Questions tagged [complexity]

Relating to the level of difficulty of a calculation or the asymptotic running time of an algorithm.

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Complexity of solving an image differential linear system

Define an "image differential linear system" as a linear system $A\mathbf{x}=\mathbf{b}$ wherein $\mathbf{x}$ contains the ($\mathbb{R}$) pixels of an image and each row of $A$ constrains ...
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53 views

Asymptotic complexity of fixed-rank SVD

According to the Wikipedia article on Singular Value Decomposition, the asymptotic complexity of computing the SVD of an arbitrary m×n matrix M with m>n by the popular Householder QR methods is O(...
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106 views

Diagonalization of Hermitian matrices vs Unitary matrices

What are the general algorithms used for diagonalization of large Hermitian matrices and Unitary matrices? ($>5000 \times 5000$) LAPACK seems to diagonalize Hermitian matrices almost 20 times as ...
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57 views

Time complexity of derivation, gradient,differential, jacobian matrix

what is the time complexity of gradient $\nabla_{f}$ using the $\mathcal O$-notation? what is the time complexity of jacobian matrix using the $\mathcal O$-notation? who knows some references to ...
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32 views

Most scalable distributed consensus mechanism based on message complexity? [closed]

One of the most challenges in distributed consensus mechanisms is both time complexity and message complexity. For example, PBFT message complexity is O(n^2) that ...
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1answer
155 views

Runtime of Gaussian elimination/row reduction on a rectangular $m \times n$ matrix

The runtime of Gaussian elimination on an $n \times n$ matrix is $O(n^3)$. What is the runtime on an $m \times n$ matrix? I am taking Gaussian elimination to mean putting the matrix in reduced row ...
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56 views

Which class does this NP problem belong to?

Suppose $n$ inputs ($x_1, x_2, x_3, \cdots, x_n$) can take on any of $m$ values, say $\{ k_1, k_2, k_3, \cdots, k_m \}$, and that there is a cost function $y = f(x_1, x_2, x_3, \cdots, x_n)$. For ...
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24 views

Hawkes Process : recursive formula for : $R'_{m,n} (k) = \sum_{ \{i : t_i^n < t_k^m \} } (t_k^m - t_i^n) \exp ( - \beta_{m,n} ( t_k^m - t_i^n ) ) $

Following the advice of a fellow mathematician, I am asking my question here from (https://mathoverflow.net/questions/365554/hawkes-process-recursive-formula-for-r-m-n-k-sum-i-t-in-t) I need to use a ...
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1answer
165 views

Implementation of a $O(n \log(n))$ method to compute eigenvalues of real symmetric tridiagonal matrices

I just came upon this paper, which details the implementation of a fast method to get eigenvalues of tridiagonal symmetric matrices : Coakley, Ed S.; Rokhlin, Vladimir, A fast divide-and-conquer ...
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65 views

Flops of the computation of symmetric matrix $A$ to the power of $p$

What is the cost in terms of flops for the computation of $A$ to the power of $p$, where $p$ is a positive integer and $A \in \mathbb R^{n\times n}$ is a symmetric matrix?
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253 views

Time complexity of numerical finite differences

I have a function $f:\mathbb R^N\to \mathbb R$ and I would like to compute all the partial derivatives of $f$ w.r.t. the $N$ input. What is the computational complexity using the (ones-sided) finite ...
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84 views

Is there an efficient algorithm for calculation of continued fraction expansion from decimal digits?

Suppose to calculate the continued fraction expansion of $\pi$, the common-sense algorithm would be to take the decimal part, perform inversion, which will give the next term as integer part, and the ...
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2answers
80 views

What is an instance (precisely) in computational complexity?

I am trying to understand the notion of reduction of a problem to another problem. As it is known this has huge impact on classifying the complexity of a problem. The definition of reduction involves ...
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109 views

Big Theta Complexity of Gaussian Elimination using Complete Pivoting

I already know the Big O for partial pivoting is $O(n^3)$ and remain the same for complete pivoting. I also know the big theta complexity for partial pivoting is $2/3 n^3$ I would like to know the ...
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195 views

Calculate amount of FLOPs for an eigenvalue problem solver

I have 2 complex, non-symmetric, matrices $A_{1000\times1000}$, $B_{1000\times1000}$ and I am using Matlab to get it's eigenvalues (functions like eig or eigs). Both matrices are different - one is ...
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1answer
50 views

Time complexity analysis

I want to know the time complexity of following code Say I have a list unique_element[] There is an array which contain elements {4,5,2,4,7,8,1,5,9,8,1} Now as per my code I want to find out the ...
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18 views

Cost functions to judge time/memory/accuracy tradeoffs

I am working on an interesting algorithm: Its absolute error is exponential in a parameter $j \in \mathbb{N}$, and for a given $j$, I have complete freedom to choose between an $\mathcal{O}(1)$ time-...
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2answers
282 views

Asymptotic Complexity of Gaussian Elimination using Complete Pivoting

I would like to know the algorithm asymptotic complexity with Complete Pivoting. With partial pivoting, it is known to be $O(n^3)$. Is it the same for complete pivoting?
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600 views

Computational complexity of Newton's method

the classical Newton's method for non-linear systems of equations is $x_{k+1} =x_k-J_F(x_n)^{-1} F(x_n)$. In pratice, rather than compute the inverse of the Jacobian matrix, one solves the systems $...
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3k views

Integer operations vs floating point operations

I have been working with an algorithm, which uses additions of floating point vectors, (sparse matrix of floats)x(dense vector of floats) dot products I recently found out that I can get the same ...
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1answer
131 views

Convergence rate and complexity for convex minimization problem

In Yurii Nesterov's Introductory Lectures on Convex Optimization, there is a description of the rate of convergence and corresponding upper bound for the analytical complexity of a minimization ...
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417 views

System of ordinary differential equations - time complexity of initial value problem

I am interested in knowing what the time complexity is (in Big-$\mathcal O$ notation) for solving system of $N$ differential equations? I am using ode15s in ...
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1answer
1k views

Time complexity of $l_2$-norm of a vector

What is the complexity (in flops, floating-point operations) of taking the $l_2$-norm of vector $\mathbf{v}\in\mathbb{R}^n$ (or $\mathbf{v}\in\mathbb{C}^n$ if a difference exists). We have the ...
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1k views

Comparing Algorithmic complexity, ODE Solvers (Big O)

I am currently using the following three methods to solve differential equations: 4th order Runge Kutta Method Euler Method Internal scipy methods: ...
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121 views

finding the growth rate from numerical data

Suppose i have a bunch of 10 data points and i have to conclude whether the increase is $n^2,n^3,\cdots,2^n,3^n, e^n,\cdots$. For example i have the image:- Now the increase is either polynomial or ...
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2answers
63 views

Efficient Representation of (spatially sparse) spatial time series

Background I have a huge dataset consisting of points (on a plane) together with a timestamp for each point. This is a collection of car GPS measures, giving us the location (latitude/longitude) of ...
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417 views

Optimisation of matrix exponential

I have a 7000x7000 sparse matrix (scipy), which I want to exponentiate. I've tried using scipy.sparse.linalg.expm, which works quite well for smaller matrices (takes a few seconds for a 1000x1000 ...
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1answer
139 views

Regarding impractical usage of direct solvers of linear systems [closed]

Since the computational complexity of direct elimilation methods for solving linear systems is $O(n^3)$, it's not practical when the number of dofs is large. But how large would you call it a large ...
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1answer
208 views

Big-O Complexity of Gini Index

What will be the complexity of finding Gini Index of a sorted vector of $N$ values, which is defined as: $Gini(\mathbf{x})=1-2\sum_{k=1}^N \frac{\mathbf{x}(k)}{\Vert\mathbf{x}\Vert_1}(\frac{N-k+.5}{N}...
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255 views

Generalization error and Sample Complexity estimation for Least Squares

I am wondering how to draw a sample complexity plot similar to the following figure which shows the estimated number of samples to incur no more than 10 percent generalisation error on average for the ...
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97 views

Is the numerical resolution of this huge sparse polynomial system tractable?

I'd like to find numerically a solution to a sparse system of 2000000 polynomial equations of degree 3 with 50000 variables and integer coefficients (or at least to decide whether or not a solution ...
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97 views

What is meant by “operations”?

In this paper, on page 243, we have $$d_{jk} = \frac{a_j}{a_k(x_j - x_k)}$$ where $$a_k = \prod_{l = 0;l\neq k}^{N}(x_k-x_l)$$ Now, $a_k$ requires evaluating N multiplications. Why does the author ...
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200 views

Efficient algorithm for a matrix product

Recall that a unit lower triangular matrix $L\in\mathbb{R}^{n\times n}$ is a lower triangular matrix with diagonal elements $e_i^{T}L e_i = \lambda_{ii} = 1$. An elementary unit lower triangular ...
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97 views

$LU$ Factorization of a nonsingular matrix with a particular pattern

Consider $S\in\mathbb{R}^{n\times n}$ whose nonzero elements have the following pattern for $n = 8$: $$\begin{pmatrix} 1 & 0 & 0 & 0 & \mu_1 & 0 & 0 & 0\\ 0 & 1 &...
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2answers
151 views

What kind of optimisation algorithm is suitable for a computationally expensive function?

I have a reference value $R$ and a modelled value $M$. $M$ is generated using a stochastic algorithm with parameters $a$ and $b$. The objective is to tune $a$ and $b$ so that $M$ is as close as $R$ ...
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192 views

Notations for algorithmic complexity in elementary operations

I am comparing several algorithms (moments and matrix products) for real-time computing in terms of numerical complexity in elementary operations. [EDIT] Algorithms are very similar in terms of ...
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1answer
94 views

Expected runtime complexity of repeated closest Point Pair search

I have to vectors $X_1$ and $X_2$ with 3 dimensional points $p_i$ and $p_j$ contained. As long as $X_1$ is not empty, I want to find the closest pair $p_i$ and $p_j$. The point $p_i$ of this pair I ...
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51 views

I'm using linear programming for production planning. Does the order in which I make products affect the cost?

I have a collection of different scrap aluminium alloys. I want to mix them together to make new alloys with customer-defined compositions. Sometimes this will involve little more than melting down ...
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87 views

computational complexity for computing perimeter of a polygon

What is computational complexity for computing perimeter of a polygon of $n$ vertices? The polygon is not necessarily regular and can be convex or non-convex.
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1answer
194 views

Probability of reconstructing a word using c substrings from a random sample

Consider a voice recording split into it's phonemes as our sample $S=(s_1,...,s_k) \in \Omega = P^k$. The number of phonemes is $|P| = 40$. Then I have a word $w = (w_1,...,w_n) \in P^n$. I want to ...
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183 views

Single Precision a x plus y (SAXPY) terminology

I've been reading books which refers to vector update operations of the form: y := y + ax, where y and x are vector variables and a is a scalar as SAXPY. I understand ax plus y part, but why "single ...
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548 views

Scientific Programming Contests

I regularly compete in so called "Programming Contests", where you solve difficult algorithmic problems with your own code and problem solving skills during a limited time-frame. For referential ...
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1answer
219 views

Mathematical Complexity of Sparse Solvers

For a system $\mathbf{x=Da}$, there exist a lot of algorithms to estimate sparse vector $\mathbf{a}$. I wish to know the big-O mathematical complexity of 1) orthogonal matching pursuit (OMP) both ...
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710 views

N-body simulation optimisation, looking for name or existing work

during the development of my N-body simulation with visualisation in WebGL, I devised an optimisation, and I'm wondering if it has a name. I find it unlikely that it has never been done before. It ...
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410 views

Storage complexity of SDP solver SCS

This is a follow up question to this question. Consider the following SDP in standard form: \begin{align} &\min_{X\in S^n, X>0} \operatorname{tr}(AX)\\ &\mbox{subject to}\; \operatorname{...
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2answers
2k views

GPU vs CPU calculation

I've been working on calculating large factorials ($N>10^9$) and I was wondering if it wasn't faster to use the GPU to run the calculations on something like openCL. What I realized however was ...
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1answer
440 views

Comparison of convex hulls [closed]

Consider a set of polytopes $P_i : i=1,2,...,k$ each of which has a structure as $P_i:= \{(x_{i1},x_{i2},..., x_{in})\; |\; x_{ij} \in [a_{ij}, b_{ij}] \subseteq [0,1]\}\;\; \text{for all}\;\; j=1,......
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1answer
36 views

When we compute the complexity of a given algorithm related to image processing does the N refers to the number of Pixels in the image?

When we compute the complexity of an image processing algorithm, we get an $O(N)$. does the $N$ refers to the number of pixels in the image or to the height/width of the image, I mean it is computed ...
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1answer
31 views

Computational complexity of chemical dynamics for biological studies [closed]

How many CPU cycle is required to simulate a complete human body from it's very initial stem cell using classical algorithms and also is it possible to use similar algorithms for simulating stem cells ...
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136 views

What is the computational cost of using complex numbers in contrast to real numbers in matrix operations, e.g. $LU$ or $LDL^T$ factorizations? [duplicate]

I am curious about how much one loses in terms of computational cost when complex numbers are used instead of real numbers? I guess the number of floating-point operations and memory doubles, but I ...