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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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Lanczos convergence for symmetric matrix with eigenvalues $1,2,\cdots,2,100$

Suppose that $A$ is symmetric with eigenvalues $1,2,\cdots,2,100 \in \mathbb{R}^{100x100}$ and $b \in \mathbb{R}^{100}$ obtained by normalizing a standard normal random vector. Show that 1 and 100 are ...
jacopoburelli's user avatar
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What is the definition of and quantifier order in consistency for one-step methods?

Consider the usual Cauchy IVP $y = f(x, y)$ with $y(x_0) = y_0$, satisfying assumptions of Picard's theorem in a rectangle $D$. A one-step method is essentially $$y_{n+1} = y_{n} + h \Phi(x_n, y_n; h),...
Linear Christmas's user avatar
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39 views

Why does it seem like two parameters $k_1$ and $k_2$ are needed to match $e^{-r}$ and $k_2 \sin(k_1*r)$ as well as their derivatives $\frac{d}{d\,r}$?

The Spherical Bessel functions that solve the Spherical Helmholtz equation in the Spherical Coordinate system come in four kinds, the Spherical Bessel Functions of the first kind, the Spherical Bessel ...
Stephen Elliott's user avatar
1 vote
1 answer
22 views

Fixed point iteration and convergence and contraction mapping

Let $p\ge 2$ be an integer, $a>0$ and $f$, $g$ be two functions defined for $x>0$ by $g(x)=ax^{1-p}$, $f(x)=x+\lambda(g(x)-x)$. Let $x_{0}$ be an appropriate initial guess, close enough to the ...
maths and chess's user avatar
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32 views

Correctness of Python implementation of numerical difference method for Cahn-Hilliard equations

Currently I am working on trying to get the Cahn-Hilliard equation solved using finite differences. I'm unsure if I am going in the right direction with my code. I have been following section 6.1 of ...
aaron clarke's user avatar
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0 answers
50 views

Compose Lambert W and exponential in a numerically stable manner

I'm looking for a numerically stable way to evaluate $$ W_0(a \exp(b)), $$ where $W_0$ is the main branch of the Lambert W function, and $a > 0$. When $b$ is large, computing $\exp(b)$ can be ...
Alex Shtoff's user avatar
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44 views

Why everyone talks SOR and nobody JOR?

Context I am considering both Jacobi and Gauss-Seidel to be well-established. Essentially, given a relaxation parameter $\omega$, we have the following two methods (we will indicate the result of ...
Simone Licciardi's user avatar
0 votes
1 answer
21 views

Work on Nonlinear ODE by Implicit Euler Method

Suppose the problem is $dx/dt=f(x,t)$ with initial value $x_0$. By Implicit Euler Method, $x_{n+1} =x_n+hf(x_{n+1},t_{n+1})$. If $f$ is linear, we can explicitly express $x_{n+1}$. If it is nonlinear ...
Wildan B. W.'s user avatar
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Is there a numerical integration method to choose the nodes wisely according to the analytically known integrand?

When solving a numerical integration such as $$ \int_a^b f(x)\,\mathrm{d}x \approx \sum_i w_i f(x_i) $$ with analytically known $f(x)$ and definite $[a,b]$. For most quadrature methods, the nodes $\{...
BALKIN's user avatar
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3 votes
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Why can we convert a power series of operators to a function, invert the function, and then work with its series expansion?

I apologize as I am still somewhat unfamiliar with infinite series and knowing when and how we are allowed to use them, and especially with working with operators in this way. I imagine that I will be ...
Vipinx's user avatar
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optimize arithmetic expression development (edited) $E=\frac{\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}}{\frac{1}{5.6}+\frac{1}{6.7}+\frac{1}{7.8}}$. [closed]

You know that I know how to do this problem, either by transforming it into rationals or approximations, but you asked me if there is a way to do it by optimizing the calculation, that is, doing the ...
Martino's user avatar
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Stability of multistep methods that contain second derivative

Consider the numerical method $y_{n+1}=y_{n}+\frac{h}{2}[\frac{dy_{n}}{dt}+\frac{dy_{n+1}}{dt}]+\frac{h^{2}}{12}[\frac{d^{2}y_{n}}{dt^{2}}-\frac{d^{2}y_{n+1}}{dt^{2}}]$ where $\frac{d}{dt}$ is the ...
maths and chess's user avatar
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48 views

Inversion formula for discrete sine and cosine transforms

$\newcommand{\wh}[1]{{\widehat{#1}}}$ $\newcommand{\R}{{\mathbb{R}}}$ I am looking for a proof of the inversion formulas for the discrete sine and cosine transforms, i.e. a proof of the fact that ...
Bettina Kraus's user avatar
-3 votes
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24 views

How I get Max error for parametric curve for example circle (cos(x),sin(x)) [closed]

How I get Max error $||u_{exact}-u_{approximate}||_L^\infty$ for parametric curve for example, a circle (cos(x), sin(x))?
Nan33's user avatar
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3 votes
1 answer
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Question About Small Angle Approximation

I'm currently going through "An Introduction to Mechanics" by Kleppner and Kolenkow. On page 38 of the text, he glosses over the small-angle approximation of $sin(x)$ and $cos(x)$. ...
Ethan Chan's user avatar
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1 vote
2 answers
65 views

Error Caused By Taylor Series Approximation In A Mechanics Problem

I'm currently going through "An Introduction to Mechanics" by Kleppner and Kolenkow. On pages 36 - 37, Kleppner discusses how to find an approximate solution to a Physics problem. I have a ...
Ethan Chan's user avatar
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1 vote
0 answers
50 views

Most efficient quadrature formula

On the integral I am looking to obtain an approximation with machine precision, I've thought about applying a Gauss-Jacobi formula or Gauss-Legendre for: $$ \int_{0}^{2}\sqrt{2-x}dx $$ I am not sure ...
Jorge Ávila Balmaceda's user avatar
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11 views

QR algorithm, equivalence of r shifts and r times 1 shift

Is it true that in general the general $QR$ algorithm applied to $A$, a multiple shift of degree $r$ is equivalent to a sequence of $r$ single shifts of degree $1$? (Assuming that no shift is an exact ...
Tora's user avatar
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0 answers
15 views

Constructing smooth paths between points in codomain

I have a differentiable non-bijective mapping f from $x \in R^T$ to $y = f(x) \in R^T$. Say I have some positions $x_s^*$ and $x_e^*$ with corresponding mappings $y_s*$ and $y_e^*$. Questions: I ...
Luke Taylor's user avatar
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28 views

How do I create conjugate gradient method out of this Result?

How do I create conjugate gradient method out of this Result? My attempt:- If $A$ is symmetric positive definite, then solving $Ax = b$ is equivalent to minimizing the quadratic form $\varphi(x) = \...
Unknown x's user avatar
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3 votes
0 answers
94 views

How to Handle a Singularity: Seeking Solutions for a Differential Equation System

I am trying to solve the following differential equation system: $\begin{aligned} & r^2 K^{\prime \prime}=K(K-1)(K-2)+\frac{1}{4} h^2 K \\ & r^2 h^{\prime \prime}=\frac{1}{2} h K^2+\left(\...
Hendriksdf5's user avatar
1 vote
0 answers
23 views

Numerical Computation of the Gamma Function for large complex numbers

I'm looking for a method to numerically compute the Gamma function $Γ(z)$ for complex numbers of the form $$z= \frac{1}{2} + it,$$ particularly for large values of $t$. Does anyone know of any ...
Felipe Oliveira's user avatar
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0 answers
57 views

linear algebra - prove that a matrix is positive definite [duplicate]

I'm given a matrix $K$ that is symmetric and positive definite. I'm asked to prove that $K_2 - (l\times u)$ is also a positive definite matrix. $K_2$ is a submatrix of $K$ so that we get rid of the ...
Itay Cohen's user avatar
1 vote
3 answers
67 views

Numerically solve a system of two equations using fourth-order Runge-Kutta

I intend to solve the following system $$ \left\{ \begin{array}{l} \frac{du}{dt} = \frac{- \cos(v) \cos(u)bc+\sin(v)a}{cab} :=f(u, v, t) \\ \frac{dv}{dt}=\frac{(\sin(v) \cos(u)bc+ \cos(v)a) \cot(u)}{...
Wrloord's user avatar
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1 vote
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36 views

Derive Lanczos algorithm by imitating the derivation of Arnoldi iteration algorithm.

If A is Hermitian, then everything above simplifies (e.g., Hessenberg matrices turn into tridiagonal), and we get what is know in the literature separately as Lanczos iteration. My attempt:- ...
Unknown x's user avatar
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1 vote
2 answers
47 views

How to transform an integro-differential equation into weak form for FEM

I have the following 2D boundary-value problem which I would like to solve numerically using FEM software: \begin{equation} a(x,y)\nabla^2 u(x,y) + \int\int K(x,y,x',y')u(x',y') dx' dy' = f(x,y), \end{...
OAN's user avatar
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0 answers
35 views

Recommended iterative numeric solver for generalized eigenvalue problem?

I have a generalized eigenvalue problem of the form $Av=\lambda Bv$, with the following conditions: $A$ is symmetric, but not sure if it is positive-definite. $B$ is diagonal with positive entries $A$...
trisct's user avatar
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2 votes
1 answer
76 views

A quadrature rule given for this $\int_0^{\infty} e^{-t} f(t) dt = a_1 f(0) + a_2 f'(0) + a_3 f(t_3) + a_4 f(t_4) + R(f)$ using Gauss-Laguerre? [closed]

Can a Laguerre polynomial be used for this problem? How does $f'(0)$ square in? Find coefficient and nodes for the following quadrature formula. $\int_0^{\infty} e^{-t} f(t) dt = a_1 f(0) + a_2 f'(0) +...
Doru Popa's user avatar
2 votes
1 answer
26 views

Substantiate the following statement " Arnoldi iteration is nothing but orthogonal projection onto Krylov subspaces"

Substantiate the following statement " Arnoldi iteration is nothing but orthogonal projection onto Krylov subspaces" My attempt Let $K_n=[\vec{b} | A\vec{b}| ... |A^{n-1}\vec{b}]$ be a ...
Unknown x's user avatar
  • 839
5 votes
4 answers
1k views

Multiply a very large integer by a very small probability

I am calculating the probability of rejecting a one-tailed hypothesis test for inequality of two proportions $p_1$ and $p_2$ (i.e., power) using $H_o: p_2 \leq p_1$ vs. the alternative $H_a: p_2 > ...
wjktrs's user avatar
  • 153
0 votes
0 answers
21 views

Recursive approximations of inverse square law

I have a toy electrostatics simulation that consists of some number of 2D point particles that each have a real-valued "charge" $q_i$, which then exert forces on each other proportional to $...
redroid's user avatar
  • 640
0 votes
0 answers
40 views

Two SDEs that share a Brownian motion

I have the system \begin{align} dX_t & = \beta(\alpha - X_t)dt + Y_t dt + dB^1_t + dB^2_t \newline dY_t & = \beta(\alpha - Y_t)dt + X_t dt + dB^2_t + dB^3_t \end{align} Now, I am creating a ...
Pero's user avatar
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1 vote
0 answers
45 views

An alternative to Levenberg–Marquardt algorithm

When trying to solve for a (over)determined non-linear least square method: $$\underset{x}{\min}||f(x)||^2_2, f: \mathbb{R}^n \rightarrow \mathbb{R}^m, x\in \mathbb{R}^n, m\geq n$$ we use the Gauss-...
William Lin's user avatar
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0 answers
20 views

Why $h_{21}=||\textbf{v}||$ ? where $h_{21}$ is the (2,1)-entry of upper hessenberg form.

I was trying to find the $\vec{q}_2$ using the Arnoldi method described here. Why $h_{21}=||\vec{v}||$ ? My attempt:- Suppose we took arbitrary vector as $\vec{q_1}$. Then $$A[\vec{q_1}]=[\vec{q_1} \...
Unknown x's user avatar
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-1 votes
0 answers
25 views

Resources for making error analysis of semi-analytical methods for solving differential equations

I want resources explaining how to perform an error analysis of semi-analytical methods for solving differential equations with details. What I found in papers is very hard to grasp. I'm looking for ...
Mohamed Mostafa's user avatar
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0 answers
42 views

Interpolation polynomial: relative error

I know that the trucation error of an interpolating at a point $a$ can be approximated with a polynomial at a higher degree evaluated at the same point, i.e. $\left|p_{n+1}(a)-p_n(a)\right|$, where $...
albin's user avatar
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0 votes
0 answers
47 views

Numerical Inverse of an integral

Let $f: \mathbb{R} \to (0, \infty)$ be a continuous function. Define $F: \mathbb{R} \to (0, \infty)$ by: \begin{equation*} F(x) = \int_{-\infty}^x f(t)dt \end{equation*} Clearly, $F$ is strictly ...
温泽海's user avatar
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2 votes
2 answers
152 views

Solving differential equations with integrals [closed]

I have a differential equation where the rate of change of a function at a point depends on the convolution of the entire function with a gaussian function: $$\frac{d}{dt}f(t, x) = f(t, x) + \int_{-\...
James's user avatar
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0 votes
0 answers
15 views

Can changing Gradient Descent step size/learing rate from constant 1/L to Armijo or exact line search change the convergence rate?

If instead of the classical $1/L$ constant step size we have adaptive step sizes chosen with exact line search or Armijo (let's say) can this alter the Big-O complexity of the convergence rate? Here: ...
ufghd34's user avatar
  • 91
0 votes
0 answers
33 views

Fit of a (discrete) Pareto distribution for prices of goods

I have goods in a price range from 200 up to a few 10 Mio. I have price bins $$ P_1 = [200,500[, \quad P_2 = [500, 3000[, \quad P_3 = [3000, \infty[ $$ I would restrict the latter interval to ...
TomS's user avatar
  • 259
0 votes
1 answer
59 views

Iteratively finding matrix inverse from a given inverted matrix.

Let's say I need to find inverse of a bunch of matrices $A_0, A_1, ... A_n$. As the matrices are large (in my use-case), I am iteratively finding the inverse of each matrix $A_i$ using Newton matrix ...
Satya Prakash Dash's user avatar
4 votes
2 answers
233 views

Approximate solution of $x^x=(x-n)^{(x+n)}$

Interested by this problem which ask for the solution of $$f_n(x)=x^x-(x-n)^{x+n}$$ that I rewrote as $$g(x)=x\log(x)-(x+n)\log(x-n)$$ After two series expansions, I obtained, for large $n$, as a very ...
Claude Leibovici's user avatar
3 votes
1 answer
157 views

Problem with Newton's method (numerical analysis)

I am not understanding how to proceed with this exercise, which asks me to solve $f(x) = 0$ by using Newton's method. It asks me to study the convergence of the sequences $x_k$ (built with Newton's ...
Heidegger's user avatar
  • 3,351
3 votes
1 answer
94 views

rational complex minmax

As the title says, I wanted to understand the following part of a proof in Guttel (Corollary 2, page 6). $f$ is an analytic function and $r_m \in \mathcal{P}_{m-1}/q_{m-1}$ is a rational function with ...
jacopoburelli's user avatar
0 votes
0 answers
46 views

How does Self-Scaling Fast Givens QR work for (Regularized) Linear Least Squares

Basic Problem I need help to understand the Self-Scaling Fast Givens QR decomposition proposed in Anda A. A. & Park H., Self-Scaling Fast Rotations for Stiff and Equality Constrained Linear Least ...
MothNik's user avatar
5 votes
0 answers
178 views
+500

Reference for Shooting Method

Consider the following setup. We have a second order boundary value problem: $$\dfrac{d^2y}{dx^2}=F(x,y,dy/dx);\qquad y(x_0)=y_0,\quad y(x_f)=y_f.$$ A numerical approach is to almost first write as ...
JP McCarthy's user avatar
  • 7,759
2 votes
0 answers
16 views

Upper bound on sum arising in the context of differential privacy

I stumbled upon the following practical problem when working on differential privacy. Define $$ \delta(\epsilon, N, F) := \sum_{k = 0}^N \binom{N}{k} \, p^k (1 - p)^{N - k}\, D(\epsilon, k), $$ where $...
Markus Hasenöhrl's user avatar
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0 answers
34 views

Insights for Outcome Function Involving Multiple Interdependent Variables

I am working on a model involving multiple interdependent variables and systems of equations, and I am trying to gain insights into the behavior and properties of a specific outcome function. Despite ...
blizzard16's user avatar
3 votes
1 answer
115 views

Trapezoidal Rule on Infinitely Differentiable Periodic Functions

If I understand it correctly, the Euler-Maclaurin summation formula states that for a periodic and infinitely differentiable function, the error of the trapezoidal rule of the numerical integration of ...
velut luna's user avatar
  • 10.1k
2 votes
1 answer
62 views

Solve the matrix equation $A = - B A B^T + C$ without matrix inversion or vectorization

Let $B$ and $C$ be two $n \times n$ matrices, which may or may not be nicely behaved. I would like to solve for the matrix $A$ in the following equation: $$ A = -B A B^T + C $$ This equation is linear ...
Zack's user avatar
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