# 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 ...
1 vote
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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 ...
48 views

### 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 ...
68 views

### 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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### 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 ...
1 vote
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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 ...
1 vote
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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 $x$ ...
173 views

### 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
106 views

### 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 ...
208 views

### 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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### 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 ...
1 vote
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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 ...