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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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Modified Euler Method: region of absolute stability

I am having trouble finding the region of absolute stability for modified Euler method: \begin{align} w^*_{i+1}&=w_i+hf(t_i,w_i) \\ w_{i+1}&=w_i+\frac{h}2\left[f(t_i,w_i)+f(t_{i+1},w^*_{i+1})\...
5 votes
1 answer
134 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 ...
0 votes
0 answers
22 views

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 $\{...
0 votes
0 answers
14 views

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 ...
-3 votes
0 answers
6 views

How to derive Adam's Moulton 3rd step using shifted legendre polynomial [closed]

Solving Adam's Moulton 3rd step using shifted legendre polynomial using maple
2 votes
0 answers
33 views

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 ...
-2 votes
0 answers
23 views

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 ...
0 votes
0 answers
44 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 ...
1 vote
2 answers
42 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{...
-3 votes
0 answers
22 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))?
4 votes
2 answers
230 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 ...
1 vote
2 answers
61 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 ...
3 votes
1 answer
91 views

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 $cos(x)$. ...
1 vote
3 answers
64 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)}{...
1 vote
0 answers
46 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 ...
6 votes
6 answers
2k views

Calculating the nth super-root when n is greater than 2?

Tetration (literally "4th operator iteration") is iterated exponentiation, much like how exponents are iterated multiplying. For example, $2$^^$3$ is the same as $2^{2^2}$, which is $2^4$ or ...
0 votes
0 answers
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 ...
0 votes
2 answers
41 views

How to approach this singular perturbation problem?

I have set myself the following singular perturbation problem: For small values of $\varepsilon > 0$ find the two roots closest to $x=0$ for the equation. $${x^4} - \,\,{x^2} + \,\,\varepsilon (x +...
0 votes
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 ...
1 vote
1 answer
114 views

Is this integral in its most simplified form?

The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$ it can ...
3 votes
0 answers
87 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(\...
0 votes
0 answers
25 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) = \...
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 ...
0 votes
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 ...
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-...
0 votes
1 answer
1k views

Is there General Formula for an nth Order Central Finite Difference

I am searching for a general formula for directly calculating the second, fourth, and sixth derivatives from time series data. Wikipedia has a formula for finding an $n$th order central finite ...
1 vote
2 answers
81 views

Finding the coefficients in fractional order polynomials with maple

I want to find coefficients in fractional order polynomials with Maple software but I'm having trouble. For example, $$ coeff(x+x^{1/2},x,1/2) $$ or $$ coeff(2x^{3/2},x,3/2) $$ Please guide me, ...
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 ...
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 > ...
1 vote
0 answers
34 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:- ...
0 votes
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 $A$...
2 votes
1 answer
75 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) +...
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 ...
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 $...
0 votes
0 answers
39 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 ...
0 votes
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} \...
16 votes
3 answers
8k views

Why does Fixed Point Iteration work?

I have searched online for an answer, but everyone gave the method, and no one explained why is it working. I'll first write what I do understand. Let $f(x)$ be a continuous function at $[a,b]$. ...
1 vote
1 answer
4k views

Numerical Integration: The degree of accuracy of a quadrature

I am in a first semester numerical analysis course and we are going over numerical integration and more specifically quadrature forms. So far we have gone over standard quadrature as well as Gaussian ...
5 votes
0 answers
536 views

Evaluating matrix-continued fractions?

I have a matrix-valued continued fraction defined in the following way: $\alpha_n$ and $\beta_n$ are matrices, and I am interested in the quantity $A_1$, where all the $A_n$, $n = 1, 2, \dots$ are ...
-1 votes
0 answers
24 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 ...
0 votes
0 answers
40 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 $...
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 ...
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 ...
2 votes
2 answers
150 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_{-\...
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 ...
0 votes
0 answers
13 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: ...
0 votes
2 answers
1k views

For the given function, find each fixed point and decide whether fixed point iteration is locally convergent to it

A theorem in my book states: Let $g$ be a function and $r$ a number fixed by the function (i.e. $g(r) = r$). Assume $g$ is continuously differentiable, $g(r) = r$ and $|g'(r)| < 1$, then the ...
5 votes
2 answers
11k views

Computing Hessian in Python using finite differences

I am computing the Hessian of a scalar field, and tried using numdifftools. This seems to work, but was quite slow so I wrote my own approach using finite differences. Here is my code for the Hessian:...
1 vote
2 answers
73 views

How exactly can we write an incomplete elliptic integral of the first kind as a sum of real and imaginary parts?

All I want to do is numerically map the upper-half plane $\mathbb H:=\{z\in\mathbb C:\Im(z)>0\}$ to the unit square $[0,1)^2$. How this can be done is described in the Wikipedia article of the ...
0 votes
0 answers
45 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 ...

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