, its default value is one (1 and 5 in the example above).
If we wanted to return the result in a list and get the number of partitions, we could do this:
def Partitions2_main(nb):
global counter, PartitionList, Stack
counter, PartitionList, Stack = 0, [], []
Partitions2(nb)
return PartitionList, counter
def Partitions2(remainder, start_number = 1):
global counter, PartitionList, Stack
if remainder == 0:
PartitionList.append(list(Stack))
counter += 1
else:
for nb_to_add in range(start_number, remainder+1):
Stack.append(nb_to_add)
Partitions2(remainder - nb_to_add, nb_to_add)
Stack.pop()
Last, a big advantage of the function Partitions shown above is that it adapts very easily to find all the compositions of a natural number (two compositions can have the same set of numbers, but the order differs in this case):
we just have to drop the variable start_number and set it to 1 in the for loop.
# get the compositions of an integer
Stack = []
def Compositions(remainder):
if remainder == 0:
print(" + ".join(Stack))
else:
for nb_to_add in range(1, remainder+1):
Stack.append(str(nb_to_add))
Compositions(remainder - nb_to_add)
Stack.pop()
Example of output:
Compositions(4)
1 + 1 + 1 + 1
1 + 1 + 2
1 + 2 + 1
1 + 3
2 + 1 + 1
2 + 2
3 + 1
4
I think the recipe here may qualify as being elegant. It's lean (20 lines long), fast and based upon Kelleher and O'Sullivan's work which is referenced therein:
def aP(n):
"""Generate partitions of n as ordered lists in ascending
lexicographical order.
This highly efficient routine is based on the delightful
work of Kelleher and O'Sullivan.
Examples
========
>>> for i in aP(6): i
...
[1, 1, 1, 1, 1, 1]
[1, 1, 1, 1, 2]
[1, 1, 1, 3]
[1, 1, 2, 2]
[1, 1, 4]
[1, 2, 3]
[1, 5]
[2, 2, 2]
[2, 4]
[3, 3]
[6]
>>> for i in aP(0): i
...
[]
References
==========
.. [1] Generating Integer Partitions, [online],
Available: http://jeromekelleher.net/generating-integer-partitions.html
.. [2] Jerome Kelleher and Barry O'Sullivan, "Generating All
Partitions: A Comparison Of Two Encodings", [online],
Available: http://arxiv.org/pdf/0909.2331v2.pdf
"""
# The list `a`'s leading elements contain the partition in which
# y is the biggest element and x is either the same as y or the
# 2nd largest element; v and w are adjacent element indices
# to which x and y are being assigned, respectively.
a = [1]*n
y = -1
v = n
while v > 0:
v -= 1
x = a[v] + 1
while y >= 2 * x:
a[v] = x
y -= x
v += 1
w = v + 1
while x <= y:
a[v] = x
a[w] = y
yield a[:w + 1]
x += 1
y -= 1
a[v] = x + y
y = a[v] - 1
yield a[:w]
# -*- coding: utf-8 -*-
import timeit
ncache = 0
cache = {}
def partition(number):
global cache, ncache
answer = {(number,), }
if number in cache:
ncache += 1
return cache[number]
if number == 1:
cache[number] = answer
return answer
for x in range(1, number):
for y in partition(number - x):
answer.add(tuple(sorted((x, ) + y)))
cache[number] = answer
return answer
print('To 5:')
for r in sorted(partition(5))[::-1]:
print('\t' + ' + '.join(str(i) for i in r))
print(
'Time: {}\nCache used:{}'.format(
timeit.timeit(
"print('To 30: {} possibilities'.format(len(partition(30))))",
setup="from __main__ import partition",
number=1
), ncache
)
)
or https://gist.github.com/sxslex/dd15b13b28c40e695f1e227a200d1646
I don't know if my code is the most elegant, but I've had to solve this many times for research purposes. If you modify the
sub_nums
variable you can restrict what numbers are used in the partition.
def make_partitions(number):
out = []
tmp = []
sub_nums = range(1,number+1)
for num in sub_nums:
if num<=number:
tmp.append([num])
for elm in tmp:
sum_elm = sum(elm)
if sum_elm == number:
out.append(elm)
else:
for num in sub_nums:
if sum_elm + num <= number:
L = [i for i in elm]
L.append(num)
tmp.append(L)
return out
F(x,n) = \union_(i>=n) { {i}U g| g in F(x-i,i) }
Just implement this recursion. F(x,n) is the set of all sets that sum to x and their elements are greater than or equal to n.
Hey guys as you've read in the question i am trying to find the element pairs in an array equal to the given sum and return the sum of their respective indices.
I was able to return the element pairs for the given sum but failed to return the sum of their indices. Here is my code:
arr = [1, 4, 2, 3, 0 , 5]
sum = 7
x = min(arr)
y = max(arr)
while x < y:
if x + y > sum:
y -= 1
elif x + y < sum:
x += 1
else:
print("(", x, y, ")")
x += 1
My output:
( 2 5 )
( 3 4 )
This is what i need to do further:
2 + 5 = 7 → Indices 2 + 5 = 7;
3 + 4 = 7 → Indices 3 + 1 = 4;
7 + 4 = 11 → Return 11;
Thanks in Advance!
you can try using a nested loop :
arr = [1, 4, 2, 3, 0 , 5]
sums = 7
tlist = []
for i in range(len(arr)):
for j in range(len(arr)-1):
if (i!=j) and ((arr[i] + arr[j+1]) == sums):
if (i,j+1) not in tlist and (j+1,i) not in tlist:
tlist.append((i,j+1))
print("index ->",i," ",j+1)
print("sum=", i+j+1)
output:
index -> 1 3
sum= 4
index -> 2 5
sum= 7
You could use itertools for easily checking sum for combinations like,
>>> import itertools
>>> num = 7
>>> for a,b in itertools.combinations(arr, 2):
... if a + b == num:
aindex, bindex = arr.index(a), arr.index(b)
... indices_sum = aindex + bindex
... print('[element sum]: {} + {} = {} [indices sum]: {} + {} = {}'.format(a, b, a + b, aindex, bindex , indices_sum))
...
[element sum]: 4 + 3 = 7 [indices sum]: 1 + 3 = 4
[element sum]: 2 + 5 = 7 [indices sum]: 2 + 5 = 7
>>> arr
[1, 4, 2, 3, 0, 5]
You could take a different approach by calculating the difference then checking if each element is present in the first array or not.
arr = [1, 4, 2, 3, 0, 5]
the_sum = 7
diff = [the_sum - x for x in arr]
for idx, elem in enumerate(diff):
try:
index = arr.index(elem)
sum_of_indices = idx + index
print("{} + {} = {}".format(idx, index, sum_of_indices))
except ValueError:
pass
output
1 + 3 = 4
2 + 5 = 7
3 + 1 = 4
5 + 2 = 7
To remove the duplicates, its always easy to take a frozenset of the indices tuple
a = [(2,1), (1,2), (3,2), (2,3)]
{frozenset(x) for x in a} # {frozenset({2, 3}), frozenset({1, 2})}
I'm trying to solve an example in my book:
Write a program that generates a sequence of 20 random die tosses in a list and that prints the die values, marking the runs by including them in parentheses, like this:
1 2 (5 5) 3 1 2 4 3 (2 2 2 2) 3 6 (5 5) 6 3 1
Use the following pseudocode:
Set a boolean variable inRun to false.
For each valid index i in the list
If inRun
If values[i] is different from the preceding value
Print).
inRun = false.
If not inRun
If values[i] is the same as the following value
Print (.
inRun = true.
Print values[i].
If inRun, print).
Here is my code:
import random
min = 1
max = 6
randomToss = []
for i in range(20):
randomToss.append(random.randint(min, max))
print(randomToss)
inRun = False
for i in randomToss:
if inRun:
if randomToss[i] != i - 1:
print(")")
inRun = False
if not inRun:
if randomToss[i] == i + 1:
print("(")
inRun = True
print(randomToss[i])
if inRun:
print(")")
When it runs, I get an output like so:
[2, 5, 1, 3, 1, 4, 4, 4, 6, 4, 2, 1, 4, 1, 4, 4, 5, 1, 6, 4]
1
(
4
)
5
3
5
1
1
1
(
4
)
1
1
5
1
5
1
1
(
4
)
5
(
4
)
1
When it should look something like:
[2, 5, 1, 3, 1, 4, 4, 4, 6, 4, 2, 1, 4, 1, 4, 4, 5, 1, 6, 4]
2 5 1 3 1 (4 4 4) 6 4 2 1 4 1 (4 4) 5 1 6 4
What is wrong?
You are doing for i in randomToss: and then doing randomToss[i]. This is wrong. i here refers to the actual value in the array and not the index.
you should use the range function to generate the indices for iteration.
for i in range(len(randomToss) - 1): # <- returns the indeices
if inRun:
if randomToss[i] != randomToss[i + 1]:
print(randomToss[i], end='') # <- you must print the number before ')'
print(")", end='')
inRun = False
else: # use else since you're changing isRun inside the loop
if randomToss[i] == randomToss[i + 1]:
print("(", end='')
inRun = True
print(randomToss[i], end='')
This is what is wrong with your current code. You have an array of integers called randomToss.You are looping over each integer in the array, when you do this, you set a variable called i, which stands for the index of the array.
[1 2 5 5 3, 1] // array
[1 2 3 4 5, 6] // index values
The index is the current array element in incremental order, 1,2,3.... While inside this loop you are checking the current integer, randomToss[i], in the array to the value of the the index value. You need to check the next and previous array elements. To do this you can use the index to get the element out of the array by adding or subtracting one (remember incremental) .
for i in range(len(randomToss)-1):
if inRun:
if randomToss[i] != randomToss[i-1]:
print(")")
inRun = False
if not inRun:
if randomToss[i] == randomToss[i+1]:
print("(")
inRun = True
print(randomToss[i])
The following code works.
import random
min = 1
max = 6
randomToss = []
for i in range(20):
randomToss.append(random.randint(min, max))
print(randomToss)
inRun = False
for i in range(20): # i is the index not the value
if inRun:
if i != 0 and randomToss[i] != randomToss[i - 1]: # check boundary conditions and compare the values
print(")", end="")
inRun = False
if not inRun:
if i != 19 and randomToss[i] == randomToss[i + 1]: # check boundary conditions and compare the values
print("(", end="")
inRun = True
print(randomToss[i], end="") # check indentation with the pseudocode
if inRun:
print(")", end="")
I'm just trying to get the logic straight and using Python to help me do it. Ultimately, I need to solve this problem using ImageJ macro language.
I have no idea if I'm using the right term, but I'd like to create a "snake" counter.
x = 1
number = 12
maxVal = 3
minVal = 1
for i in xrange(number):
%do something
x = incrementSnakeCounter(x, maxVal, minVal)
print("i = ", i)
print("x = ", x)
The "snake" part is making the counter go up only to the maxVal, repeating that number on the next iteration, counting down to the minVal, repeating that value on the next iteration, and repeating the process.
For instance, in the above
I'd like the following to happen :
i = 0
x = 1
i = 1
x = 2
i = 2
x = 3
i = 3
x = 3
i = 4
x = 2
i = 5
x = 1
i = 6
x = 1
i = 7
x = 2
i = 8
x = 3
i = 9
x = 3
i = 10
x = 2
i = 11
x = 1
You will find some useful utils in itertools:
from itertools import chain, cycle
def snake(lower, upper):
return cycle(chain(range(lower, upper+1), range(upper, lower-1, -1)))
> s = snake(1,3)
> [next(s) for _ in range(10)]
[1, 2, 3, 3, 2, 1, 1, 2, 3, 3]
Here's a silly mathematical solution:
def snake(low, high, x):
k = (high-low+1)
return k - int(abs(x % (2*k) + low - k - 0.5))
[snake.snake(1,3,x) for x in range(8)]
[1, 2, 3, 3, 2, 1, 1, 2]
Add a conditional to determine if x should be increasing or decreasing at any given point within the loop.
x = 1
number = 12
maxVal = 3
minVal = 1
for i in xrange(number):
%do something
if(xIsIncreasing)
x = incrementSnakeCounter(x, maxVal, minVal)
else
x = decrementSnakeCounter(x, maxVal, minVal)
print("i = ", i)
print("x = ", x)
Then inside your incrementSnakeCounter() change the value of xIsIncreasing to false when x == maxVal and inside your decrementSnakeCounter() to true when x == minVal (you'll have to do some work to make sure that you're staying at the same value twice in a row, I don't have time right now to solve that part for you).
You can write a little custom generator.
The key is to create a list of the pattern you want to repeat [1, 2, 3, 3, 2, 1] and then index that with the modulo of the length to get the repeating behavior:
def snake(x, max_v=3, min_v=1):
cnt=0
sn=list(range(min_v, max_v+1,1))+list(range(max_v, min_v-1,-1))
while cnt<x:
yield cnt, sn[cnt%len(sn)]
cnt+=1
Then:
for i,x in snake(12):
print("i=",i)
print("x=",x)
print()
Prints:
i= 0
x= 1
i= 1
x= 2
i= 2
x= 3
i= 3
x= 3
i= 4
x= 2
i= 5
x= 1
i= 6
x= 1
i= 7
x= 2
i= 8
x= 3
i= 9
x= 3
i= 10
x= 2
i= 11
x= 1