Skip to main content

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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
19 views

Zeros of Legendre Polynomial and Quadrature rule

Comsider the quadrature rule $I_{n}(f)=w_{1}f(x_{1})+w_{2}f(x_{2})+\ldots +w_{n}f(x_{n})$ for approximating $I(f)=\int_{-1}^{1}f(x)dx$ a)Suppose $x_{1},x_{2},\ldots x_{n}$ are the zeros of the ...
maths and chess's user avatar
0 votes
0 answers
25 views

given $x_0,...,x_{n}\in\mathbb{R}$ and $y_0,...,y_n\in\mathbb{R_{+}}$ then there exists a nonnegative polynomial with deg $\leq 2n$ s.t. $p(x_i)=y_i$

I have the following question: let $n \in \mathbb{N}$ and $x_0,x_1,\dots,x_n \in \mathbb{R}$ different points and $y_0,\dots,y_n \in \mathbb{R}$ nonnegative points show that there exists non-negative ...
oneneedsanswers's user avatar
2 votes
1 answer
36 views

numerical integration quadrature rule Radau formula

Consider a numerical integration rule of the form $\int_{0}^{1} f(t)dt \approx af(0)+bf(c)$ a)Find a,b,c such that this quadrature rule has highest order of precision. b)The quadrature rule above has ...
maths and chess's user avatar
1 vote
1 answer
39 views

How do I get the Jacobian of numerical intengration-methods with half steps?

in my book, a series of proofs for numerical integration works with the definition $$ F_\epsilon= \frac{\mathrm{d}}{\mathrm{d} \tilde{t}}\left(\boldsymbol{X}_{i}, \boldsymbol{V}_{i}\right)=\left[\...
alo bre's user avatar
  • 15
0 votes
0 answers
40 views

Gaussian quadrature degree of precision on arbitrary interval

Consider the quadrature $I_{n}(f)=w_{1}f(x_{1})+w_{2}f(x_{2})+\ldots +w_{n}f(x_{n})$ for approximating the integral $I(f)= \int_{-1}^{1}f(x) \,dx$. Prove that for any choices of distinct real numbers $...
maths and chess's user avatar
1 vote
0 answers
73 views

Computing the root of a polynomial that has the lowest imaginary part

Suppose that we have a polynomial $P$ of degree $n$ whose roots are known to be all complex and with distinct imaginary parts. With such conditions, $P$ should have a unique root $z_0$ such as $|\Im(...
edrezen's user avatar
  • 243
2 votes
0 answers
37 views

Solving differential equations with sharp corners in the form of Dirac Delta function

Suppose we have the following well behaved differential equations of time $t$ $$ \ddot y + (-\frac{\dot w}{w}+2w)\dot y + w^2y = w^2x.$$ By well behaved it is meant that change in $w$ and scaling the ...
Magemathician's user avatar
0 votes
0 answers
10 views

Hessenberg matrix and Householder transformations

Let $A$ be a real unreduced upper Hessenberg matrix of order $n$. a)Show that A can be factored into A=QR, where R is upper triangular matrix and $Q=H_{1}H_{2}\ldots H_{n-1}$ orthogonal with each $H_{...
maths and chess's user avatar
0 votes
1 answer
73 views

Numerical methods derivation

a) Derive a method in the form of $y_{n+1}=\alpha_{0}y_{n-1}+\alpha_{1}y_{n}+h\beta f(t_{n+1},y_{n+1})$ for the numerical solution of the initial value problem $y'=f(t,y), y(0)=y_{0}$, where $a_{0}\...
maths and chess's user avatar
1 vote
1 answer
42 views

Gaussian-quadrature-rule and composite

Consider a numerical integration rule of the form $ \int_{-1}^{1} f(x) \,dx\approx Af(-\sqrt{\frac{3}{5}})+Bf(0)+Cf(\sqrt{\frac{3}{5}}) $ Derive the composite quadrature rule resulting from the ...
maths and chess's user avatar
0 votes
1 answer
27 views

Why is the determinant of the Jacobian of symplectic integrators always 1?

My numerics books says that a symplectic integrator has the property that the determinant of $det \frac{\partial F}{\partial \xi}=1$ for the state vector $\xi = (X,V)$ for $F_\epsilon: \xi _t \...
alo bre's user avatar
  • 15
0 votes
1 answer
41 views

Convergence of Regula Falsi (false position) method

Let $f$ be a continuous function such that $f(a)f(b)<0$.Let $a_0=a$ and $b_0=b$ $$x_1:=a_0-\frac{b_0-a_0}{f(b_0)-f(a_0)}f(a_0)$$ If $f(a_0)f(x_1)<0$, set $a_1=a_0$ and $b_1=x_1$ If $f(x_1)f(b_0)&...
Arshdeep Sandhu's user avatar
-1 votes
3 answers
105 views

numerical analysis convergence, Newton iteration [closed]

Consider Newton s method for finding a root of $f(x)=x^{4}-9$. Prove that for any initial point $x_{0}\in (\sqrt[4]{6},100)$, the number sequence ${x_{n}}$ generated by Newton s iteration converges to ...
maths and chess's user avatar
0 votes
0 answers
42 views

How to integrate division of a multivariate/multivariable polynomial with respect to its variables?

as the question suggests, I have a multivariate/multivariable polynomial and I have them in a division. In example, the first multivariate polynomial is in the dividend and the second multivariate ...
aaparker's user avatar
0 votes
0 answers
29 views

Two definitions of order of convergence

Let ${x_n}$ be a sequence which converges to $x$. According to Richard L. Burden, the order of convergence is defined as follows: A positive real number $\alpha$ is said to be order of convergence of $...
Arshdeep Sandhu's user avatar
1 vote
0 answers
63 views

Estimation of a sum by an integral in Shor's algorithm

I am reading Peter W. Shor's original paper on his algorithm for integer factorisation. On p.17 of the paper he is trying estimate a sum by an integral. He claimed that the expression inside the ...
eurekamath's user avatar
0 votes
0 answers
60 views

Numerical differentiation of sharp function while solving PDE

Sorry for a silly question. I'm calculating a second-order partial differential equation of probability density function. My equation is a form of $$ \frac{\partial}{\partial t}P(x,t)=P(x,t)+x\frac{\...
Juhee Lee's user avatar
1 vote
0 answers
16 views

Computing the multiple-integral of a $n$ variable symmetric function

I'm interested in evaluating the following integral analytically, numerically, or using a combination of both. Let $$ I_n(\rho) = 2^n\int_{\mathbb{R}_+^n} \sqrt{1+ \rho^2 \left(\prod_{k=1}^n x_k^{-2}\...
P.S. Dester's user avatar
  • 1,053
2 votes
1 answer
136 views

Lanczos convergence for symmetric matrix with eigenvalues $1,2,\cdots,2,100$

Suppose that $A$ is symmetric with eigenvalues $1,2,\cdots,2,100 \in \mathbb{R}^{100x100}$ and $b \in \mathbb{R}^{100}$ obtained by normalizing a standard normal random vector. Show that 1 and 100 are ...
jacopoburelli's user avatar
2 votes
0 answers
94 views

How to simplify the following stability criteria?

I'm trying to understand the mathematics explained in the following video. Basically, we want to identify the constraints on the parameters $k_1$ $k_2$ and $T$ to have a stable second-order system ...
Breeky's user avatar
  • 33
0 votes
0 answers
71 views

Laplace's Approximation for Integral from 0 to Infinity

Background I am working on a problem where I need to evaluate an integral from 0 to infinity and compare the numerical solution to an analytical approximation using Laplace's method. However, I am ...
Alireza's user avatar
  • 309
1 vote
0 answers
87 views

Hard/Interesting analytical problem to solve: $0 = a_{1} \ln(1+a_{2}e^{a_{3}x^{2}+a_{4}x+a_{5}})+a_{6}x^{2}+a_{7}x+a_{8}$

Is there some trick or approach I'm missing in attempting to solve \ref{1}, or better \ref{2}, analytically. $$0 = a_{1}\ln(1+a_{2}e^{a_{3}x^{2}+a_{4}x+a_{5}})+a_{6}x^{2}+a_{7}x+a_{8} \tag{1}\label{1}$...
Mitternachtian's user avatar
0 votes
0 answers
53 views

Is there a numerical integration method to choose the nodes wisely according to the analytically known integrand?

When solving a numerical integration such as $$ \int_a^b f(x)\,\mathrm{d}x \approx \sum_i w_i f(x_i) $$ with analytically known $f(x)$ and definite $[a,b]$. For most quadrature methods, the nodes $\{...
BALKIN's user avatar
  • 1
0 votes
0 answers
19 views

Stability of multistep methods that contain second derivative

Consider the numerical method $y_{n+1}=y_{n}+\frac{h}{2}[\frac{dy_{n}}{dt}+\frac{dy_{n+1}}{dt}]+\frac{h^{2}}{12}[\frac{d^{2}y_{n}}{dt^{2}}-\frac{d^{2}y_{n+1}}{dt^{2}}]$ where $\frac{d}{dt}$ is the ...
maths and chess's user avatar
1 vote
0 answers
24 views

Local approximation of geodesic distance

Consider a surface $S\subset \mathbb{R}^3$ which is the graph of a function $Z(x,y)$ and two points $p,q\in S$ which are "very close to each other". Is there a simple (and computationally ...
UCL's user avatar
  • 381
10 votes
1 answer
553 views

Numerically compute and clear divergence of discrete vector field

I have a fluid simulation that represents velocity as a vector field in a grid of cells. The cells all have the same width and the same height, but the height is not necessarily equal to the width. I ...
Gyoo's user avatar
  • 409
0 votes
0 answers
53 views

Numerical Inverse of an integral

Let $f: \mathbb{R} \to (0, \infty)$ be a continuous function. Define $F: \mathbb{R} \to (0, \infty)$ by: \begin{equation*} F(x) = \int_{-\infty}^x f(t)dt \end{equation*} Clearly, $F$ is strictly ...
温泽海's user avatar
  • 2,497
1 vote
1 answer
56 views

Double integrals: strange accuracy behavior

In double integrals: when x-y region is rectangular as in $$\int_0^3\int_0^3(x^2+y^2) \, dy \, dx $$ accuracy of Simpson 1/3 rule is better than that of Trapezoidal rule which in turn is better than ...
Ahmed's user avatar
  • 81
0 votes
2 answers
60 views

Explicit formula representing the integral $\int_{-\infty}^\infty x^2\Phi(bx+a)\varphi(x)\mathrm d x$

Let $\varphi$ (resp. $\Phi$) be the pdf (resp. cdf) of the standard normal distribution and let $a$ and $a$ be real numbers with $b > 0$. Question. Does the integral $$ \int_{-\infty}^\infty x^2\...
dohmatob's user avatar
  • 9,575
3 votes
1 answer
160 views

Problem with Newton's method (numerical analysis)

I am not understanding how to proceed with this exercise, which asks me to solve $f(x) = 0$ by using Newton's method. It asks me to study the convergence of the sequences $x_k$ (built with Newton's ...
Heidegger's user avatar
  • 3,492
3 votes
1 answer
126 views

Trapezoidal Rule on Infinitely Differentiable Periodic Functions

If I understand it correctly, the Euler-Maclaurin summation formula states that for a periodic and infinitely differentiable function, the error of the trapezoidal rule of the numerical integration of ...
velut luna's user avatar
  • 10.1k
3 votes
2 answers
88 views

Books on numerical integration

I am looking for a book on numerical integration [ numerical analysis ] that reflects the current state of the art. I have a copy of Davis and Rabinowitz , but it was written more than 50 years ago. ...
Arin Chaudhuri's user avatar
1 vote
0 answers
50 views

Numerical integration with modified Bessel function of second kind

I am working with the so-called screened Poisson PDE, whose solutions in two-dimensions are given in terms of the modified Bessel function of the second kind, $K_0$, for Dirichlet boundary conditions ...
Woe's user avatar
  • 111
1 vote
0 answers
31 views

Integrating a function over the surface of a unit hypersphere

Suppose I have a function $f(\mathbf{x})$ across the surface of the unit hypersphere, where $\mathbf{x}=(x_1,...,x_d)'$ are the hyperspherical coordinates of a point on the surface of the unit ...
Ron Snow's user avatar
  • 265
0 votes
0 answers
48 views

Consistency of Runge-Kutta methods

Consider the Runge-Kutta method given by \begin{equation*} y_{n+1} = y_n + \Delta t \phi(t_n,y_n,\Delta t), \end{equation*} with \begin{equation*} \phi(t_n, y_n, \Delta t) = \sum_{i=1}^s b_i ...
Somestudent01's user avatar
1 vote
0 answers
69 views

Help understanding proof of error bound on Simpson's quadrature rule

I have found the following proof of the error bound for Simpson's quadrature rule: Using Newton's interpolation method, we derive a cubic polynomial $p_3(x)$ that interpolates $f(x)$ at the points $a, ...
codeing_monkey's user avatar
1 vote
1 answer
38 views

Limit of a Succession [closed]

$\lim _{n\to \infty }\left(\left(1+\sqrt{16n-3}-4\sqrt{n}\right)^{2\sqrt{n+7}}\right)$ Anyone knows if its ok the result $\frac{1}{\sqrt[4]{e^3}}$ This is my resolution First Image Second image Thanks ...
lucasg638's user avatar
0 votes
1 answer
39 views

Green's Formula for vector fields in the Navier Stokes Weak Formulation

I am currently studying the weak formulation of the Navier-Stokes equations and came across the following equation: \begin{equation} \int_{\Omega} \mathbf{v} \cdot \Delta \mathbf{u} \, dx = -\int_{\...
Luigi's user avatar
  • 75
3 votes
0 answers
52 views

Is Hutchinson's trick $\operatorname E\left[\langle V,AV\rangle\right]=\sigma^2\operatorname{tr}A$ of any practical use?

Let $d\in\mathbb N$ and $A\in\mathbb R^d$. Hutchinson's trick is the easy to prove observation that if $(\Omega,\mathcal A,\operatorname P)$ is a probability space and $(V_1,\ldots,V_d)$ is a real-...
0xbadf00d's user avatar
  • 13.9k
1 vote
1 answer
51 views

Question on Lagrange series inversion proof

I am stuck in understanding a passage of a proof I have found for Lagrange series inversion formula. The theorem is proven in the following form: Consider the equation $x=y+\epsilon f(x)$ where $f$ ...
ebenezer's user avatar
  • 121
1 vote
0 answers
36 views

Stable timestep criterion for variable density acoustic wave equation

As part of my PhD, I am implementing the isotropic variable density acoustic wave equation numerically with finite-difference and I have a question regarding its stable timestep criterion. The ...
Anon's user avatar
  • 11
0 votes
0 answers
33 views

Numerical solution of Perona-Malik equation: How to handle the boundary properly?

In the paper Perona-Malik equation and its numerical properties, the following PDE is considered: The $u_0$ I'm (and so is the author) interested in is given by an image and hence decomposes into ...
0xbadf00d's user avatar
  • 13.9k
1 vote
0 answers
44 views

Number of arithmetic operations (multiplications and divisions) needed for the Jordan elimination method

I have an assignment to find the number of operations (only multiplications and divisions) needed to solve $N \times N$ linear equation while using Jordan Elimination method. I thought about it in two ...
Nabil El Houssein's user avatar
0 votes
0 answers
29 views

Spatial discretization of the PDE $\partial_tu=\nabla\cdot\kappa(t,\nabla u)\nabla u$ for image processing

The question is basically in the title. I want to evolve an image $u_0$ (i.e. a $n_1\times n_2$ resolution set of discrete values in $[0,1)$, which is a discrete function on $\{0,\ldots,n_1-1\}\times\{...
0xbadf00d's user avatar
  • 13.9k
3 votes
1 answer
56 views

How to choose initial (x0,y0) point to approximate a solution of a system of non-linear equations using newton method?

I'm studying Newton's method for solving a system of non-linear equations. A system of non-linear equations, whose (one of the) solutions is ($x^*$, $y^*$): \begin{equation*} \left\{ \begin{alignedat}{...
imensy dy gordy's user avatar
0 votes
2 answers
89 views

How to calculate the ratio of convergence for Euler's, Gauss' and Viète's approximation of $\pi$?

Let $\sqrt{6\sum_{k=1}^\infty{\frac{1}{k^2}}}$ be Euler's approximation of $\pi$; $\lim_{n\rightarrow\infty}\frac{2}{g_n}$ Gauss approximation of $\pi$; and $2\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\...
Marina's user avatar
  • 1
2 votes
0 answers
39 views

Derive implicit method from Butcher tableau

\begin{align*} \begin{array}{c|cc} 0 &0 &0 \\ 1 & \frac{1}{2} & \frac{1}{2} \\ \hline & \frac{1}{2} & \frac{1}{2} \end{array} \end{...
uga09492's user avatar
2 votes
1 answer
72 views

FEM for non linear PDEs

I am looking for an easy but rigorous reference for FEM methods for non linear PDEs like the p-laplace equattion or non linear heat equation ect. Can one recommend me a good exposition of this topic ...
Furkan's user avatar
  • 69
2 votes
0 answers
65 views

Help understanding how to apply IMEX methods to one-dimensional PDEs

I need to compute a solution of the following PDE: $$\frac{\partial u}{\partial t} + v \frac{\partial u}{\partial x} = 0$$ For didactic purposes, I need to use an IMEX method. The point is no one ever ...
EleDan's user avatar
  • 21
1 vote
0 answers
42 views

Solving the double compounding integral

I have the following double integral which I would like to solve: $$\int_{0}^{100} f_S(\sigma) \sigma \int_{-\infty}^{\infty} r e^{-\frac{(r-k\sigma)^2}{2\sigma^2}} \frac{1}{\sigma\sqrt{2\pi}} \, dr \,...
Dmitry's user avatar
  • 11

1
2 3 4 5
15