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

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57 views

To calculate a derivative of a set of points, is it more correct to interpolate finite differences or to derivate the interpolation?

I have a series of points extracted from numerical simulations. I also recently discovered the amazing power of finite differences. Nevertheless, I was used to estimate my derivatives from the ...
3
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1answer
158 views

Numerical computation of continuous Fourier transform

Are there any algorithms that numerically compute the continuos Fourier transform of a given function f? I find plenty of implementations of the discrete Fourier transform, using FFT, but, if I´m not ...
3
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1answer
557 views

Steady State Solution Non-Linear ODE

I'm working through some problems studying for a numerical methods course, but I'm stuck on how to answer the following question analytically. It says to find the steady state solution of the ...
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2answers
67 views

If the points $x_1,x_2,\ldots,x_n$ are distinct,then…

I am stuck on the following problem that says: If the points $x_1,x_2,\ldots,x_n$ are distinct,then for arbitrary real values $y_1,y_2,\ldots,y_n$, prove that the degree of the unique ...
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1answer
211 views

Formula for Adding Time expressed as decimal

If I record the duration of an event that was 97 minutes as 1.37 and then record the duration of another event that was 162 minutes as 2.42 and then add the two together to get 3.79, is there a ...
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1answer
467 views

Spline Interpolation

I have four questions about splines. Any help would be greatly appreciated. 1) Boundary conditions for cubic spline interpolation to a set of data $a=x{}_{1}<x2<...<x_{m} , $ like for ...
2
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1answer
473 views

Gauss Kronrod quadrature rule

Given the abscissae and weights for 7-point Gauss rule with a 15-point Kronrod rule (Wikipedia); Can anyone provide me a working example how to numerically integrate a function given below: ...
2
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1answer
674 views

What tutorial videos is best for numerical methods 1?

I'm doing this course and we have only been told to work through the prescribed textbook. No study guide was given and I'm not sure how/where to start with the assignment questions. Please advise.
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1answer
57 views

Prefactoring to solve many similar linear systems

I am designing an algorithm that needs to solve many (large) linear systems of the form $$\Phi^\top D_i\Phi \vec x_i=\vec r_i,$$ where $\Phi\in\mathbb{R}^{m\times n}$ with $m>n$ is fixed. We will ...
2
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1answer
2k views

Numerically solving a system of nonlinear ODEs with boundary conditions

I have a system of 6 second-order nonlinear ODEs involving 5 different functions of a variable $t$. Every function has a boundary condition at $0$. I've never taken a differential equations class and ...
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1answer
106 views

How to calculate a big combination $\binom nr$

How to calculate a big combination such as $$\binom {10^{80}}{10^{10}}$$, using software or by hand, or at least can we get an acceptable approximation.
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1answer
55 views

Numerical Analysis, build a contractive function

I have a question regarding Numerical Analysis. I've never been asked these sorts of questions before and don't even know where to begin. The goal of this exercise is to find a value alpha such that: ...
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1answer
66 views

Approximating a complicated multi-variable function over an interval?

Consider $$ F(\mathbf{r})=F(x,y,z) = \frac{2z^2 - x^2 - y^2}{(x^2+y^2+z^2)^{5/2}} $$ where $x,y,$ and $z, $ are all $n^{\text{th}}$ order polynomial functions of a parameter $t$ with arbitrary ...
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0answers
40 views

correctness of functional iteration and contraction proof

I need to prove: If $F$ is contractive from $[a, b]$ to $[a, b]$ and $x_{n+1} = F(x_n)$ with $x_0\in[a,b]$ then $|x_n -s| \leq C\lambda^n$ for an appropriate $C$ where $s$ is a fixed point of $F(x)$. ...
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0answers
60 views

understanding derivation of golden ration convergence for the secant method

I have just worked through the error analysis of the secant method to find: \begin{equation} e_{n+1} \approx \dfrac{f''(r)}{2f'(r)}e_ne_{n-1} \end{equation} Where $e_n = |x_n - r|$ $r$ is a zero of ...
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1answer
49 views

Graphing a function

A function is defined in a article equation 37, $$A(t)= 0.456+\frac{2.58}{(1+0.136 t)^{0.249}}$$ The simulation of the equation is: But I have tried like by this: ...
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0answers
86 views

Nodes in spherical equations and graph matching

The graph for this spherical equation is, (equation no 44) $$\frac{d^2 S}{d \rho^2}+ \frac{D-1}{\rho}\frac{dS}{d \rho}-S +S^3=0.$$ What I didn't understand here is $S_0, S_1,S_2$ and $S_3$. ...
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0answers
103 views

Mysterious term in semi-implicit Euler scheme

In the paper i'm currently working with I don't understand the role of the term $C^m$ in the following semi-implicit numerical Euler scheme they use, which consists of following two recurrence ...
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1answer
48 views

How to fit a formula to three data points?

I need a very basic formula that will be used to determine a CSS line-height based on a provided font pixel size. So in essence, I need the formula to covert ...
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1answer
163 views

Numerical methods for a second order PDE boundary value problem

Is there any numerical method (like the method of splitting of variables for the equation of 2D-diffusion) for solving the following boundary value problem $$\left\{\begin{aligned}&\frac{\partial ...
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1answer
1k views

Problem to find the intersection of a exponential and linear function

I have the problem to find the intersection of a exponential and linear function. My math teacher can't help me, but I'm interested how I can solve this. I tried to use the equating method, but it ...
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1answer
60 views

Numerical solutions to wave equation

Does the wave equation always have an analytical solution given well-behaved boundary/initial conditions? If not, under what conditions does the wave equation need to be solved numerically? This ...
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0answers
101 views

Discretisation of the Perona Malik anisotropic diffusion Equation

The perona malik diffusion equation mentioned in this paper is given as $I_t = C(x,y,t) \nabla ^2 I + \nabla C . \nabla I$ The discretisation is given as $I_{i,j} ^{t+1} = I_{i,j} ^{t} + ...
14
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2answers
371 views

how to compare $\sin(19^{2013}) $ and $\cos(19^{2013})$

how to compare $ \sin(19^{2013})$ and $\cos (19^{2013})$ or even find their value range with normal calculator? I can take $2\pi k= 19^{2013} \to \ln(k)= 2013 \ln(19)- \ln(2 \pi)=5925.32 \to k= ...
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1answer
149 views

Diffusion Advection equation discretization scheme

I am looking for a good reference to understand the basic discretization schemes applied to the Stationary Diffusion Advection equation. $$-\epsilon \frac{d^2u(x)}{dx^2}+\beta \frac{du(x)}{dx}=0$$ ...
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0answers
50 views

How to conserve probability using a numerical integration scheme?

I have an iterative operator that conserves probability given by $P_{n+1}(z_j) = \int_a^b P_B(x+z_j)P_n(x) dx$, where $P_n$ is the PDF at time step $n$ and $P_B$ is a PDF that is fixed with compact ...
4
votes
2answers
300 views

Plotting graphs using numerical/mathematica method

From the author's equation 13, 14 We can write by inserting V''(A)=0, Solving for R we get, $$R= \frac{6^{D/4} \sqrt{D}}{\sqrt{-2^{1+\frac{D}{2}} 3^{D/2}+3 2^{1+D} A-3^{1+\frac{D}{2}} A^2}}$$ Now ...
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0answers
38 views

$H^1$ estimate by $L^2$-norm

Let $(\tau_h)$ be a shape regular triangulation. Prove that there exists a constant $c>0$ such that $$\|v\|_{H^1(\Omega)}\leq \frac{c}{\min_{T\in\tau_h}} \|v\|_{L^2(\Omega)}$$ for all $v\in V^h$ ...
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0answers
202 views

Gradient Descent in a 3D parameter space

I'm trying to computationally implement a gradient descent algorithm in 3D to find the maxima of a function. I want to use a recursion scheme like $$\textbf{x}_{k+1} = \textbf{x}_k + \alpha \nabla ...
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275 views
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1answer
268 views

Is Householder orthogonalization/QR practicable for non-Euclidean inner products?

The question Is there a variant of the Householder QR algorithm to orthonormalize a set of vectors with respect to an inner product if no orthonormal basis is known a priori? Background Let's ...
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1answer
69 views

How to normalize these numbers for better visualization?

My dataset is like this: a1 4565380 a2 676477 a3 359939 ... b1 222431 b2 12222 ... g1 139 ... h1 134 i1 10 j1 11 and goes on.. The problem is when I ...
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2answers
198 views

square root of a symetric matrix

I have a symmetric matrix which positive-definite, but it contains zero as eigen value. So the method of Cholesky does not work, could someone give another method to do this? I do not want an ...
2
votes
1answer
252 views

How do I apply the Fast Multipole Method to Thin Plate Spline interpolation?

I have n scattered measurements of elevation, z, as a function of x and y coordinate: $ \{(x_i,y_i,z_i)\}_{i=1..n}$ that I want to interpolate so that I find z(x,y) for all x and y. Using Thin Plate ...
4
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1answer
456 views

constructive canonical form of orthogonal matrix

For every orthogonal matrix $Q$ over the reals there is an orthogonal matrix $P$ and a block diagonal matrix $D$ such that $D=PQP^{t}$. Each block in D is either $(1)$, $(-1)$ or a two dimensional ...
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2answers
146 views

solution of the equation with exponential function

Let $m, n$ be an integers. Let $b \in R$. Solve the following equation for $n$. $$ \exp(m-\frac{2}{\pi}n)=n^{b/{\pi}}. $$ Thank you.
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1answer
46 views

Convergence of expression involving normal c.d.f.

I have derived the following expression for the error of some approximation: for any $\epsilon > 0$ $|u_{precise} - u_{approx}| \leq C_1 \epsilon + C_2 \cdot \Phi \left ( -\sqrt{-2 \ln ...
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0answers
38 views

Chebyshev spectral differentiation matrix demonstration

How to demonstrate that $d_{i,j}=\dfrac {c_i(-1)^{(i-j)}}{(c_j(x_i-x_j))}\;,i\ne j$ $d_{0,0}=-d_n,n=-2(n^{2})+\dfrac 16$ $d_{j,j}=\dfrac {-x_j}{2(1-x_j^{2})}$ with $c_i$ and $c_j=2,$ for $i=0$ ...
2
votes
2answers
114 views

an $L^2$-estimate for finite element solutions

Let $(\tau_h)_h$ be a quasi-uniform family of triangulations. Show that for $u_h\in V^h$ the following holds: $$c||u_h||_{L^2(\Omega)}^2 \leq h^2 \sum_{i=1}^N |u_h(z_i)|^2 \leq C ...
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1answer
192 views

Finite Difference without boundary conditions

I'm working through the paper where the Finite Difference method is employed to solve the PDE $\displaystyle \frac{\partial u(x,t)}{\partial t} = a \cdot \frac{\partial^2 u(x,t)}{\partial x^2} + b ...
2
votes
1answer
72 views

shape regular triangulations and Zlamal's condition

I'm trying to show that a triangulation $\tau_h$ is regular if and only if there exists $\theta_0>0$ such that for all $T\in\tau_h$ we have $\theta_T\geq \theta_0>0$, whereas $\theta_T$ is the ...
2
votes
1answer
201 views

Numerical Analysis best estimate on polynomial order

I need to determine the best integer value of $k$ for the equation: \begin{equation} \arctan(x) = x + O(x^k) \text{ as $x\to 0$} \end{equation} Taylor's Theorem with Lagrange Remainder would ...
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3answers
70 views

Numerical Analysis and Big O

How can I show that $e^x -1$ is not $O(x^2)$ as $x\to0$ I'm not sure where to start. We can use Taylor's Theorem with remainder: \begin{equation} e^x = \sum\limits_{k=0}^n\dfrac{x^n}{n!} ...
3
votes
1answer
2k views

ODE solving in Scilab

I have a certain ODE problem which needs to be solved using Scilab. dx(1)/dt=k*x(1)-x(5) dx(2)/dt=k2*x(2)-k1*x(1) dx(3)/dt=k1*[x(2)-x(3)] dx(4)/dt=k1*[x(3)-x(4)] ...
1
vote
1answer
211 views

How should I interpolate between values in a logarithmic series?

What's the best way to interpolate between 2 values of a logrithmic series? More specifically, I have a process where we encode values as $b = \text{floor}(\log(x, k))$. We discard the original ...
0
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0answers
116 views

Numerical methods for solving nonlinear equation

I need to solve some nonlinear equation which looks like this: $$x = \frac{L\cos(Wx)}{1+LW\sin(Wx)}$$ I have listened about some method of the bruteforce of the roots. Can you help me to find some ...
0
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0answers
48 views

How to choose a numeric approach for derivates

I would like to find the derivate from some combined logistic and exponential functions that all describe the same data numerically. $$f'(t)=\frac{f(t+h)-f(t)}{h}$$ seems not the best choise for ...
2
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1answer
41 views

Question about matrix discretisation numerical methods

Tomorrow I have an exam about Numerical Methods, and I came up with the following question. Let $$-\frac{d}{dr} \left ( \frac{1}{r} \frac{dy}{dr} \right ) = 1 $$with $r\in [1,2], y(1) = 1 \mbox{ and ...
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2answers
97 views

Recommendations for website/journal/magazine in applied mathematics

Which website/journal/magazine would you recommend to keep up with advances in applied mathematics? More specifically my interest are: multivariate/spatial interpolation numerical methods ...
2
votes
4answers
108 views

Chaotic iterative example needed

I'm using a very simple numerical method to find solutions to an equation. Start with an equation $\operatorname{f}(x)=0$ that you need to solve. Rearrange to give $x=\operatorname{g}(x)$ and then use ...