Questions tagged [numerical-methods]

Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

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Bachelor's level research work.

For next year I have most important event for my mathematical career, I should write good Bachelor's work. For these reasons I am preparing some research about subjects of work. I have interesting for ...
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10 views

Showing that Newton's divided difference formulas are invariant to permutations of points by swapping neigbhoring points

This is not the right way to prove that the Newton's divided difference formula is invariant to permutations, I just want to see if this kind of proof could be made to work. We have a set of points $...
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1answer
12 views

Basis for CPL$\left\{0,1,2,3 \right\}$

I have the following question from a linear algebra textbook: Let $V = CPL \left\{0,1,2,3 \right\}$, where this denotes the continuous piecewise linear functions on $[0,3]$, ie. the set of ...
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1answer
13 views

Critical Points of Vector Function

Given a matrix $A$ that is $n \times d$ and a $n$-dimensional vector $w$, define the vector function $f$ as: $f(v) = \frac{w^TAv}{||Av||^2}$ where $v$ is a $d$-dimensional vector. How can I find all ...
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11 views

Constructiong an implicit 2-step process of order 4

Construct an implicit 2-step process of consistency order 4 and check its stability As stated, I want to construct such a process. I start by definition of an implicit 2-step process: $\sum_{j=0}^...
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13 views

Why bounded solution and convergence imply a stability for nonlinear finite difference scheme

I have been review the finite difference for long wave family equation (RLW, Rosenau-kawahara, etc) and see that the boundedness and convergence always imply the stability of the proposed difference ...
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32 views

What is the average value of $x + \sqrt{2}\, y$ over all integer points in the ellipse $x^2 + 2y^2 < N$?

Consider the oval $x^2 + 2y^2 < N$, the number of integer points is approximately $2\pi N$. Now consider the average: $$ \frac{1}{2\pi N} \sum_{\{ (x,y): x^2 + 2y^2 < N\}} (x + \sqrt{2}\, y\...
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38 views

Exploring mathematical relationship bewteen time-series of three variables and their ratios

I am trying to evaluate three parameters ($V_{1}$, $V_{2}$, and $V_{3}$) (3D of V1 V2 and V3) related to a physical phenomenon and trying to find a mathematical connection between them. Time-series of ...
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1answer
17 views

How many Chebyshev nodes are necessary to approximate the function $\sin(xπ)$

I am trying to understand Chebyshev Interpolation, and having trouble understanding this problem. How many Chebyshev nodes are necessary to approximate the function $\sin(x\pi)$ to an error of at ...
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1answer
22 views

Steps to derive a weak formulation (streamline diffusion)

I'm studing ADR equations and now the book I'm following (Quarteroni - Numerical models for differential problems) wants to add artificial diffusion in a 2-D problem. This corresponds to add the ...
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19 views

Solving a PDE by wavelets

I know how to solve a PDE by wavelets. I found a lot of articles about it. But, if the PDE includes Dirac Delta distribution $( \delta)$, how to solve it by wavelets? (Haar, Legendre, Chebyshev ...
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18 views

Show that a certain system preserves the weighted area $ (dx \wedge dy)/xy$

I already told few questions ago that I'm currently reading an abstract about the Lotka Volterra differential equations. But now I have a proof, where I need explanations. Consider: $$ \dot{x} = -xy\...
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14 views

Numerically stable transformation of logistic mixture to unimodal logistic distribution

I have an $n$-dimensional random vector $\pmb{v}$, where each element is distributed according to some 10-component logistic mixture-distribution. I want to transform each component of the vector into ...
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24 views

Interpolation polynomial of data with errors

Good day, everybody! I have an assumption that I've tried to proof for a fortnight, but still have no results. Let's say we have a function $f(x)$ and a set of evenly distributed data points $\{x_0,...
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2answers
49 views

The role of finite precision arithmetic in the precision of numerical methods

Every book on numerical methods studies the precision of the algorithms as if they will be executed on a machine with infinite precision. Apparently, the effects of using floating point arithmetic (...
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25 views

Convergence of a multi-step process

We have the linear multi-step process $-y_k-y_{k+1}+y_{k+2}+y_{k+3}=4hf(t_k,y_k)$ for the initial value problem $y'(t)=f(t,y(t)), y(0)=0$ Show that for $f=0$ and $y_0=y_1=y_2=0$ ...
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1answer
15 views

Criterion of Dahlquist and consistency of a process

We have the linear multi-step process $-y_k-y_{k+1}+y_{k+2}+y_{k+3}=4hf(t_k,y_k)$ for the initial value problem $y'(t)=f(t,y(t)), y(0)=0$ Check the consistency of the process and the ...
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1answer
21 views

Characteristical polynomial of a multi-step process

I have the following question. I have shown that for coefficients $\alpha_k, \beta_k$ of a linear multi-step process with characteristic polynomial $\rho$ and $$C_j=\sum_{k=0}^m \alpha_k\frac{k^j}{j!...
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1answer
30 views

Laplacian in space with non uniform step

I am trying to find the laplacian of a point in 3D but the major issue is that the distances between my points not constant. For example in 1D : Illustration, the problem is the same for each axis. ...
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2answers
51 views

How can I numerically implement $\delta(f(x,y))$

I would like to numerically implement a Dirac Delta function whose argument is another 2 variable function. I know that I can model a Dirac Delta numerically using a Gaussian. What can I do if I want ...
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1answer
54 views

Understanding stability of numerical ODE methods from just linear problem

So in general for ODE we come up with a numerical method for $y'=f(t,y)$ Then apply the numerical method to $y'=\lambda y$, where $\lambda$ is some scalar My question is how can we claim the ...
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1answer
53 views

How to solve a linear difference equation?

We have the linear multi-step process $-y_k-y_{k+1}+y_{k+2}+y_{k+3}=4hf(t_k,y_k)$ for the initial value problem $y'(t)=f(t,y(t)), y(0)=0$ Give every solution of the homogeneous ...
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0answers
15 views

Formula for Area of a Triangle - nodal basis function

Let T be a triangle with corners $P_1, P_2, P_3$ and the nodal basis function $\lambda_1, \lambda_2, \lambda_3$ and $\alpha, \beta, \in \mathbb{N}_0$. I want to show that $$ \int_{T}^{} \lambda_1^\...
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1answer
31 views

Gauss error function of square root function

Why does the following equation hold? $$\sum_{i=0}^{\infty}\frac{2^i}{\sqrt{\pi}\prod_{j=1}^{i}(2j-1)}x^{\frac{2i-1}{2}}=e^x Erf(\sqrt{x})$$ in which $Erf$ is the Gauss error function.
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12 views

Composite trapezoidal quadrature error convergence

Let $\hat{T_n}(f)$ be the composite trapezoidal quadrature for the integral $I(f)=\int_{a}^{b}f(x)dx$ based on $n$ equal subintervals of $[a,b]$. We define: $$\hat{Q_n}(f)=\hat{T_n}(f)- \frac{(b-a)^...
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1answer
32 views

Natural linear spline function representation

Let $s: \mathbb{R} \to \mathbb{R}$ be a natural spline function of degree one (that is it is piecewise a polynomial of degree at most 1) and let $x_0 \leq x_1 \leq ...\leq x_n$. Show that $s$ can be ...
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16 views

What does the Simpson coefficient matrix for 3D integration look like and how can I generate this matrix?

I would like to numerically integrate a 3 variable function within some finite limits for the 3 variables. I know how to generate the 2D Simpson coefficient matrix for numerical integration. How does ...
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24 views

Solving a system of non linear equations (which can be solved in some special cases)

I am trying to solve the following system of $n^2$ equations for $X$ $c_i^\top (A_{i,i} B + X)^{-1}(A_{j,j} B + X)^{-1} c_j = I_{i,j}, \forall i,j\in\{1,\dots,n\}$ where $c_i$ is an $n$-dimensional ...
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29 views

How to determine the following function at $x=0.1$ by four-digit rounding arithmetic?

$$f(x) = \frac{x\cos x - \sin x}{x - \sin x}$$ The answer from the back of the book is $-1.941$. But I got $1$. $\cos(0.1) ≈ 1$ by four-digit rounding arithmetic. $\sin(0.1) ≈ 0.001571$ So, $$\...
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1answer
31 views

Sufficient conditions for bounded output of an explicit linear multi-step method

The system: \begin{align} \dot{x}(t) &= f(x(t))\\ x(t_0) &= x_0 \end{align} is being solved by an explicit linear multi-step method (assume perfect initialization): \begin{equation} \widetilde{...
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1answer
21 views

Trapezoid method for system

For $\alpha \geq 0$ and $\beta \in \mathbb{R}$, we consider the system $$x'(t)+\alpha x(t)=-\beta y(t), t \geq 0, \\ y'(t)+\alpha y(t)=\beta x(t), t \geq 0, \\ x(0)=1 \\ y(0)=2.\\$$ I want to state ...
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10 views

Show that the maximum value is less than $|y_0-z_0|$

$f$ satisfies Lipschitz condition. For given initial values $y_0, z_0\in \mathbb{R}$ we consider the following problems: \begin{align*}&y'=f(t,y), \ \ a\leq t\leq b \\ &y(a)=y_0\end{align*} ...
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2answers
59 views

Show that sequence converges of $r_2$

We have the function $f(x)=x^2-x-12=0$ with roots $r_1=-3$ and $r_2=4$. We consider the sequence $x_{n+1}=g(x_n), \ n=0,1,2,\ldots $ where $g(x)=\sqrt{x+12}$. We want to show that $x_n\rightarrow ...
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23 views

Numerical methods for PDE- solve partial integral differential equation with time dependant input

I am a bit a newbie when to PDEs and especially the numerical solution of such. I am looking for some advice on how to integrate an equation which has derivative terms, integral terms and some ...
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2answers
42 views

Norm-preserving implicit numerical method

Consider the implicit single-step method $$x_1 = x_0 + h A(x_0 + x_1)$$ where matrix $A$ is skew-symmetric. Show that this method preserves the norm, i.e., $\|x_1\| = \|x_0\|$. I don't really know ...
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2answers
37 views

How do I construct a second order convergent fixed point iteration?

Say I am dealing with $f(x) = x^3+2x+1$ and I need to come up a fixed point iteration to find its root, that has a second order convergence. How should one approach this question? Should I just ...
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1answer
24 views

Computational complexity of $x x^H$

Assume $x\in \mathbb{C}^{n\times n}$. What is the computational complexity (cost) of $x x^H$ where $H$ is the conjugate transpose? I know this gives a symmetric matrix and we can divide by $2$ the ...
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1answer
49 views

Riemann sum not converging

I am trying to find the volume of a hemisphere by numerical integration. I have a set of points equally spaced over x-y plane. I am trying to calculate its volume by trying to evaluate Riemann sum. $$...
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17 views

What are possible ways to update the Hessian if the calculated gradient is negative in BFGS algorithm (Quasi-Newton method)?

When applying the globalized BFGS algorithm (Quasi-Newton Method, optimization, minimization) to approximate the minimum of a function using the Quasi-Newton-Method, sometimes one can get a negative ...
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1answer
36 views

Show: $||u_n||_X \leq \gamma^{n-1} ||u_0||_X$ $\mbox{ } \mbox{ }$ $\forall n \geq 1$

Let $H$ be a Hilbert space with norm $||.||_X$ and scalar product $(.,.)_X$ , which is the direct sum of the subspaces $V$ and $W$. Regarding $V$ and $W$ it holds the intensified Cauchy Schwarz ...
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Averaging higher order term

$d\theta/dt =-1-\epsilon z \cos\theta \sin\theta$ $dz/dt=\epsilon(r^2-T)$ $dr/dt=-\epsilon rz\sin^2(\theta)$ After averaging,I got : $d\phi/dt=-1+O(\epsilon^2)$ $d\zeta/dt=\epsilon(\rho^2-T)+O(\...
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39 views

Solving this problem with SQP method [closed]

Hi everyone I want to Minimize this function with (SQP) method with these conditions : Initial Starting Points : $$ x_1=2.4 $$ $$ x_2=1 $$ $$ Tolerance = 0.1 $$ $$ \operatorname{Min} F(x)=40 - ...
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2answers
37 views

derivation of order 3 method for differential equations

I am stuck with a big problem. Trying to understand the proof that the numerical method of solving differential equation $x_{i+1} = x_i + \tau_iF(t_i+\frac{\tau_i}{2}, x_{i+\frac{1}{2}})$ $...
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1answer
25 views

Newton method non-linear systems envolving curves

I have to solve the following system: $$ x'(t) \times y'(s) = 0 $$ $$ (y(s) - x(t)) \times y'(s) = 0 $$ Where: $$ x(t) = (cos(t), sin(t)) $$ $$ y(s) = (s, s^2) $$ using Newton's method for non-...
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12 views

solving rotation matrix angles using Gauss-Newton

I am attempting estimate the three angles of a rotation matrix that has been applied to a known set of vectors using the Gauss-Newton method through direct observation the rotated vectors that have ...
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1answer
134 views

Crank-Nicolson code for in-situ combustion model (MATLAB)

I need to solve in MATLAB the following system (1): \begin{align} &\frac{\partial \theta}{\partial t} + u \frac{\partial (\rho\theta)}{\partial x} = \frac{1}{\text{Pe}_T} \frac{\partial^2 \...
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11 views

Trying to evaluate high variance in large dataset

So I have an large active dataset, recording speed tests of connections. I am looking for the best approach to identify possible problem connections. To do this I started by looking at averages etc. ...
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1answer
61 views

Error Propagation Using an Exponential Function.

Consider the programming problem of estimating the correct value of $2^{x}$, where $x$ is an irrational number entered with a precision of 500 binary bits. (1) How accurate (in terms of bits) is the ...
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22 views

How to find norm $||U_n||$ of Chebyshev polynomials of the second kind?

how to find norm $||U_n||$ and the values $U_n(\pm)$ of the Chebyshev polynomials of the second kind?
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38 views

Want to compute product of Bessel and Hankel function in Matlab - $J_m(x)\cdot H_m^{(1)}(y)$ - but large order or argument returns 'inf'?

I am trying to compute (in Matlab) the product of a Bessel function and a Hankel function where the order $m$ of the functions may be very large and/or the arguments $\alpha x$ and $\alpha y$ of the ...