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

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41 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
43 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 ...
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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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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 ...
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1answer
71 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 ...
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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
49 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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32 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
43 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
31 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 $x_k$ ...
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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
122 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 ...
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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
44 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 ...
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1answer
27 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
12 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
162 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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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
42 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
55 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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29 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
46 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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27 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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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
31 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
80 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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40 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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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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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
46 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 ...