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Questions tagged [numerics]

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2
votes
0answers
108 views

Why would someone use empirical sum instead of numerical integration methods?

In the context of a scientific computing application, using data coming from (powerful) embedded systems, acquiring raw data (but from calibrated acquisition electronics), I have been asked to ...
2
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1answer
1k views

Applying Neumann boundaries to Crank-Nicolson solution in python

Consider the heat equation $$u_t = \kappa u_{xx}$$ with boundary conditions of $$u(x,0)=0\\ u(0,t)=100\\ u(l,t)=0$$ Numerical analysis by pyton can be done with ...
2
votes
1answer
111 views

Calculation of the EFIE integral

I need help computing the following integral: $$ \int_{}\frac{(1+jk|\vec{r}-\vec{r}^\prime|)e^{-jk|\vec{r}-\vec{r}^\prime|}}{|\vec{r}-\vec{r}^\prime|}d\vec{r}^\prime $$ in this integral $\vec{r}$ ...
4
votes
1answer
175 views

Why do I get “estimated error” -1.#IND when doing BICGSTAB linear solver using ILUT perconditioner in eigen

I'm using Eigen (a C++ library for numerical linear algebra) to solve a linear equation with the the bi-conjugate gradient BICGSTAB algorithm with Incomplete LU preconditioner. However, the result <...
3
votes
1answer
173 views

Numerical Lax-Wendroff scheme order of convergence on Burgers equation

I was suggested to move that question here. The question to be as follows. Statement of the problem Is it possible to achieve the second order of convergence (OOC) of Lax-Wendroff (LxW) scheme ...
16
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1answer
887 views

How do you debug numerical code, what could be source of this oscillatory error?

Quiet a lot of insight can be gained form experience, I was just wondering if anybody has seen something similar to this before. The plot shows the initial condition (green) for the advection-...
1
vote
1answer
50 views

Problems with deriving an equation for a finite-difference scheme given in the journal paper

I'm reading this paper and trying to follow everything that the author has done. A Comparative Study of Finite Volume Method and Finite Difference Method for Convection-Diffusion Problem But there ...
2
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0answers
109 views

Meaning of a symbol in a research paper

The 2014 paper "Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights" by I. Bogaert (https://doi.org/10.1137/140954969) contains the following expression in Appendix A: What is ...
3
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1answer
476 views

Finite difference method basic implementation on Octave

Trying to study the error of FDM for a second order derivative versus step size I calculated the coefficients and validated them, but the output has errors for small step sizes. The function in ...
2
votes
1answer
125 views

Implementation details for high order IMEX methods by Kennedy and Carpenter

This question is a continuation of Fourth order IMEX Runge-Kutta method, concerning the implementation. Is seems to me that the first implicit stage value involves a direct evaluation, rather than ...
1
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0answers
43 views

Apply flux-limiter to nonlinear hyperbolic equation

I am trying to solve the LWH traffic flow equation, which is a nonlinear hyperbolic equation $$\frac{\partial \rho}{\partial t}+\frac{\partial (v\rho)}{\partial x}=0,$$ where $$v=v_0(1-\frac{\rho}{...
2
votes
1answer
302 views

LSA, SVD and the Frobenius norm

In Latent Semantic Analysis one uses the SVD to perform a dimensional reduction of the term-document matrix, via the Eckart-Young theorem. Now, the rank $k$ approximation obtained by E-Y is proven to ...
1
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1answer
114 views

Questions about iterative projection methods in Saad book

I am reading Chapter 5 of Saad's iterative methods book, and I don't understand section 5.2.1 about the two propositions of optimality results. In the statements of the propositions, what does it mean ...
1
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1answer
397 views

What is a relative condition number of a sum of positive values?

We want to compute the relative condition number of: $$x_1+x_2+x_3+\cdots$$ We assume all values are positive, and we will do a limit of a large $x_1=10^{8}$, and smaller values for all the other ...
3
votes
1answer
79 views

Fabry-Perot Interferometer with Frequency-Dependent Refractive Index

I am looking for assistance with calculating the fabry-perot standing modes in a resonator which has a non-static refractive index. For a resonator with perfectly reflective mirrors only the standing ...
9
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2answers
482 views

How much regularization to add to make SVD stable?

I've been using Intel MKL's SVD (dgesvd through SciPy) and noticed that results are are significantly different when I change precision between ...
3
votes
1answer
325 views

Numerical calculation sum of exponential functions

I have to repeatedly calculate a function which contains a sum of a large number (~100) of exponential terms: $f(x) = \sum_{r=1}^{100} C_r e^{b_r \cdot x}$ There is no relation between the $C_r$ ...
1
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0answers
63 views

Stable Method of orthogonal projection onto a subspace with the help of Moore-Penrose inverse,

Projection of a vector $v$ onto the column space of a matrix $A$ is given by $AA^\dagger v$. From the definition of Moore-Penrose Inverse we know that $AA^\dagger v = (A^T)^\dagger A^T v $. Below is ...
3
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0answers
84 views

Stabilizing online average calculation

In Knuth, the following method for computing an average is presented: \begin{align*} M_{n} = M_{n-1} + (x_{n} - M_{n-1})/n \end{align*} (See here, if you don't have TAOCP.) Assuming the samples all ...
1
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0answers
89 views

Eigenvalue problem (LAPACK)

I am working on a project in numerical analysis which I have to program in C (using Lapack and Blas). Matrix is given which is tridiagonal and "almost" symmetric (one element is to be changed to make ...
0
votes
1answer
554 views

Crank–Nicolson method for nonlinear differential equation

I want to solve the following differential equation from a paper with the boundary condition: The paper used the Crank–Nicolson method for solving it. I think I understand the method after googling ...
3
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0answers
161 views

Convergence of Gauss quadrature for a discontinuous function

Is there a known error estimate for Gaussian quadrature when applied to a discontinuous function? For simple one-dimensional experiments, the error appears to be bounded by $C h$, where $C$ is some ...
3
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0answers
76 views

Backward stable algorithm to get orthogonal projection onto the column space of a matrix

I have to find the orthogonal projection of a vector $b$ onto the matrix $A$ of size $m \times n$. In my application, I don't have the luxury of calculating the QR factorization. All I have are ...
0
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2answers
142 views

Nédélec Elements and Newton-Methods

If you want to develop numerical algorithms for variational inequalities, you often choose a Semismooth Newton Method. In many cases, this approach involves derivatives of $\max$ or $\min$ functions ...
2
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0answers
147 views

What could be causing multi-dimensional numerical integration inconsistency?

I'm trying to numerically integrate a multi-dimensional expression. The integrand is complicated; for example this is the integrand for $N=4$: $$\begin{aligned}&x_1^6x_2^5x_3^3x_4^2(x_1-x_1x_2)(...
1
vote
1answer
199 views

FFT Poisson Solver for non-uniform grid

I have a 3D solver for the incompressible Navier-Stokes equations which uses a FFT library for the Poisson equation with a uniform grid on all directions. In 2D the Poisson equation is given by: $$ ...
2
votes
3answers
272 views

Matrix vector multiplication performance

I have been learning about the impact of cache size on code performance. I wrote a small code to see how using a column major loop in MATLAB would be better than using a row major loop, since MATLAB ...
1
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0answers
89 views

(Approximate) Incremental Projection Method for Navier-Stokes equations

I am trying to implement an incremental projection method for the 2D incompressible Navier-Stokes. The type of projection method I am trying is $$ \frac{u^{*} - u^{n}}{dt} = - \nabla p^{n} - u \cdot ...
1
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2answers
317 views

Polynomial approximation

Is there any universal method to fill this matrix for any $n$ value: $\textbf{A} = \left[ \matrix{n & \sum x_i & \sum x_i^2 & \cdots & \sum x_i^n \cr \sum x_i & \sum ...
3
votes
3answers
126 views

How to calculate $\arg(z_1z_2\cdots z_n)$ to minimize results error?

As in title, which method is the most optimal for numerical calculating value of: $\arg(z_1z_2\cdots z_n)$? Method 1: one can first calculate $Z=z_1z_2\cdots z_n$ and then calculate $\arg(Z)$. ...
4
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0answers
158 views

How to apply an integrated constrain condition in FEM?

I'm running some simulation using FEM. In my model I need to apply a constraint condition to the governing equation. My governing equation similar to the diffusion equation as below: $$\frac{\...
1
vote
1answer
139 views

Integrating over $\mathbb{R}^{3}$ without a convex subset

I am working on a problem (solid state physics, I am stripping all the details for brevity but if more details can help I'll elaborate) where I need to numerically calculate an integral of the form: $$...
2
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0answers
106 views

What is Chebfun `eigs` doing

What is this doing? Looks like the original eigenvalue problem is converted into generalized eigenvalue problems with different dimensions of collocation points. Can someone explain more about this? ...
0
votes
1answer
80 views

How to treat non-linear term in finite difference solution of $T''_x+T''_y+aT^2=0$?

Can we linearize $T^2$ When solving $T''_x+T''_y+aT^2=0$ by finite difference? I solved $T''_x+T''_y=0$ in Matlab using a finite difference explicit scheme. But when there is a source term, I come ...
3
votes
1answer
168 views

Rounding errors in images of Julia sets

One typically computes Julia sets by iterating a complex function, such as a polynomial or rational function. How do rounding errors affect the results? I'm looking for references on this issue, ...
2
votes
1answer
202 views

How to show that Gauss-Seidel iterative method is equivalent to successively setting each component of residual vector to zero?

As stated in the title, it's said in the book that Gauss-Seidel iterative method is equivalent to successively setting each component of residual vector to zero. After rearranging G-S scheme, I got ...
0
votes
3answers
301 views

Why there are people that still prefer fortran 77 over new versions?

I am reading some notes from a course in numerical analysis for physical sciences and it is my impression that there are still people that prefer Fortran 77 over new version due to the implicit ...
0
votes
1answer
101 views

Normalization of MATLAB HermiteH

I was wandering - what kind of normalization does Matlab use in hermiteH, its implementation of the Hermite polynomials? It is certainly not the case that they use ...
1
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0answers
309 views

Vectorised root finding in Python

I have an array of size (254, 80) which I am trying to use Scipy's fsolve on. I have found that the speed of using fsolve on a vector is quicker than it is in a for loop but only for vectors upto ...
1
vote
0answers
331 views

Newton - Raphson method : maxima of function in 2 variables

I am computing the maximum of a function (with two-variables) using Newton-Raphson method. The function is : $e^{-(x \ - x_0)^2 - (y \ - y_0)^2}$, whose maxima exists at $(x_0,y_0)$. The Jacobian ...
8
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2answers
305 views

Lagrange multipliers space is too rich in a mathematical view

Background: Lagrange multiplier method has been employed in numerous fields, such as contact problems, material interfaces, phase transformation, stiff constraints or sliding along interfaces. It is ...
1
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0answers
200 views

the augmented global stiffness matrix is not positive semi-definite using Lagrange Multipliers method within FEM

The augmented global stiffness matrix is not positive semi-definite when using Lagrange Multipliers method to enforce boundary constraints on a simple square domain of integral form: I am considering ...
1
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1answer
73 views

finite difference for a second order ode

I saw in a code for discretization of something like $\frac{d^2T(x)}{d^2x}$ , ( $x = sin(\theta)$ ) tries ...
2
votes
2answers
134 views

How does Mathematica compute real and complex solutions to single, non-polynomial equations?

This question on StackOverflow has led me to ask myself what would be involved in solving an equation like this one using tools available to the Python programmer. $$ \frac{0.125567841}{d^{2.25}} = \...
3
votes
1answer
72 views

SIRS Model doesn't depend on initial conditions?

So i have been working (as an undergrad, by working i mean "Redoing a few things my professor does") in a SIRS model for epidemies. SIRS stands here for: Susceptible -> Infected -> Recovered -> ...
3
votes
1answer
278 views

How to compute the Frobenius norm of matrices whose entries are either too large or too small?

While implementing in Matlab the Frobenius norm of a matrix $$\| A\|_{\text F} := \sqrt{ \sum_{i=1}^m \sum_{j=1}^n a_{ij}^2 },$$ a problem arises when numbers are too big or too small: If a number ...
0
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1answer
137 views

Error measure for a simple finite difference scheme

I am responsible for assisting with certain error measurements for an FE program. The idea is to set up some benchmark comparisons which we can use for future development of the program. To simplify ...
1
vote
0answers
118 views

Definition of CFL number in Arbitrary Lagrangian-Eulerian framework

In an Eulerian frame of reference, the CFL number is defined as $$\sigma=\frac{u \Delta t}{\Delta x}$$ with $u$ the magnitude of the fluid velocity. A restriction such as $\sigma<1$ for time ...
2
votes
1answer
144 views

Function similar to erf that is fast at scale and allows for changing the slope at 0?

I'm interested in a function that would allow me to weight my system similar to using the error function; however computing the error function at scale would be a bottleneck. Is there something like ...
2
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0answers
109 views

How to compute the sum of a power series in a more robust way? [closed]

In order to compute the sum of a power series, we can use for loop, while loop or the analytic formula. I am wondering what is difference between those algorithms and how to improve the robustness of ...