# Questions tagged [complexity]

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

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### Why is it that SVD routines on nearly square matrices run significantly faster than if the matrix was highly non square?

In Python / Matlab, if you run a routine for SVD on a significantly non-square matrix, X, such as X.shape = (2,15000) you will ...
107 views

### Algorithm about finding a combination that the sum is closest to a given number

Given a matrix $P\in \mathbb{R}^{n*k}$ (just for ease of notation, no matrix or linear algebra is actually needed; bound to $(0,1)$ if necessary), select one number from each row and compute the sum. ...
1 vote
162 views

### Space complexity of a semidefinite program

What is the space complexity of a semidefinite program (SDP)? What is the answer to the same question for convex optimization problems in general?
29k views

### Computational Complexity of 2D Convolution

I am using image filtering for an image processing algorithm I'm developing. I'm using a predefined Matlab function to do the convolution, but I'd like to know what the computational complexity is for ...
1k views

### How does a Sparse Direct Solver know about dimensionality of a problem being solved?

It is claimed that the time and memory complexities of sparse direct solver are $O(N^2)$ and $O(N^{4/3})$ for 3D problems and $O(N^{1.5})$ and $O(N \log N)$ for 2D, respectively. But how does a ...
234 views

### Complexity of Branch-and-cut algorithm in terms of "Big O"

How can I compute the Big O complexity of the Branch and cut algorithm? I am solving an integer linear program using MOSEK that includes $M$ binary variables, but I do not know how to calculate the ...
287 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 ...
353 views

### Complexity of recovering all roots of a polynomial

Given a polynomial of degree n and a list of putative roots $\{r_i\}_{i=1}^{n}$, we can verify that all the putative roots are indeed correct by $n$ applications of Horner's method. Hence verifying ...
362 views

### How does the number of function calls in BFGS scale with the dimensionality of space?

Question Is there any estimate for the scaling of the number of function calls in BFGS-optimization with the dimensionality of the search space? Specifically I am assuming a (free) expression for the ...
75 views

### Eigen-decomposition one eigenpair by one eigenpair?

Is it possible to conduct an Eigen-decomposition of a matrix one eigenpair by one eigenpair? And related to this question, what is the time complexity of truncated eigendecomposition? I am trying (...
714 views

### Poisson solver on unstructured mesh

For the 2D Poisson equation, there exist on finite-difference mesh, some code taking $O(n \log(n))$ operations to solve it on a mesh with $n$ nodes. They rely on Fast Fourier Transform or Block Cyclic ...
91 views

### Worst Case complexity of a search engine algorithm

Computer make it possible to find information in large databases. However, the results are often too large to be returned in their entirety to the user who requests them. Computer therefore sort the ...
129 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 ...
66 views

### Is there a unit of measure for computational complexity; through quantum computers? [closed]

I'm concerned with trying to determine whether the same computational processes on a Turing computable algorithm can be ascertained for a quantum computer in some form of actual 'metric' for how many ...
7k views

### Why are log and exp considered 'expensive' computations in ML?

In many resources/videos I see comments being made along the lines of "and we can see here that we have a logarithm/exponential so this will be an expensive computation to make." (such as ...
71 views

### Comparison of computational complexities of MD versus MC simulations

In my humble understanding MD simulations of systems with short-range(like LJ interactions) and long-range(electrostatic) has a computational complexity $O(N . log(N))$. What will be the computational ...
3k 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 ...
1 vote
156 views

### 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 ...
1 vote
280 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(...
323 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 ...
171 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 ...
1 vote
994 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 ...
1 vote
50 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 ...
60 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 ...
4k 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: ...
437 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 ...
145 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 ...
1 vote
101 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?
228 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 ...
288 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 ...
203 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 ...
6k views

### How can the gravitational n-body problem be solved in parallel?

How can the gravitational n-body problem be solved numerically in parallel? Is precision-complexity tradeoff possible? How does precision influence the quality of the model?
468 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 ...
21 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-...
827 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?
5k views

### Is the Thomas algorithm the fastest way to solve a symmetric diagonally dominant sparse tridiagonal linear system

I am wondering if the Thomas algorithm is the fastest way (provably?) to solve a symmetric diagonally dominate sparse tridiagonal system in terms of algorithmic complexity (not looking for ...
3k views

249 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 ...
7k views

### How does the computational cost of an mpi_allgather operation compare with a gather/scatter operation?

I'm working on a problem that can be parallelized by using a single mpi_allgather operation or one mpi_scatter and one mpi_gather operation. These operations are called within a while loop, so they ...
142 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 ...
70 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 ...
540 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 ...
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 ...
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 ...