# Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

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### Projection estimate for finite element basis?

Let $\mathcal{T}_n$ be equidistant mesh of $(0,1)$ with $n+ 1$ nodes, $X_n$ be corresponding finite element space, $P_n$ be the orthogonal projection mapping $L^2(0,1)$ onto $X_n$. We ...
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### One-sided Jacobi SVD and Divide&Conquer SVD stability and cost [closed]

I'm studying SVD, in particularly the Jacobi SVD and Divide&Conquer SVD algorithms. I can't find anything on the stability and error analysis on these methods. Also can someone show show me what's ...
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### Rate of convergence of Padé approximants

Let $f$ be an entire function of order $1$. Two questions: 1) Can one assert that the diagonal Padé approximants converge to $f$ (pointwise or uniformly over compacts of $\mathbb C$)? 2) if yes, can ...
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### Discrete curve-shortening flow – numerical implementation

I need to investigate the properties of open curves which evolve according to the standard curve-shortening flow (Wikipedia link), but with fixed extremes as boundaries (si it should converge to the ...
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### Intrinsic numerical methods on Riemannian manifolds

I am interested in numerical methods for ordinary differential equations on a Riemannian manifold $M$. The general form of such an equation is $\dot x(t)=V(x(t)), x(0)=x_0 \in M$, where $V$ is a ...
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### Failure in numerical experiment of singular integral equation?

Define \begin{equation} G(t,s) := - \frac{1}{2\pi} \left[\ln \left(4 \sin^2 \frac{t-s}{2}\right) -1 \right] \quad (t \neq s) \end{equation} and \begin{equation} K_0 \Psi := \int^{2\pi}_0 G(t,s) \Psi(...
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### Solving numerically a linear ODE

I start by saying that I do not have a strong background in numerical analysis, so I may miss some basic things or make trivial mistakes. Motivated by some problems in digital signal processing, I ...
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### ADMM for solving linear systems

I would like to use ADMM for solving $Mx=b$, where $M\in \mathbb{R}^{R\times R}$ is symmetric and positive definite. I know that a lot of methods will do for me in this case, but I'm specially ...
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### Consistency results of optimal control solutions using direct transcription

I have read a number of books on discrete time solutions to continuous time optimal control problems. What is not clear to me is what consistency results exist with respect to showing the discretized ...
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### Compute weights (analytically or numerically) so that they minimize an integral

I have a measurable function $f:E\to[0,\infty)$, measurable weights $w_1,\ldots,w_k:E\to[0,1]$ with $\sum_{i=1}^kw_i=1$, positive probability densities $q_1,\ldots,q_k:E\to[0,\infty)$ and measurable ...
I am trying to solve an ordinary differential equation (ODE) using an operator splitting approach: $\frac{\partial f}{\partial t} = A(f) + B(f)$ Let's assume that $A$ and $B$ are very simple: $\... 0answers 600 views ### I found a (probably new) family of real analytic closed Bezier-like curves; is it publishable? Given$n$distinct points$\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$in the plane$\mathbb{R}^2$, I associate a real analytic map:$f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$with the following ... 0answers 74 views ### Existence of the inverse Fourier transform, Carr Madan I have a function$C_T(k)$that is not$L_1$, because its limit in negative infinity is a constant. So I dampened it by$ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ... 0answers 204 views ### Numerical analysis with p-adic numbers How should one go about doing numerical analysis with$p$-adic numbers? By that I mean, how should one go about implementing numerical integration (using analogues of Newton-Cotes or perhaps Gaussian ... 0answers 117 views ### Rigorous error estimate for semi-discrete heat equation in bounded domain Let$\Omega$be a bounded Lipschitz domain in$\mathbb R^N$and$u_hbe a solution of \begin{cases} \partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\ u_h=0 &\text{ in } \... 0answers 33 views ### Bounding the working precision required in Spouge's Approximation Spouge's approximation for the gamma function is \Gamma(z+1) = (z+a)^{z+\frac{1}{2}}e^{-z-a} \left(c_0 + \sum_{k=1}^{a-1} \frac{c_k}{z+k} + \epsilon_a(z) \right) where the coefficients are given ... 0answers 86 views ### Hardness results for approximating Hölder continuous functions Let f \in \mathrm{Lip}^{L,\alpha}[a,b], and let f_{h} \in C^{L} be a spline which interpolates f at a + ih. Then standard theorems show that \begin{align*} \left\| f - f_{h} \right\|_{\infty} \... 1answer 105 views ### History- calculating convolution by tabular method I often see a trick for calculating convolution of discrete data by a so-called Tabular method. There are a lot of Youtube videos and many Indian textbooks on Signal Processing [Books].1 Basically, ... 0answers 24 views ### Reference request: numerical methods for HJB free boundary problems Suppose r: \mathbb{R}^{d+1}\to \mathbb{R}, \ g: \mathbb{R}^d \to \mathbb R,\ b: \mathbb{R}^{d+1}\to \mathbb{R}^d and \sigma: \mathbb{R}^d \to \mathbb R^d, d \ge 2, and consider an optimal ... 1answer 157 views ### Non-polynomial splines, a non-linear problem I'm looking for references on how to construct spline-like functions from a basis that does not include piecewise polynomials. To be specific, given a class of functions such as "decaying ... 0answers 188 views ### analytic approximations of the min and max operators Question: What is the state of the art on analytic approximations of \min and \max? My hunch is that numerical analysts probably have a better solution than the one I propose here. For any \... 1answer 143 views ### Numerical problems floating-point arithmetic I am trying to calculate the following function in floating-point arithmetic.f(c,z)=\frac{(c-1)z}{(z-1)^2}\left( \sum_{k=2}^{c-1}\frac{1}{c-k}\left(\frac{z-1}{z}\right)^k-\left(\frac{z-1}{z}\right)... 0answers 82 views ### Smallest eigenvalue for large kernel matrix I am interested in the the asymptotics of the minimum eigenvalue\lambda_n^n$of a class of kernel matrix$P = [ K(x_i - x_j) ]_{i,j}$, with$x_i$equally spaced in the unit cube of$\mathbb{R}^d$. ... 1answer 149 views ### A numerical calculation for an integral I am interested in the numerical calculation of $$F(\eta)=\frac{2}{π}\int_0^{+\infty}\sin(t\eta-t^3)\frac{dt}{t}\quad\text{for \eta\ge 0}.$$ I believe that the function$F$is bounded, but I do ... 0answers 125 views ### Accuracy of Richardson's error estimate in the presence of rounding errors Richardson extrapolation is a well known technique in scientific computing. It is used to compute error estimates when our approximations can be viewed as a function of a parameter$h$. Common ... 1answer 103 views ### summation of oscillating functions Consider series of the form$S=\sum_{n\ge1}f(n)P(n)$, where$f$is some smooth function, and$P$is a periodic or quasi-periodic function (e.g.,$P$can be a trigonometric function, so$S$a Fourier ... 0answers 111 views ### Why is this identity about commutators of Lie derivatives true? I am reading the paper "On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations" by Lubich. On page 2147 the author claims $$[T,V](\psi) = T'(\psi) V(\psi) - V'(\psi) T(... 0answers 51 views ### Convergence of Quasi-Newton method with fixed derivative Consider the Newton iteration x^{(k+1)} = x^{(k)} - DF( x^{(k)} )^{-1} \cdot F( x^{(k)} ) to find a zero of a function F : \mathbb R^k \rightarrow \mathbb R^k. If we freeze the first derivative,... 1answer 92 views ### Non-trivial examples of regular Lagrangian flow in BV case What is a concrete example of BV vector field v with \mathrm{div}\, v = 0 that makes Ambrosio's theory of regular Lagrangian flow relevant? With concrete I mean that we can compute the flow ... 1answer 338 views ### Taylor expansion of exponential of a Lie derivative In this paper on page 8 the author claims that the Taylor expansion for the expression e^{tD_V} where D_V is the Lie derivative with respect to a vector field V (defined by (D_VG)(x) = \frac{d}{... 1answer 286 views ### Questions about a return map Consider the following map in the interval u\in[-1,1] (U\in[-1,1] also)$$U=u-\frac{64}{3}u^{3}+64u^{5}-64u^{7}+\frac{64}{3}u^{9}$$It has 3 fixed points at$u=0,\pm 1$. If we compute the ... 1answer 84 views ### Numerical method for simultaneous computation of eigenvalues of a family of commuting matrices I have a problem where I have$n$commuting matrices$M_1,\dots,M_n$. It is a well-known fact that commuting matrices are simultaneously diagonalizable/triangularizable. I need to find the eigenvalues ... 0answers 89 views ### Is there an easy way to solve an “almost quadratic” equation? I have an equation of the form:$ax^{k_1} + bx^{k_2} + cx^{k_3} + dx^{k_4} + e = 0$where$a$,$b$,$c$and$d$are arbitrary real numbers;$k_1$and$k_2$are positive reals in the range of 1.92...... 1answer 70 views ### Convergence of Chebyshev interpolation in L^1 Let$f\in C^0([-1,1])$and$P_n(f)$its interpolation polynomial at the Chebyshev nodes. I would be interested to know about any existing results (positive or negative) about the convergence of$P_n(...
Let $(E,\mathcal E,\lambda)$ be a measure space; $f:E\to[0,\infty)^3$ be $\mathcal E$-measurable with $\|f\|\in\mathcal L^2(\lambda)$; $\tilde p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ for some \$\...