Questions tagged [numerical-methods]
Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.
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Connection between collocation methods and quadrature rules
I know what a collocation method is and how it can be useful for the direct transcription of optional control problems, using specific collocation points.
I also know what a quadrature rule is and how ...
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20
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Numerically stable form for vector rejection between two nearly parallel vectors?
Given two vectors $\vec{M}$ and $\vec{N}$, I want to compute the rejection between them : $oproj(\vec{N}, \vec{M}) = \vec{N}-(\vec{M}\cdot\vec{N})\vec{M}$. For my purposes both input vectors are unit ...
- 573
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Transform 2d finite element into standard 2d element
A certain three node triangular element has nodes with coordinate as follows:
$$1:(0.13,0.01)\quad2:(0.25,0.06)\quad3:(0.13,0.13)$$
Deduce a transformation that will map this element into the standard ...
- 97
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Principal components of Vandermonde matrix
Recall that the Vandermonde matrix of a collection $\{x_0,\ldots,x_m\}$ of points is
$$ V = \begin{pmatrix} 1 & x_0 & x_0^2 & \cdots & x_0^n \\
1 & x_1 & x_1^2 & \cdots &...
- 269
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Alternating-direction implicit method for Laplace's initial value problem
Problem 1. Consider the Laplace's initial value problem
$$
\begin{gathered}
\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0,0<x, y<4 \\
u(x, 0)=u(0, y)=u(4, y)=0, u(x, 4)=\...
- 97
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Weird hexagons distribution of singular vectors of random matrices.
I ran the following numerical experiment:
Sample 10⁶ random Gaussian 2×2 matrices ($A_{ij}∼𝓝(0, 1)$)
Compute SVD $A = UΣV^⊤$. Let $u_\max, v_\max$ be the singular vectors of the maximum singular ...
- 11.3k
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Calculating $\cos(1.1)$ with an absolute error smaller than $10^{-4}$
My goal is to calculate $\mathrm{cos(1.1)}$ with an absolute error smaller than $\mathrm{10^{-4}}$ using the following relations:
$\mathrm{cos^{2}(x)+sin^{2}(x)=1}$
$\mathrm{sin(x)=\sum_{k=1}^{\...
- 45
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How to "speed up" a series $\displaystyle \sum_{k=0}^{\infty}a_k=\ell$
Introduction
I was numerically calculating some limits of functions and series and I noticed a disparity:
I have a function $f(x)$ such that: $$\displaystyle\lim_{x\to\infty}f(x)=\ell$$ and I want to ...
- 2,391
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Solve the system of equations $x_1+10x_2-x_3=3,2x_1+3x_2+20x_3=7,10x_1-x_2+2x_3=4$ using the Gauss-Elimination with partial pivoting.
Solve the system of equations $$x_1+10x_2-x_3=3,$$$$2x_1+3x_2+20x_3=7,$$$$10x_1-x_2+2x_3=4$$ using the Gauss-Elimination with partial pivoting.
I tried solving the problem as follows:
We have the ...
- 1,887
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Existence of paths, homotopy and functions
Consider the homotopy $h(t,x)=tf(x)+(1-t)g(x)$ in which $f(x)=x^2-5x+6$, and $g(x)=x^2-1$.Show that there exists no path connecting a root of $g$ to a root of $f$.
I assumed there exists a path ...
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Are there any resources for numerical methods in extracting coefficients from generating functions?
Oftentimes in combinatorics, one can derive a closed form for a generating function, say $F(z)$, describing some scenario. From this function, we can extract the coefficient of $z^k$ to arrive at an ...
- 107
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Series with more terms converges more slowly than with lesser terms
I was working with the Laplace differential equation with certain boundary conditions, problem that yields the following analytic solution:
$$
\mathbb{V}(x,y)= \mathbb{V}_o\frac{4}{\pi}\sum^\infty_{n=...
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Finding groups of number that satisfys certian conditions and minimize a certain function of numbers in these groups.
Say that we have $3$ groups of $10$ numbers:
$$
g_{1}= \left\{x_1,x_2,...,x_{10} \right\}, g_{2}= \left\{y_1,y_2,...,y_{10} \right\}, g_{3}= \left\{z_1,z_2,...,z_{10} \right\}.
$$
The sum of each ...
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Global truncation error of backward Euler method
It's often found in books that the global truncation error of the forward Euler method applied to $\dot{y} (t) = f(t, y(t))$ is given by something like
$$ \frac{\exp(LT) -1}{L} \frac{Mh}{2},$$
with $L$...
- 3,170
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Numerical method for calculating large discrete harmonic function
I am working on a problem, and for testing purposes, I want to do the following.
Let be given a sparse graph $G(V,E)$ with edge weight matrix $W$, and $|V|$ large (anywhere from $10^3$ to $10^6$). We ...
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Generating uniform grids
I have the following first order differential equation:
$$0.05u'+2xu=0.05^2e^{5-20x^2}$$ for $x \in [0,1]$, and $u(0)=0$.
then I want to approach the differential equation with a Taylor series at ...
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+50
Flaw in proof about an exact quadrature method
Consider the quadrature method
$$
\int_{-1}^1f(x)dx \approx \sum_{k=0}^N w_kf(x_k),
$$
where $x_0=-1, x_N=1$, and $x_1,\ldots,x_{N-1}$ are the roots of the derivative of the degree-$N$ Legendre ...
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How does one converge to an SDE from a discrete approximation?
It is well known that Euler approximation of an ODE
$$dx(t) = f(t,x(t))dt$$
can be defined as follows, for a uniform partition of coarseness $1/N$ over $[0,1]$,
$$ x_{t_{n+1}} = x_{t_n} + \frac{1}{N} ...
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Simple iterative method to solve $X-\alpha X^{-1} = A$?
The is a simple iteration to compute $A^{-1}$ by applying Newton's method to
$$X^{-1} = A,$$
namely the iteration
$$X_{n+1} = X_n(2I - AX_n).$$
Is there is similarly simple iteration to solve the ...
- 11.6k
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how to find out the matrix for parametric cubic spline for natural spline boundary condition. [closed]
Write Python programs to approximate the sine curve between 0 and 2 using curve fitting by
(a) a cubic spline;
For the case of the cubic spline, you may consider the use of “natural splines” which ...
- 1
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41
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Product between matrix-polynomial and vector [closed]
I was wondering if it is possible to optimize the evalutaion of the product of a matrix polynomial and a vector.
$$ \vec{y} = \left( \sum_{i=0}^{n}a_iM^i \right)\vec{x}$$
Matrix size is maybe ...
- 31
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How to approximately diagonalize a special symmetric hermitian matrix?
Given a hermitian matrix $H$ as follows:
\begin{equation}
H =
\begin{bmatrix}
H^1 & V^{12} \\
V^{21} & H^2
\end{bmatrix}.
\end{equation}
Here, $H^1,H^2\in\mathbb{C}^{N\times N}$ ...
- 1
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Proof of the convergence of the Rayleigh-Ritz Method
In this article The convergence of the Rayleigh-Ritz Method in quantum chemistry by Bruno Klahn & Werner A. Bingel they have at page 11
Let $H_B$ be that Hilbert space which can be obtained as the ...
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Computing ratio of two sums
I'm interested in computing
$$ f(s) := \frac{ \sum_{n=0}^{\infty} a_n s^n } { \sum_{n=0}^{\infty} b_n s^n } $$
for some given $s \in \mathbb{C}$, where the power series in the numerator and ...
- 2,118
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Centered, forward and backward approximation
For these approximations, do we simply plug in the values? If someone could confirm, it would be highly appreciated
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Combine 1D FEM tent basis piecewise functions to a single function
The tent basis functions
$$\phi_i(x) = \begin{cases} \frac{x-x_{i-1}}{x_i - x_{i-1}}; \quad x \in [x_{i-1}, x_i]\\ \frac{x_{i+1}-x}{x_{i+1} - x_{i}}; \quad x \in [x_{i}, x_{i+1}]\\ 0; \quad \text { ...
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Fast addition of logarithmic values
Given two values $\log(a)$ and $\log(b)$ of complex values $a$ and $b$. Is there a numerically fast way to compute $\log(a + b)$ (on a CPU)?
I'm aware that, $\log(a + b) = \log(a) + \log(1 + \exp(\log(...
- 577
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How to prove a Big-O bound
In "Numerical Analysis: Mathematics of Scientific Computing" (3rd ed), chapter 1.2, the author says that $sin x = х - \frac{x^3}{6} + \mathcal{O}(х^5) \quad (х \rightarrow 0)$. This means ...
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numerical integration of a function satisfying a ode
I need to numerically approximate an integral of the form $$\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.$$...
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What does "anchoring one of the coefficients" in an optimization problem with a polynomial basis mean?
I am trying to solve an optimization problem, and I am looking for coefficients of a polynomial basis function. The problem that I am working on is of the following form:
$$F(y) = \inf_{a\in\mathbb{R}^...
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Computational limits for nonlinear solver accuracy
Recently I've been curious about the following question. Suppose that $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is a nonlinear convex function and we seek to minimize it by Newton's method. That is we ...
- 2,080
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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 ...
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Cholesky Decomposition of sum of matrix
Consider a positive definite matrix $A=B+C$, where $B$ and $C$ are both semi-positive definite and $Im(C)=Im(B)^{\perp}$. If $A$ has a Cholesky decomposition $A=LL^{T}$, then whether the matrix $L$ ...
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Does $\approx$ have a formal definition?
I recently came across a problem in a Numerical Algorithms (Computer Science) class that asked me to prove the Trapezoidal rule of integration.
Derive the trapezoidal rule for numerical integration ...
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Nonlinear second order partial differential equation Integration
I have the following nonlinear second order partial differential equation:
$\dfrac{\partial^2}{\partial t^2} \log(1 + u) = \nabla^2 u.$
My question is how can i use a finite differences scheme to ...
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Why Spurious Eigenvalues comes from mixed states?
In this book Many-Electron Approaches in Physics, Chemistry and Mathematics they have the following definition at page 36
Definition 1 (Spurious spectrum) A real number $\lambda \in(-1,1)$ is called ...
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Does the mean area of triangles with equal perimeter $p$ and circumradius $R$ have a local minima at $p=4R$?
Definition: Isoperimetric triangles are triangles which have the same perimeter and the same circumradius.
Isoperimetric area curve: The largest perimeter of a triangle that can be inscribed in a ...
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Convergence of the Jacobi method
Consider this question
The function $u(x)=x(x-1), 0 \leq x \leq 1$, is defined by the equations $u^{\prime \prime}(x)=2,0 \leq x \leq 1$, and $u(0)=u(1)=0$. A difference approximation to the ...
- 2,072
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Rate of convergence of fourier series for non periodic functions
Consider
Suppose that an analytic function $f$ on $[-1,1]$ is not periodic, yet $f(-1)=f(+1)$ and $f^{\prime}(-1)=f^{\prime}(1)$. Integrating by parts the Fourier coefficients $\hat{f}_n$ show that $\...
- 2,072
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Clarification on what the question wants me to do
Consider
Let the Gauss-Seidel method be applied to the equations $A \boldsymbol{x}=\boldsymbol{b}$ when $A$ is the nonsymmetric $2 \times 2$ matrix
$$
A=\left[\begin{array}{cc}
10 & -3 \\
3 & ...
- 2,072
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Suggestion for the numerical solution of a nonlinear pde system.
I am faced with the following system of coupled nonlinear partial differential equations
$$
\begin{array}{lcccl}
\varphi_{tt} &-& a_1\varphi_{xx} &+& a_2\varphi_t &=& a_3\sin{\...
- 183
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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
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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 ...
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Interpolation, advantages and disadvantages
So during our numerics course we learned a few interpolation methods Aitken/Neville, divided differences,Lagrange and the Vandermonde matrix. How these work is clear to me for the most part, I'm just ...
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Approximating $\frac{1}{(1+x)^2}$ in $L^2$ norm with polynomial of degree $1$
I'm currently looking at an exercise where I'm supposed to get the best approximation of $ \frac{1}{1+x^2}$ regarding $ ||f||= \sqrt{\int_{-1}^{1}f(x)^2dx}$ with a polynomial $p(x)=ax+b$. So my idea ...
- 11
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2
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57
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Runge Kutta Method for in $1+1$ dimension
Given a partial differential equation
$\partial_t u(t,x) = F(t,x,u,u')$
Suppose I know the functions $u(t_0,x)$ and $u'(t_0,x)$ at some point $t_0$ for all $x$. In order to obtain the function $u$ at ...
- 612
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Choosing correct parameters in a model with hamiltonian equation
I am working on the hamiltonian of a system related to the extension of the Potts Model which is Cellular Potts Model. The total hamiltonian of the system is:
$$
H = H_1 + H_2
$$
$$
H_1 = - J \sum_{\...
- 3
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1
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41
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interpolation numerical methods
I have translated this question into English so excuse me if it is not proper.
I am struggling to figure out how to solve the following question any directions would be greatly appreciated
Let $F$ be ...
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Does this method theoretically can help to estimate the optimal step size in a neural network training? [closed]
In my previous post Questions regarding backward propagation. It is understood the error term of a 3 node neural network can be represent by taylor series with adjustable w1 and w2 and
$\text{error}=\...
- 15
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2-dimensional Gauss-Hermite quadrature?
I have to implement a numerical algorithm that can calculate following integral as fast as possible:
$$
I=\iint_{\mathbb{R}^2}dx \,dy\, e^{-\left(x^2+y^2\right)} f\left(x,y\right)
$$
where the ...
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