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

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1answer
30 views

What is the relation between analytical Fourier transform and DFT?

First of all let me state that I searched for this topic before asking. My question is as follows we have the Analytical Fourier Transform represented with an ...
2
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2answers
29 views

Evaluating differential entropies with Matlab: NaN issue

With Matlab I am trying to evaluate differential entropies. These are integrals like $$\int_\mathbb{R} p(x) \log (p(x)) \mathrm{d}x$$ where $p(x)$ is a probability density function. My $p(x)$ is ...
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1answer
14 views

Derivation of $f(x)=\prod_{m=0}^{n}(x-x_m)^{m+1}\tan(x), x_m=m\pi, M>0$

I have the following function: $$f(x)=\prod_{m=0}^{n}(x-x_m)^{m+1}\tan(x), x_m=m\pi, M>0$$ I would like to calulate the numeric root of: $n\pi, n\ge0.$ In order to do that, I want to use ...
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0answers
28 views

Calculate Derivative while Runge Kutta

I am thinking about writing a C++ code to solve an ODE using Runge Kutta method. As you know, RK method calculates the state space vector $X'$ in a few mid-points and uses these mid-points for ...
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1answer
10 views

Using divide difference formula find the value of $f\left[x_{0},x_{1},x_{2},…,x_{10}\right]$

Consider the polynomial $f(x)=x^{10}+x-1$ , $x\in \mathbb R$ & let $x_{k}=k$ for $k=0,1,2,...,10$. Then the value of the divide difference $f\left[x_{0},x_{1},x_{2},...,x_{10}\right]=$ (a) $-1$ ...
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2answers
22 views

Give an estimate for the error.

Use the first three nonzero terms of Taylor’s formula for $\sin x$ to find an approximate value for the integral $\int_0^1 \frac{\sin x}{x}$ and give an estimate for the error.(It is understood that ...
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0answers
27 views

Fourier series question - represent $x$ as a series of $\cos$

I was asked to represent $f(x)=x$ in $(0,\pi)$ as a sum of $\cos$ functions, using fourier series. I couldn't solve it on my own, but here is what the teacher did, and I don't fully understand why ...
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0answers
11 views

Is there a formula of coefficients of Newton-Cotes Method in numerical intergation?

We know the coefficients of Newton-Cotes method in numerical integration are: 2-points $ 0.5$ , $0.5$ 3-points $ 1/6$, $2/3$, $ 1/6$ 4-points $1/8$, $3/8$, ...
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0answers
26 views

Lipschitz Constant (Burden and Faires Exercise)

There's an exercise in Burden & Faires Numerical Analysis book, Section 5.1 #2a, where they appear to want the reader to verify that a Lipschitz constant exists for the following ODE: ...
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1answer
20 views

polynomial approximation - basic chebyshev question

I was asked to find the best linear approximation to $f(x)=x^2$ in $x \in [0,1]$ using chebyshev polynomials, meaning, using the known property that $2^{1-n}T_n(x)$ is the best approximation to $0$ at ...
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0answers
21 views

How to improve stability of numerical solutions to partial differential equations

This is a quite general question, but I am working with a system of partial differential equations in two variables. There is one time direction $t$ and one spatial direction $z$ and the numerical ...
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0answers
24 views

Name of method which includes Taylor linearization inside fixed point iteration

I read paper about Horn-Schunck multiscale method for computing optical flow Core part of this algorithm is minimizing some functional. One part of functional contains nonlinear term inside L2 norm. ...
2
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0answers
27 views

What is the best method to calculate the square root when I know that the root is always an integer?

I have been through the wikipedia page, but wanted to know if there was a preferred (most efficient) method when there is an exact solution to find?
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0answers
41 views

Power method convergence explanation.

A sufficient condition for the power method to converge for a given diagonalizable matrix A is that the eigenvalues of A satisfy: $|\lambda_{1}|>|\lambda_{2}|\geq...\geq|\lambda_{n}|$ If this ...
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0answers
116 views

Can gradient descent solve this problem $\text{argmin}_x \|Ax-[Var(Ax)]^{\frac{1}{2}}-b\|^2$?

How can I find the (approximate) solution to the following problem: $$\text{argmin}_x \|Ax-[Var(Ax)]^{\frac{1}{2}}-b\|^2,$$ where $Var(.)$ denotes the variance? $A$ is matrix and $b$ and $x$ are ...
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2answers
18 views

The formula of the order of multistep methods

How can I derive this $$(1+\xi) \left(1+\frac{1}{2}\xi-\frac{1}{12}\xi^{2}\right)+O(\xi^3)$$ from $$\frac{1+\xi}{1-\frac{1}{2}\xi+\frac{1}{3}\xi^{2}}+O(\xi^3)$$ ? The whole formula is below. This is ...
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0answers
21 views

With the secant method, how can we ensure the constraint to prove super=linearity?

I know that as long as the first derivation does not equal to 0, then the secant method is super-linear. However, we're not typically given the derivative in things such as MATLab. How are we ...
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0answers
23 views

Efficient method for refining parameters in nonlinear curve fitting

I have time-series electrical current data $i(t)$ with transient steps in it which are convoluted with the hardware filter used in data acquisition. As a result, the real steps in current, which would ...
0
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1answer
26 views

spline derivation

Assume the following representation for cubic splines with $T$ interior knots is given. Let $g(Y)=\sum_{j=0}^3 \alpha_j Y_j+\sum_{t=1}^T \gamma_t (Y-\zeta_t)_{+}^{3}$ where $(Y-\zeta_t)_{+}:= ...
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0answers
30 views

Numerical Integration of $\int^{t_i}_{t_i-\Delta t}\frac{e^{-\frac{a}{t_n-\tau}}}{\sqrt{t_n-\tau}}d\tau $ for heat conduction problem

I am looking for a quadrature method to accurately evaluate the integral: $$I=\int^{t_i}_{t_i-\Delta t}\frac{e^{-\frac{a}{t_n-\tau}}}{\sqrt{t_n-\tau}}d\tau $$ Where $a$ is a positive constant of the ...
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1answer
41 views

Calculate the divide difference $f[1,2,3,4]$

Let, $f:[0,4]\to \mathbb R$ be a three times continuously differentiable function. Then the value of the divide difference $f[1,2,3,4]$ is (a) $\frac{f'(\xi)}{3}$ , for some $\xi \in (0,4)$ (b) ...
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0answers
15 views

LU-factorisation of a square matrix

I need to show that the following matrix cannot be factor into the product LU. \begin{equation} A=\begin{bmatrix}1&2&-1\\2&4&0\\ 0&1&-1\end{bmatrix} \end{equation} I did the ...
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0answers
21 views

Saturation Modeling in ODE45

I have a machine with an arm that can move in a linear one dimensional way. There are 3 limits on the arm: The arm has boundary for its location $(x_{min},x_{max})$ The arm has limit on its velocity ...
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1answer
68 views

How to solve the equation $\int_0^{t}\frac{1}{200+4(x+1)\arctan{\left(\frac{x+1}{100}\right)}}dx=1$

Let $l(x)=200+4(x+1)\arctan{\left(\frac{x+1}{100}\right)}$. I want to find real number $t>0$ such that $s(t)=l(t)$, where $s'(x)=\dfrac{l'(x)}{l(x)}s(x)+1$, $s(0)=0.$ It is a first order linear ...
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1answer
47 views

Matlab numerical integration involving Bessel functions returns NaN

I need to numerically compute integrals such as this (some parameters omitted for simplicity): $$ \int_{0}^{\infty} e^{-x^2} I_{0}(x) K_{0}(x) \mathrm{d}x $$ where $I_{0}$ and $K_{0}$ denote the ...
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0answers
25 views

Search Direction in Conjugate Gradient

Could you help me with a Conjugate Gradient question? In using CG to solve $Ax = b$, why is the search direction $p_{k+1}$ in CG chosen as a linear combination of the residual $r_k$ and previous ...
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1answer
26 views

Proving $\Delta^nx^n=n!h^n$.

How can I prove $\Delta^nx^n=n!h^n$. Here $\Delta$ is forward difference and h is the step size. I used induction . When $n=k$ assume the result is true. $$\begin{align}\Delta^{k+1}x^{k+1} &= ...
1
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1answer
44 views

Use Newton's method to find root for the following equations

I have to use Newton's method to find the roots with accuracy $10^{-5}$ of the following equation : $e^{x} + 2^{-x} +2\cos x -6 =0$ in the interval $(1,2)$ So $f'(x)= e^x - [2^{-x}]*[\log(2)] ...
0
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1answer
70 views

Help for Integral and evaluating - Eikonal equation

Hy guys I'm reading a paper of "Finding Exact Solutions to the Two- Dimensional Eikonal Equation" - E.D. Moskalensky. link for the paper: ...
1
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1answer
65 views

Numerical convergence depending on summation order

I'm looking for an example of convergent series such that the numerical convergence depends on the order of summation? Or perhaps a series of positive terms where the partial sums value depend on the ...
0
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1answer
32 views

Heat equation in 1D with collocation method

I want to use the collocation method to solve $u_t=u_{xx}$. I impose the PDE pointwise and expand the solution in Fourier Series: $$ \partial_{t}\sum_{k=-K}^{K}\hat{u}_{k}(t)\ ...
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2answers
32 views

How should terms be scaled by finite dx and dt in numerical integration of 1D diffusion?

I am familiar with numerically integrating systems of ordinary-differential equations, but I feel that I am missing something important in terms of how numerically integrating ODEs differs from ...
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0answers
22 views

Chebyshev polynomials approximation - Is there a way to generalize this

In an exam I was given this question: let $f(x)=x^3$. We want to find the best linear approximation (best in the sense that the maximal error is minimized) of $f$ in the interval $[-1,1]$ using ...
1
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1answer
28 views

A problem about lub and glb of matrix

For any matrix $A\in \mathbb{C}^{n\times n}$, define $$lub_K(A):= \inf\{\alpha\geq 0: AK\subset \alpha K\},$$ and $$glb_K(A):= \sup\{\alpha\geq 0: \alpha K\subset AK\},$$ where $K$ is a equilibrated ...
5
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1answer
117 views

How can one find intermediate digits of a root of an algebraic equation?

I was wondering whether there is a way to find intermediate digits of an algebraic equation. For example, if I have $$234x^{\frac{1}{12345}}-24621x^{\frac{1}{3456}}=1$$ And I want to find the ...
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0answers
42 views

Is the Taylor Expansion a good approximation

Say I use a computer to sum the first 26 terms of $e^{-5}$ (degree 25), will this taylor expansion provide a good approximation? It summed to $.0067$ To me this seems like a good approximation, but I ...
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2answers
33 views

Taylor Series approximation

Let $f(x) = (1-x)^{-1}$ and $x_0=0$. Find the $n$-th Taylor polynomial $P_n(x)$ for $f(x)$ about $x_0$. Find a value of $n$ necessary to approximate $f(x)$ within $10^{-6}$ on $[0,0.5]$. I am ...
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1answer
14 views

Selecting denominator for relative error margins

When looking at this page: http://floating-point-gui.de/errors/comparison/ there are values a, and b that are being compared ...
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0answers
10 views

Subset of density one

A subset S of positive integers will be said of density 1 if the following expression D_(N )(S) :=1/N #(S∩[1,N]) tends to 1 as N tends to infinity. The question is: Prove the equivalence of: 1) S ...
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1answer
45 views

Evaluate derivative of Lagrange polynomials at construction points

Assume, that we have points $x_i$ with $i=1,...,N+1$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align} Now according to my ...
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0answers
14 views

Embedding an Implicit Runge-Kutta Algorithm

I am interested in implementing a fourth-order RK method with variable step-size. However, in order to test for accuracy, the solver needs to periodically utilize a higher-order method to see if the ...
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0answers
53 views

How to solve Energy Balance equation by numerical method

Good Day I am new to heat transfer technique please give me some suggestion on solving energy balance equation $$a \frac{\partial T_p}{\partial t}=\frac{\partial}{\partial x}\left(b\frac{\partial ...
1
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1answer
60 views

Looking for fast computation method of $Ax=b$ ($A$ is sparse matrix)

I am looking for fast method to solve linear equation $$Ax=b$$ In which A is sparse matrix. Could you suggest to me some current method for this task. Thank in advance
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0answers
31 views

Generalize an average to a sum

Any help would be appreciated! Thanks so much!
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0answers
42 views

High dimensional Differential Evolution

I want to minimize a cost function with Differential Evolution (DE) algorithm and I have 55 unknown parameters as an input for DE algorithm. Therefore, the DE should search in high-dimensional space ...
0
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1answer
28 views

Runge Kutta Method

Here,$$y'(x)=x^2+y^2,y(0.9)=14.3$$ I calculated the value of $y(1.0)$ using step sizes of h=0.1 and then h=0.05. However,my result for the different step sizes are very different. I got ...
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1answer
16 views

minimum order of bspline curve for C2 continuity

Given a control polygon with five pairwise different points $d_0,...,d_4$ what is the minimum order of B-Spline curve for this polygon such that it is $C^2$ continuous ?
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0answers
32 views

Question about forward slash and set theory with matrices and vectors …

Hi People I'm new to this forum and this is my first question,(Please forgive me for my improper posting method.) In the Below equation which is where we start with a linear equation such as ...
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0answers
64 views

Error estimation absurd for Simpson's rule

If $f:[a,b]\to \mathbb R$ is a $\mathrm C^{3}([a,b])$, Simpson's Rule (or Newton-Côtes for $a$, $(a+b)/2$ and $b$) gives that if $P$ interpolates $f$ in those points, $$ \int_a^b f \approx \int_a^b P ...
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1answer
21 views

Bisection method

I know how to use the Bisection Method to find the roots, however I have never used it to find the point of interconnection on two graphs. I looked it up and found that you can just subtract the two, ...