# Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

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### Cubic spline interpolation without a constant term

Two main questions: I am wondering if it is possible to construct a cubic spline that interpolates data WITHOUT a constant term $a$. That is, the polynomial takes the form $f(t) = bt + ct^2 + dt^3$, ...
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### Probability finite precision random matrix has distinct eigenvalues

copied from math stack exchange There is a theorem which says the probability/size of a random matrix having repeated eigenvalues is 0 and this result is used in many fields. What I am wondering is, ...
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### How does Mathematica do symbolic integration?

I suppose there was at least once in our lifetime the point where we resorted to mathematica for help with an integral.-Unless you chose not to have the pleasure of using the continuum in your ...
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### Quadrature methods for high-dimensional Gaussian integration

Suppose that $f$ is the density of a high(-$d$)-dimensional Gaussian measure with mean $\mu$ and non-singular covariance matrix $\Sigma$. Let $g:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous ...
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### Assignment problem with priorities and scores

I have run into a real problem that is actually a sort of assignment problem. I am describing it here because I am interested in knowing whether this problem already has a name (and whether there is ...
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### Did human computers use floating-point arithmetics?

Before the proliferation of computers in the 1950s, did human computers use floating-point formats for their computations? Floating-point calculation was reportedly implemented already in the 1910s (...
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### Numerically differentiated values and their corresponding x-coordinates

If we numerically differentiate a given time series data consisting of N points by finite forward difference method, we will have N-1 points corresponding to first derivative. If it is a second ...
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### Recursive formula for integral of Chebyshev-type integral

Define $$I_{m,n}(x,y,r) = \int_a^b T_m(x + r \sin(\gamma)) T_n(y-r \cos(\gamma)) d\gamma$$ where $T_m(x)$ are the Chebyshev polynomials of the first kind, and $a$ and $b$ are constants. Assume that ...
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### How to constrain the integral of the control function to a fixed value?

In the following, I am referring to the general "Hamiltonian control theory" using the conventions defined here. I am working on a very simple S($x_1$)I($x_2$)R($x_3$) model for infectious ...
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### Can we improve the error bounds for spline interpolation if the interpolated function is smooth?

Let me first state the original problem I want to solve: Given a closed curve $C:[a,b]\to\mathbb R^2$ that is smooth ($C^\infty$), a partition in the parameter space $a=t_0<t_1<\cdots<t_n=b$,...
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### Robust estimation of $Ax=b$

Problem setting : $\underset{x}{\text{min}} \|Ax-b\|$, where $A \in \mathcal{R}^{m \times n}, m\gg n$, full rank. L1 loss is used for robust estimation using IRLS. The corresponding equation to ...
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### Newton-Raphson with multiple root [closed]

To approximate the root of a function, which also happens to be of multiplicity greater than 1, how do I choose the starting point of the algorithm? For example, I am trying to approximate the root $0$...
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### Numerical methods for evaluating singular integrals

The Helmholtz decomposition for a vector field B contains both volume integrals and two boundary integrals (https://en.wikipedia.org/wiki/Helmholtz_decomposition). For brevity I show just one of the ...
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I am looking for book recommendations or hints on numerical integration over infinite intervals. I am particularly interested in integrals of the form $\int\limits_{-\infty}^{+\infty} g(x) \exp(p_d(x))... 1answer 108 views ### Complexity of solving$\sum_i A_i X B_i = C$Is anything known about computational complexity of finding$X$which satisfies the following matrix equation? $$\sum_i^n A_i X B_i = C$$ With$A_i,B_i,C$dense$d\times d$matrices. Any literature ... 1answer 147 views ### Estimate for computing the$L^2$-norm of a function from its data Let$f:\mathbb{T}^m \to \mathbb{R}$is a function of bounded variation(BV). Let$D=\{\boldsymbol{p}_i,i=1,2,3\ldots\}$be a countable dense subset of$(0,1)^m$. Let$E_n, n = 1,2,3\ldots$be a ... 0answers 36 views ### Solving nonlinear equations involving expectations Let$X$be a random variable and$g(x,y)$be a function of two variables. Consider the equation $$\mathbb{E}_Xg(X,y) = 0$$ Are there any specialized techniques for solving such equations (... 0answers 45 views ### Computing the stabilizer of a specific vector in a Lie group representation Let$x$be a fixed vector in the carrier vector space of an irrep$\rho$of a compact Lie group$G$, and let$G_x$be the stabilizer subgroup of$G$with respect to$x$. Assume that$x$is not ... 1answer 191 views ### Computing$(AA\otimes BB + AB \otimes BA)^{-1}$Can anyone suggest a way to numerically compute the following matrix vector product? $$u=A^{-1}b=(AA\otimes BB + AB \otimes BA)^{-1}\operatorname{vec}(C)$$ Here$AA,BB,AB,BA$and$C$are$d\times d$... 0answers 46 views ### Optimal approximation of circles with sum of logarithms By playing around, I found that $$\left\|\frac{\log{(a\cdot(x+1)+1)}+\log{(1+(1-x)a)}-\log{(2a+1)}}{2\log{(a+1)}-\log{(2a+1)}}- \sqrt{1-x^2}\right\|_\infty\lt 0.12$$ indicating that the fraction quite ... 1answer 59 views ### Solving equation for higher degree of composition Given this function$f(x) = x - 1/x$, the equation$f(f(x)) = x$has two solutions:$\frac{1}{\sqrt{2}}$,$\frac{-1}{\sqrt{2}}$. But how about solving this equation for a higher degree of composition, ... 0answers 104 views ### A method for extracting a condition to check whether a feature is related to an object Let we have the object$\bf S$. This object has some properties such as length, temperature and other features. Assume that for the object$\bf S$we selected$n$features.For example the vector${\bf ...
Take a continuous function $f:[-1,1]\to\mathbb{R}$ and a sequence of independent random variables $X_1,X_2,\ldots$ uniformly distributed in $[-1,1]$. Define $Y_n=\max\{f(X_1),f(X_2),\ldots,f(X_n)\}$. ...
I am trying to get an estimate for the induced 2-norm condition number $\kappa_{2}(M)$ of this matrix $M$: M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} ...