Questions tagged [na.numerical-analysis]

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What infinite sums cannot be approximated in polynomial time?

The following is from the book Geometric algorithms and combinatorial optimization: It shows an infinite sum that has an FPTAS (= an $\epsilon$-approximation can be computed using poly($1/\epsilon$) ...
Erel Segal-Halevi's user avatar
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113 views

Complexity of, given any elementary function $f$ and a natural $n$, compute $n$ digits of $f(x)$

We define problem $A$ as follows. Each instance of the problem consists of: (a) some succinct codification of an elementary function, that is, a function constructed by composing arithmetic operators, ...
EXPTIME-complete's user avatar
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What is the complexity of the "characteristic bisection" method?

The characteristic bisection method is an algorithm for finding approximate zeros of multi-dimensional functions. It is a generalization of the bisection method; it is described briefly here. ...
Erel Segal-Halevi's user avatar
4 votes
1 answer
185 views

Computing an approximate root of a two-dimensional monotone function

Let $f$ be a Lipschitz-continuous function from the square $[-1,1]^2$ to itself, satisfying the following conditions: For all $y\in [-1,1]$: $~~~~f(-1,y)_1\leq 0\leq f(1,y)_1$, and $f(x,y)_1$ is ...
Erel Segal-Halevi's user avatar
4 votes
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51 views

Error analysis of Estrin's method

Estrin's Method is an alternative to Horner's method for evaluating polynomials. To evaluate a polynomial $P(x)=\sum_{i=0}^7 a_i x^i$ at a point $x\in\mathbb R$, it first computes the powers $x^2$ and ...
Thomas Ahle's user avatar
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34 views

Interval arithmetic adapted to backwards stable problems

In numerical analysis, there are algorithms which are either forwards stable or backwards stable. Forwards stability is strictly stronger, and is more desirable. Unfortunately, it is in many instances ...
wlad's user avatar
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2 answers
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Can finite difference methods approximate the space/time complexity of given programs?

While benchmarking a language prototype, I realized that I had a superlinear implementation of a test program, but wasn't sure if it was quadratic or cubic. I stayed up too late and wrote half a page ...
Corbin's user avatar
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Smallest nonzero eigenvalue of a sum of +1/-1 rank-one matrices?

Suppose we have a $k\times k$ matrix $A = \sum_{i=1}^{n} a_i a_i^T$ where $n \leq \mathrm{poly}(k)$ and each $a_i\in\{-1,1\}^{k}$. It is easy to prove that the largest eigenvalue of $A$ is at most $\...
Octopus's user avatar
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3 answers
139 views

Cost of Numerically Solving a System of P polynomials, each of V variables, and degree D to a Specific Accuracy

Let there be a set of $P$ polynomial equations $f_j(x_1,x_2...x_V)=0$ where $1\leq j\leq P$. For each $f_j$ the coefficients are real and every variable goes up to degree $D$. It is also guaranteed ...
Ken Robbins's user avatar
2 votes
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Inverting a sum of easy to invert linear terms?

Below are some examples of linear equations with $d\times d$ matrices $Y,A,B,C,D$ which can be solved for unknown $d\times d$ matrix $X$ in $O(d^3)$ time. Repeated indices in each term are summed over:...
Yaroslav Bulatov's user avatar
2 votes
1 answer
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Can the Banach-Tarski paradox be "realized" by floating-point round-off?

The Banach-Tarski paradox says that a ball in $\mathbb{R}^3$ can be partitioned into a finite number of pieces, whose rearrangement has a larger volume than the original. It occurred to me that it ...
Joseph O'Rourke's user avatar
3 votes
0 answers
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Monotonic and bounded sequences throughout computer science

When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure ...
wlad's user avatar
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How can I understand the Coppersmith–Winograd algorithm?

I want to do research on matrix multiplication algorithms. I glanced at the Coppersmith-Winograd algorithm paper, but I didn't understand anything. How can I complete the background to read this paper?...
Huseyin Okan Demir's user avatar
9 votes
1 answer
332 views

Proof for Upper Bound of Sum of Square Roots Problem

In [1], Garey et al. identify what would later be known as the Sum of Square Roots Problem in the course of working out the NP-completeness of Euclidean TSP. Given integers $a_1, a_2, \ldots, a_n$ ...
mhum's user avatar
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Complexity of low-rank matrix factorizations with rows in a simplex and outliers

Our goal is to obtain a matrix factorization in form of $M = U V'$, where $U\in\mathbb{R}^{d\times r}, V \in\mathbb{R}^{N\times r}$ and each row of $V$ satisfies $$ \sum_{j}(V)_{ij}=1, (V)_{ij}\ge 0 $$...
Yufeng Zhang's user avatar
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Complexity of checking $a > br^m + cr^n$, with $r$ rational

I'm wondering if the following problem is decidable in P-time (or even NP): Given $a, b, c \in \mathbb{Z}$ and $m, n, p, q \in \mathbb{N}$ all in binary, decide if $a > br^m + cr^n$, where $r = {p ...
Michaël Cadilhac's user avatar
2 votes
1 answer
146 views

Graph sparsification and eigenspaces

I am currently trying to understand whether I can make some claims about the relation between eigenspaces of a sparsifier and the original matrix. In this context, let me first define a couple of ...
LeoW.'s user avatar
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3 votes
1 answer
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What is the time complexity of increasing the precision of finding matrix eigenvalues?

There are various algorithms that output the eigenvalues of an $n \times n$ matrix in time $O(n^3)$. However, I can't find anywhere that tells me about the precision of the output of the algorithm. ...
Hans Schmuber's user avatar
3 votes
0 answers
874 views

Is there any algorithm to find just the largest eigenvalue with subquadratic time complexity?

SVD or PCA can be used find the largest eigenvalue, but at a cost of $O(n^3)$ complexity. Lanczos algorithm runs much faster on a sparse matrix with complexity $O(dn^2)$ where $d$ is the average ...
River's user avatar
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3 votes
1 answer
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Density of multiples

I have an infinite collection of positive integers $n_1,n_2,n_3,\ldots$ and I would like to find the density of the numbers divisible by one or more of these.* If the density does not exist, the ...
Charles's user avatar
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What is the computational complexity of sin and cos for floating point inputs?

What is the computational complexity of the problem INPUT: $\;\;\;$ integers $x$ and $y \:$ (both in binary) OUTPUT: rational approximations to $\: \sin\hspace{-0.03 in}\left(x\hspace{-0.04 in}\...
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2 votes
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the confusion about 'with high probability (w.h.p.)'

w.h.p. can often be seen in the analysis of randomized algorithms. It's definition can be seen here https://en.wikipedia.org/wiki/With_high_probability. However my confusion is that: Assuming we ...
olivia's user avatar
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118 views

Computing log of sum of positive integers

As input, we are give $k$-bit approximation (after the decimal point) of $\log(a)$ and $\log(b)$ for positive integers $a$ and $b$, i.e, we are given $\alpha$ and $\beta$ (as binary strings) as input ...
Gorav Jindal's user avatar
1 vote
0 answers
60 views

Computational complexity of Initial Value Problems of ODEs

Are there known results on computational complexity of initial value problems of ODEs? As my question may be somewhat vague, I want to mention that I'm mainly interested for results on the ...
Kapoios's user avatar
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8 votes
1 answer
492 views

The exponential function over algebraic numbers

Given an algebraic number $\alpha$, I am interested in finding an approximation of $\Re(e^\alpha)$ up to a given precision, where $\Re()$ refers to the real part of the complex number. Formally, I ...
user22166's user avatar
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36 votes
1 answer
2k views

Computational complexity of pi

Let $L = \{ n : \text{the }n^{th}\text{ binary digit of }\pi\text{ is }1 \}$ (where $n$ is thought of as encoded in binary). Then what can we say about the computational complexity of $L$? It's ...
Scott Aaronson's user avatar
2 votes
2 answers
200 views

Multiple independent random number streams

Having multiple streams of pseudo-random numbers known to be independent and with a uniform distribution I want to do Monte Carlo simulations in parallel. In other words, one thread will have a full-...
mrei's user avatar
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12 votes
5 answers
452 views

Motivation for volume estimation

What are some concrete and compelling applications for estimating the volume of convex polyhedra of the sort considered in the more recent papers on random walk methods? These papers on volume ...
user avatar
10 votes
1 answer
2k views

Integer roots of a polynomial

What algorithm can we use to find all integer roots of a polynomial $f(x)$ with integer coefficients? I observe that Sage can find the roots within a few seconds even when all coefficients of $f(x)$ ...
user12290's user avatar
  • 203
26 votes
2 answers
15k views

Universal Approximation Theorem — Neural Networks

I posted this earlier on MSE, but it was suggested that here may be a better place to ask. Universal approximation theorem states that "the standard multilayer feed-forward network with a single ...
Matt Munson's user avatar
1 vote
1 answer
110 views

Delimiting the roots of a polynomial

Given a polynomial of degree $n$, namely, $y=a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$, with $a_i\in \mathbb{R}$, I would to know if it is possible to compute a rational number $K$ such that the ...
LORE81's user avatar
  • 111
13 votes
1 answer
881 views

Numerical stability of Simplex method

The simplex algorithm is often treated either within real arithmetic, or in the discrete world with exact computations. However, it seems to be implemented most often with floating-point arithmetic. ...
shuhalo's user avatar
  • 1,165
7 votes
1 answer
931 views

Numerically stable fast convolution algorithm?

Suppose you have two vectors of real numbers $\langle a_1, \dots, a_n \rangle, \langle b_1, \dots, b_n \rangle$, with $a_i, b_i \geq 0$, and wish to compute the convolution $$ c_i = \sum_{j \leq i} ...
David Harris's user avatar
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3 votes
1 answer
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Research papers regarding disinterval analysis

I'm trying to implement a disinterval analysis similiar to what is described in Clousot: Static Contract Checking with Abstract Interpretation (under section 5.2) - and I'm trying to find some papers ...
kastermester's user avatar
3 votes
0 answers
80 views

Efficiently Detecting "edges" in the time frequency plane

Given a signal $y(t)\in\mathbb{R}$ I wish to detect edge patterns. $s(f,t)$ is a time-frequency decomposition of $y(t)$ in some window $(t-n,t+n)$ so that $f$ loosely corresponds to a local frequency....
Mark S's user avatar
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8 votes
0 answers
367 views

Efficiently approximating derivative of a well-behaved function

I need an algorithm for adaptive sampling a well-behaved function and computing its derivative in the sampling range with prescribed accuracy. The function has no more than one minimum in the sampling ...
Alexey Popkov's user avatar
16 votes
1 answer
2k views

How to compute powers of square matrices?

Suppose we are given a matrix $A \in \mathbb R^{N\times N}$, and let $m \in \mathbb N_0$. How fast can we compute the power $A^m$ of that matrix? The next best thing in comparison to computing $m$-...
shuhalo's user avatar
  • 1,165
2 votes
1 answer
112 views

Finding mapping between two spatial representations of the same objects

I have two matrices $U$ and $V$. $U$ is $n \times n$ and $V$ is $n \times m$. (Both are empirical results of an experiment.) I would like to find a linear transformation $A$, $m \times n$, such that $...
aaronsteven's user avatar
7 votes
0 answers
129 views

What's the state of the art for matrix nuclear/trace norm optimization

I am interested in simple matrix optimizations with nuclear/trace norm: $\min_X \left(f(X) + \|X\|_*\right)$ where $\|X\|_*$ stands for the trace norm of the matrix $X$, and $f$ is a convex smooth ...
Jonathan's user avatar
1 vote
1 answer
268 views

Most optimal parallel method for calculating the integral of a 2D function

I posted already this question to SO but got no answer so I try it now here: In some crunching number program, I have a function which can be just 1 or 0 in three dimensions. I do not know in advance ...
Open the way's user avatar
1 vote
0 answers
134 views

Optimal parallel algorithm for finding roots of a function [closed]

In some problem I need to find the zeroes (multiple real solutions) of some functions in 1D and 2D. I wonder which is the best parallel algorithm for this, which can provide the highest accuracy and ...
Open the way's user avatar
1 vote
0 answers
129 views

High Dimensional Volume (HDV) estimator for Entropy estimation

I am writing a program using high-dimensional volume (HDV) estimator to estimate entropy and mutual information for variable selection. Let $ D = (x^i_1, x^i_2, ..., x^i_M)$, N is the number of data ...
JYJ's user avatar
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7 votes
0 answers
140 views

What's new in sparse eigensystems solution

As a part of other work I need to solve relatively large (~1E5x1E5) and sparse (~100 non-zero elements in each raw in few blocks) hermitian eigensystems. Usually only few eigenvalues+vectors are ...
Misha's user avatar
  • 171
2 votes
2 answers
432 views

LU factorization of a 0-1 matrix

I have a rather naive question on LU factorization which probably should be easy to answer. Say I have a matrix with entries only from $\{0,1\}$. When can we expect to get an LU factorization of such ...
Turbo's user avatar
  • 12.9k
6 votes
2 answers
561 views

Iterative algorithms in algebraic complexity (Blum-Shub-Smale-Model)

I know the Blum-Shub-Smale model. It is claimed to provide a theoretical framework for algorithms in real and complex algebra and analysis. A very general question: Most algorithms compromise of ...
shuhalo's user avatar
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1 vote
1 answer
110 views

Adapted FEM mesh

Is there a technique where a square grid is used to as a mesh for time dependent partial differential equations (PDEs) but which the points are permuted in such a way as to minimize error? e.g., ...
AbstractDissonance's user avatar
2 votes
1 answer
146 views

Scaling procedures to address false 0's after multiplying probabilities

I need to translate a training algorithm that involves sums and multiplications of probabilities to actual code. For that I need some sort of scaling procedure that allows me to avoid underflows, that ...
juanmirocks's user avatar
11 votes
2 answers
4k views

Complexity of Finding the Eigendecomposition of a *Symmetric* Matrix

This is a specialized version of a previous question: Complexity of Finding the Eigendecomposition of a Matrix . For NxN symmetric matrices, it is known that O(N^3) time suffices to compute the ...
Lihong Li's user avatar
  • 111
46 votes
8 answers
21k views

Complexity of Finding the Eigendecomposition of a Matrix

My question is simple: What is the worst-case running time of the best known algorithm for computing an eigendecomposition of an $n \times n$ matrix? Does eigendecomposition reduce to matrix ...
Lev Reyzin's user avatar
  • 12k
27 votes
6 answers
2k views

How are real numbers specified in computation?

This may be a basic question, but I've been reading and trying to understand papers on such subjects as Nash equilibrium computation and linear degeneracy testing and have been unsure of how real ...
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