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

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Backwards Stability of systems

Let $A$ be a nonsingular matrix, let $x_{k+1}$ be an approximation to the solution of $Ax=b$, and let $r^{k+1}=b-Ax^{k+1}$. Show that $x^{k+1}$ is $\epsilon$-backward stable approximate of ...
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23 views

Numerical scheme for system of PDEs

I'm trying to solve the following coupled PDE system for my master thesis: \begin{align} \kappa_0\frac{\partial p}{\partial t}&=- \nabla \cdot v \\ \rho_0\frac{\partial v }{\partial t} &= ...
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46 views

Convergence Newton method convex function

Is the Newton method convergent for convex differentiable functions and system of convex differentiable functions? Assuming you don't start with a stationary point and there exists a root. If not are ...
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22 views

Looking for polynomial to represent approximate 2D matrix.

I am looking for a polynomial that similars Legender polynomial(a set of orthogonal polynomial basis function. Could you suggest to me some polynomial? Because my goal is that I want to approximate 2D ...
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25 views

Even least squares approximation

Can anyone help me with this problem or give me a tip on where to start. Let's consider $\theta_n$ a class of approximations with the following properties: all functions $\varphi \in ...
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40 views

How to solve this ODE numerically?

I have a question about how to solve this ODE numerically: $$\frac{C}{4}y'^2+\frac{C}{4}y''y+(0.098)^2y''y'''=0$$ where $C$ is a constant and the initial conditions are $y(0)=y''(0)=0$ and $y'(0)=1$. ...
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14 views

sparse, complex, unymetric test-matrix

Can anybody recommend me a sparse, complex, unsymmetric test-matrix (maybe from MartixMarket) which is solvable with a transpose-free QMR without preconditioning in under 1000 iterations?
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26 views

Order of convergence of nonlinear iterative solver

I'm given a sequence $x_n \rightarrow \alpha$ of a nonlinear solver such that $$\lim_{n\rightarrow\infty}\frac{x_{n+1}-\alpha}{x_n-\alpha}=c$$ converges linearly (i.e. $c\in(0,1)$). Now, I need to ...
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17 views

How to calculate u1-modified parametric iteration method

Please refer to paper "A modified parametric iteration method for solving nonlinear second order BVPs, link: http://www.scielo.br/pdf/cam/v30n3/a02v30n3. In example $5.1$, I am unable to understand ...
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25 views

Region of absolute stability

We have the problem $$\left\{\begin{matrix} y'=\lambda y &, t \in [0,+\infty), \lambda \in \mathbb{C}, Re(\lambda)<0 \\ y(0)=1 & \end{matrix}\right.$$ Applying the Backward Euler method ...
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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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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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33 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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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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24 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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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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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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10 views

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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24 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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17 views

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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11 views

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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29 views

Shooting method with non-robin (or derivative) boundary conditions

I am trying to solve a problem in which I have to find all the values of $\lambda$ for which the boundary value problem has just one solution for each $a,b\in\mathbb{R}$. The problem is the following: ...
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27 views

Convergence of the Midpoint (Leapfrog) method when applied to $u'(t)=\lambda u(t)$?

So, I am trying to solve this question: where example 7.7 can be found here: http://i.stack.imgur.com/PVCIC.png My approach: Forward Euler (FE) method is given by: ...
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27 views

Newton-Raphson for Discontinuous Spring Moment Balance

I am trying to solve the following problem where there should exist an equilibrium between spring forces and moment applied to a cylinder: To do so I am solving the equality: $ M_a - \sum_{i=0}^n ...
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27 views

How do I make a Maclaurin series expansion faster?

Suppose I want to approximate to e using the Maclaurin series. In this case, increased accuracy comes with at trade off of computation time. How do I make the Maclaurin series expand faster/ using a ...
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42 views

Basic Corrected trapezoidal Rule for a Cubic Hermite Polynomial

The basic trapezoidal rule for approximating $$ I_f=∫^b_af(x)dx $$ is based on linear interpolation of f at x0=a and x1=b=a+h. Consider now a a cubic Hermite polynomial, interpolating both f and its ...
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9 views

Accurate numerical integration for “data times an analytical function”

The Question is as follows: I have an algorithm/data that provides me the value of a function $f(x,y,z)$ on the points of a grid. On the other hand I have an analytical function ...
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38 views

How do I write this equation as a tridiagonal matrix to write the $n+1$ implicit formula?

I am doing a homework problem for my Applied Numerical Methods class, and I've worked the problem up to this point: $$ \large \frac{u_m^{n+1} - u_m^n}{k}=\frac{u_{m+1}^{n+1} - 2u_{m}^{n+1} + ...
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15 views

Simpson's rule and Hermite interpolant

For a uniform grid, $$x_n = -1 + nh$$ where $h = \frac{2}{N}$, I need to show that Simpson's rule is an $\mathcal{O}(h^5)$ integration rule. So far, I know to let $p(x)$ be the Hermite polynomial from ...
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20 views

Local Truncation error of Gaussian Quadrature

We have error estimate formula for Gaussian quadrature is: $$ \frac{(b-a)^{2n+1}(n!)^4}{(2n+1)[(2n)!]^3}f^{(2n)}(\xi) \; \; \; a < \xi < b$$ Suppose that we have 10 Gaussian points, so how can ...
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19 views

Solving System of Boundary Value problem

The boundary value problem: $$y'' + Q(t)y = f(t)$$ satisfying $$Ay(a) +By(b) = g$$ where A, B and Q are the matrices of order n. After calculation, we can get the form of solution will be $$y(x) = ...
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14 views

Taking a Derivative after a linear transformation

Maybe I'm overthinking this since I know d(L*f)/dx = L * df/dx... Anyway, if you know df(x,y,z,w)/dx of a function f at a (4d) point p, how could you find d(q.z/q.w)/dx if you know that q = Ap (where ...
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26 views

Computing integrals in order to find an approximation function

For a project in scientific computing I am trying to find an approximation of an unknown function $f(x)$. Given: data points $(x, f(x))$ A basis with which we can approximate $f(x)$ consists ...
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35 views

Numerical Triple integral with three other parameters in R

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
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53 views

Runge kutta 4th order computation of force solving 2nd order ode

\begin{equation} \frac{dx}{dt} = v \end{equation} \begin{equation} m .\frac{dv}{dt}= F_{p }(x)+F_{g}(v,x) \end{equation} Conside I am solving the above two equations using runge kutta 4th order ...
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21 views

How to make a 9-point two-dimensional stencil for a elliptic operator?

I want use a finite difference schem to discretizate the elliptic operator: $$ \nabla \cdot \left( k(x, y) \nabla p\right), $$ where $k(x, y)$ a positive scalar function and $p$ is the unknow. We can ...
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9 views

How to optimize this types of problems?

Given that $min [ t_{f} - t_{0} ]$ such that $x(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $y(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $z(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $x(t_{f}) = ...
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16 views

Help Required in eigenvectors for sparse matrix?

I have a large sparse matrix A(~400000,~400000) . If I randomly remove few rows from the matrix will there be considerable change in the eigenvalues and the eigenvector's compared to eigenvector's of ...
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12 views

What is the difference between newton interpolation polynomial and interpolation polynomial with Neville scheme?

I am trying to find the interpolation polynomial by using Neville scheme. It looks like divided difference . What is the difference between newton interpolation polynomial and interpolation polynomial ...
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36 views

Finite Difference Method for Heat Equation with Neumann Boundary

I have read the book of Morton and Mayers. In chapter6, it said that the explicit finite difference scheme of a heat equation, $\frac{U^{n+1}_j-U^{n}_{j}}{\Delta ...
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33 views

Selecting nodes for spline interpolation

Is there a general method to determine the best sample points for spline interpolation (whether for piecewise linear or piecewise cubic Hermite) given $x$, $f(x)$, and estimating $f^\prime(x)$? Does ...
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40 views

Finding a function using first derivative

I have some data about just first derivative of a function. Also, I know a point of this function(e.g. (x1,y1)). How can I obtain the function? All my date are numerical. dev f(x)=[ 580.00 , 479.7308 ...
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29 views

What exactly are divided Differences?

I'm reading a numerical analysis textbook, and we have a definition of something called "divided differences". I have read the notation and know what the definition is, but what exactly is it ...
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19 views

How to solve constrained ode problem

Currently I'm facing question in which let's say I have 3 coupled eqn. \begin{align} x = f(x', z', y', t) \\ y = f(x', y', z', t) \\ z = f(x', y', z', t) \\ \end{align} There is initial ...
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48 views

How would you integrate numerically this function?

How do you understand this integral ignoring the rect function? dx and dz are the pixel size of the numerical grid.
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40 views

What is he easiest way to approximate γ as a decimal number?

What is the easiest way to give the numerical value of the Euler-Mascheroni constant? The mathmetical way to give that value? Thanks lot!
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25 views

How to transform $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^3 u}{\partial x^3}$ into a system of first order PDE's and finite difference matrix

So I have this equation: $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^3 u}{\partial x^3}$ and I need to transform it into a system of first order PDE's. I was thinking like this: ...
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13 views

Finding an analytic form of a function that satisfies asymptotic conditions

I have a family of functions that I obtain numerically. They depend on $x$ and parametrically also upon a certain parameter $L$. I would like to find an analytical form for this family of functions so ...
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16 views

Approximating an integral with a change of integral

(I have previously found out $x_1 = -\frac{1}{\sqrt{3}}$ and $x_2 = \frac{1}{\sqrt{3}}$ ) Approximate an integral using the 2-point rule, with an appropriate change of integral, to approximate ...
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18 views

(xλ)→(Ax−λxxxT−1) write down Newton's method for this equation

this question is taken from Rainer Kress (numrical Analysis). i coud not translate this question into newton's method form. Because there is matrix and vector. it is hard to take derivation of this ...