Journal•ISSN: 0029-599X
Numerische Mathematik
Springer Science+Business Media
About: Numerische Mathematik is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Numerical analysis & Finite element method. It has an ISSN identifier of 0029-599X. Over the lifetime, 4733 publications have been published receiving 242168 citations. The journal is also known as: Numerische Mathematik (Print).
Topics: Numerical analysis, Finite element method, Discretization, Boundary value problem, Partial differential equation
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TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to
22,704 citations
TL;DR: In this article, the authors present two families of non-conforming finite elements, built on tetrahedrons or on cubes, which are respectively conforming in the spacesH(curl) and H(div).
Abstract: We present here some new families of non conforming finite elements in ?3. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spacesH(curl) andH(div). We give some applications of these elements for the approximation of Maxwell's equations and equations of elasticity.
3,049 citations
TL;DR: The decomposition of A is called the singular value decomposition (SVD) and the diagonal elements of ∑ are the non-negative square roots of the eigenvalues of A T A; they are called singular values.
Abstract: Let A be a real m×n matrix with m≧n. It is well known (cf. [4]) that
$$A = U\sum {V^T}$$
(1)
where
$${U^T}U = {V^T}V = V{V^T} = {I_n}{\text{ and }}\sum {\text{ = diag(}}{\sigma _{\text{1}}}{\text{,}} \ldots {\text{,}}{\sigma _n}{\text{)}}{\text{.}}$$
The matrix U consists of n orthonormalized eigenvectors associated with the n largest eigenvalues of AA T , and the matrix V consists of the orthonormalized eigenvectors of A T A. The diagonal elements of ∑ are the non-negative square roots of the eigenvalues of A T A; they are called singular values. We shall assume that
$${\sigma _1} \geqq {\sigma _2} \geqq \cdots \geqq {\sigma _n} \geqq 0.$$
Thus if rank(A)=r, σ r+1 = σ r+2=⋯=σ n = 0. The decomposition (1) is called the singular value decomposition (SVD).
3,036 citations
TL;DR: In this paper, a method for estimating the optimum amount of smoothing from the data is presented, based on smoothing splines, which is well known to provide nice curves which smooth discrete, noisy data.
Abstract: Smoothing splines are well known to provide nice curves which smooth discrete, noisy data. We obtain a practical, effective method for estimating the optimum amount of smoothing from the data. Deri...
2,799 citations
TL;DR: In this article, the 8th International Meeting of the Institute of Management Sciences, Brussels, August 23-26, 1961, the authors presented a paper entitled "The International Journal of Management Science and Management Sciences".
Abstract: Paper presented to the 8th International Meeting of the Institute of Management Sciences, Brussels, August 23-26, 1961.
2,782 citations