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QUESTION
I am a computer science student; I am studying the Algorithms course independently.
During the course, I saw this question:
Given an n-bit integer N, find a polynomial (in n) time algorithm that decides whether N is a power (that is, there are integers a and k > 1 so that a^k = N).
I thought of a first option that is exponential in n: For all k , 1
For example, if N = 27, I will start with k = 2 , because 2 doesn't divide 27, I will go to next k =3. I will divide 27 / 3 to get 9, and divide it again until I will get 1. This is not a good solution because it is exponential in n.
My second option is using Modular arithmetic, using ak ≡ 1 mod (k+1) if gcd(a, k+1 ) = 1 (Euler's theorem). I don't know if a and k are relatively prime.
I am trying to write an algorithm, but I am struggling to do it:
...ANSWER
Answered 2022-Mar-15 at 10:07Ignoring the cases when N is 0 or 1, you want to know if N is representable as a^b for a>1, b>1.
If you knew b, you could find a in O(log(N)) arithmetic operations (using binary search). Each arithmetic operation including exponentiation runs in polynomial time in log(N), so that would be polynomial too.
It's possible to bound b: it can be at most log_2(N)+1, otherwise a will be less than 2.
So simply try each b from 2 to floor(log_2(N)+1). Each try is polynomial in n (n ~= log_2(N)), and there's O(n) trials, so the resulting time is polynomial in n.
QUESTION
I have wrote some code in Python to solve problem 10 of Project Euler: 'Find the sum of all the primes below two million'. My code works, but not optimal. It takes a lot of time to show the output.
When I am trying to do the problem with primes below 20000 it works. When I change it to 200 000 or 2 million, it doesn't show the output and it is just calculating and it takes a lot of time, whereas I expected it to be faster, since the calculations aren't too difficult. I've never had the output. It almost seems like it's an infinite loop. Does anyone know what the problem is?
This is my code:
...ANSWER
Answered 2022-Mar-26 at 14:06As mentioned in the comments try a faster algorithm such as Sieve of Eratosthenes:
QUESTION
I've tried this problem from Project Euler where I need to calculate the sum of all primes until two million.
This is the solution I've come up with -
...ANSWER
Answered 2022-Mar-23 at 10:04With two little modifications your code becomes magnitudes faster:
QUESTION
For starters, I'm reinventing the wheel here. I know there is a constant in C for Euler's number. As I find it easier to create scientific programs than any other type of problem, I use these problems to practice and become a good developer.
...ANSWER
Answered 2022-Feb-26 at 07:21You can avoid calculating the factorial completly, if you store this 1.0/(fat(y)) in a variable, and divide it by progressing ys.
That way you should only hit an obstacle when the precision of your datatypes starts failing.
QUESTION
Consider an Euler integrator that solves a stochastic differential equation:
...ANSWER
Answered 2022-Feb-22 at 10:14Seeding a PRNG if often costly and should usually only be done once during the whole program run so, no, this is not efficient.
I suggest that you break the creation of the PRNG out into a separate function that has a static PRNG (only initialized once).
QUESTION
I'm trying to have an object oscillate between -45 and 45 degrees, and the following is the code I am using. Note that direction is initialized to 1, zRotation to 360, and speed to 100. This works perfectly fine. However, when I change speed to 1000, the object does a complete 360 rotation before continuing to oscillate correctly. I'm confused why that happens.
This is my code(inside the Update method):
ANSWER
Answered 2022-Feb-20 at 14:22The problem is that you do not limit the zRotation values between your desired values. In the current state of your code you only rely on your computer being fast and having little Time.deltaTime values (the time elapsed since the last frame).
If you have any kind of hiccup (which happens often in the first couple of frames after starting up), the Time.deltaTime value is going to be relatively large, resulting in a zRotation that is so large that it takes longer to return into the [-45...45] range. You can check this by adding a line of
Debug.Log($"Time: {Time.deltaTime} - Rotation: {zRotation}");
to your update method.
This way the following scenario can occur (each line is the log of a frame):
- Time: 0 - Rotation: 360
- Time: 0,02 - Rotation: 380
- Time: 0,02 - Rotation: 420
- Time: 0,333 - Rotation: 733,33
In the last step there was a deltaTime value that was so big that caused the zRotation to get loose and go really far from your desired range.
If you limit the rotation values with
zRotation = Mathf.Min(Mathf.Max(zRotation, 315), 405);
You would not go out of the desired range. (Of course you have to update your if statements from < to <= or you could leave some kind of threshold in the Min(Max()) part.
I would also advise you use the modulo operator (%) and initialize the value of zRotation to 0 so that you don't have to constantly keep track of the 360-minAngle, 360+minAngle values, you could just use the angles in a much easier way and the modulo operator helps you stay in the [-360...360] range.
I updated your code:
QUESTION
I am doing a project to which I predict the behaviour of COVID using a SEIRDV model. I have created the differential equations and have set initial parameters. I am solving these ODE's using the forward Euler method. When I create a program to find the S, E, I, R, D, V at each day for 180 days and display it using matplotlib. I get an error which can be seen below. I believe this is due to the following operation.
...ANSWER
Answered 2022-Feb-04 at 16:09You never update N[i], so it's 0 after the first step .. it should probably be set at each iteration and perhaps filled to begin with
EDIT: As @Lutz Lehmann notes in a comment, directly setting N[i] isn't a trivial "add this line" and requires doing more work
- to get the previous
D[i-1](perhaps starting at index 1) - switching
Nto be cumulative living - switching
Dto immediate fatalities
QUESTION
Scenario
I'm using unity c# to re-invent a google-earth like experience as a project. New tiles are asynchronously loaded in from the web while a user pans the camera around the globe. So far I'm able to load in all the TMS tiles based on their x & y coordinates and zoom level. Currently I'm using tile x,y to try and figure out where the tile should appear on my earth "sphere" and it's becoming quite tedious, I assume because of the differences between Euler angles and quaternions.
- I'm using the angle of
Camera.mainto figure out which tiles should be viewed at any moment (seems to be working fine) - I have to load / unload tiles for memory management as level 10 can receive over 1 million 512x512 tiles
- I'm trying to turn a downloaded tile's x,y coordinates (2d) into a 3d position & rotation
Question
Using just the TMS coordinates of my tile (0,0 - 63,63) how can I calculate the tile's xyz "earth" position as well as its xyz rotation?
Extra
- in the attached screenshot I'm at zoom level 4 (64 tiles)
- y axis 0 is the bottom of the globe while y axis 15 is the top
- I'm mostly using
Mathf.SinandMathf.Costo figure out position & rotation so far
I've figured out how to get the tile position correct. Now I'm stuck on the correct rotation of the tiles.
The code that helped me the most was found with a question about generating a sphere in python.
I modified to the code to look like so:
...ANSWER
Answered 2021-Dec-07 at 21:20For the positioning and rotation of the planes, you can do that in c#:
QUESTION
I'm trying to write a MATLAB code for the Forward Euler method, but I don't think the output is completely correct.
This is my code:
...ANSWER
Answered 2022-Jan-02 at 19:10A minimal straightforward implementation could look like
QUESTION
I'm trying to build a controller for a unity gameobject to move it similary to a normel quadrotor drone. So the drone should be able to ascend and descend by mobing the left stick up and down, turn around by moving the left stick sidewards and move horizontally by using the right stick.
I tried implementing it with the unity inputSystem, but unfortunately, it doesn't really work the way it is supposed to. Movements are mostly not smooth and the rotation causes the horizonzal movement to move in wrong directions.
Here is my code:
...ANSWER
Answered 2021-Dec-25 at 14:37There are a few mistakes I'd say
Your rotation and your ascending are frame-rate dependent. Here you didn't use
Time.deltaTimeIn both
OnClockwiseRotationandOnCounterclockwiseRotationyou do
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