Python: Calculate the last digit of a large Fibonacci Number with less time - python-3.x

# Uses python3
# Compute the Last Digit of a Large Fibonacci Number
def Fib_Last_Digit(n):
if n == 0 : return 0
elif n == 1: return 1
else:
a,b = 0,1
for i in range(1,n):
c = a + b;
a = b;
b = c;
# Calculate the last digit of the final number
lastdigit = int(repr(c)[-1]);
print(lastdigit);
n = int(input(""));
Fib_Last_Digit(n);
This code works very well. However, I want to revise the algorithm to save more time and memory. By the way, the input and output should be kept the same with the previous version.


Only keeping the last digit during calculation saves a lot of time:
def fib_last_digit(n):
if n < 2: return n
else:
a, b = 0, 1
for i in range(1,n):
a, b = b, (a+b) % 10
print(b)
n = int(input())
fib_last_digit(n)
Handling numbers that fit in fewer bytes saves time.
When you're working with huge numbers, you can save a lot of time using the answer described here, slightly modified to only keep track of the last digit:
def fib_last_digit(n):
v1, v2, v3 = 1, 1, 0 # initialise a matrix [[1,1],[1,0]]
for rec in bin(n)[3:]: # perform fast exponentiation of the matrix (quickly raise it to the nth power)
calc = (v2*v2) % 10
v1, v2, v3 = (v1*v1+calc) % 10, ((v1+v3)*v2) % 10, (calc+v3*v3) % 10
if rec == '1': v1, v2, v3 = (v1+v2) % 10, v1, v2
return v2
And finally, based on the concept described in Eugene Yarmash's answer (the last digits repeat every 60 steps) we can make an even faster solution (O(1)):
def fib_last_digit(n):
return (
[1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0]
[n % 60 - 1]
)


The series of final digits of Fibonacci numbers repeats with a cycle length of 60. Therefore, the Nth Fibonacci number has the same last digit as the (N % 60)th, which should be pretty fast to calculate. As an additional optimization, you can keep only the last digit of each term:
def fib_last_digit(n):
a, b = 0, 1
for i in range(n % 60):
a, b = b, (a + b) % 10
return a
print([fib_last_digit(n) for n in range(1, 11)])
Output:
[1, 1, 2, 3, 5, 8, 3, 1, 4, 5]


def fib(n):
phi = (1 + 5 ** 0.5) / 2
fib_n = round(((phi** n) - (phi**-n) )/(5 ** .5))
return fib_n % 10
Phi is your friend.


def fib_digit(n):
f=[1,1]
for i in range(2,n):
f.append((f[i-1]+f[i-2]) % 10 )
return f[-1]
n = int(input())
print(fib_digit(n))
This is one of the simplest answers,i'm sure, there is a faster algorithm.
Here is what I found:
f1, f2 = 0, 1
for i in range(int(input())-1):
f1, f2 = f2, (f1+f2)%10
print(f2)


It took only --- 0.002832174301147461 seconds --- to complete the code.
import time
n = 100000000000000000000000000000000000000000
def printfib(previous, latest, n):
if(latest > n):
return
print(', ', latest, end='')
printfib(latest, previous + latest, n)
start_time = time.time()
print(0, end='')
printfib(0, 1, n)
print(" ")
print("--- %s seconds ---" % (time.time() - start_time))

Related

why code does not giving the right answer it suppose to to create a function that finds the middle item from a list of values

def middle_element(lst):
if len(lst) % 2 == 0:
midindex1 = int(len(lst) / 2)
print(midindex1)
average = (lst[midindex1 - 1] + lst[midindex1]) / 2
return average
else:
midindex2 = int(len(lst) / 2)
return lst[midindex2]
print(middle_element([5, 2, -10, -4, 4, 5, 8]))

Seeing your code I think you are trying to implement Median on the list
Mean Median Mode Formula <- Link to formula
For median you first need to sort the list
Method 1:
def middle_element(lst):
lst.sort() #Sorting the list
if len(lst) % 2 == 0:
midindex1 = int(len(lst) / 2)
print(midindex1)
average = (lst[midindex1 - 1] + lst[midindex1]) / 2
return average
else:
midindex2 = int(len(lst) / 2)
return lst[midindex2]
print(middle_element([5, 2, -10, -4, 4, 5, 8]))
This will output:
4
Method 2:
You can use statistics library
from statistics import mean, mode, median
lst = [5, 2, -10, -4, 4, 5, 8]
print(mean(lst))
print(mode(lst))
print(median(lst))
This will output:
1.4285714285714286
5
4

How to print list?

So i'm writing a code, listing square numbers lower than n.
I want the output to be a list. like so: [1, 4, 9, 16] etc.
n=int(input("input number: "))
counter = 1
while counter * counter < n:
for counter in range(1,n):
a = counter*counter
print (a)
if a < n:
break
I would be very grateful if I can get some help.

you can remove the need for the while loop and utilise pythons range function to generate your numbers up to n and use a for loop to iterate over them. you also need to initalise a list to store them in before the loop starts. and then each iteration of the loop append the square number to the list.
n=int(input("input number: "))
squares = []
for num in range(1, n):
squares.append(num * num)
print(squares)
OUTPUT for n=10
[1, 4, 9, 16, 25, 36, 49, 64, 81]
Although this can be simpfied using pythons list comprehension style as
n=int(input("input number: "))
squares = [num * num for num in range(1, n)]
print(squares)
OUTPUT for n=10
[1, 4, 9, 16, 25, 36, 49, 64, 81]
UPDATE based on comment from B. Go
user B. Go pointed out that my answer is printing all the results of squaring number up to the value of N. But the question was actually print all the square numbers less than N. Below code to print squares less than N
n=int(input("input number: "))
squares = []
for num in range(1, n):
result = num * num
if result >= n:
break
squares.append(num * num)
print(squares)
OUTPUT N=50
[1, 4, 9, 16, 25, 36, 49]

Is there a way to find the sum of the number after the ending of 9?

The question is:
Return the sum of the numbers in the array, except ignore sections of numbers starting with a 6 and extending to the next 9 (every 6 will be followed by at least one 9).
sixty_nine([4, 5, 6, 7, 8, 9]) --> 9
sixty_nine([4, 5,7,8, 6, 7, 8, 9,34]) --> it should return -- 58
but it returns -- 24(by ignoring 34)
i am not able to understand how to take the sum of numbers after 9
arr = [4, 5,7,8, 6, 7, 8, 9,34]
def sixty_nine(arr):
sums = 0
l = len(arr)
for i in range(l):
if arr[i] == 6 and 9 in arr[i:]:
for j in arr[0:i]:
sums = sums + j
return sums
sixty_nine(arr)
t=The output of this program is --- 24

You code contains various errors, among them you are returning inside your for-loop which prevents it from traversing the whole list.
I suggest treating this problem in two steps. First write a generator which filters out sublists of the form [6, ..., 9].
Then use sum which can sum any iterable, generators included.
def ignore_between(lst, a, b):
i = 0
while i < len(lst):
if lst[i] == a:
try:
i = lst.index(b, i) + 1
continue
except ValueError:
pass
yield lst[i]
i += 1
lst = [4, 5, 7, 8, 6, 7, 8, 9, 34]
output = sum(ignore_between(lst, 6, 9))
print(output) # 58

Python - assigning digits of a number to variables [duplicate]

This question already has answers here:
How to split an integer into a list of digits?
(10 answers)
Closed last month.
I'm writing a program which calculates the check digit of an ISBN number. I have to read the user's input (nine digits of an ISBN) into an integer variable, and then multiply the last digit by 2, the second last digit by 3 and so on. How can I "split" the integer into its constituent digits to do this? As this is a basic homework exercise I am not supposed to use a list.

Just create a string out of it.
myinteger = 212345
number_string = str(myinteger)
That's enough. Now you can iterate over it:
for ch in number_string:
print ch # will print each digit in order
Or you can slice it:
print number_string[:2] # first two digits
print number_string[-3:] # last three digits
print number_string[3] # forth digit
Or better, don't convert the user's input to an integer (the user types a string)
isbn = raw_input()
for pos, ch in enumerate(reversed(isbn)):
print "%d * %d is %d" % pos + 2, int(ch), int(ch) * (pos + 2)
For more information read a tutorial.

while number:
digit = number % 10
# do whatever with digit
# remove last digit from number (as integer)
number //= 10
On each iteration of the loop, it removes the last digit from number, assigning it to digit.
It's in reverse, starts from the last digit, finishes with the first

list_of_ints = [int(i) for i in str(ISBN)]
Will give you a ordered list of ints. Of course, given duck typing, you might as well work with str(ISBN).
Edit: As mentioned in the comments, this list isn't sorted in the sense of being ascending or descending, but it does have a defined order (sets, dictionaries, etc in python in theory don't, although in practice the order tends to be fairly reliable). If you want to sort it:
list_of_ints.sort()
is your friend. Note that sort() sorts in place (as in, actually changes the order of the existing list) and doesn't return a new list.

On Older versions of Python...
map(int,str(123))
On New Version 3k
list(map(int,str(123)))

(number/10**x)%10
You can use this in a loop, where number is the full number, x is each iteration of the loop (0,1,2,3,...,n) with n being the stop point. x = 0 gives the ones place, x = 1 gives the tens, x = 2 gives the hundreds, and so on. Keep in mind that this will give the value of the digits from right to left, so this might not be the for an ISBN but it will still isolate each digit.

Convert it to string and map over it with the int() function.
map(int, str(1231231231))

Recursion version:
def int_digits(n):
return [n] if n<10 else int_digits(n/10)+[n%10]

Converting to str is definitely slower then dividing by 10.
map is sligthly slower than list comprehension:
convert to string with map 2.13599181175
convert to string with list comprehension 1.92812991142
modulo, division, recursive 0.948769807816
modulo, division 0.699964046478
These times were returned by the following code on my laptop:
foo = """\
def foo(limit):
return sorted(set(map(sum, map(lambda x: map(int, list(str(x))), map(lambda x: x * 9, range(limit))))))
foo(%i)
"""
bar = """\
def bar(limit):
return sorted(set([sum([int(i) for i in str(n)]) for n in [k *9 for k in range(limit)]]))
bar(%i)
"""
rac = """\
def digits(n):
return [n] if n<10 else digits(n / 10)+[n %% 10]
def rabbit(limit):
return sorted(set([sum(digits(n)) for n in [k *9 for k in range(limit)]]))
rabbit(%i)
"""
rab = """\
def sum_digits(number):
result = 0
while number:
digit = number %% 10
result += digit
number /= 10
return result
def rabbit(limit):
return sorted(set([sum_digits(n) for n in [k *9 for k in range(limit)]]))
rabbit(%i)
"""
import timeit
print "convert to string with map", timeit.timeit(foo % 100, number=10000)
print "convert to string with list comprehension", timeit.timeit(bar % 100, number=10000)
print "modulo, division, recursive", timeit.timeit(rac % 100, number=10000)
print "modulo, division", timeit.timeit(rab % 100, number=10000)

After own diligent searches I found several solutions, where each has advantages and disadvantages. Use the most suitable for your task.
All examples tested with the CPython 3.5 on the operation system GNU/Linux Debian 8.
Using a recursion
Code
def get_digits_from_left_to_right(number, lst=None):
"""Return digits of an integer excluding the sign."""
if lst is None:
lst = list()
number = abs(number)
if number < 10:
lst.append(number)
return tuple(lst)
get_digits_from_left_to_right(number // 10, lst)
lst.append(number % 10)
return tuple(lst)
Demo
In [121]: get_digits_from_left_to_right(-64517643246567536423)
Out[121]: (6, 4, 5, 1, 7, 6, 4, 3, 2, 4, 6, 5, 6, 7, 5, 3, 6, 4, 2, 3)
In [122]: get_digits_from_left_to_right(0)
Out[122]: (0,)
In [123]: get_digits_from_left_to_right(123012312312321312312312)
Out[123]: (1, 2, 3, 0, 1, 2, 3, 1, 2, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 2, 3, 1, 2)
Using the function divmod
Code
def get_digits_from_right_to_left(number):
"""Return digits of an integer excluding the sign."""
number = abs(number)
if number < 10:
return (number, )
lst = list()
while number:
number, digit = divmod(number, 10)
lst.insert(0, digit)
return tuple(lst)
Demo
In [125]: get_digits_from_right_to_left(-3245214012321021213)
Out[125]: (3, 2, 4, 5, 2, 1, 4, 0, 1, 2, 3, 2, 1, 0, 2, 1, 2, 1, 3)
In [126]: get_digits_from_right_to_left(0)
Out[126]: (0,)
In [127]: get_digits_from_right_to_left(9999999999999999)
Out[127]: (9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9)
Using a construction tuple(map(int, str(abs(number)))
In [109]: tuple(map(int, str(abs(-123123123))))
Out[109]: (1, 2, 3, 1, 2, 3, 1, 2, 3)
In [110]: tuple(map(int, str(abs(1412421321312))))
Out[110]: (1, 4, 1, 2, 4, 2, 1, 3, 2, 1, 3, 1, 2)
In [111]: tuple(map(int, str(abs(0))))
Out[111]: (0,)
Using the function re.findall
In [112]: tuple(map(int, re.findall(r'\d', str(1321321312))))
Out[112]: (1, 3, 2, 1, 3, 2, 1, 3, 1, 2)
In [113]: tuple(map(int, re.findall(r'\d', str(-1321321312))))
Out[113]: (1, 3, 2, 1, 3, 2, 1, 3, 1, 2)
In [114]: tuple(map(int, re.findall(r'\d', str(0))))
Out[114]: (0,)
Using the module decimal
In [117]: decimal.Decimal(0).as_tuple().digits
Out[117]: (0,)
In [118]: decimal.Decimal(3441120391321).as_tuple().digits
Out[118]: (3, 4, 4, 1, 1, 2, 0, 3, 9, 1, 3, 2, 1)
In [119]: decimal.Decimal(-3441120391321).as_tuple().digits
Out[119]: (3, 4, 4, 1, 1, 2, 0, 3, 9, 1, 3, 2, 1)

Use the body of this loop to do whatever you want to with the digits
for digit in map(int, str(my_number)):

I have made this program and here is the bit of code that actually calculates the check digit in my program
#Get the 10 digit number
number=input("Please enter ISBN number: ")
#Explained below
no11 = (((int(number[0])*11) + (int(number[1])*10) + (int(number[2])*9) + (int(number[3])*8)
+ (int(number[4])*7) + (int(number[5])*6) + (int(number[6])*5) + (int(number[7])*4) +
(int(number[8])*3) + (int(number[9])*2))/11)
#Round to 1 dp
no11 = round(no11, 1)
#explained below
no11 = str(no11).split(".")
#get the remainder and check digit
remainder = no11[1]
no11 = (11 - int(remainder))
#Calculate 11 digit ISBN
print("Correct ISBN number is " + number + str(no11))
Its a long line of code, but it splits the number up, multiplies the digits by the appropriate amount, adds them together and divides them by 11, in one line of code. The .split() function just creates a list (being split at the decimal) so you can take the 2nd item in the list and take that from 11 to find the check digit. This could also be made even more efficient by changing these two lines:
remainder = no11[1]
no11 = (11 - int(remainder))
To this:
no11 = (11 - int(no11[1]))
Hope this helps :)

Similar to this answer but more a more "pythonic" way to iterate over the digis would be:
while number:
# "pop" the rightmost digit
number, digit = divmod(number, 10)

How about a one-liner list of digits...
ldigits = lambda n, l=[]: not n and l or l.insert(0,n%10) or ldigits(n/10,l)

Answer: 165
Method: brute-force! Here is a tiny bit of Python (version 2.7) code to count'em all.
from math import sqrt, floor
is_ps = lambda x: floor(sqrt(x)) ** 2 == x
count = 0
for n in range(1002, 10000, 3):
if n % 11 and is_ps(sum(map(int, str(n)))):
count += 1
print "#%i: %s" % (count, n)

Just assuming you want to get the i-th least significant digit from an integer number x, you can try:
(abs(x)%(10**i))/(10**(i-1))
I hope it helps.

how to fix python fibonacci sequence script?

My assignment is to create a function that produces 3 lists of the numbers in the fibonacci sequence starting at 0. Here is my code so far.
def fibList(n):
a = 0; b = 1; fibList = []
if n <= 0:
return
elif n == 1:
fibList = [a]
elif n == 2:
fibList = [a,b]
else:
for i in range(0,n):
a, b = b, a + b
fibList.append(b)
return fibList
def main():
print (fibList(4))
print (fibList(10))
print (fibList(-4))
what i want my output to look like is [0,1,1,2] for 4, [0,1,1,2,3,5,8,13,21,34,55] for 10, and [] for -4
My issue begins with fibList(4) currently giving an output of [1, 2, 3, 5] and fibList(10) gives an output of [1, 2, 3, 5, 8, 13, 21, 34, 55, 89] and for -4 I get "None" instead of a [].
If I type in fibList(1) I get [0] and for fibList(2) I get [0, 1], but when i test fibList(3) the first 0 and 1 are lost, giving me [1,2,3]
How would I go about making it so any number above 3 starts with [0, 1, 1, 2...]? My main issue is getting the 0 and 1 to be the first two numbers in the sequence and getting fibList(-4) to produce a [].
any help or tips would be greatly appreciated :-)

All that you are missing is to add an empty list in the case of less than or equal to zero, and recurse correctly over your range of Fibonacci numbers greater than 2. Making those small changes like so:
def fibList(n):
if n <= 0:
fibnums = []
elif n == 1:
fibnums = [0]
elif n >= 2:
fibnums = [0, 1]
for i in range(2,n):
fibnums.append(fibnums[i-1]+fibnums[i-2])
return fibnums
Note that this recursive method can get quite slow for large numbers, if that is of concern to you with your program. Best of luck!
With these changes,
print (fibList(4)) => [0, 1, 1, 2]
print (fibList(10)) => [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
print (fibList(-4)) => []

You did not quite define your function. Should the resulting list have n values [f(0), ..., f(n-1)] or n+1 values [f(0), ..., f(n)]? Your examples are contradictory: the 'expected' output for 4 has 4 values ending with f(3) while that for 10 has 11 values ending with f(10).
I am going to assume that the latter is correct. Here is a revised version of your fast iterative solution. (If my assumption is wrong, stop the range at n instead of n+1.)
def fibs(n):
"Return [fib(0), ..., fib(n)."
ret = [0, 1] # fib(0), fib(1)
a, b = ret
if n <= 1:
return ret[:n+1]
else:
for i in range(2, n+1):
a, b = b, a+b # b = f(i)
ret.append(b)
return ret
print(fibs(-4), fibs(0), fibs(2), fibs(4), fibs(10))
#
[] [0] [0, 1, 1] [0, 1, 1, 2, 3] [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

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