We propose a feasible primal-dual path-following interior-point algorithm for semidefinite least squares problems (SDLS). At each iteration, the algorithm uses only full Nesterov-Todd steps with the advantage that no line search is required. Under new appropriate choices of the parameter $\beta$ which defines the size of the neighborhood of the central-path and of the parameter $\theta$ which determines the rate of decrease of the barrier parameter, we show that the proposed algorithm is well defined...

We present a method that in certain sense stores the inverse of the stiffness matrix in $O(NlogN)$ memory places, where $N$ is the number of degrees of freedom and hence the matrix size. The setup of this storage format requires $O\left(N^{3/2}\right)$ arithmetic operations. However, once the setup is done, the multiplication of the inverse matrix and a vector can be performed with $O(NlogN)$ operations. This approach applies to the first order finite element discretization of linear elliptic and parabolic problems in triangular...

We consider the multiple ellipses detection problem on the basis of a data points set coming from a number of ellipses in the plane not known in advance, whereby an ellipse $E$ is viewed as a Mahalanobis circle with center $S$, radius $r$, and some positive definite matrix $\Sigma$. A very efficient method for solving this problem is proposed. The method uses a modification of the $k$-means algorithm for Mahalanobis-circle centers. The initial approximation consists of the set of circles whose centers...

The conjugate gradient method is one of the most effective algorithm for unconstrained nonlinear optimization problems. This is due to the fact that it does not need a lot of storage memory and its simple structure properties, which motivate us to propose a new hybrid conjugate gradient method through a convex combination of ${\beta }_k^{RMIL}$ and ${\beta }_k^{HS}$. We compute the convex parameter ${\theta }_k$ using the Newton direction. Global convergence is established through the strong Wolfe conditions. Numerical experiments...

The adaptive $s$-step CG algorithm is a solver for sparse symmetric positive definite linear systems designed to reduce the synchronization cost per iteration while still achieving a user-specified accuracy requirement. In this work, we improve the adaptive $s$-step conjugate gradient algorithm by the use of iteratively updated estimates of the largest and smallest Ritz values, which give approximations of the largest and smallest eigenvalues of $A$, using a technique due to G. Meurant and...

Assume $f_1,...,f_N$ a finite set of functions in $H^{\infty }\left(D\right)$, the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function $f$ in $H^{\infty }\left(D\right)$ to belong to the norm-closure of the ideal $I(f_1,...,f_N)$ generated by $f_1,...,f_N$, namely the property $$\left|f\right(z\left)\right|\le \alpha \left(\right|f_1\left(z\right)|+...+|f_N\left(z\right)\left|\right)\text{for}z\in D$$ for some function $\alpha$ : ${\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ satisfying ${lim}_{t\to 0}\alpha \left(t\right)/t=0.$ The main feature in the proof is an improvement in the contour-construction appearing in L. Carleson’s solution of the corona-problem. It is also shown that the property $$\left|f\left(z\right)\right|\le C\underset{1\le j\le N}{max}\left|f_j\left(z\right)\right|\text{for}z\in D$$ ...

Every real number $x,0\<x\le 1$, has an essentially unique expansion as a Pierce series : $$x=\frac{1}{x^1}-\frac{1}{x^1{x}^2}+\frac{1}{x^1{x}^2{x}^3}-\cdots$$ where the $x_i$ form a strictly increasing sequence of positive integers. The expansion terminates if and only if $x$ is rational. Similarly, every positive real number $y$ has a unique expansion as an Engel series : $$y=\frac{1}{y^1}-\frac{1}{y^1{y}^2}+\frac{1}{y^1{y}^2{y}^3}+\cdots$$ where the $y_i$ form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi...

We show that if $\alpha >1$, then the logarithmically weighted Bergman space $A_{{log}^{\alpha }}^2$ is mapped by the Libera operator $ℒ$ into the space $A_{{log}^{\alpha -1}}^2$, while if $\alpha >2$ and $0<\epsilon \le \alpha -2$, then the Hilbert matrix operator $H$ maps $A_{{log}^{\alpha }}^2$ into $A_{{log}^{\alpha -2-\epsilon }}^2$.We show that the Libera operator $ℒ$ maps the logarithmically weighted Bloch space ${ℬ}_{{log}^{\alpha }}$, $\alpha \in ℝ$, into itself, while $H$ maps ${ℬ}_{{log}^{\alpha }}$ into ${ℬ}_{{log}^{\alpha +1}}$.In Pavlović’s paper (2016) it is shown that $ℒ$ maps the logarithmically weighted Hardy-Bloch space ${ℬ}_{{log}^{\alpha }}^1$, $\alpha >0$, into ${ℬ}_{{log}^{\alpha -1}}^1$. We show that this result is sharp. We also show that $H$ maps ${ℬ}_{{log}^{\alpha }}^1$, $\alpha \ge 0$,...

Let We prove that $F$ is completely monotonic on $(0,\mathrm{\infty })$. This complements a result of Miller and Moskowitz (2006), who proved that $F$ is positive and strictly decreasing on $(0,\mathrm{\infty })$. The sequence $\{S(k)\}$ $(k=1,2,\mathrm{\dots })$ plays a role in information theory.