Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

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fractional heat equation and spectral method

I want to apply a spectral method for the weak formulation of the equation $(-\Delta)^su=f$ $s>0$ with zero Dirichlet boundary conditions, where $(-\Delta)^s$ shall be the fractional laplacian on ...
Furkan's user avatar
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Numerical method for second order O

Here is a question I got to thinking about while reading speculation about the world being discrete. Suppose I want to solve a second order ODE numerically. Let's say $y''+ay'+by =f(x)$ for ...
Johan's user avatar
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11 votes
3 answers
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Probability that $b^2 - 4ac \geq 0$ where $a,b,c$ are normally distributed (numerical integration)

I would like to determine the probability that a random quadratic polynomial has positive discriminant, where the 3 coefficients $a, b, c$ are normally distributed and independent: That is, given $a,...
Alphonse's user avatar
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9 votes
2 answers
683 views

Is the notion "If a polynomial has small coefficients (relative to the exponent), then it has small roots" true?

Basically I'm trying to find good starting values for algorithms that determine the roots of a polynomial (e.g. newton method). Obviously we are trying to get as close as possible to the root as we ...
Zedssad's user avatar
  • 478
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1 answer
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Maximizing non-linear model input under constrained outputs

I have a non linear model with input A that can be in the range [A1, A2] and outputs B, C, D. The outputs have the following constraints: $ B_1 \le B \le B_2 $ $ C_1 \le C \le C_2 $ $ D_1 \le D \le ...
max's user avatar
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2 votes
0 answers
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Derive implicit method from Butcher tableau

\begin{align*} \begin{array}{c|cc} 0 &0 &0 \\ 1 & \frac{1}{2} & \frac{1}{2} \\ \hline & \frac{1}{2} & \frac{1}{2} \end{array} \end{...
uga09492's user avatar
2 votes
1 answer
35 views

FEM for non linear PDEs

I am looking for an easy but rigorous reference for FEM methods for non linear PDEs like the p-laplace equattion or non linear heat equation ect. Can one recommend me a good exposition of this topic ...
Furkan's user avatar
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Positive-semidefinite-constrained nuclear-norm-regularized optimization problem

I want to solve the following optimization problem: Let $S \in \mathbb{R}^{p \times p}$ be a symmetric matrix, and fix $n \in \mathbb{R}$ such that $n$ is significantly less than $p$ and the rank of $...
Eco-nometrician's user avatar
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Is the algorithm $f(a,b)=\dfrac1{1-ab}$ numerically stable? [closed]

I am trying to understand the different definitions of numerical stability (I am using [1] as a reference on the topic). Is the algorithm $$ f(a,b)=\frac{1}{1-ab}$$ forward/backward/mixed stable? [1]- ...
Lorenzo's user avatar
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2 answers
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Finding period of pendulum through interpolation

I’m looking for finding an efficient answer to this problem, which is to find the period time of a pendulum using interpolation. The pendulum behavior was given using the equations $\phi’’+\frac g L \...
albin's user avatar
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Numerical method,how to do it? [closed]

Let $f(x)=x^4-x^2+17x+1$. Let $p\in\mathcal{P}_{20}$ interpolates f at $x_j=2^j(j=0,...,20)$,Compute $p(0)$
fyx_mather's user avatar
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Which root does the bisection method prefer in case of multiple roots existing in the given interval?

Let $f$ be a continuous function over the interval $[a, b]$ such that $f(a)f(b) \leq 0$. Suppose $f(x)=0$ has greater than $1$ root in this interval. Which root does the bisection method eventually ...
BDS's user avatar
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1 answer
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How to transform this expression to a numerically stable form?

I have this function $$f(x, t)=\frac{\left(1+x\right)^{1-t}-1}{1-t}$$ Where $x \ge 0$ and $t \ge 0$. I want to use it in neural network, and thus need it to be differentiable. While it has a ...
yuri kilochek's user avatar
2 votes
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Help understanding how to apply IMEX methods to one-dimensional PDEs

I need to compute a solution of the following PDE: $$\frac{\partial u}{\partial t} + v \frac{\partial u}{\partial x} = 0$$ For didactic purposes, I need to use an IMEX method. The point is no one ever ...
EleDan's user avatar
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Weak convergence of derivatives - Backward Differentiation Formula - BDF 2 method

I am using a space-time discretization to discretize an initial value problem, using P1-FEM for space discretization and BDF-2 for time discretization. To fix the notations: $\Omega\subsetneq\mathbb{R}...
GCMA's user avatar
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1 answer
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Relationship between Laplace Method and Gaussian Distribution?

I am trying to learn about the Laplace Approximation of Integrals (https://en.wikipedia.org/wiki/Laplace%27s_method). Supposedly, this is closely related to "Gaussian Functions". Here is my ...
firstpassage's user avatar
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1 answer
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Why is easier to get inverse of mass matrix?

On my lecture notes, I came across the statement: 'Inverting the mass matrix is significantly easier than solving a linear system involving the stiffness matrix.' Can someone explain why this is the ...
Ariel So's user avatar
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Standard way to do Runge-Kutta (4th order) for coupled ODE's in Python?

I am somewhat familiar with using RK4 for coupled ODE's, I found a very elegant way (in my opinion) to utilize it in Python, like so: ...
Vox Winters's user avatar
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1 answer
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SVD and least square solution

Let $K \in \mathbb{R}^{m,n}$, $u \in \mathbb{R}^n$, and $f \in \mathbb{R}^m$. Assume that $m < n$ and $K$ have full rank so a solution exists but is not unique. I want to understand why this ...
endeavor's user avatar
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2 votes
1 answer
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Solve Determinant Equation

Identify $t\in\mathbb{R}$ such that (basically it is a determinant of a block matrix of size $n$): $$ \left| \begin{array}{cc} \mathbf{x}+t\mathbf{y} & \mathbf{B} \\ \mathbf{c} & \mathbf{D} \...
Karbo Lei's user avatar
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How can I fix my understanding of Numerical Analysis?

I am an undergraduate student taking Numerical Analysis. I’m having a hard time understanding some of the material because it feels as though my instructor is jumping all over the place. When it comes ...
Dr. J's user avatar
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Implicit Numerical method 1D unsteady state heat transfer problem [closed]

I want to apply either backward euler or crank nicholson to see the effect on stability of the heat transfer problem. The explicit euler do not work if I put large values of thermal conductivity. ...
beschichtung346's user avatar
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Is it possible to compute the projection of a linear system solution on a constraint set instead of computing the solution of the constrained problem?

I try to solve the following problem for a personal project : $M$ is a wide matrix, $M \in \mathbb{R}^{m\times n}$, with $m << n$, such that $MM^t = D$, a diagonal matrix, and $A$ is a square, ...
Baptiste GENEST's user avatar
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Investigating the numerical accuracy of a truncated Legendre polynomial expansion of an unknown function

I have an integral equation involving an unknown function $f(x)$, of the most basic form $$ \int_{-1}^{1} e^{iω(t) x} f(x) \ dx = g(t) $$ I am solving for an approximation of $f(x)$ by substituting in ...
Silver Pages's user avatar
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Higher-order optimal Runge-Kutta methods?

I am aware of Ralston's paper on optimal (minimum truncation error) Runge-Kutta methods for orders up to 4 with the minimum number of stages for those orders. I have copied the Butcher tableaux of the ...
Mel's user avatar
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15 views

Need help with proving energy stability of forward Euler for heat equation

I have the 1D heat equation $u_t= \alpha u_{xx}$ on $x \in [0,1]$. It has homogeneous Dirichlet boundary conditions. I intend to use the forward Euler numerical scheme. $$\frac{u^{n+1}-u^{n}}{k} = \...
laplacian18's user avatar
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Taylor coefficients and termwise integration

Many special function are calculated using its Taylor series, and is an efficient method of estimation the values of a function. Nevertheless, for some function evaluation of a formula of Taylor ...
poeaqnwgo's user avatar
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Finding Roots of (Large) Determinants

I have a matrix of the form $$\mathbb{A}_{i,j}(\omega) = \sum_n \frac{\sin(nx_i)\sin(nx_j)}{-\omega^2+C(n)}+\sum_{r_x}\sum_{r_y}\frac{\sin(r_xx_i)\sin(r_xx_j)\sin(r_y\alpha)}{-\omega^2+D(r_x,r_y)}.$$ ...
Alex Vaughan's user avatar
1 vote
0 answers
26 views

Applicability of Ritz method,weak formulation

Im trying to find values of c, when I can use Ritz method for this differential equation.$ L_cu=-u^{\prime \prime}+c u=f(x) ; \quad u^{\prime}(0)=0 ; \quad u(1)=b $ for $ u \in C^{2}([0 ; 1]) $ (this ...
VadimStacheff's user avatar
1 vote
2 answers
110 views

Where to find proof for the remainder formula of the interpolation in two variables

Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book ...
Juan's user avatar
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1 answer
54 views

Difference between finite difference approximation and differential quadrature approximation

As a student of numerical analysis, I understand that a finite difference approximation (FDM) of the derivative '$u_x$' of a desired solution '$u$' at the point $x_n$ in the domain is just a linear ...
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Numerical Analysis of Differential Equation with Boundary Conditions

Note I have decided to edit this post so it is more specific and also because I was incorrect the first time. I didn't wanna make another post about the same subject, but if this is not allowed then ...
Need_MathHelp's user avatar
1 vote
0 answers
24 views

Quadrature of a random function

I would like to numerically compute the following integral $$ \int_a^b f(x) p(x) \mathrm{d} x $$ for some known smooth function $f(x)$ and unknown smooth function $p(x)$. I can estimate $p(x)$ by ...
Radost's user avatar
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51 views

Where does 2.3.5 come from and how to prove it?

I was looking over this proof in Atkinson’s Numerical Analysis 2nd Edition and I am having a hard time seeing where 2.3.5 is coming from. What is Atkinson trying to show with 2.3.5 and the line below? ...
Dr. J's user avatar
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1 answer
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Take invertible matrix $A$, and set a diagonal entry of $A$ to $\infty$. Can we use $A^{-1}$ to find the pseudo inverse of this new matrix?

Consider having $A$, with $A^{-1}$ previously calculated. Say I apply a rank 1 update on $A$ using $v = (\infty,0,0,0,0)$: $${B} = vv^T A $$ Clearly A is no longer invertible, but the pseudo inverse ...
j bloggs's user avatar
1 vote
0 answers
44 views

Techniques to Place a Lower Bound on the Real Zeros of a Polynomial

I am working with a family of polynomials with some particularly nasty coefficients. Unfortunately, I cannot give very many details as to what these polynomials look like, but, suffice it to say that ...
peabody's user avatar
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0 answers
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Runge-Kutta 2 Model problem Estimate (Calculations problem)

The RK2 implicit method is given by the scheme \begin{equation} \begin{cases} &y_{n+1}=y_n+h\Phi(x_n,y_n;h),\\ &\Phi(x_n,y_n;h)=\sum_{i=1}^2 b_ik_i=b_1k_1+b_2k_2,\\ &k_r=f\big(x_n+c_rh, ...
Hamdiken's user avatar
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How to prove Preconditioned Richardson - Steepest Descend converges

Hi, I am trying to solve this problem but I am feeling like I am missing some prerequisites. Can help me solve this ? or at least a reference with a book where this is solved ? $\text{Let } A \in \...
johnyy's user avatar
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Numerical method for a free boundary problem

I am consider the following free boundary problem: $u_t + \mathcal{L}u = 0, 0<x<s(t)$, with the boundary conditions: $u(T,x) = f_T(x)$, $u(t,s(t)) = g(t)$ and $u_x(t,s(t))=C$. Here, $\mathcal{L}...
Kenneth Ng's user avatar
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1 answer
96 views

How to find a matrix $A$ such that $\langle x,y\rangle = x^T A y$ for the $L^2 (D)$ inner product

So I know that I should have some matrix $A$ such that $\langle x,y\rangle = x^T A y$ for the $L^2 (D)$ inner product. I would like to find this $A$. My domain $D$ is a $3 \times 3$ square grid in $\...
k12345's user avatar
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0 answers
29 views

Applications of highly oscillatory integrals

I was reading a series of articles on numerical integration of highly oscillatory functions, e.g., S. Olver, Numerical approximation of highly oscillatory integrals S. Xiang, H. Wang, Fast ...
Vl F's user avatar
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1 answer
99 views

Choose start vector for gradient descent so that it converges with one step

I'm currently looking at an old exam and I have encountered the following task: $ A=\left( \begin{array}{rrr} 2 & -1 \\ -1 & 2 \\ \end{array}\right)$ $b=\left( \begin{array}{rrr} 1 \\ 1 \\ ...
LostInTheSauce's user avatar
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What is the difference between H-Adaptive Integration and adaptive quadrature algorithms

I am a little bit confused about the difference between the H-Adaptive Integration and adaptive quadrature algorithms, AFAIK: Both break down the entire region of integration into smaller sub-regions....
Rubem Pacelli's user avatar
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36 views

Simpsons Integral or the like for Integrating over Triangular Surfaces

I have several triangles in 3D space where I know the value of the function at the three vertices of the triangle (Could be determined in between at any point as well). I wanted to know if there ...
uzairphd's user avatar
0 votes
1 answer
31 views

Imaginary components in Discrete Fourier Transform of Gaussian?

I am trying to understand Discrete Fourier Transform (DFT), after only having experience with the continuous transformation. The natural idea was to try to understand DFT on the simplest function, ...
Szgoger's user avatar
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0 answers
33 views

How to find out the Runge-Kutta 4 constants for numerically evaluating an nth order IVP?

I know how to use the RK 4 method for a first order differential equation of the form: $$y' = f(x, y(x))$$ $$y_{k+1} = y_k + (G_1 + 2G_2 + 2G_3 + G_4)*dx/6$$ where $G_1 + 2G_2 + 2G_3 + G_4$ are ...
Ajaykrishnan R's user avatar
-1 votes
0 answers
33 views

How to find pdf of a random variable x if $x$ is $f(\text{some parameters})$, and $f$ is not known straight forward.

In the original question cited here: Intersection of two ellipses at exactly 2 points, if we find the 2-dimensional solution vector $x$, I am curious on how we can extract a probability density ...
Aravind Muraleedharan's user avatar
0 votes
1 answer
33 views

Euler Backward method order of accuracy

I solved a question with Euler's backward method with 4 different step lengths. Then I calculated the error and drew a graph using loglog. Now the task is to find the order of accuracy using the graph....
Need_MathHelp's user avatar
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0 answers
27 views

Help with numerical simulation of a backwards ODE.

This is a toy problem of a much more complicated ODE system. Suppose I have the following ODE $$\frac{dx}{ds} = kx-c, \quad x(0) = 1$$ Assume that $s<t$ and that $x(s)$ is the value of some ...
Nima's user avatar
  • 51
1 vote
1 answer
47 views

Numerically solving ODEs that can't be expressed explicitly

I'm an engineer and therefore not as knowledgeable most of you, so please don't lynch me if I ask something stupid. How can we efficiently solve a nth order ODE which can not directly be expressed as ...
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