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

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21 views

Trying to model a substance settling in water using an advection equation?

I am trying to model a substance dispersed in a container of water gradually settling at the bottom. I am considering only one dimension. The top is at $z = 1$, and the bottom is at $z = 0$. So at $t ...
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0answers
17 views

Is there a big difference between runge kutta 4th for ODEs vs SDEs?

I was working on 2nd, 4th order runge kutta method for stochastic differential equations. I saw 2nd formula for ODEs and SDEs. There is some difference between their formulas . Unfortunately I can't ...
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3answers
23 views

What should the initial guess be for the Bablyonian method of calculating square roots?

You can use any value as the initial guess for the Babylonian method of calculating a square root (other than 0), but the closer the guess to the root, the more accurate your result per iteration. Of ...
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0answers
20 views

Efficient approximation to integration of analytic expressions involving product of four bessel functions

I have to take many integrals of the form $$ \int_0^\infty \!dx\,\,e^{-x}\,x^{\gamma - 2\beta - 2\alpha} j_\alpha ( u_1 x)j_\alpha (u_2 x)j_{\beta}(u_3 x)j_\beta (u_4 x),$$ where $\gamma$ is an ...
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0answers
23 views

How can I modify this simple code to include the pressure term? (1-D Navier Stokes)

I have a mathematical model that involves a cylindrical container that is being modeled with a one dimensional simplification as the system is isotropic with respect to the z-axis. As part of the ...
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1answer
15 views

newton raphson method convergence problem

My problem is: An iterative method to find $n$-th root of a positive number $a$ is given by $x_{k+1}=\frac{1}{2} \left[x_k +\frac{a}{x_k^{n-1}}\right]$ Find the value of $n$ for which this ...
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0answers
52 views

Reducing or avoiding the Gibbs phenomenon.

What is your favourite method which would help reduce the Gibbs phenomenon in Fourier Series and Fourier Transforms. This could mean pre-processing or post-processing or altering the transform. With ...
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0answers
65 views

Contour integral mystery: why is the answer different from Maple/Matlab?

The mystery is that here is a fairly standard contour integral which can be done by the residue theorem. Yet when I tried to evaluate it using numerical softwares like Maple or Matlab, the answer is ...
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0answers
108 views

Radial Basis Functions Interpolation

$ \let\oldcdot\cdot \renewcommand{\cdot}{\!\oldcdot\!} \newcommand{\e}{\varepsilon} \renewcommand{\p}{\varphi} \renewcommand{\p}{\varphi} \renewcommand{\vp}{\vec{\boldsymbol\p}(x)} ...
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1answer
22 views

Godunov scheme for advection equation

I'm trying to solve the advection equation $$m_t+(\alpha m)_x=0$$ with $m(0,\cdot)=m_0$ numerically using the first order Godunov scheme. Hence I write ...
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1answer
58 views

Finite difference method works for $\frac{\partial u}{\partial t} = \frac{du}{dz}$ but not for $\frac{\partial u}{\partial t} = - \frac{du}{dz}$?

I am using the method of lines with forward differences to solve the transport equation $$\frac{\partial u}{\partial t} = \frac{du}{dz}$$ with initial condition $u(z, 0) = z$ and boundary condition ...
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1answer
23 views

How to find X, Y Coordinates on wooden plane

I'm working on a wooden cnc machine. How to calculate the x, x coordinates? I have a square 300mm x 300mm I want my point to be on a circle and to find the x, y to drill a hole. Can you help me?
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0answers
8 views

von Neumann stability of the implicit downwind scheme

I want to investigate von Neumann stability of the implicit downwind scheme for this PDE: $u_t-u_x=0$. I got $\frac{\Delta t}{h} \geq 1$. It seems odd. I also checked the CFL condition of implicit ...
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0answers
19 views

Fluid dynamics: mesh resolution close to the origin in spherical co-ord system

Suppose you have a spherical implosion calculation (e.g. ICF etc.) in which you have a material interface that you want to apply some sort of perturbation to. There are two possible configurations in ...
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0answers
59 views

How to I approximate $I = \int_{-1}^{1} \sqrt{1-x^2}\cos(x)dx$ s.t. the error is bounded?

Edit: Because the original question was pretty trivial, I want to ask the same question but with:$I = \int_{-1}^{1} \sqrt{1-x^2}\cos(x)dx$. How to I approximate $I = \int_{-1}^{1} ...
2
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0answers
13 views

variable transformation in optimization

I have an optimization problem with two sets of parameters, $x_i \in [0,1]$ and $y_k \in [-\frac{\pi}{2},\frac{\pi}{2}]$ where $i,k \in \{1...n\}$ are indices. One way to solve this problem is using ...
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0answers
13 views

How to obtain the stability function of $y_{n+1}=y_n+h[\theta f(y_n)+(1-\theta)f(y_{n+1})]$?

I am going through a past exam paper but I don't know how to obtain the stability function in this case, I know how to do it when I have the matrix or when I have an explicit method, but not in this ...
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0answers
10 views

how to project optimal parameters on to feasible region

Hi: I'm trying to understand the concept of projection and I created a toy example that might help me to do that. Suppose that I have a non-linear optimization with 3 parameters theta_1, theta_2 and ...
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0answers
42 views

Why does this “incorrect” Chebyshev function approximation work better than the correct one?

I recently had the need to approximate this function $$f\left(x\right)=\begin{cases} \log\left(\frac{\pi+2\arcsin\left(x\right)}{\pi}\right), & x<0\\ ...
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1answer
44 views

What method of numerical integration is this?

I am trying to update some old code that finds the area under a curve from $17$ evenly spaced discrete data points. I'd like to update it to calculate from $65$ data points. I'd like to use the same ...
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1answer
19 views

Determine the order of consistency of $y_{n+1}=y_n+(h/2)(y_n'+y_{n+1}')+(h^2/12)(y_n''-y_{n+1}'')$ (I want to improve my answer)

I can solve this problem but I was wondering if there is a quicker way to do it since time will be tight during the exam... I would really appreciate your tips and advice on how to calculate this in a ...
2
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1answer
31 views

What type (explicit, Runge-Kutta, Taylor series, one-step, etc.) is the numerical method $y_{n+1}=y_n+(h/2)(y_n'+y_{n+1}')+(h^2/12)(y_n''-y_{n+1}'')$?

This exam question is asked every year, but I am struggling to understand the difference between numerical methods even though I can solve all the exercises. Thanks a lot in advance for your help! ...
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0answers
24 views

Intuitive explanation for error in Newton's Divided Differences?

When interpolating a smooth function $f$ using $n+1$ points, the error in the interpolation is bounded by $e(x) \leq$ $f[x_0,\ldots,x_n,x] \cdot \prod_{i=0}^n(x-x_i)$. This seems kind of interesting ...
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1answer
35 views

Is my textbook solution wrong or Am i missing something?

My doubt here is that in last row of table where $x_n$ is 2.7984 is there, f(2.7984)=1 approximately and no way near zero. So is this misprint or i am missing something here. This has happened with ...
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0answers
21 views

Finding Local Linear Basis Functions

Seeking the linear basis functions for a finite element solution, I was given the paper shown by my professor and was asked to find the remaining local basis functions and then compute all the global ...
7
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1answer
73 views

How to shift two CDF's to maximize the number of crossings

So suppose I have two continuous, monotone increasing function $F$ and $G$ defined on an interval $I_F=\{x:0<F(x)<1\}=(l_F,u_F)$ and $I_G=\{x:0<G(x)<1\}=(l_G,u_G)$ which can be computed ...
1
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1answer
27 views

Order of a corrector-predictor method

Given an explicit method: $$ x_{i+1} = x_i+ h \Phi(t_i,x_i,h) $$ as predictor method and an implicit method: $$ x_{i+1} = x_i + h \Psi(t_i,x_i,x_{i+1},h) $$ as corrector method, it follows that $$ ...
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1answer
52 views

Finite difference differentiation formula

I'm trying to understand how the co-efficients of finite differences are calculated. In particular I'm interested in the first derivative for a uniform grid of unit width. I found this document ...
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33 views

Parametrization of the $Ax=b, x \geq 0$ domain for Monte-Carlo simulation

I have a linear system, $n=15$, with $6$ constraints. There's no problem finding a single solution or establishing the null space; so I can see the full solution space. But I'm only interested in ...
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1answer
19 views

Fixed point iteration, finding g(x)

I have struggle on finding this function g(x). Assume function $f(x) = 5x^3 -20x + 3$ and it is specified to find root in [0, 1]. So I guess, first thing is to find function g(x). $$g_1(x) = ...
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1answer
47 views

Runge Kutta Method Matlab code

So I have a programming assignment with the following instructions: Consider the nth-order differential equation $$Ax^n (t) = x ^{(n-1)}(t) + x^{(n-2)}(t) + ... + x(t)$$ where $A$ is a ...
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0answers
15 views

When do I use a specific interpolation method?

I am having a course on Numerical Analysis and I was wondering if I can use any interpolation method to interpolate any data, or one method has some specific advantages over another. Here are some of ...
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1answer
33 views

Find the nodes and coefficients of Gauss-Lobatto Quadrature with $n=4$

I am stucked at this problem: Gauss-Lobatto quadrature is defined as: $\int_{-1}^1 f(x)dx\approx w_1 f(-1)+w_n f(1) + \Sigma_{k=2}^{n-1}w_k f(x_k)$ ($2\leq n\in\Bbb{N}$) Where the nodes $x_k$ ...
2
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2answers
41 views

Calculating $f'(x)$ with $f(x)$ and a relative error?

I want to calculate $f'(x)$ using the formula: $$ f'(x) = \frac{f(x+h) - f(x)}{h}$$. Of course the error here is $o(h)$. However, what if in measuring $f(x)$ and $f(x+h)$ I have a relative error of ...
2
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1answer
123 views

Numerically Calculating the solution of very complex equations

I wanted to confirm a question of my own, and I figured out if there is a solution of the following equations such that every variable is real and $x,y\ge 0$, my question could be partially verified. ...
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1answer
39 views

Trapezodial Rule Error Proof (taylor)

I search for a proof of the (local) error of trapezodial rule using taylor series. I can only find proofs for the error of the rectangle rule and for trapezodial it's always just "similar" whatever ...
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0answers
18 views

Little Doubt in Secant Method

Given Question is : A root of equation $xe^{x}-1=0$ lies in interval $(0.5,1.0)$. Determine this root correct to three decimal places using secant method DOUBT I know method, but my problem is how ...
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1answer
15 views

Explanation of the difference operator $\mathscr{N} \textbf{y}(x_n)$ used in numerical analysis

In books about numerical methods one can get across the difference operator (methods for numerical solution of ODE's): $\mathscr{N} \textbf{y}(x_n)$. However in all the books I have only the example ...
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0answers
35 views

To determine the interval of unit length which contains the smallest positive root of $x^{3}-5x-1=0$

I am doing Bisection method of numerical analysis. The question I encountered is as follows To determine the interval of unit length which contains the smallest positive root of $x^{3}-5x-1=0$. Hence ...
2
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2answers
46 views

Probability calculation with large numbers

I do have a probability measure: $P = 1 - \dfrac{k!\, \binom{2^{32}} {k}}{(2^{32})^k}$, where $k$ is an positive integer. Yet, I do have trouble evaluating it in terms of a numerical plot, as the ...
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2answers
45 views

Find to how many digits is the value 355/113, an accurate approximation to $3.1415929204$

Find to how many digits is the value 355/113, an accurate approximation to $3.1415929204$ What i did was i computed using calculator value of 355/113 which came out to be $3.14159265$ Now i see up ...
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0answers
13 views

Universal polynomial approximation algorythm

I would like to ask, is there any universal algorythm to fill this matrix for any n value? $\textbf{A} = \matrix{n & \sum x_i & \sum x_i^2 & \cdots & \sum x_i^n \cr \sum ...
1
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1answer
28 views

Gaussian Integration verification

I have the following problem: For the formula $$\int_0^1 f(x) dx\approx w_1f(0)+w_2f(x_2)$$ determine the weights $w_1, w_2$ and the node $x_2$ so that the formula is exact for all polynomials of as ...
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0answers
13 views

Gauss Numerical Integration Verification and Help

I have the following problem: Determine constant $c_1$ and $c_2$ in the formula $$\int_0^1 f(x)dx \approx c_1f(0)+c_2f(1),$$ so that it is exact for all polynomials of as large degree as possible. ...
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1answer
123 views

Is anyone talking about “ball bundles” of metric spaces?

In differential geometry: Each smooth manifold $M$ is equipped with a tangent bundle $TM,$ which is a manifold equipped with a projection back to $M$ Given a smooth map $f : M \rightarrow N$ between ...
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4answers
163 views

Exam question on fixed point iteration

I am solving the following exam problem. Problem: An iterative scheme is given by $$ x_{n+1}= \frac{1}{5}\left(16-\frac{12}{x_n} \right).$$ Such a scheme with suitable initial approximation $x_0$ ...
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0answers
33 views

Can I “squeeze” the x-axis when I solve a diff. eq?

I am trying to solve a (rather ugly) differential equation numerically. (If you're curious, the equation is $\frac{3}{2}\left(\frac{\partial_x f(x)}{f(x)}\right)^2+\frac{\partial_x ...
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1answer
23 views

Confused with an interpolation problem using Lagrange.

I'm really confused about the following interpolating problem.Not sure if this is the right method. For $n =3$, explain why $$ x_0^jL_o(x) + x_1^jL_1(x) + x_2^jL_2(x) + x_3^jL_3(x) = x^j, \ \ j \leq ...
2
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1answer
44 views

Expressing unit quaternions in three degrees of freedom

Short version of question: I am trying to use quaternions to avoid gimbal-lock, but I don't know how to express unit quaternions using three degrees of freedom without re-introducing Euler angles and ...
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0answers
59 views

Please guide me books and online materials for this course

I have recently taken Course on Numerical Analysis. It is correspondence course. So i to do self study. I will be glad if someone mentions online videos and elementary books which contains following ...