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

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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 ...
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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 ...
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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 ...
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0answers
31 views

Why does the pressure term complicate numerical methods for Navier-Stokes Equations?

I'm looking to code a solver for the Navier-Stokes equations. I will be using finite differences with the method of lines. Two questions: What is the significance of the pressure term in the full ...
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0answers
47 views

sum of reciprocal of roots

Say we are given $$\alpha \tan \sqrt x =(1+\alpha) \sqrt x$$ where $\alpha > 0$ ad are after its positive roots. In particular, I am interested in estimating the following ...
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0answers
29 views

Numerical method to solve a first-order differential equation when derivative is available

$$dy/dt+y=f(t)$$ where $f(t) = \sum_{\omega}cos(\omega t)$. Assume that derivative can be calculated easily at any $t$, and $f(t)$ at any $t$ can also be easily calculated, but not higher-order ...
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0answers
11 views

Example Of Initial Value ODE For Stability Check Of Linear Multi-Step Method

We are attempting to provide an example of an initial value ordinary differential equation to show that the following "linear multi-step method" is unstable. ...
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0answers
34 views

(Numerical) Integration in log space

I have some function $f(x)$, which I'd like to integrate to find, $F(r) = \int_r^\infty f(x) dx $. Is there a way to do this using the values parametrized in log-space? I.e. some function $G(r, ...
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2answers
81 views

Numerical solution to a system of secon order differential equations

I'm writing a sort of physical simulator. I have $n$ bodies that move in a two dimensional space under the force of gravity (for instance it could be a simplified version of the solar system). Let's ...
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0answers
45 views

Can I apply Newton-Raphson Method in the Systems of Trigonometric Equations?

can I apply the Newton-Raphson Method in the given systems $$ \left\{\begin{aligned} A\sin x + B\cos y &= c\\ x + y &= d \end{aligned}\right.?$$ Can you show me the derivation on getting the ...
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0answers
35 views

Solving Linear equations using Conjugate gradient method

Given this two linear equations $$\begin{cases} 3x-y=1 \\ -{ x }+2y=-1 \end{cases}\\ $$ How can this system be solved iteratively with the Conjugate Gradient method?
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0answers
41 views

Most stable algorithm to solve a system of linear equations?

I am doing some image processing involving solving a system of linear equations. I am getting some errors and bits of the image looks corrupted. I would like to know what is the most stable way to ...
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1answer
14 views

FEM for PDE on curves: describing interpolation inequality

in page 312 Lemma 4.3 (interpolation) of the monograph Finite element methods for surface PDEs is stated as follows:\ for $n \leq 3$ and given $\eta \in H^2(\Gamma)$ (where $\Gamma$ is a surface) ...
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1answer
34 views

Explaining Non-Uniquuness of an Interpolation Polynomial

I am stucked at this problem: If $f\in C^1[a,b]$ and $x_0,...,x_n$ are $n+1$ distinct points in $[a,b]$, Then there exist unique polynomial $H_{2n+1}$ of degree at most $2n+1$ that satisfies the ...
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2answers
47 views

Looking for a Finite Difference scheme of the following form…

I'm having trouble deriving a finite difference scheme that calculates the second derivative of a function on the boundaries of a non-uniform grid and makes use of a known first derivative at the ...
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2answers
26 views

Does logarithm of Gaussian image still gaussian distribution?

I have an image 2D that pixel intensity follows multi Gaussian distribution such as $$p \left( I(x) \in \Omega_i \mid (I(x)\right)=\frac{1}{2\pi \sigma_i}\exp\left(-\frac ...
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0answers
21 views

How to discretize in space to get discrete curvature?

I have a free boundary problem. At each time I have two domains $\Omega_+(t)$ and $\Omega_-(t)$ in the plane, and an interface $\Gamma(t)$ separating them. In each domain I have a diffusion. At the ...
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0answers
27 views

two point block method for solving ODE

How to solve the ordinary differential equation $$y'(t) = -1000 y(t)+ 999 e^{-t}, \hspace{10mm} 0≤t≤5.$$ $y(t)=e^{-t}$, for $t<0$. Using two point block method $$hf_{n+1}= \frac{1}{3} (hf_{n+2} ...
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0answers
17 views

block method to solve ODE

How to solve the equation $$y'(t)=-1000y(t)+999e^{-t},\quad 0 \leq t \leq 5\\ y(t)=e^{-t},\quad t \leq 0 $$ using block method $$ hf_{n+1}=\frac{1}{3}\left(hf_{n+2} - 2y_n + 2y_{n+1}\right)\\ ...
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0answers
51 views

Average of two Newton's forward-difference polynomials gives Stirling formula

I am stucked at this problem: Let $x_{-n},x_{-n+1},...,x_{-1},x_{0},x_{1},...,x_{n-1},x_{n}$ be some real numbers such that $\forall k\in\{-n,-n+1,...,-1,0,1,...,n-1,n\}, x_k=x_0+k\times h$ for ...
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1answer
46 views

Fourier transform of $f(x)=x$ if $0<x\leq 1$ and $f(x)=0$ otherwise

What is the Fourier transform of the function defined by $f(x)=x$ on $[0,1]$ and $f(x)=0$ otherwise, i.e., $\hat f(\xi) = \int_\mathbb{R} { e^{-iu\xi} f(u) du }$? Is there a closed-form? Else, how ...
0
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1answer
19 views

Draw a tanh-ellipsoid

The function $f(x,y)=\tanh(R-\sqrt{x^2+y^2})$ for a given $R>0$ draws a circle with radius R with maximum and minimum of +1 and -1. If I want to do the same, but draw an ellipsoid with major and ...
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0answers
27 views

Why truncation error diverges if step is too small for $(x+1)^3(x-2)^2?$

Why truncation error diverges if step is too small for $(x+1)^3(x-2)^2?$ I have calculated the integral with trapezium rule on interval $[-1, 2].$ EDIT: I publish needed code to test, maybe it ...
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0answers
24 views

analytical solution of non-linear least square problem

I am implementing a trust region optimization algorithm and I would like to compare it against already done similar work, where authors measures performance on this problem. $$ \min_{u,\gamma}\Bigg\{ ...
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0answers
36 views

How to scale polynomial coefficients for root-finding algorithms?

I've implemented the Jenkins Traub algorithm in c++ (Github repo). While the majority of the solutions work well, it seems that a small portion of the roots are unstable. Here is a link to a ...
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2answers
75 views

What types of polynomials cause root finders to fail?

I am under the impression there are certain types of polynomials that root finders have trouble with. In other words, multiple real roots, complex roots very near to each other, etc. I am not ...
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1answer
27 views

what step should i choose to get needed precision in trapezoidal method

What step should i choose to get needed precision in trapezoidal method? ...
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1answer
54 views

Alternatives / Extensions to the Thin Plate Splines method

Thin Plate Splines are a great method to find a smooth interpolating surface given scattered data. Essentially, the method involves calculating weights for a radial basis function centred around each ...
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1answer
20 views

Numerical analysis-secant method

I am trying to solve the secant problem here but i do not know how to derive the f(x). Question: Use secant method to approximate ln(2) to 3 decimal place, x_0= 0.6, X_1= 0.7 I need help.
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2answers
39 views

Find $x_i$ and weights such that the following integration rule is exact for all polynomials of degree $\leq 5$

I'm going over an exam I failed. I was told that I can't use the method I used to solve the following question, and I don't know why. Can you please explain and suggest a correct solution? Question ...
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0answers
14 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
5
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1answer
38 views

Gauss Hermite Integration of 1/(1+x^2)

I'm trying to learn Gauss Hermite Integration and was manually try to calculate the value of integral of $\frac{1}{1+x^2}$ from $-\infty$ to $+\infty$ The exact answer is simply $\pi$ ($\approx$ ...
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0answers
23 views

Time advance in Adaptive Mesh Refinement method

I am working on solving complex system of 2D PDEs governing the behaviour of plasma in a gas lamp during discharge. Recent tests have shown that because of steep gradients in temperature field and ...
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0answers
26 views

How can we calculate the tensor product of Lagrange basis polynomials?

Given data points $(x_i,y_i)$, the Lagrange basis polynomials are $$\mathcal l_j(x):=\sum_{i\ne j}\frac{x-x_i}{x_j-x_i}\;.$$ I'm reading a text targeting Smolyak's algorithm. In this text, they use ...
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2answers
76 views

Looking for finite difference approximations past the fourth derivative

I scanned the internet and could not find further representations of the central difference approximations past the fourth derivative. Are there published results past the fourth derivative? Ideally ...
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0answers
49 views

Newton's method on a surface

I am trying to use Newton's method to find the stationary solutions of the integro-differential equation of the form $$\frac{\partial u(r,t)}{\partial t} = -u(r,t) + \int_{\mathbb{R}^{2}}w(r - ...
3
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1answer
40 views

Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of ...
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0answers
36 views

Runge-Kutta methods that satisfy row condition produce same solutions for equivalent autonomous problems

Given a IVP $y'(x) = f(x,y)$ $y(a)=\eta$ in $[a,b]$, it can be written as an autonomus IVP by increasing the space dimension: $$ \tag{*} \bar y(x) = \begin{pmatrix} x\\ y(x)\end{pmatrix},\quad \bar ...
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2answers
41 views

Find an algorithm to calculate the following function

I'm solving questions from an exam I failed, and I would love some help with the following question: Question We want to calculate the following function in Matlab: $$ f(x) = \frac{e^{x^2} - (1 + ...
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3answers
74 views

Showing that the sequence $x_n=\frac{1}{3}x_{n-1}(4+x_{n-1}^3)$ where $x_0=-0.5$ quadratically converges

I am stuck at a point in solving this problem: Show that the sequence defined by: For all $$n\in\mathbb{N}, x_n = \begin{cases} -\frac{1}{2}, & \text{if $n=0$} \\ ...
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1answer
25 views

Product rule in discrete derivative in finite difference scheme.

Suppose we are on real line and I want to discretize the usual derivative operator. Take a smooth function $u$ and step size $h$. Then I could define $$ \Delta_+u(i) = \frac{u(i+1)-u(i)}{h} $$ as the ...
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0answers
18 views

For what $p$ is the condition number of a given matrix $A$, using the $p$ norm on matrices, minimal? [duplicate]

For what p is the condition number of a given matrix $A$, using the p norm on matrices, minimal? The condition number on $A$ is given as: $$K(A,p) = \|A\|_p \times \|A^{-1}\|_p$$ I tried ...
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0answers
22 views

Error estimate of polynomial quadratures missing some terms

Normally, for trapezoid rule and simpson's rule, etc, error analysis is done by using the error formula for interpolation. However, if the polynomial is restricted to some terms, for example, a ...
4
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1answer
42 views

Numerically evaluate Gauss' hypergeometric function ${}_{2}F_{1}(a,b;c;x) $ for large $|a|$ or $|b|$ and $x\ll 0$ or $ x \approx 1$?

I need to compute Gauss' hypergeometric function $${}_{2}F_{1}(a,b;c;x)$$ for the case where one of $|a|$ or $|b|$ is large and $x\ll 0$ or $ x \approx 1$. By employing some linear transformations, I ...