Tagged Questions

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

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0
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0answers
24 views

Trying to solve this system with Gauss-Seidel

I'm trying to solve this system: $$ \begin{cases} {-x}+5y+3z=2\\ 7x+4y+2z=7\\ 3x-y+5z=5 \end{cases} $$ I have to use Gauss-Seidel, but no matter how I try the system does not converge. So my question ...
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0answers
10 views

Finite element solution

I need to obtain the solution of the following finite element formulation: "given $A_h^{n+1}$ and $\hat{Q}_h^{n+1}$, find $\tilde{Q}_h^{n+1} \in V_h^0$ such that: $\Bigg( ...
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1answer
23 views

Numerically solve integral with a function as variable of integration

I want to use a function as variable of integration for example in evaluating the integral: $\int_0^1 e^{\cos x}f(\sin x)d\cos x$ in which $f(x)$ is an arbitrary function.
1
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1answer
12 views

estimation of condition number for column equilibration

I have trouble with the following problem: Let $A$ be an invertible square matrix. Let $D$ be the diagonal matrix with entries $d_j=\dfrac{||A||_1}{\sum_i |a_{i,j}|}$. Show that $||D||^{-1}_\infty ...
4
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3answers
251 views

Finding the all roots of a polynomial by using Newton-Raphson method.

Is there a general formulation for finding all roots of a polynomial, especially the complex ones, by using the Newton-Raphson Method?
1
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1answer
28 views

Solving trancendental with variable argument. $20 = ax\sin(ax)$

Approaching transcendental equations is in general new to me. My experience with numerical methods is limited, and this equation seems to require such a method. But there's a catch - it contains an ...
2
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0answers
14 views

Can the Lanczos algorithm converge very fast by choosing initial guess smartly?

Suppose I have the two lowest eigenvectors $v_1$, $v_2$ of a matrix $M$. If slightly change $M$ to $M'$. Can I use $v_1$ or $v_2$ as an initial guess for $M'$? If so, which one should be used, $v_1$ ...
0
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1answer
26 views

Which numerical method is more accurate than Range-Kutta. [on hold]

I need a numerical method is more accurate than Range_Kutta to solve the differential equation especially for third-order and more.
2
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1answer
30 views

Spectral convergence of coefficients of a Fourier series

I have seen claims that if a smooth function $f(x)$ is represented by its Fourier series, $f(x)=\sum_{n=-\infty}^\infty a_ne^{i(nt)}$, then as $|n|\rightarrow\infty$, then $|a_n|\rightarrow 0$ ...
0
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0answers
17 views

Help with SVD computation

Suppose B is $n\times n$ nosingular square matrix and the QR iteration with shift for $B^TB$ and $BB^T$ are given by $$B^TB-\rho I=QR$$ and $$BB^T-\rho I=PS$$ where Q,P: orthogonal and R,S: upper ...
0
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1answer
17 views

Determining unknown coefficients of cubic splines

The problem : Find $c$ in the following cubic spline. $S \scriptstyle{1}$$(x)$ = $\large4 - \large\frac{11}{4}x + \large\frac{3}{4}x^3$, on $[0,1]$ $S \scriptstyle{2}$$(x)$ = $\large2 - ...
1
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1answer
38 views

Floating point arithmetic: $(x-2)^9$

This is taken from Trefethen and Bau, 13.3. Why is there a difference in accuracy between evaluating near 2 the expression $(x-2)^9$ and this expression: $$x^9 - 18x^8 + 144x^7 -672x^6 + 2016x^5 - ...
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0answers
23 views

Composite Trapezoidal Rule for $\int_0^{\pi} \sin x\, dx$

Use the Composite Trapezoidal rule to find the approximation to $\int_0^\pi \sin x\,dx$ with $m = 1, 2, 4, 8, 16.$ Progress The Comp-Trap rule states: $$\int_a^b f(x)\,dx\approx ...
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0answers
16 views

Is there any direct method for Lagrange multiplier based domain decomposed problem?

In elastic problem, we often solve K * u = f, where K is the stiffness matrix, f the external force vector and u the displacement vector. I'm trying decompose the mesh to domains, using Lagrange ...
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0answers
23 views

Trapezoidal Method

Use Taylor Expansion to show that the implicit Trapezoidal Method $Y_{k+1} = Y_k+ ∆t/2 (f(t_{k+1}, Y_{k+1})+f(t_k, Y_k))$ has a local truncation error of order $∆t^2$. My understanding: The ...
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0answers
22 views

Does it really matter that we are using the Taylor polynomial and remainder?

Assuming that the quadrature rule $I_n$ integrates all polynomials of degree less than or equal to N exactly: $I_n(p)$=$I(p)$ for all p $\epsilon$ $P_N$. Using this it could be proved that for any ...
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0answers
23 views

In interpolation, why does my choice of $x_0…x_n$ matter?

This is more of a theoretical question regarding my choice of x's for my interpolation. I'm wondering if someone can explain to me why when I choose different x's for approximating a value at a point, ...
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0answers
15 views

How to solve this matrix equation with Hadamard product?

I am having trouble in solving $X$ in the following equation: $AX+B\otimes X=C$ where the first product is the usual matrix product and the second is a element-wise multiplication (aka Hadamard ...
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0answers
38 views

How can I solve this PDE?

$\dfrac{\partial \hat{Q}}{\partial t} - \dfrac{Am}{\rho} \dfrac{\partial ^3Q}{\partial t \partial z^2} = 0$ I really do not know which method could I use to solve it!
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0answers
19 views

Determine for what scalar a function has multiple intersections

I am unfamiliar with numerical analysis, and would like some help figuring out how to find when two functions intersect on multiple points. In particular, I would like to determine for what $\lambda ...
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2answers
23 views

Cubic convergence of itearative method

thank you for your time at first! It's my homework, so I don't expect answer with result, only some hint. With given iteration method $$x_{n+1} = \frac{x_n(x_n^2 + 3U)}{3x_n^2 + U} $$ show cubic ...
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0answers
22 views

Finding the inverse of an integral

I'm looking for a computational approach here, since I don't think there is a closed-form solution. I have the following: $$ s(x) = \rho + \int_{\rho}^{x} \sqrt{ 1 + (\alpha \cos t - k)^2 } \, dt $$ ...
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0answers
13 views

Numerical Stability

In my numerical analysis class we have been working on approximating functions with Maclaurin Series. I am sort of confused by the definition of what makes an algorithm numerically stable. I ...
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2answers
26 views

Numerical problem

The value of 1001 to the power 3?. Any trick for quick answer?
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1answer
25 views

Taylor Expansion on Newton's Method

I get that the Newton's Method is can be derive from the first Taylor Expansion, which is $$ f(r) = f(x_{0})+f'(x_{0})(r - x_{0}) + \frac{f''(\xi)}{2}(r - x_{0})^{2} $$ with $r$ is the root of ...
1
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1answer
18 views

RK4 stability problem.

I'm trying to solve the following problem using RK4: $$y'=y^{\frac{1}{3}}, \quad y(0)=0, \quad t\in[0,6].$$ The exact solution is $y(t)=\sqrt{\left( \frac{2t}{3} \right)^{3}}$. I wrote the following ...
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0answers
26 views

Newton's Method Convergence [duplicate]

Let r be the zero of multiplicity 2 of the polynomial p(x), how do I prove that $x_{n}$ converges quadratically to r? I only know the basic Newton's Method which is $x_{n+1} = x_{n} - ...
1
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0answers
14 views

Scheme for mixed partial derivative from Taylor

I need to deduce a scheme(finite difference) for the partial derivative: $$\frac{\partial^3 u}{\partial t \partial x \partial x} $$ How can I deduct it from Taylor polynomial? Thanks for your help
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0answers
26 views

Quantum Mechanics in Electric Field

I asked this problem in Physics SE but I did not get any useful answers except one. I believe asking this question here would be more beneficial owing to the Mathematical nature of the problem. I am ...
0
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0answers
22 views

WKB approximation for multiple turning points

I'm working on a numerical program which approximates the eigenvalues of a Schrödinger equation by making use of the WKB approximation formulas. For example, if the Schrödinger equation is $$ y''(x) = ...
0
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0answers
18 views

How to find how many steps are required to obtain the root of f(x)

I'm curious about how to find how many steps are required to get the $10^{-n}$ accuracy of f(x) in Newton's Method, I believe we can implement this method in computer way, and this is my pseudo-code ...
0
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0answers
8 views

finite subspace of $L_0^2 $- Stokes Problem

Consider $L^2_0(\Omega)$ be the $L^2(\Omega)$ functionspace with mean-value $\int_\Omega u \, dx$. For numerical reasons I search a finite subspace of $L^2_0$ on a triangulation, but I don't know how ...
3
votes
1answer
60 views

Proving that $\lim\limits_{n \to \infty} \frac{E_{n+1}}{E_n}=2^{-2/3}$

$$\def\ut#1{\underline{\text{#1}}}\def\vec#1{\mathbf{#1}} \def \d{\mathrm{d}} \def \p{\partial } \def \[{\left[} \def \]{\right]} \def \({\left(} \def \){\right)} \def \n{\boldsymbol{ \nabla}} ...
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0answers
18 views

Explanation of the Leibniz formula

I am reading the book Solving Ordinary Differential Equations I - Nonstiff Problems (1987) by Hairer et al. My question is from Section II, chapter 2 (Order conditions for RK methods), equation 2.4. ...
0
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1answer
28 views

Time and work.. Problem [closed]

A, B and C together can do a piece of work in 40 days. After working with B and C for 16 days, A leaves and then B and C Complete the remaining work in 40 days more. A alone could do the work in?
0
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1answer
46 views

Solve Bratu problem using Python

I am going crazy trying to solve the Bratu problem using Python: $$y''(x)+ e^{y(x)} = 0, \quad \lambda = 1, \quad x \in(0,1),$$ $$y(0) = y(1) = 0$$ I have to solve this using the tridiagonal ...
0
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2answers
47 views

What's the general procedure to analyse a function to find roots using numerical methods and sketch the graph?

When I'm facing functions for which no formula exist to calculate the roots directly, what can I do with calculus to analyse it so that I can obtain information about the function's behavior? Suppose ...
1
vote
1answer
41 views

Trying to re-write Simpson's Rule: mistake?

Pre-Question (edited): Thanks Arthur Orignal Problem: The standard form of Simpson's Rule requires an even value of n so that you can make a series of parabolas Parabola 1 has area ...
0
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0answers
39 views

Are there ever cases where it's easy to get coefficients for the series representation for an integrand, but hard to approximate the integral?

WHY I'M ASKING THIS I'm working on a faster way to approximate integrals of series. So I'd like to know if this could be useful. THE QUESTION If we suppose that we can get a formula for the ...
3
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0answers
26 views

Roots of characteristic polynomial have negative real parts implies positive coefficients of the polynomial

Can you help me prove that if all the eigenvalues $\lambda_i$ of a square n-dimensional matrix $A$, have a strictly negative real part then prove that all the coefficients $a_j$ of the characteristic ...
0
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0answers
33 views

solve complicated system of non-linear equations numerically

I have two algebraic equations I am trying to solve in MAPLE. They are: $14\,{a}^{26}{b}^{2}-91\,{a}^{24}{b}^{4}-364\,{a}^{22}{b}^{6}-1001\,{a} ...
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0answers
32 views

question in Numerical analysis

please guide me how to start and I will continue the another steps
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0answers
9 views

Numerical Overflow in Dirichlet Boundary Value Problem On High Dimensional State

I am using multigrid methods to solve a quasilinear parabolic pde with Dirichlet boundaries. The problem is too long to reproduce here, but my question is more practical than theoretical: The state ...
0
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2answers
39 views

When $a\ll b$, how to approximate $f = \int_0^a \sqrt{b^2+x^2}/\sqrt{a^2-x^2} \, \, dx$?

Suppose $a\ll b$. How do I then approximate $$\int_0^a \frac{\sqrt{b^2+x^2}}{\sqrt{a^2-x^2}}dx$$ ? I think that maybe Taylor approximation may help, but I am not sure how to proceed. My physics ...
0
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1answer
15 views

Gauss Curvature…Product of Minimum and maximum values

The function g(ϑ ) = cos2 (ϑ ) fxx (x0 , y0 ) + 2 cos(ϑ )sin(ϑ ) fxy (x0 , y0 ) + sin2 (ϑ ) fyy (x0 , y0 ) represents the Gauss curvature of the surface f (x, y) at the critical point (x0 , y0 ) in ...
2
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1answer
37 views

Find the lowest value of a parameter for which two functions intersect

I am attempting to an equation to determine the lowest value of $\lambda$ for which $f(x) = \lambda \sin ( \pi x)$ and $y = x$ intersect outside of 0 on the interval $[0,1]$ for some numerical ...
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0answers
25 views

A Free Boundary Problem

Is there any special way to solve such a problem. Any idea would be appreciated. At least does anybody know which method is useful to solve this problem numerically? Is it even solvable numerically? ...
0
votes
1answer
21 views

intersection of an ellipsoid and cylindrical plane.

I need to understand if an ellipsoid and a cylindrical arc intersect, what will be the general equation of the cutted ellipse? How can I solve for that equation? I know in 3D, the equation of an ...
0
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1answer
23 views

Compute the following norm derivatives

I was wondering if anyone can explain me how to compute the derivatives of the following norms: $\frac{d}{ds}||x+sp||^2_q$ for $x,p\in\mathbb{R^n}$ and $1<q<\infty$ $\bigtriangledown ...
1
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0answers
40 views

Trajectory With Air Resistance

For a video game, I am trying to calculate the angle needed for a projectile to hit coordinates x,y (both non-zero) with air resistance, i used equations from this site, and derived a function of y ...