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

Gradient descent with linear perturbation

Given a convex, differentiable function $f$ (from a Hilbert space to $\mathbb{R}$) with a minimum (say $x^*$), I know you can find $x^*$ using gradient descent. Suppose now that you apply gradient ...
1
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0answers
55 views

3Dimensional runge kutta and Euler method ( help to verify the idea and proposition)

I been discussing this idea with a tutor for sometime. However it turn out that the proof is not comprehensible.Can someone please help to verify the the proof for 3D Euler method and runge kutta ...
2
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1answer
45 views

Stability analysis of Ralston's method

Ralston's method is given by: $$y_{n+1} = y_n + \frac{h}3(f(t_n,y_n)+2f(t_n+\frac34h, y_n + \frac34h f(t_n,y_n)))$$ carry out a stability analysis of this method to determine the condition for ...
2
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3answers
130 views

How to prove $\det(I+uv^\intercal)=1+v^\intercal u$

Let be $u,v\in\mathbb{R}^n$, then $\det(I+uv^\intercal)=1+v^\intercal u $ where $I$ denotes the identity matrix of order $n$. How to prove this? what I did: let be ...
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0answers
26 views

How to apply Runge-Kutta to an implicit scheme?

I see there are some differences in the solution as I increase the resolution of my grid. I'm using Operator Splitting to solve Diffusion Reaction equation \begin{equation} \frac{\partial ...
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0answers
18 views

Integral on an region defined by a regular grid of points

I'm trying to evaluate a multidimensional integrand $f$ that I know the major contribution is restricted to a specific region around its maxima (to be concrete, imagine a 2D gaussian function). What ...
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2answers
57 views

Newton Raphson Step Size

I am solving old exams and I came across the following question: Let $$ x_{n+1} = x_{n} - \alpha\frac{f(x_{n})}{f'(x_{n})} \;\;,\;\; f(x_{n}) \gt0 \;\;,\;\; f'(x_{n}) \neq0 $$ Is it true ...
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0answers
52 views

Numerical mathematics, Lagrange interpolation

I am trying to solve this problem, but I don't have any idea. Maybe it doesn't look at first sight that Lagrange interpolation can be used, but I found this problem in that chapter of Numerical ...
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0answers
9 views

Numerically/Computationally estimating parameters

I have a function $f(x)$ and I have an estimating function $\hat f(a,b,c,d;x)$ Say, I also have a scoring function $S(f,\hat f,x)$ (which could very well be mean square error) And I have some ...
1
vote
1answer
59 views

Quadrature formula on triangle

I am looking for a quadrature formula on the triangle, with points at the vertices and at the mid-edges, so 6 points, and that is exact for polynomials of degree at least 2, with weights strictly ...
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0answers
25 views

non-linear PDE finite difference approach

How to approach this equation using finite difference method ...
4
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1answer
184 views

Singularities of an integral

We have the integral : $$I(t)=-i\int_0^\infty \frac{\log\left[\frac{\sin(t\log\sqrt{1+ix})}{\log(1+ix)} \right ]-\log\left[\frac{\sin(t\log\sqrt{1-ix})}{\log(1-ix)} \right ]}{e^{2\pi x}-1} \, dx$$ I ...
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0answers
16 views

Numerically solve: system of non-linear complex valued equations

I need to solve a system of equations numerically. Definitions: $$ -1\leq \epsilon_1,\epsilon_2\leq 1 $$ $$ E_1, E_2, \omega\geq0 $$ $$ E_0 < 0 $$ $$ n=1,2,3,... $$ $$ \frac{1}{t_f-t_i} \left( ...
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0answers
34 views

Numerical Laplace Transform

I want to compute the Laplace transform of data vectors. I have tried the usual numerical software and I'm surprised to see that does not have this operation available. I wonder if there is a straight ...
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2answers
42 views

Numerical integration of divergent function

I am having trouble with the numerical integration of a divergent function. For example, \begin{equation} n= \int f(x)\,dx = \displaystyle\int \dfrac{\Theta(x-\varepsilon)\,dx}{\sqrt{x-\varepsilon}} ...
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0answers
19 views

Coefficients and synthesis of Associated Legendre Polynomials

First of all, all the Associated Legendre Polynomials (ALP) I'm mentioning below are NORMALISED according to the convention of Spherical Harmonics, and the ALPs can be accessed in Mathematica using ...
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0answers
13 views

How to optimize the repartition of samples in weighted channels?

This is more like an applied mathematics question, so my apologies if I am at the wrong place. Let S(n) be an infinite sequence of real numbers strictly growing from 0 to 1 (asymptotically). Let P be ...
2
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3answers
76 views

Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on ...
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0answers
59 views

How to do polynomial composition/substitution? (Vincent-Alesina-Galuzzi)

For the polynomial $$ p(x) = \sum_{i=0}^n c_i x^i, $$ of real coefficients and real variable, obtain the coefficient of $$ q(x) = \left(1 + x\right)^n p\left( \frac{a + b x}{1 + x} \right), $$ as ...
0
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1answer
30 views

Stepping backwards with Forward Euler?

Let us say I want to use Forward Euler scheme to solve the heat equation $$ \frac{\partial u}{\partial t} = -\frac{\partial^2 u}{\partial x^2} $$ in the domain $t \in (0, 1)$, $x \in (0,1)$ but ...
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1answer
18 views

Numerical Integration by Undetermined coefficients

The important part of my question is after the bold "Now" The method of undetermined coefficients is defined so that the error of a function $f(x)$ to be integrated is zero. I.e. $E=\int_{a}^{b} ...
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1answer
18 views

order of convergence PDE

If I have the generic PDE \begin{equation} u_t + u_x = f, \end{equation} approximated with a first order in time and a second order in space numerical scheme, how can I show that the solution ...
1
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1answer
31 views

ODE, Picard approximation of a second order equation: How do I make sure that this is correct.

I have the following problem: $$\ddot{x} + \dot{x}^2-2x=0$$ and I.V are: $x(0)=1 \qquad$ $\dot{x}(0) = 0$. and I need to find two first "Picard" approximations. I first arranged it in the form ...
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0answers
38 views

Adi-method for Diffusion-reaction equation in 2d

i'm trying to solve this pde using an adi-method (alternating-direction-implicit). $\frac{d f}{d t}=D\nabla^2_{x,y} f+Q(x,y)f+C$ After discretizing, the equation looks like this. Implicit in ...
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1answer
21 views

$O(\text{polylog}(1/\epsilon))$-time Algorithm for Numerical Integration to Within Additive $\epsilon$?

I'm trying to approximate a 1D definite integral to within an additive $\epsilon$ for a given $\epsilon$. I was wondering whether there is an $O(\text{polylog}(1/\epsilon))$-time algorithm for this. ...
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0answers
37 views

Name of numerical methods for second-order differential equation

Numerical methods that try to solve first-order differential equations of the form: $$ \frac{\partial}{\partial t} y = f(y,t) $$ are often Runge-Kutta methods, and there is a whole family of ...
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0answers
30 views

Ruling span derivation?

I have recently read a paper about the ruling span for electrical wires and they have an approximation that looks like it can be derived with mathematical intuition only. I'd like to find a derivation ...
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1answer
25 views

Finding approximation of root

AS for newton approximation of the reciprocal of the square root of 5. Does the function f(x)=1/x - 5^1/2 apply for newton method
2
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3answers
73 views

How do i solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x$?

How do I solve this equation ${\mathbb{R}}$: $3 \sin^3x+2 \cos^3x=2 \sin x+\cos x $? Note : I have tried using trigonometric transformation but it seems very complicated to get the result .. may ...
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0answers
21 views

Direct computation of $\operatorname{log}(\operatorname{cdf})$ for a normal distribution

This question is linked to the normal distribution for a random variable. The probability density function (pdf) is expressed as: \begin{equation} \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - ...
0
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1answer
54 views

Determine the roots of equation if possible

How to determine the roots of equation using numerical methods? I have this particular equation: $$\arctan(e^x)=\ln \left(\sqrt{\frac{e^{2x}}{e^{2x}+1}}\right)$$ In my solution I have that this ...
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0answers
41 views

How to take derivative of $F(u)=\sum_{i=1}^{N} \int f^2(x) u_i^q(x) dx $

I have to find the derivative of a function. Could you help me to find it $$F(u)=\sum_{i=1}^{N} \int_{\Omega} f^2(x) u_i^q(x) dx $$ where $q \ge 1$, $f(x): \Omega \to R$, $u_i$ is membership ...
1
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1answer
42 views

Underdetermined vs Overdetermined Problem

I'm trying to create a model which is of the form $$y = (a_0 + a_1l)[b_0+\sum_{m=1}^M b_m\cos(mx-\alpha_m)] [c_0 +\sum_{n=1}^N c_n\cos(nz-\beta_n)]$$ In the above system, $l$,$x$ and $z$ are ...
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0answers
16 views

Implicit numerical integration: error bound

Suppose I'm solving this equation numerically with a time step $h$: $$x''(t) = f(x)$$ Discretizing it and using implicit integration: $$x^{n+1} - 2x^n + x^{n-1} = h^2f( x^{n+1})$$ $x^{n-1}$ and ...
1
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1answer
34 views

Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$ E\leq K\frac{(b-a)^3}{24 n^2} $$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
2
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1answer
36 views

If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
1
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5answers
69 views

How to determine if 2 line segments cross?

Give two line segments, each defined by $2$ points in $x,y$ space, such as $L_1 = (x_1,y_1)-(x_2,y_2)$ and $L_2 = (x_3-y_3)-(x_4,y_4)$, and that these points are the result of sampled data (they are ...
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0answers
31 views

The equivalent of least squares, but for vectors

Given a set of poins, one can use a fitting method such as least squares to find the straight (or the parabola, or the 3rd grade equivalent) that's closest to all points at the same time (via ...
1
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0answers
23 views

How can I solve this specific set of equations?

Here are the equations: $$\sum_{k = 1}^n i_k + Y_n u_n = J \quad \quad (1)$$ $$i_1 + Y(u_1 - u_2) = J \quad \quad (2)$$ $$i_k - Y(u_{k - 1} -2u_{k} + u_{k + 1}) = 0, \quad \quad k = 2, ..., n - 2 ...
1
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1answer
74 views

Runge-Kutta method for PDE

I consider certain partial differential equation (PDE). For example, let it be heat equation $$u_t = u_{xx}$$ I want to apply numerical Runge-Kutta method for solving it. As a first step I ...
1
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0answers
42 views

Numerically stable SVD

In this question regarding SVD, it is explained why eigen decomposition of $ A^tA $ is not numerically stable compared to "direct SVD algorithms". Since the former is the algorithm I'm most familiar ...
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0answers
38 views

Using Multipule Scale Analysis to solve a non-linear differential equation

I would like to know if there are other methods to solve equations such as this one below. I don't really understand the theory behind the multiple scale analysis and why it works I understand some of ...
1
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1answer
49 views

Numerically find a potential field from gradient

I know that the gradient of a potential field/scalar field is a vector field, and given the function of the gradient I know how to integrate each component to get back the original scalar field. But ...
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0answers
21 views

Gauss-Legendre quadrature error

I'm trying to evaluate the error in Gauss-Legendre quadrature formulae on $[a,b]$. So far I have that the error is less or equal to $$ \frac{f^{(2n)}(\xi)}{(2n)!}\langle p_n,p_n \rangle, \enspace ...
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0answers
13 views

Composite trapezoid rule and trigonometric functions

I am trying to solve the problem talked about in: Trapezoid rule over trigonometric polynomials Show that the composite trapezoid rule over an equidistant partitioning with interval size ...
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0answers
27 views

How to select the number of nodes in a spline interpolation?

I am writing a program to test the precision of different methods for imputing missing data in a time series. One of the methods I am going to test is a natural cubic spline interpolation. I'll be ...
1
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1answer
21 views

Quadrature on segment

Is there a quadrature formula on the segment [0,1], such that on [0,1/2] the points and weights are symetric with respect to 1/4, on [1/2,1] they are symetric with respect to 3/4, and such that the ...
0
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1answer
14 views

Iteration Scheme Converging

I am in a class called Numerical Analysis and I have a quick question regarding iterative schemes. How would I go about finding out whether or not a certain iteration scheme converges to a unique ...
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0answers
25 views

Finding area by integration, increasing inaccuracies with complex functions?

I am looking for an explanation as to why the method of integration to find the area of function using limits provides a greater % difference between other methods (In this example Simpsons) with ...
0
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0answers
21 views

Error in Gauss-Legendre quadrature

I've tried Googleing this, but so far I haven't succeeded. Can someone point to me a webpage or book in which I can find the error estimate (detailed, not just the final formula) of the Gauss-Legendre ...