Questions tagged [complex-numbers]
This challenge involves the manipulation of complex numbers, including parsing and printing them as well as performing complex arithmetic. This tag also encompasses generalised complex numbers like quaternions.
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Convert a Gaussian integer to its positive form
We already have a challenge for computing the GCD of two Gaussian integers, but that challenge explicitly allows to choose any of the four possible values.
Your task now is to bring the Gaussian ...
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The rectangle spanned by two numbers
One way to generalize the concept of a range from the integers to the Gaussian integers (complex numbers with integer real and imaginary part) is taking all numbers contained in the rectangle enclosed ...
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Convert to base i - 1
Given \$ i = \sqrt{-1} \$, a base-\$ (i - 1) \$ binary number \$ N \$ with \$ n \$ binary digits from \$ d_{0} \$ to \$ d_{n - 1} \$ satisfies the following equation.
$$ N = d_{n - 1} (i - 1) ^ {n - 1}...
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Greatest Common Gaussian Divisor
Gaussian integers are complex numbers \$x+yi\$ such that \$x\$ and \$y\$ are both integers, and \$i^2 = -1\$. The norm of a Gaussian integer \$N(x+yi)\$ is defined as \$x^2 + y^2 = |x+yi|^2\$. It is ...
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Simplify ijk-string
Related: Multiply Quaternions
Challenge
Given a string made of ijk, interpret it as the product of imaginary units of quaternion and simplify it into one of the ...
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Floor of complex number
Background
Complex floor is a domain extension of the mathematical floor function for complex numbers. This is used in some APL languages to implement floor ⌊, ...
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Complex Fibonacci numbers
The Binet formula is a closed form expression for the \$n\$'th Fibonacci number:
$$F_n = \frac {\phi^n - (1-\phi)^n} {\sqrt 5}$$
where \$\phi = \frac {1 + \sqrt 5} 2\$ is the golden ratio. This ...
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Counting shortest paths on a triangular grid
Background
An Eisenstein integer is a complex number of the form \$ z = a + b\omega \$ where \$a, b\$ are integers and \$\omega\$ is the third root of unity \$\frac{1-\sqrt3i}{2}\$. The Eisenstein ...
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Gaussian integer division reminder [closed]
Gaussian integer is a complex number in the form \$x+yi\$, where \$x,y\$ are integer and \$i^2=-1\$.
The task is to perform such operation for Gaussian integers \$a,b\$, that
\$a=q \cdot b+r\$ and \$|...
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The qvolume of an integer
It is ancient knowledge that every non-negative integer can be rewritten as the sum of four squared integers. For example the number 1 can be expressed as \$0^2+0^2+0^2+1^2\$. Or, in general, for any ...
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Quaternion square root
Background
Quaternion is a number system that extends complex numbers. A quaternion has the following form
$$ a + bi + cj + dk $$
where \$ a,b,c,d \$ are real numbers and \$ i,j,k \$ are three ...
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Hermitian matrix?
Note that this challenge requires no handling or understanding of complex numbers.
Given a non-empty square matrix where every element is a two-element (Re,Im) integer list, determine (giving any ...
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Eigenvalues of a Matrix
Given a square matrix, output the matrix's eigenvalues. Each eigenvalue should be repeated a number of times equal to its algebraic multiplicity.
The eigenvalues of a matrix ...
user45941
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Exponent of complex numbers
Given two integers, which may be negative, zero, or positive, \$a\$ and \$b\$ (taken in any reasonable format, including inputting a plain complex number), convert it to \$a + bi\$ where \$i\$ is the ...
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Factorize a Gaussian integer
A Gaussian integer is a complex number whose real and imaginary parts are integers.
Gaussian integers, like ordinary integers, can be represented as a product of Gaussian primes, in a unique manner. ...
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Compute a complex power
The Rundown
Given any input x and y, perform a complex operation, and print a corresponding result.
How your program should work
Given an input x and y in the form z = x+yi, find zi-z
If the ...
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Draw a graph of \$y=(-n)^x\$
Challenge
Given an input of an integer, \$n\$ (where \$0<n<50\$), output the graph of \$y=\mathrm{Re}((-n)^x)\$ from \$x = -3\$ to \$x = 3\$ inclusive.
Where \$\mathrm{Re}(p)\$ is the real part ...
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Imaginary Parts of Non-Trivial Riemann Zeroes
Introduction
According to the Riemann Hypothesis, all zeroes of the Riemann zeta function are either negative even integers (called trivial zeroes) or complex numbers of the form ...
user45941
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Complex Binary Numbers
Let's create a simple, surjective mapping from positive integers to Gaussian integers, which are complex numbers where the real and imaginary parts are integers.
Given a positive integer, for example ...
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Codegolf the permanent
The challenge is to write codegolf for the permanent of a matrix.
The permanent of an \$n\times n\$ matrix \$A = a_{i,j}\$) is defined as
$$\text{perm}(A) = \sum_{\sigma \in S_n} \prod^n_{i=1} a_{i,\...
user9206
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Gauss to Eisenstein
Given a Gaussian integer \$a+bi\$ where \$a\$,\$b\$ are integers and \$i = \exp\left(\pi i/2\right)\$ is the imaginary unit, return the closest (w.r.t to the Euclidean distance) Eisenstein integer \$k+...
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Counting Eisenstein primes
Introduction
Eisenstein integers are complex numbers of the form
a+bω
Where a,b are integers, and
...
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Parse a Quaternion
If you don't know already, a quaternion is basically a 4-part number. For the purposes of this challenge, it has a real component and three imaginary components. The imaginary components are ...
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Addition in base -1+i
Gaussian integers are complex numbers of the form a+bi where a and b are both integers. In ...
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Evaluate the Riemann Zeta Function at a Complex Number
Introduction
I found this question that was closed because it was unclear, yet it was a nice idea. I'll do my best to make this into a clear challenge.
The Riemann Zeta function is a special function ...
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Primitive Roots of Unity
Let z be a complex number. z is an nth primitive root of unity if for a certain positive integer ...
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Output quater-imaginary base numbers in binary
Write a function or program that outputs Quater-imaginary base displayed as binary digits. The number base is 2i, where i is the square root of -1. See Complex Number for more details on i. Each digit ...
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Add and multiply perplexing numbers
The split-complex numbers, also known as "perplex numbers" are similar to the complex numbers. Instead of i^2 = -1, however, we have ...
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What's the voltage over each component?
The picture below shows a RLC circuit. A RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. (1)
In order to ...
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Multiply Pauli Matrices
The Pauli matrices are a set of 2x2 matrices which appear very commonly in quantum physics (no, you don't need to know any quantum physics for this challenge). If we include the identity in the set, ...
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Solve a 2x2 Eigensystem
For those with a little linear algebra background, the challenge is as simple as this: determine the eigenvalues and eigenvectors of a given complex 2x2 matrix. You may skip ahead to The Challenge for ...
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Wait, but what's e^N?
Objective:
The objective is to calculate e^N for some real or imaginary N (e.g. 2, -3i, 0.7, 2.5i, but not 3+2i). This is code golf, so, shortest code (in bytes) wins.
So, for example:
N = 3, e^N =...
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Find i^n, given n
The Challenge
In as few characters as possible, find the value of \$i^n\$, given \$n\$, a positive integer greater than 0. This should be outputted as a String.
For those that don't know, \$i\$ is ...
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Multiply Quaternions
Write a named function or program that computes the quaternion product of two quaternions. Use as few bytes as possible.
Quaternions
Quaternions are an extension of the real numbers that further ...
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is_gaussian_prime(z)?
Task
Write a function that accepts two integers \$a,b\$ that represent the Gaussian integer \$z = a+bi\$ (complex number). The program must return true or false depending on whether \$a+bi\$ is a ...
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Too Fast, Too Fourier: FFT Code Golf
Implement the Fast Fourier Transform in the fewest possible characters.
Rules:
Shortest solution wins
It can be assumed that the input is a 1D array whose length is a power of two.
You may use the ...
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Perform basic operations on complex numbers in a language without native support for complex numbers
Your code has to provide the following functions:
read(x)
Reads a complex number from the standard input. It has to accept and evaluate something in the form ...
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Resolve quadratic equation
Challenge:
Write the smallest program (in characters) that resolves quadratic equations
i.e. ax² + bx + c = 0
Rules:
Given 3 numbers in R comma-delimited on ...
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Generate Newton fractals
You all know the Newton method to approximate the roots of a function, don't you? My goal in this task is to introduce you into an interesting aspect of this algorithm.
Newton's algorithm converges ...
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