Issues related to the representation of numerical quantities in a finite representation in a given base differing from their exact mathematical value.

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1
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2answers
87 views

C++ libraries for Fast Fourier Transform in high precision

I am looking for a C++ library for Fast Fourier Transform (FFT) in high precision (e.g., using high precision real data types similar to mpfr_t in MPFR or ...
-2
votes
2answers
114 views

$\exp(\ln(x))-x\neq0$?

Analytically, the expression $$\exp(\ln(x))-x \enspace,$$ should give 0. However, in Matlab, it does not. x = linspace(1, 10, 10); exp(log(x)) - x; for $x \in ...
0
votes
1answer
23 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 ...
1
vote
1answer
78 views

Addition and subtraction of two floats in Python

Yesterday I was wondering how floats are handled in a computer and what they look like in binary... I learnt about the single-precision floating-point and I tried to see the limit of that format... I ...
4
votes
2answers
118 views

Known issues with eigenvalue numerics?

Are there any known issues (such as precision issues) with $\mathsf{MATLAB}$ eig and charpoly functions for large enough $\{-1,0,+1\}$ matrices? Even if I change $1$ or $2$ entries between matrices ...
2
votes
1answer
76 views

Templated Numerical Linear Algebra in Parallel

I have to invert large, but densely populated matrices with higher precision arithmetic. Therefore I am looking for something like the PLASMA library, which can do Cholesky or LU factorization in ...
3
votes
1answer
131 views

Are there tasks in machine learning which require double precision floating points?

Double-precision calculations are significantly slower or more expensive than single-precision calculations. For example, the NVidia Tesla which performs well on doubles is much more expensive then ...
3
votes
1answer
122 views

GSL linear algebra LU/determinant precision

I am working with symmetric matrices of order $n \times n$ where $n \leq 50$. The diagonal elements of my matrices are a fixed number $d$ and the off diagonal elements are limited to two small numbers ...
4
votes
2answers
103 views

Under what circumstances can two (nearly) identical sparse matrices give different solutions to Mx = b?

Suppose I have two sparse matrices, $A$ and $B$, of size $N \times N$. They each have the same sparcity pattern ("footprint"). They each also have values which in theory should be identical, but ...
7
votes
1answer
364 views

What is the numerical difference between abs(z)^2 and z x z*, where z is a complex number

I am doing research in electromagnetics (a branch of physics) and I deal with complex numbers in my computing tasks (I use MATLAB). Let's say $z$ is a complex number. To calculate the square of the ...
4
votes
1answer
61 views

Error propagation in recurrence relation

I have a recurrence relation $$P_{n} = A_{n} P_{n-1} - B_{n}P_{n-2}$$ with given $P_{0}$ and $P_{1}$. Numerically, each $A_{n}$ and $B_{n}$ is calculated with some precision. The same applies to ...
9
votes
2answers
116 views

How can I numerically solve an ODE to $N$ provably correct digits?

Suppose we have an initial value problem of the form $$ \frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t} = f(\mathbf{x}) \qquad \mathbf{x}(0) = \mathbf{x}_0 $$ where $\mathbf{x}_0 \in \mathbb{R}^n$ is known ...
2
votes
1answer
124 views

precision loss in non-trigonometric, periodic functions using FFTW and NaNs after marching forward in time (Fortran)

I have developed a pseudospectral solver of the Navier-Stokes equations using FFTW. I tested my formulation of right hand sides (RHS) of the NS equations against standard trigonometric functions ...
7
votes
1answer
201 views

Why should I renormalize physical variables?

I am working with legacy physical codes and I develop new ones based on the output of them. They all use their own internal normalization of variables (for example all distances are divided by the ...
2
votes
1answer
85 views

Testing 1D root-finding procedures for robustness

How can I test whether a given 1D root-finding procedure is robust? I know that there are data sets and resources online for different kinds of optimization, but I have yet to find anything with ...
1
vote
1answer
344 views

Calculate the machine epsilon in Matlab

How can I calculate the machine epsilon for two numbers then calculate the theoretical limit for machine epsilon in Matlab ?
3
votes
1answer
134 views

Compute hypergeometric function ratio: $\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$?

I need a numerically stable way to compute the following ratio: $$\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$$ All the parameters are real numbers, with $a< 0$,$\ $ $b,c > 0$ and ...
0
votes
1answer
295 views

Overcoming floating point issues when Inverse does not exist but determinant provides nonzero result

I have an expression (let's say determinant of matrix A) expressed in symbolic form in terms of 2 decision variables x1, x2 and 2 parameters q1 and q2. I'm minimizing this using fmincon for different ...
1
vote
3answers
395 views

Are scaled equations still needed?

If one wants to solve a problem in physics, one often has to deal with very small numbers because of the units, e.g. the energy range of interest of semiconductors lies in the region $eV \approx ...
8
votes
2answers
191 views

Representing Eisenstein numbers without floats

I have a project where I need to use quadratic fields Specifically numbers of the form $a + b \sqrt{-3}$ with $a,b \in \mathbb{Q}$. For example here are the prime numbers in Eisenstein integers: ...
1
vote
1answer
304 views

What is Precision-Recall Curve? [closed]

I have a data mining assignment where I make a content-based image retrieval system. I have 20 images of 5 animals. So in total 100 images. My code returns the 10 most relevant images to an input ...
3
votes
1answer
439 views

Numerical derivative and finite difference coefficients: any update of the Fornberg method?

When one want to compute numerical derivatives, the method presented by Bengt Fornberg here (and reported here) is very convenient (both precise and simple to implement). As the original paper date ...
3
votes
2answers
1k views

Number of equations and precision of SciPy's integrate.odeint()

Is there any reason why SciPy's integrate.odeint() should become less precise when the number of equations increase? I'm trying to solve these two sets of differential equations: $\frac{dy_1}{dx} = ...
7
votes
0answers
187 views

Updating matrix diagonal with Woodbury matrix identity and maintaining numerical accuracy

I have a dense matrix A and its corresponding inverse $A^{-1}$. The Woodbury matrix identity states: $$ (A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1} $$ I wish to perform small ...
5
votes
2answers
103 views

Odd accuracy barrier in C/PETSc regarding finite elements

I’m implementing a finite element code (translating from a working MATLAB version, so I have results to compare to) and for some odd reason, some of my computations are only accurate to around 6 ...
1
vote
2answers
230 views

Determine the step size in a differential equation numerical solver

How can we define the precision we require in a numerical differential equation solver? What is it that I have to optimize to know? And how do I know that I'm at a sufficient time-step value? For ...
4
votes
1answer
4k views

Machine epsilon (eps)

The wiki for machine epsilon says: "Machine epsilon gives an upper bound on the relative error due to rounding in floating point arithmetic" If machine epsilon is the upper bound on the relative ...
1
vote
3answers
1k views

Matlab output in scientific notation [ERROR: Subscript indices must either be real positive integers or logicals.]

I have X, Y and Z variables in matrix form, each of size n x 1. Eg.: X = [-38.0400, -38.6700, -38.9300, -39.4500...] Whenever I run the code below: ...
13
votes
1answer
571 views

Scientific computing with Python with modern GPUs with double precision

Has anyone here used double precision scientific computing with new generation (e.g. K20) GPUs through Python? I know that this technology is rapidly evolving, but what is the best way to do this ...
6
votes
4answers
1k views

computing the determinant of a dense nonsymmetric 100x100 matrix having very big and very small eigenvalues

The motivation for my question is the following: in one of Project Euler questions there is a need to count the spanning trees of a rectangular grid graph of dimension 100x500. By the Matrix-Tree ...
5
votes
2answers
947 views

Need for quad precision in scientific computing?

Even if quad precision is not directly supported by most CPUs, many Compilers (GNU, Intel) support them. Also some software packages allow to compile with quad precision, e.g. PETSc. But is there ...
8
votes
2answers
444 views

Higher precision floating-point arithmetic in numerical PDE

I have the impression, from very different resources and talks with researches, that there is a growing demand for high precision computations in numerical partial differential equations. Here, high ...
10
votes
2answers
455 views

Diagonalization of Dense Ill Conditioned Matrices

I am trying to diagonalize some dense, ill-conditioned matrices. In machine precision, results are inaccurate (returning negative eigenvalues, eigenvectors do not have the expected symmetries). I ...
7
votes
5answers
436 views

What algorithm for solving a set of stiff ODEs would be easiest to port to high precision floating point arithmetic?

I want to solve a relatively small system of stiff ODEs (< 10 first-order equations) using high precision floating point arithmetic (using MPFR or alike). What would be the easiest algorithm to ...
12
votes
3answers
3k views

Single versus double floating-point precision

Single precision floating point numbers take up half the memory and on modern machines (even on GPUs it seems) operations can be done with them at almost twice the speed compared to double precision. ...
20
votes
6answers
2k 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?
20
votes
3answers
2k views

What's the state-of-the-art in highly oscillatory integral computation?

What's the state-of-the-art in the approximation of highly oscillatory integrals in both one dimension and higher dimensions to arbitrary precision?
6
votes
3answers
379 views

Does there exist an arbitrary-precision convex optimization solver?

I have a relatively simple convex optimization problem that involves less than 100 variables but contains a terribly ill-conditioned matrix. I have tried CVX and CPLEX; even though both can typically ...