Questions tagged [numerical-calculus]
This tag is for various question on numerical calculus / numerical analysis which concerned with all aspects of the numerical solution of a problem, from the theoretical development and understanding of numerical methods to their practical implementation as reliable and efficient computer programs.
698
questions
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5
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numerically compute eigenfunctions of $a(u,v)=\langle f(\nabla u)\nabla u,\nabla v\rangle_{L^2}$
Let $D:=(0,1)^2$ and consider the nonnegative form $$a(u,v):=\langle f(\nabla u)\nabla u,\nabla v\rangle_{L^2(D;\:\mathbb R^2)}$$ for $u,v\in L^2(D)$ where $f:\mathbb R^2\to(0,\infty)$ is a smooth ...
- 13.3k
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votes
1
answer
24
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diverging determinate of numpy eigenvalues
I am trying to solve the equation
$u''=\lambda u $
via discretized matrix scheme. Therefore, as a first step, I need to compute the eigenvalues $\lambda$ using the numpy.linalg.eigvals function.
If I ...
- 3
-2
votes
0
answers
24
views
Help with theory part to be able to implement Advection diffusion into MATLAB code.
I am solving the 1-D advection-diffusion equation, but I am stuck on how to implement the formula into my code. I understand how to the the next time step when the advection term and diffusion term ...
1
vote
0
answers
23
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Quasi-Monte Carlo Integration of Probability Density With Light Tails
I want to integrate $$f(x) = (2\pi)^{-\frac{p}2}\operatorname{det}(\Sigma)^{-\frac 12}\exp\left(-\frac 12 (x-\mu)'\Sigma^{-1}(x-\mu)\right)$$ over the $p$-dimensional hypercube $[a,b]^p$. Since $p$ ...
- 517
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votes
0
answers
43
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Point in use of Taylor Series to approximate functions in an age with computers?
I hope this doesn't sound too vague or like I'm dismissing the use of Taylor Series entirely, I'm just curious about any proper real-world applications.
Many times Taylor Series are shown-off as a ...
- 53
0
votes
1
answer
46
views
Numerically compute an oscillating series
I would like to compute in a numerically stable way an oscillating series. Imagine I have a signal $C(n)$, $n\in\mathbb{N}$ which decays exponentially with $n$ e.g. $C(n) = e^{-2n}$. Also, imagine I ...
0
votes
0
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30
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Approximation of product or product of approximations?
I came to an interesting question that I cannot easily answer.
There is a uniform grid $x = {x_1, \dots, x_{i-1}, x_{i}, \dots, x_n}$. Values of two functions that are not known analytically are given ...
- 103
0
votes
1
answer
61
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A question on fixed point iteration scheme
Consider the function $f:[0,1] \to \mathbb{R}$ defined by
$$
f(x)=
\begin{cases}
2^{- \bigg\{ 1+ \bigg( \log_2 \big( \frac{1}{x}\big) \bigg)^{\frac{1}{\beta}} \bigg\}^{\beta} }& \text{for }x \in (...
- 424
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votes
0
answers
34
views
Help with Simpson rule
I have a function I need to evaluate numerically.
$$ \int^T_t c(s)^2 e^{-2 \alpha (T-s)}\mathrm{d}s$$
I know the values of $c(s)$ but I don't know its exact functional form. I believe I can use ...
- 77
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votes
0
answers
31
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Finite Difference Method for Poisson's Equation
I'm writing a Python program to solve Poisson's equation
$$ \nabla^2 u = f \quad \mathrm{on} \quad \Omega $$
with
$$ \frac{\partial u}{\partial n} = 0 \quad \mathrm{on} \quad \partial \Omega. $$
Here,...
1
vote
1
answer
59
views
A question on the order of a numerical scheme
Consider the first order intial value problem $y'(x)=-y(x),x>0,y(0)=1$ and the corresponding numerical scheme $$4 \bigg( \frac{y_{n+1}-y_{n-1}}{2h}\bigg)-3\bigg( \frac{y_{n+1}-y_{n}}{h}\bigg)=-y_n,$...
- 424
-1
votes
0
answers
43
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How to use the Young inequality to derive the following inequality？
$$\begin{aligned}
&c\|\mathbf{H}\|_{1,2}^{1/2}\|\mathbf{H}\|_{2,2}^{1/2}\|\mathbf{curl}\mathbf{H}\|_{0,2}\|A\mathbf{u}\|_{0,2} \\
&\leq\frac\nu8\|A\mathbf{u}\|_{0,2}^2+\frac1{8\sigma}\|\Delta\...
- 1
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votes
0
answers
33
views
Show that the elements of the LR decomposition for tridiagonal matrices T can be determined by the following recursive relation
Show that the elements of the LR decomposition $ T=L R $ can be expressed by tridiagonal matrices $ T $ with
can be determined by the following recursive relation:
How could you prove this statement ...
- 109
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votes
0
answers
23
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Question about divergence of fixed point iteration
Let $\phi \in C^1([a,b]), x^* \in (a,b)$ such that $\phi(x^*)=x^*$ and $|\phi^{'}(x^*)| > 1$.
Show that: $\exists \delta > 0 : 0 < |x^* - x_0| < \delta \implies |x^* -x_0| < |x^*-x_1|$ (...
- 673
1
vote
1
answer
26
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Equalise point-distance when optimising points on graph
Recently I have been learning about optimisation-techniques and built a simple "gradient-descent brachistochrone solver thingy" to try out some methods.
One thing currently still hurting the ...
- 111
0
votes
1
answer
47
views
A question that was related to the evaluation about the errors between Riemann sum and double integrals
I have a question about the error evaluation between the double integral and its' riemann sums. It seems that this formula is apperantly not zero:
$$\displaystyle\lim_{n\to\infty}n[\int_0^1 dx\int_0^1 ...
- 3
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votes
0
answers
23
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Convergence of a sequence with order of convergence $ > 1$
If $\{x_n\}_{n \geq 1} \subset \mathbb{R}, x_n \xrightarrow{n \to \infty}x^*$ with order of convergence $r > 1$, then $\{x_n\}_{n \geq 0} \subset \mathbb{R}$ converges to $x^*$ with superliniar ...
- 673
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votes
0
answers
36
views
Error Lagrange interpolation on $\mathbb{R}^2$
The Lagrange interpolation on $\mathbb{R}^2$ of $f\in\mathcal{C}^{2}([a,b]\times[c,d])$ is defined as:
$$f(x,y)\approx\sum_{m=0}^M \sum_{n=0}^{N}L_{n,m}(x,y) f(x_n,y_m),\qquad N,M\in\mathbb{N},$$
...
- 63
1
vote
1
answer
55
views
Error bound for Gaussian Quadrature
Suppose that x$_1$ , x$_2$ , ... x$_N$, are the roots of a polynomial ortoghonal to the measure w(x) in the interval [a,b] $\subset \mathbf{R} $. We know then that gaussian quadrature formula is exact ...
0
votes
0
answers
58
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Global truncation error of backward Euler method
It's often found in books that the global truncation error of the forward Euler method applied to $\dot{y} (t) = f(t, y(t))$ is given by something like
$$ \frac{\exp(LT) -1}{L} \frac{Mh}{2},$$
with $L$...
- 3,252
2
votes
0
answers
124
views
numerical integration of a function satisfying a ode
I need to numerically approximate an integral of the form $$\int_0^\tau f(X_t)\:{\rm d}t,\tag1$$ where $(X_t)_{t\ge0}$ is the solution of a SDE $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag2.$$...
- 13.3k
0
votes
0
answers
22
views
Anger-Weber function for an integer value of the order
The Anger-Weber function is defined by
$$
A_{\nu}(z) = \int_0^\infty \exp\bigl(-\nu t - z \sinh(t)\bigr) \mathrm{d}t
$$
where $\nu, z \in \mathbb{C}$ with $\Re(z) > 0$.
I am not able to numerically ...
- 1,716
2
votes
0
answers
31
views
Relation between Poisson equation and Wilson lattice gauge invariance theory
I've recently started writing a library of numerical solvers for elliptic partial differential equations, with particular focus on the Poisson equation. If one considers typical Poisson equation in ...
- 51
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0
answers
55
views
Numerical evaluation of the Schläfli integral
I'm trying to numerically evaluate
$$
S_{\nu}(z) = \int_0^\infty \exp\bigl(-\nu t - z \sinh(t)\bigr) \mathrm{d}t
$$
where $\nu, z \in \mathbb{C}$ with $\Re(z) > 0$. This integral is a part of the ...
- 1,716
0
votes
1
answer
53
views
BVP problem with conjugate gradient method
Life and death situation. I have the differential equaction $$1.1 \dfrac{d^2T}{dz} - 0.01\dfrac{dT}{dz} = 0, T(-1000) = 0.12, T(0) = -1.55$$ that im trying to solve with conjugate gradient method.
So ...
- 1
0
votes
0
answers
24
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Problem of numerical analysis of conditioning and PVI 2
"Let $a\in \mathbb{R}$, consider the PVI: $u'(t)=u_0e^{at}(a \cos(t) − \sin(t))$, $t > 0$ $u (0) = u_0$ with $u (t)=u_0e^{at}\cos(t)$ being the exact solution. Study the conditioning with ...
0
votes
1
answer
52
views
What happens to the $O(\cdot)$ terms when we implement the Leapfrog Integration algorithm in a programming language?
Integration Algorithms
The Verlet leapfrog algorithm is an economical version of the basic algorithm, in that it needs to store only one set of positions and one set of velocities for the atoms, and ...
- 1,601
1
vote
0
answers
29
views
To find a interpolating cubic spline to the points
My problem is to find a interpolating cubic spline to the points
$$\left\{(-2022,8043), (-4, 1989), (-2,1983), (0, 1977), (1, 1974), (3, 1968),(2022,-4089)\right\}$$
I know I can use a formula here or ...
- 83
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votes
0
answers
37
views
Numerically approximating the differential map?
Say I have a manifold $M$ and another manifold $N$ and a map from one to the other $\phi$.
Say I am given a point $p \in M$, I can get $\phi(p)$. Now say I can numerically approximate tangent vectors ...
- 3,250
1
vote
0
answers
33
views
Finite difference formula of second order for a fourth-order mixed partial derivative
I derived this by repeatedly using the second-order finite difference formula for the first derivative.
Two questions:
1- ) Is the following finite difference formula correct?
2- ) If so, is the order ...
- 276
0
votes
0
answers
14
views
Stop criterion for an iterative solver for laplace equation
Suppose I am iterating with Euler's forward method on finite differences like so
$$
z_{n+1}(k, l) =
\begin{cases}
z_n(k, l) + \frac{1}{4}\left(z(k - 1, l) + z(k, l -1) -4z(k, l) + z(k, l+1) + z(k + 1, ...
- 199
1
vote
0
answers
40
views
How to calculate the kernel of an integral given the original function and its product [closed]
I am trying to solve for the kernel of the following integral.
$\int_{-\infty}^{\infty}K(x,t)f(t)dt = g(x)$
I know g(x) and I know f(x) but I am unsure of how I may solve for the kernal. I am trying ...
2
votes
0
answers
30
views
How to estimate $\int_0^\tau f(X_t)\:{\rm d}t$ when $X$ is a diffusion process?
Say we have Markov processes $\left(X^{(i)}_t\right)_{t\ge0}$ with lifetime $\tau_i$ such that $\left(\left(X^{(i)}_t\right)_{t\ge0},\tau_i\right)_{i\in\mathbb N}$ is independent and identically ...
- 13.3k
15
votes
6
answers
684
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About $I(a,b)=\int_{a}^{b}\sqrt{1+x+x^2+x^3+x^4}\text{ d}x$
The following is an MCQ question, one should answer it without a calculator, within $3$ minutes.
Consider the expression
$$I(a,b)=\int_{a}^{b}\sqrt{1+x+x^2+x^3+x^4}\text{ d}x.$$
Which of the ...
- 4,878
1
vote
1
answer
58
views
If $p$ is bounded and bounded away from $0$, can we find upper and lower bound for $\|\nabla\hat p\|^2+\Delta\hat p$?
Let $d\in\mathbb R^d$ and $p:\mathbb R^d\to(0,\infty)$. Moreover, let $\sigma>0$, $$\tilde p(x):=p(\sigma x)\;\;\;\text{for }x\in\mathbb R^d$$ and $$\hat p:=\frac12\ln\tilde p.$$
Question: Can we ...
- 13.3k
0
votes
1
answer
94
views
Proving that the semi-implicit Euler method is symplectic
I'm having trouble understanding the following proof that the semi-implicit Euler method
$$\begin{cases} p^{k+1} = p^k-\Delta t \frac{\partial H}{\partial q}(p^{k+1},q^k) \\ q^{k+1} = q^k + \Delta t \...
- 1
0
votes
2
answers
77
views
Tricks in the floating point operations for better numerical results
I'm attempting to comprehend a passage from the book "Computational Modeling and Visualization of Physical Systems with Python" which I may be mentally fatigued to grasp. Here's the issue: ...
- 15
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votes
0
answers
28
views
Taylor Series - when to stop the accumulation or orders? - Numeric Analayze - rate of convergence
Let us say we have the sequence:
$$\epsilon_{n+1}=\epsilon_n-\frac{1+\epsilon_n-e^{2\epsilon_n}}{1-2e^{2\epsilon_n}}$$
In order to find the rate of convergence, I have to do taylor series here.
...
- 179
0
votes
0
answers
17
views
How to do deconvolution but with exponential cross terms?
I have an equation
$$
g(x_0, y_0)=\int\int dxdy[h(x_0-x, y_0-y)f(x, y)e^{i\omega_0y(x_0-x)}]
$$
in the equation $g$ and $h$ are known complex functions, $\omega_0$ is a known real constant and $f$ is ...
3
votes
1
answer
229
views
Approximating ODE solution with polynomial function
I am trying to decipher the method for generating non-linear frequency-modulated signals described in this paper by A. W. Doerry. On page 5, there is an algorithm described that boils down to ...
- 105
0
votes
1
answer
81
views
Why is in Matlab exp(pi * sqrt(163)/3) - 640320 = -2.3283e-10
Why is in Matlab
$$
e^{\pi\cdot\frac{\sqrt{163}}{3}} - 640320 = -2.3283 * 10^{-10}
$$
exp(pi * sqrt(163)/3) - 640320
ans =-2.3283e-10
I know that it does have ...
- 11
0
votes
1
answer
39
views
Limits & Derivatives introduction | Get the instantaneous velocity from the position function in real life
I am currently relearning calculus. (Since I did not understand the fundamentals very well).
We've just talked about how limits were relevant in the real world.
The teacher showed us a case where he ...
- 3
0
votes
0
answers
15
views
Determine numerically centrum of symmetry of sampled function
I have a samped function $f$ represented as a vector f of length N, which shall be approximately even/symmetric with respect to ...
- 348
0
votes
0
answers
21
views
Complete with continuity a piecewise function
Good morning, I have been working with piecewise functions lately.
I am trying to completethis piecewise function, in order to make it smooth, with both first and second derivative smooth as well (...
- 41
1
vote
3
answers
80
views
Rate of convergence of the given sequence
Consider the sequence: $x_n=\frac{1}{n+1}+\frac{1}{n+2}+\cdots +\frac{1}{n+n}$. Find the rate of convergence of the sequence $\{x_n\}$.
For this, first we find the limit of the sequence $\{x_n\}$.
We ...
- 12.9k
0
votes
0
answers
13
views
RK 4 simultaneous method
Help with using the Runge-Kutta 4th order simultaneous method on a system of 12 first order ODE's.
i have 12 ODE and want to solve them using RK simultaneous method.
0
votes
0
answers
64
views
How to solve this system of ODE by series expansion?
I have the following system of ordinary differential equations up to fourth order (one of the equations is like a restriction for the other) for the one variable real functions $a(t)$ and $b(t)$:
$0=\...
0
votes
0
answers
34
views
How to obtain the quadrature weights for the cotangent DVR scheme?
I would like to apply the so-called cotangent DVR method (https://doi.org/10.1016/j.chemphys.2010.07.006) for numerical integration, which uses the following basis functions:
$\chi_n(\theta) =
\begin{...
- 488
0
votes
0
answers
29
views
How to numerically solve the eigenvalues of a partial differential operator？
I have a partial differential operator of the following form:
$$L_X=A(X) \frac{\partial}{\partial X} + \frac{1}{2}D(X)\frac{\partial^2}{\partial X^2}$$
where A and B allow to be nonlinear.For example:
...
0
votes
0
answers
51
views
Solving a system of two coupled ordinary differential equations up to fourth-order
I have the following (and terrible) system of ordinary differential equations up to fourth order (one of the equations is like a restriction for the other) for the one variable real functions $a(t)$ ...