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

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Error of linear Interpolation with intermediate points obtained from an explicit RKM

For the initial value problem $y'(t)=f(t,y(t))$ $(f\in C^\infty(\mathbb{R^2}))$ with $t\in [a,b]$ and $y(a)=y_0$ let $u_k, k=0,...,n$ be the approximation of $y(t_k)$ obtained from an explicit ...
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54 views

Tangent vector for a curve defined by a discrete set of points

I have a curve defined by a discrete set of points (x,y). How can I approximate the tangent vector at a point for such a curve?
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19 views

V-shape of error function of numerical derivative vs. analytical derivative

I'm given the following function: $$f(x) = \frac{x^2}{\sin(x)}$$ and I'm supposed to derive the derivative numerically at the point $x=1$ with the following central difference quotient: ...
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17 views

SOR and Conjugate Gradient Method

Let $H_n=[H_{ij}]\in\mathbb{R}^{n\times n}$ be Hilbert Matrix, define $h_{ij}=\frac{1}{i+j-1}$ and $x=\left(1\quad 1\quad\cdots\quad 1\right)\in\mathbb{R}^{n\times n}$ such that $b_n=H_nx$. Use SOR ...
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38 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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19 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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27 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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27 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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29 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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32 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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22 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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22 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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37 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) = ...
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27 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 ...
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24 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. ...
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43 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 ...
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37 views

question in Numerical analysis

please guide me how to start and I will continue the another steps
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84 views

Solving Duffing equation by Matlab ode23

How can I use Matlab to solve numerically this duffing equation with known $\kappa, \Gamma, \omega$..thanks.. $$x'' +\kappa x' +x -x^3 =\Gamma \cos\omega t$$ I have only few knowledge of Matlab..
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18 views

Multiplication of polynomials in Chebyshev basis

For polynomials in the monomial basis like $p_n(x) = \sum_{k=0}^N a_k x^k $, the product of 2 polynomials is can be either found though the convolution of the 2 corresponding polynomial vectors or ...
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36 views

Crank-Nicolson method for solving nonlinear parabolic PDEs

Is the Crank-Nicolson method appropriate for solving a system of nonlinear parabolic PDEs like $\partial u/\partial t - a\Delta u + u^4 = 0$ ? I tried to apply this method for solving such system but ...
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33 views

Polynomial division/deflation with FFT

There is a need to divide a polynomial $p(x)$ by polynomial $q(x)$, whereas it is known that the remainder will be zero (i.e. the question is about polynomial deflation). A known method is to use the ...
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44 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 ...
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15 views

Integration by parts applied to weak form of boundary value proble

In my finite element textbook the proof for strong and weak form equivalence is determined as such: $$\int_0^1w_{,x}u_{,x}dx = \int_0^1wfdx + w(0)h$$ Integrating by parts and making use of the fact ...
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28 views

Machine Floating Point Theorem

Completely stuck on this floating point question. Let $x \in \mathbb{R}$ have the following floating point representation: $$ x = (-1)^s[0.a_1a_2\dots a_ta_{t+1}\dots]\cdot \beta^e $$ [Where $\beta$ ...
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35 views

Numerical method of lines for solving PDEs

Could you please advise some literature about the numerical method of lines (MOL) for parabolic PDEs? It is a method of solving PDEs with discretizing only by space but not by time. A system of ODEs ...
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30 views

Uniqueness of a differential equation

Let $I_o=[t_0,t_0+T]\subset\mathbb R$, where $T>0$, $f\in C^0(I_0\times\mathbb R;\mathbb R)$ and satisfying Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
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19 views

Minmax approximation

Let $f(x)=a_nx^n+....+a_1x+a_0, a_n\neq0.$Find the minmax approximation to $f(x)$ on $[-1,1] $by a polynomial of degree$\leq n-1 ,$and also find the error $\rho_{n-1}(f).$ This problem is from one of ...
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22 views

Calculating availability of a system

Honestly I don't really know whether i should post this here or on cs.stackexchange.com! This is the question i have : Last year, a company providing web application services needed an ...
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34 views

MATLAB standard deviation

How do I calculate standard deviation using a for loop in Matlab? This is what I have but it seems too easy so I don't know if I am doing it correctly: ...
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29 views

Linear Interpolation adjacent nodes

I have a table of values which are $(x_i,y_i)$: (0.0,2.00),(1.0,2.1592), (2.0,3.1697),(3.0,5.4332), (4.0,9.1411),(5.0,14.406),(6.0,21.303). I am supposed to use linear interpolation between adjacent ...
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24 views

modifying plot in matlab

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69 views

MATLAB linear interpolation

I'm trying to write a MATLAB program to do linear interpolation and to check its accuracy. I have to input $x_0$ and $x_1$ and then generate the data values using $y=e^x$. Then, for a variety of ...
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17 views

for loop standard deviation

I'm very new to MATLAB programming and thus I doubt myself when doing things with matlab. I just wanted to confirm I am doing this correctly. I am supposed to complete this program: function ...
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How to compare the speed of convergence of infinitesimal of the same order?

Given two iterated function $f(x)$ and $g(x)$. Both of them converge to A, that is, both $\{p_n=f(p_{n-1})\}_{n=1}^\infty$ and $\{q_n=g(q_{n-1})\}_{n=1}^\infty$ converge to A. And the order of ...
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How to estimate the error of a numerical multiple integration

I'm integrating over the wholes space the function $$f(\vec{r_1},\vec{r_2})=\exp{\bigg[-(r_{1\alpha}+r_{1\beta}+r_{2\alpha}+r_{2\beta})\bigg]} \cdot 1/r_{12}$$ where $\vec{r_\alpha}=(-R/2,0,0), ...
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37 views

Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation by multistep methods?

I am reading the book Discrete Variable Methods in Ordinary Differential Equations (1962) by Peter Henrici. I am confused about the accuracy definition in multistep methods for the second order ...
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28 views

Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation?

I am reading the book Discrete Variable Methods in Ordinary Differential Equations (1962) by Peter Henrici. I am confused about the accuracy definition in multistep method for the second order ...
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30 views

Numerical solution of SDEs with fractional Brownian motion

I am trying to numerically solve some SDEs representing a nonlinear circuit (possibly chaotic) driven by noise: $$ dX = f(X) dT + \sqrt{P_{w}} dW + \sqrt{P_{f}} dC $$ where $X$ is my circuit state, ...
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28 views

The weak formulation of Navier Stokes Equation

I come across this problem in the weak formulation in Navier-Stokes equation. In the book, it let $D(\Omega_T)=\{\vec{\phi}\in C_0^{\infty}(\Omega_T),div\vec{\phi}=0 $, where ...
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330 views

Real world applications of numerical methods, for a mathematics project

I'm doing a mathematics project and I have been given 3 areas to have a look at and choose from. There's plenty of information on the academic side but not a lot of information on there real world ...
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34 views

Derive Runge-Kutta matrix with known weights and nodes

Can I derive a Runge-Kutta method by choosing freely the weights and the nodes? what are my constraints? So, if this is the general form of the explicit RK method: $$ y_{n+1} = y_n + \sum_{i=1}^s b_i ...
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14 views

Degrees of freedom of differential algebraic equations (DAE)

I have a set of DAEs that seem to be giving the solver difficulty (the solver is APMonitor, a web service), and I suspect I haven't formulated them correctly. The physical system is a pair of rigid ...
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26 views

derivative of function equal to zero in newton raphson method

In Newton’s Method, if derivative of the function is made equal to zero instead of the function itself at a particular x. What would you call that value of x?
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34 views

Use fixed point iteration to find root of equation $2x-\tan x=0$

I've tried $g(x) = \tan(x)/2$ and $g(x) = \operatorname{arctan}(2x)$, but neither of them satisfied the convergence condition. I guess I have a misunderstanding of the convergence condition. I ...
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19 views

Looking for a motivating example for backward error analysis

For many people, it seems self-evident that backward error is a powerful tool in numerical analysis. But for me, it is hard to imagine a situation in which backward error analysis provides any useful ...
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54 views

Finite difference method for nonlinear partial differential equations

I have the following partial differential equation (PDE) $ \forall (x,t)\in(0,L)\times(0,\infty) $ \begin{equation} \begin{split} m_{z}\ddot{w}&+EIw'''-Tw''-f+c_{1}\dot{w}-EAv''w'-EAv'w'' ...
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46 views

Scilab : simulating model of general equilibrium equations

Hi i'm new to scilab !!! I have a static general equilibrium model with 8 endogenous variable and 8 independent equations.Can anyone guide me how to do simulation of such model in scilab ? I want to ...
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32 views

Newton Raphson Method Overestimating Parameters

I have implemented an almost plain vanilla algorithm to find the MLE estimates of 3 parameters in a log-likelihood function (in R.) When I test my algorithm with some simulated data it does pretty ...
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32 views

How fast can we approximate the sum of the tangent?

So Wikipedia gives the sum of the tangent on this page as: \begin{align} \sum_x{\tan{(x)}} &= ix - \psi_{e^{2i}}(x + \pi/2)+C \\ &= -\sum_{k=1}^\infty{ \psi(k \pi - \pi/2 + 1 - x)} \\ &- ...
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27 views

how to prove this sparse coding equation

How can I prove the following? $\sum_i \frac{1}{2} \|\mathbf{x}_i - D\mathbf{\alpha_i}\|^2 = \frac{1}{2}Tr(D^TDA_t) - Tr(D^TB_t)$ where, $A_t = \sum_{i=1}^T \mathbf{\alpha}_i\mathbf{\alpha}_i^T\\ ...