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

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Reduction of matrix $A$ to $B$ to find eigenvalues by Power method [duplicate]

How to reduce matrix $A$ to $B$ such that it has all eigenvalues and eigenvectors of $A$ but the dominant eigenvalue (eigenvalue with largest magnitude) is replace by $0$ ? I am using Power method to ...
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1answer
27 views

Why does the midpoint method have error $O(h^2)$ [duplicate]

In solving an ode $$ y'(t) = f(t, y(t)), \quad y(t_0) = y_0 $$ the midpoint method estimates $$y_{n+1} = y_n + hf\left(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n, y_n)\right)$$ But why is the error ...
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17 views

How to define a unimodal function?

I am doing MATLAB assignment in which I need to find the minimum of a given 3D function. It is more or less guided, but I'm confused on what one step means: a. First, write a function that will ...
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1answer
31 views

Integrate Using Gauss Laguerre Quadrature

Can we integrate a function F(x) using Gauss Laguerre Quadrature when it is not of the form: $$\int_0^{+\infty} F(x) dx$$ An example in my numerical methods book seems to imply that it can; however, ...
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17 views

Implementing specific SVD algorithms

My goal is to learn to implement the two-sided Jacobi SVD, a method of SVD for bidiagonal matrices, and a method of SVD for tridiagonal matrices. Can anyone recommend a place to learn about these, or ...
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18 views

Linearization around an equilibrium point.

I am trying to understand linearization around an equilibrium point. This is new to me. So I would like to 'see' how it works basically and see how important it is to choose a right equilibrium point ...
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32 views

contour integration and analytically

I am trying to compute the following contour integration but am quite stuck I have to evaluate it analytically, by extending it to the complex plane and solving an appropriate integral involving a ...
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1answer
28 views

How do I turn this Crank-Nicolson type equation into three vectors representing the middle, upper, and lower diagonals in a tridiagonal matrix?

I have the following homework problem: I have calculated the Crank-Nicolson equation to be Equation 1 $$ -200.05u_{m-1}^{n+1}+400.9995u_{m}^{n+1}-199.95u_{m+1}^{n+1} = ...
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21 views

How to implement QR method for bidiagonal matrices?

My goal is to take the singular value decomposition of a (not necessarily square) matrix. I have a method to do bidiagonalization of a matrix, and I can chop the bottom rows of zeros. In order to find ...
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1answer
21 views

Backward-Euler implicit integration for multiple variables

I'm a bit confused how the Backward-Euler implicit integration method works for multiple variables (i.e., systems of differential equations). For single variables, we solve the implicit equation ...
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15 views

Do quasi-Newton methods check the second-order optimality condition?

I have a practical question about quasi-Newton methods. In quasi-Newton methods, Hessian matrix is approximated. It seems to be impossible for them to check the second optimality condition. In ...
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47 views

N body ODE simulation

I have been staring at the below matlab code, which simulates planetary motion, for a while now and I could really use a fresh set of eyes. I cannot understand why it is giving straight line ...
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1answer
20 views

Observe approximation order of numerical solution of a Partial Differential Equation

When solving a Partial Differential Equation numerically, I estimated the approximation orders theoretically as follows, $$ u(x,t)= u_{h,k} + C_1 h^{p} +C_2 k^{q}, $$ where $ u_{h,k} $ is a numerical ...
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22 views

Estimates of $L^2$-orthogonal projection in $H^1$ and $H^{-1}$-norm

suppose we have a finite element space $M_k$ of $L^2(\Omega)$ and the orthogonal projector $Q_k$, defined by $(Q_k w,v)=(w,v)$ for all $w \in L^2(\Omega)$ and $v\in M_k$. My aim is prove the ...
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19 views

Ideas on how to improve stability in solving PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
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1answer
48 views

Inhomogeneous eigenvalue problem, the shooting method and constraints

In trying to solve a problem occurring in QM calculations I've encountered the following pickle, with which I hope you could help me. I am trying to solve an inhomogeneous eigenvalue differential ...
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2answers
46 views

How to compute ratios when the numbers are extremely small and numerical issue might arise?

Consider that a set of numbers that can be written in the form $b^{x}$ (for some base): $$A = \{ a_1, ..., a_n\}$$ for example maybe $\{ 2^{-2000}, 2^{-2002}, 2^{-2500}\}$. Also assume we can ...
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29 views

Numerical Methods for ODE

I have been working through Iserles books A first course in Numerical Analysis of Ordinary Differential Equations. I am trying to figure out how to prove that the error constant for an s-step BDF is ...
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1answer
52 views

Local Truncation error

We have error formulas for Gaussian Quadrature and polynomial interpolation with 10 Gaussian points, use them to derive formulas for the local truncation error, where $$P = ...
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2answers
15 views

convergence of iterative methods for linear system

Here is a theorem about convergence of iterative methods for linear system in Burden and Faires' book "Numerical Analysis" For any $x_0 \in \mathbb{R}^n$, the sequence defined by $x^k = Tx_{k-1} + c$ ...
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1answer
95 views

weak convergence lim inf sequence example

I have been stumped by the following problem , I was able to answer the first part of the problem which involved the inequality by using weak convergence. But when it came to looking for such an ...
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Set $T$ is Countably Infinite [closed]

How can it be shown that $$T = \{\,(i, j, k) \mid i, j, k \in\mathbb N\,\} $$ is countably infinite?
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27 views

Finding the roots and the rescaling of an equation

This question is taken from Hinch's book on perturbation. I need to find the rescalings $x=\delta X$ and the roots of the equation $\epsilon^2x^3+x^2+2x+\epsilon=0$ I have found to possible ...
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25 views

What is the intuition behind matrix splitting methods (Jacobi, Gauss-Seidel)?

Descent Methods, like Gradient and Conjugate Gradient ones, have a nice geometric interpretation and I really love them. What about Jacobi, Gauss-Seidel or other matrix splitting methods? I can't see ...
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16 views

Spectral radius and convergence of fixed point iteration

Let $F: \mathbb{R}^n \to \mathbb{R}^n$ be a differentiable map. -editted- Let $x^\star$ be a fixed point of $F$. Then, is it true that the fixed point iteration $x_{n+1} = F(x_n)$ converges locally ...
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82 views

Numerical series

Consider the series below that consist of 2 different formula $P_aV_a^{1.4}=P_bV_b^{1.4} $ and $P_aV_a=P_bV_b$ that keep repeating itself in the whole sequence. By assuming $P_1$ and $V_1$ both=1, ...
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Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
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35 views

Derivative for numerical models.

I am working in Mechanical engineering and Computer vision, in which I use a matlab code (or codes) to represent a specific system or process. Of course such model has an input , an implimented ...
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13 views

LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
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1answer
34 views

Show that S is a cubic spline (natural or clamped)

Please see question. I believe the answer should be: $S_0(2)=\frac12(x^3-3x+2)=2$ $S'_0(2)=\frac12(3x^2-3)=\frac{9}{2}$ $S''_0(2)=\frac12(6x)=6$ $S_1(2)=\frac12(x^3-12x^2+45x-46)=2$ ...
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Generating volume constrained splines

Suppose I have a set of points in $\mathbb{R}^3$, $\{\vec{r}_1,\vec{r}_2, ...,\vec{r}_n\}$, suppose between points $\vec{r}_i$ and $\vec{r}_{i+1}$ there is an associated volume $V_i$. I want to ...
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70 views

Find the smallest $N$ such that $\sum_{k=1}^N\frac{1}{p_k}>\pi$. (The $p_k$'s are the prime numbers.)

How to solve the following problem? Let $\{p_k\}_{k=1}^\infty$ be the set of primes (in increasing order). What is the smallest integer $N$ such that $$\sum_{k=1}^N \frac{1}{p_k}>\pi?$$ We ...
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What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
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Central Difference Method

Solve the following using the central difference method: $y(x)= y'+ y + 2x$ where $0 < x < 4$ with $n=4$ subintervals (thus $h=1$). Given that $y(0)=0$ and $y(3)=1$, find $y(1)$. Really ...
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quadrature schemes for integral equations fredholm

I am trying to solve this fredholm integral equation using numerical schemes. Can anyone please suggest a quadrature rule for this. The singularity makes it difficult here. $f(x) = \int_0^1 ...
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28 views

Problem with Broyden update: Divide by a matrix?

I am implementing a maximum likelihood method (the EM algorithm) for which I'm using Broyden's method at each iteration. Here is the formula: $\Delta A = \frac{(\Delta \theta - A ...
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convergence of a numerical method

given a function $f:\mathbb R\to\mathbb R$ in of class $C^3$. We suppose that there exists $s\in \mathbb R$ such that $f(s)=0$ and $f'(s)\neq 0$. Let $\beta$ be a real number s.t. $\beta \neq 0$. We ...
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1answer
49 views

Newton-Raphson Method used in a real engineering/physical/mathematical situation

I've been using the Newton-Raphson Method in my Numerical Methods course for a while now, blindly solving non-linear equations and systems of equations . This makes me somehow lose motivation, as I ...
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1answer
64 views

Find desired root (based on certain constraints) while using Newton-Raphson method

My function is a vector of dimension 6, and on using Newton Raphson method, the solution usually converges to the nearest root. However, I know that my function has multiple roots (it's an Inverse ...
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49 views

Did I do something wrong solving this PDE in MATLAB?

I have the following PDE problem on a practice exam: I have completed the problem using MATLAB to the best of my ability. Here is the code I used ...
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26 views

Can trigonometric functions for double precision be implemented in terms of those for single precision?

In some program environments like GLSL there is support for single and double precision numbers for arithmetic and square roots computation, but only single precision trigonometric functions are ...
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How to define boundary conditions for a sphere to run reaction-diffusion equations on its surface?

I'm in a Biology lab, and we managed to simulate reaction-diffusion equations on a torus using periodic boundary conditions for a 2D matrix. We want to try doing the same on a sphere, but I'm a ...
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23 views

prove this using lagrange and newton divided difference error!

suppose f(x) is polynomial with degree of three.prove $f[{x}_{0},{x}_{1},{x}_{2}] = \frac{1}{2}{f}^{(2)}(\frac{{x}_{0}+{x}_{1}+{x}_{2}}{3})$ and ${x}_{0},{x}_{1},{x}_{2}$ are distinct point. I ...
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A question about the condition of quadrature formula

I am reading through my numerical mathematics script and I am currently in the chapter 4 (see listing) computer arithmetic direct solution of linear systems of equations polynomial interpolation ...
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55 views

Finding $ \max_{x \in [2,4]} \left| 2 x \cos(2 x) - (x - 2)^{2} \right| $.

This is a problem taken from Burden’s and Faires’ Numerical Analysis. Define $ f: \Bbb{R} \to \Bbb{R} $ by $$ \forall x \in \Bbb{R}: \quad f(x) \stackrel{\text{df}}{=} 2 x \cos(2 x) - (x - 2)^{2}. $$ ...
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Distribution of SDE numerically from Fokker-Planck.

I'm aware of some numerical methods related to SDEs such as Euler-Maruyama, Milstein etc. However, couldn't one also simulate the equivalent Fokker-Planck equation via finite element methods? This ...
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26 views

Is there a meaningful distinction between “direct” and “iterative” methods for solving equations?

I'll motivate this question with an example. The Abel-Ruffini theorem states that there is no general "formula" for the roots of polynomials of degree greater than 4. (Specifically it states that ...
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1answer
73 views

Name of function $(1+x)^n-1$

Is there any name for this formula $$(1+x)^n-1$$ When working with floating point numbers this can be calculated with much better precision for very small $|x|<1$ values using Taylor series ...
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24 views

How to find the order of accuracy of this implicit RK method (using Taylor series)?

I want to get the order of accuracy (local truncation error - LTE) of this implicit 2-step method. The first step is Backward Euler to determine an approximation to the value at the midpoint in time, ...
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46 views

Runge-Kutta force at each time-step

Consider that I am solving a second order ODE using RK2/RK4. The ODE represents simple equations of motion: Equations of motion I am trying to solve: \begin{align} \frac{dx}{dt} &= v \\[.3em] ...