Questions tagged [na.numerical-analysis]
Numerical algorithms for problems in analysis and algebra, scientific computation
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Substitution vs elimination in solving system of linear equations [closed]
I believe that elimination is generally the preferred method to solve a system of linear equations compared with substitution.
To be precise, by substitution method on a system of linear equations, it ...
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Characterization of the behavior of the residuals in conjugate gradient
In conjugate gradient method for solving symmetric positive definite linear system $Ax=b$, which can also be regarded as a convex optimization problem $\dfrac{1}{2} x'Ax - x'b$, the $A$-norm of the ...
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Finding a branch cut or a branch point
Is there a way to find a branch cut or a branch point, through which a curve over a complex function goes, or in general in some region of complex function, say $\ln(f(z))$, using numerical methods or ...
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An adaptive stepsize approach to solve numerically ODE with stiffness using complementarity conditions
Let us considered the following system of ODEs
\begin{align*}
\dfrac{dX}{dt} = f(X), \tag{1.1}
\end{align*}
where the unknown $X\in \mathcal{D} \subset \mathbb{R}^l$ and it is stiff. However, for ...
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finding weak form of nonlinear differential equation for FEM simulation
The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
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Can the best constants in harmonic analysis be approximated in principle?
Consider the trivial example of Holder's inequality $\|f\|_p\,\|g\|_q\geq |fg|_1$ if $\frac{1}{p}+\frac{1}{q}=1, p,q\geq 1$ and $f,g$ are functions on $\mathbb{R}^n$. Let's suppose we don't know how ...
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Numerical Method Simulation for 2D Advection Diffusion Equation on Python [closed]
Here it is an Advection-Diffusion equation in 2D:
$$
\frac{\partial C}{\partial t}+U \frac{\partial C}{\partial x}+V \frac{\partial C}{\partial y}=D\left(\frac{\partial^2 C}{\partial x^2}+\frac{\...
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Numerical analytic continuation/asymptotics
I posted this question, quite a while ago, on math.stackexchange.com, here. I received an interesting answer but not sufficiently accurate for my purposes, so I'm trying here.
I have a class of ...
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Rigorous definition of space and time order of accuracy of numerical PDEs
Suppose that we are solving numerically a PDE (with a numerical scheme like this one) which involves space $x$ and time $t.$ It is a commonly seen expression in the literature that "the method ...
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Functions that have their second derivatives as a lipschitz function [migrated]
What would be an example of a function that is not $C^3$ on $[0,1]$, but has its second derivative as a Lipschitz function?
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starting methods for general linear methods
I'm self studying general linear methods and are looking for more detailed worked examples of creating starting methods following section 533 of Butchers Numerical Methods for Ordinary DiffEquations.
...
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Fast numerical integration of $\int_{[0,\:1)^d}\left|f_x(y)-g(y)\right|^p\:{\rm d}y$ for varying $x\in[0,1)^d$
Let $k\in\mathbb N$ and $y_1,\ldots,y_k\in[0,1)^d$ with $$\frac1k\sum_{i=1}^kh(y_i)\approx\int_{[0,\:1)^d}h(y)\:{\rm d}y\tag1$$ for every nice enough function $h:[0,1)^d\to\mathbb R$.
Now let $p\ge1$, ...
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Approximate piecewise linear function with finite incontinuities with polynomial on discrete points
I wish to use single polynomial to approximate a piecewise linear discontinuous function. I wish to minimize the $L^\infty$ norm between such function and original function at every discrete point of ...
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Finite difference approximation
I'm trying to find formulas for the finite difference approximation "Five-points-stencil" of the first derivative for non-constant grid spacing. It's needed for the outermost left and right ...
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Lagrange's interpolating polynomial
Let $f:[a,b]\rightarrow R$ be a function that is not $C^{(n+1)}$ on $[a,b]$ but its $n$-th derivative is a Lipschitz function? How does the Lagrange's interpolating polynomial formula change? How does ...
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What's a good approximation for the first derivative at the endpoints of given datapoints for a cubic spline interpolation?
I'm using a cubic spline interpolation for given data points. The boundary condition for the spline is that $f'(a)$ and $f'(b)$ are given (I'm using a finite difference formula $\frac{y_1-y_0}{x_1-x_0}...
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Numerical approaches to functional equations
I'm interested in finding numerical approaches to solving functional equations such as
f(xy)=f(x)+f(y),
where the equations had no derivatives or integrals, and contains arguments involving x
and y
.
...
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answers
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Find a vector in the null space of a large dense matrix, where elements in the matrix are not directly accessible
I am working with Conjugate Gradient method to solve for 𝐴𝑥=𝑏, where 𝐴 is an extremely large PSD and Singular matrix. I cannot directly access the elements of 𝐴. The only thing I can do is ...
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Linear system with sum of Kronecker products
Here and here, specific ways to address the equation in $x$, for $N=2$, are given:
$$\sum_{i=1}^N (A_i\otimes B_i)x=c$$
Is anything know about the case $N>2$?
I am looking in fact for an efficient ...
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Efficiently determining surface intersections along a line segment
Background
In general, I know how to determine the points of intersection between a surface and a line. In my case, I may have a large number of defined surfaces that may (or may not) intersect each ...
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Implementable numerical scheme for the equation $a=\text{Erf}\big(z/\sqrt{2N_{a}}\big)$
Let $z>0$ be fixed and $A$ be the set of non-increasing functions from $\mathbb R_+$ to $[0,1]$ with norm $\|\cdot\|:=\|\cdot\|_\infty$. Define by $F$ the operator on $A$ by
\begin{equation*}
F(...
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Numerical solution to some functional equation
Let $z>0$ be fixed. Consider the function $p_a: \mathbb R^2_+\to\mathbb R_+$ given as
$$
p_a(t,x):=\frac{1}{\sqrt{2\pi N_a(t)}}\left[\exp\left(-\frac{(x-z)^2}{2N_a(t)}\right)-\exp\left(-\frac{(x+z)^...
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The backward error of tridiagonal linear system $Ax=b$ by Gaussian elimination without pivoting
Let $A$ be an $n \times n$ nonsingular tridiagonal matrix having an $LU$
factorization. It can be shown that the computed solution of the linear system
$Ax = b$ using Gaussian elimination without ...
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Optimal Truncation of LDL-factorization to improve conditioning
Suppose I factored real symmetric quasi-definite $ A_0= L_0 \cdot D_0 \cdot L_0^T$ and the factorization exists, with $D$ diagonal and $L$ unit lower-triangular; and suppose $L$ and $D$ are badly ...
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Slope assertion in Cholesky on digital computers
For a real symmetric positive definite linear system
$$ A \cdot x = b, $$
solved using Choelsky with forward- and backward-substitution, we know it for the numerical approximation $\tilde{x}$ to $x$ ...
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open problem in numerical analysis [closed]
I am interested in open and current issues in numerical analysis, there are good references in this respect. Thanks for your response
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Practical symmetric equivalent to QR factorization updates
As we know, the QR-factorization $Q\cdot R=A$ of any real symmetric $n \times n$ matrix $A$ with full rank is unconditionally numerically stable. Further, when A is rank-1-updated, the factorization ...
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Locating the maximum point $x_n$ of $f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$ in $(0,1)$
I am trying to observe the behavior of $x_n \in (0,1)$ defined such that the function
\begin{equation}
f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)
\end{equation}
attains its maximum inside the ...
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Recursive formula for approximate multiple Wiener integrals
Given $m$ $d$-dimensional Brownian motion and a multi-index $(j_1,...,j_l)$ with $j_i \in \{0,1,...,m\}$ we can define the multiple Stratonovich integral
$\int_0^t \circ dW_{s_1}^{j_1}...\int_0^{s_{l-...
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What is the computational complexity of Arnoldi algorithm for diagonalization?
What is the space and time computational complexity of finding $k$ eigenvalues of an $N\times N$ matrix using the iterative Arnoldi algorithm?
I know that exact diagonalization scales like $O(N^3)$, ...
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Pressure integrated by parts in finite element method
Most FEM texts or tutorials apply FEMs on diffusion equations where the 2nd spatial derivative is integrated by parts during weak formulation. For convection diffusion equations, there is a first ...
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Optimal polynomial approximation of rational function $\frac{1}{1-x}$
I've been working on the following polynomial approximation problem. I want to find the optimal Chebyshev approximation of the rational function $\frac{1}{1-x}$ on the real interval $x\in[-\rho, \rho]$...
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1
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Numerical methods for systems of trilinear polynomials
I have some large system of particular non-linear polynomial equations:
each equation mentions at most three variables
no variable appears with a degree larger than 1.
I'm not an expert in this area ...
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1
answer
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Finding minimax approximation of a permutation equivariant polynomial
Is there any known method to approximate a given permutation-equivariant smooth function $f: \mathbb{R}^{n} \to \mathbb{R}^{n}$ as multivariable polynomial function $p: \mathbb{R}^{n} \to \mathbb{R}^{...
- 1,633
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Interpolation on Sobolev space on $[0, 1]^d$ over finite meshes
Let $\Omega = [0, 1]^d$ and suppose that $f \colon \Omega \to \mathbb{R}$ lies in order $m > d/2$ Sobolev space; i.e.,
$$
\|f\|_{H^m(\Omega)}^2 = \sum_{|\alpha| \leq m} \|D^\alpha f\|_{L^2(\Omega)}^...
- 353
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answers
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Error bounds for Sobolev space norm approximation on a finite grid
Suppose that $f : [0, 1] \to \mathbb{R}$ is an element of the order-$k$ Sobolev space,
\begin{multline}
f^{(k - 1)}~\text{is absolutely continuous},\quad \|f\|_{W^k}^2 := \int_0^1 f^{(k)}(x)^2 \, dx &...
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How to maximize certain function of hundreds variables related to correlations between sets vectors ? (and win Kaggle :))
It might be helpful for data science/bioinformatics challenge.
Consider for simplicity three rectangular matrices $Y_{true}$ , $Y_{predict0},Y_{predict1}$ of the same sizes say 70000*140.
Let us ...
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How do I know sparse identification of nonlinear dynamics (SINDy) is correct?
Here is the algorithm/technique in question, https://www.youtube.com/watch?v=NxAn0oglMVw.
How do I know that this algorithm is correct? No where in any of the papers do they talk about a proof? What ...
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A good estimation for the norm of a matrix
For a given natural number $n$, let us consider $E_n=\{0\leq k\leq n-1 : k\equiv_41 \}$. Suppose that $E_n$ including $k_1<k_2\cdots < k_m$. Consider the following matrix $A$:
$$A=\left(\cos\...
- 3,764
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Kolmogorov $\epsilon$-entropy, $n$-width, and $\epsilon$-capacity and applications
What is the relationship between Kolmogorov $\epsilon$-entropy, Kolmogorov $n$-width, and Kolmogorov $\epsilon$-capacity of a set $M$ in a metric space $X$? (The $\epsilon$-capacity here is the ...
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Uniform bound on the measure of $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ if $\Omega$ is an open bounded set with Lipschitz boundary
Let $\Omega \subset \mathbb R^d$ be an open bounded set with Lipschitz boundary. Let us consider $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ for $\delta >0$. I want to say that the measure of $\...
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How to numerically compute the operator norm of an operator acting on a matrix algebra?
Let $M_n(C)$ denote the $n\times n$ matrices with complex entries acting on the Hilbert space $C^n$. As norm on $M_n(C)$ we take the operator norm, i.e. the largest eigenvalue of its absolute value. ...
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Mathematics of sustainable development and energy sobriety in the classroom
Faculty members are encouraged to highlight the connection between the courses we teach and climate change, and raise awareness of the issue in our lectures, across subjects in my university. I am ...
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Rate of uniform approximation by piecewise constant functions
Definitions and Notation:
Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$.
For every positive integer $N$, define the ...
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Approximation in Bochner spaces
Is there any result like the Bramble-Hilbert lemma for Bochner spaces?
More specifically: let $H$ be a (e.g.) Hilbert space, $I\subset \mathbb R$ a bounded interval, and $L \in \mathcal L(H^k(I;H), Y)...
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answers
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Chebyshev-like polynomials [closed]
In some approximation problems I'm working on, the errors turned out to be polynomials of various degrees whose graphs on the interval $[-1,1]$ look like this:
As you can see, these things look a bit ...
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answers
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Direct (first-order ?) algorithm to minimize $u(x) := \|x-a\|_C + r\|x\|_p$
Fix $a \in \mathbb R^n$, $r \ge 0$, $p \in \{1,2\}$, and a positive-definite matrix $C$ of order $n$. Define $u:\mathbb R^n \to \mathbb R$ by $u(x) := \|x-a\|_C + r\|x\|_p$, where $\|z\|_C := \sqrt{z^\...
- 6,338
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Unbounded solution but bounded Euler discretization
Is there an ordinary differential equation in $\mathbb{R}^d$ induced by a gradient vector field with unbounded solutions, for which the difference equations obtained by using the forward Euler method ...
- 309
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votes
0
answers
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Identifying redundant vectors In non-negative matrix bases
I have a target non-negative matrix $X$ that I would like to factor.
I have two non-negative matrices $W$ and $H$ such that $WH = X$. In this formulation, the rows of $H$ are $L^2$ normalized and ...
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Numerically expanding a function in a rational-power "basis"
I have some scientific code which interfaces with a library which accepts real functions specified as any number of additive terms with exponential powers. For instance, it is capable of accepting ...
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