scholarly journals Double condensation of singularities for product-quadrature formulas with differentiable functions

2012 ◽  
Vol 28 (1) ◽  
pp. 83-91
Author(s):  
ALEXANDRU I. MITREA ◽  
Keyword(s):  
Quadrature Formulas ◽  
Triangular Matrices ◽  
Differentiable Functions ◽  
Equidistant Nodes ◽  
The Matrix

The main goal of this paper is to emphasize the phenomenon of double condensation of singularities for spaces of differentiable functions with respect to product-quadrature formulas associated to a class of node triangular matrices in [−1, 1], including a Gegenbauer node matrix and the matrix of equidistant nodes in [−1, 1].

scholarly journals Closed expressions for coefficients in weighted Newton-Cotes quadratures

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 649-658 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir Milovanovic ◽  
M.A. Jafari
Keyword(s):  
Short Note ◽  
Stirling Numbers ◽  
Quadrature Formulas ◽  
Numerical Examples ◽  
Quadrature Formulae ◽  
Equidistant Nodes

In this short note, we derive closed expressions for Cotes numbers in the weighted Newton-Cotes quadrature formulae with equidistant nodes in terms of moments and Stirling numbers of the first kind. Three types of equidistant nodes are considered. The corresponding program codes in Mathematica Package are presented. Finally, in order to illustrate the application of the obtained quadrature formulas a few numerical examples are included.

Solving Systems of Linear Algebraic Equations with the Use of an Selective Regularization

2021 ◽  
pp. 11-15
Author(s):  
Vladimir N. Lutay
Keyword(s):  
Diagonal Element ◽  
Decomposition Process ◽  
Algebraic Equations ◽  
Original Matrix ◽  
Triangular Matrices ◽  
First Case ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Gauss Method ◽  
Scalar Products

The solution of systems of linear algebraic equations, which matrices can be poorly conditioned or singular is considered. As a solution method, the original matrix is decomposed into triangular components by Gauss or Chole-sky with an additional operation, which consists in increasing the small or zero diagonal terms of triangular matrices during the decomposition process. In the first case, the scalar products calculated during decomposition are divided into two positive numbers such that the first is greater than the second, and their sum is equal to the original one. In further operations, the first number replaces the scalar product, as a result of which the value of the diagonal term increases, and the second number is stored and used after the decomposition process is completed to correct the result of calculations. This operation increases the diagonal elements of triangular matrices and prevents the appearance of very small numbers in the Gauss method and a negative root expression in the Cholesky method. If the matrix is singular, then the calculated diagonal element is zero, and an arbitrary positive number is added to it. This allows you to complete the decomposition process and calculate the pseudo-inverse matrix using the Greville method. The results of computational experiments are presented.

scholarly journals Common Factors in Fraction-Free Matrix Decompositions

2020 ◽  
Author(s):  
Johannes Middeke ◽  
David J. Jeffrey ◽  
Christoph Koutschan
Keyword(s):  
Normal Form ◽  
Common Factors ◽  
Reduction Process ◽  
Triangular Matrices ◽  
Specific Data ◽  
Matrix Decompositions ◽  
The Matrix ◽  
The Common

AbstractWe consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauß–Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, while statistical factors depend on the specific data. We relate the existence of row factors in the LU decomposition to factors appearing in the Smith–Jacobson normal form of the matrix. For statistical factors, we identify some of the mechanisms that create them and give estimates of the frequency of their occurrence. Similar observations apply to the common factors in a fraction-free QR decomposition. Our conclusions are tested experimentally.

COMPUTING THE LAW OF A FAMILY OF SOLVABLE LIE ALGEBRAS

2009 ◽  
Vol 19 (03) ◽  
pp. 337-345 ◽  
Author(s):  
JUAN C. BENJUMEA ◽  
JUAN NÚÑEZ ◽  
ÁNGEL F. TENORIO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Computational Data ◽  
Solvable Lie Algebras ◽  
Final Output ◽  
The Matrix ◽  
The Law

This paper shows an algorithm which computes the law of the Lie algebra associated with the complex Lie group of n × n upper-triangular matrices with exponential elements in their main diagonal. For its implementation two procedures are used, respectively, to define a basis of the Lie algebra and the nonzero brackets in its law with respect to that basis. These brackets constitute the final output of the algorithm, whose unique input is the matrix order n. Besides, its complexity is proved to be polynomial and some complementary computational data relative to its implementation are also shown.

scholarly journals The best, by coefficients, weighted quadrature formulas of the form $\sum\limits_{k=1}^n \sum\limits_{i=0}^{r-1} A_{ki} f^{(i)}_{jk}$ for some classes of differentiable functions

1987 ◽  
pp. 37
Author(s):  
Ye.Ye. Dunaichuk
Keyword(s):  
Quadrature Formula ◽  
Sharp Estimate ◽  
Quadrature Formulas ◽  
Differentiable Functions ◽  
Derivatives Of ◽  
Weighted Quadrature

For the quadrature formula (with non-negative, integrable on $[0,1]$ function) that is defined by the values of the function and its derivatives of up to and including $(r-1)$-th order, we find the form of the best coefficients $A^0_{ki}$ ($k = \overline{1, n}$, $i = \overline{0, r-1}$) for fixed nodes $\gamma_k$ ($k = \overline{1, n}$) and we give the sharp estimate of the remainder of this formula on the classes $W^r_p$, $r = 1, 2, \ldots$, $1 \leqslant p \leqslant \infty$.

scholarly journals Optimized Cramer’s Rule in WZ Factorization and Applications

2020 ◽  
Vol 13 (4) ◽  
pp. 1035-1054
Author(s):  
Olayiwola Babarinsa ◽  
Azfi Zaidi Mohammad Sofi ◽  
Mohd Asrul Hery Ibrahim ◽  
Hailiz Kamarulhaili
Keyword(s):  
Sparse Matrices ◽  
Schur Complement ◽  
Matrix Group ◽  
Lu Factorization ◽  
Triangular Matrices ◽  
Performance Time ◽  
The Matrix ◽  
Cramer’S Rule ◽  
And Performance ◽  
Matrix Norms

In this paper, W Z factorization is optimized with a proposed Cramer’s rule and compared with classical Cramer’s rule to solve the linear systems of the factorization technique. The matrix norms and performance time of WZ factorization together with LU factorization are analyzed using sparse matrices on MATLAB via AMD and Intel processor to deduce that the optimized Cramer’s rule in the factorization algorithm yields accurate results than LU factorization and conventional W Z factorization. In all, the matrix group and Schur complement for every Zsystem (2×2 block triangular matrices from Z-matrix) are established.

scholarly journals Solving Separable Nonlinear Equations Using LU Factorization

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yun-Qiu Shen ◽  
Tjalling J. Ypma
Keyword(s):  
Nonlinear Equations ◽  
Iterative Process ◽  
Full Rank ◽  
Original System ◽  
System Of Equations ◽  
Lu Factorization ◽  
Numerical Implementation ◽  
Direct Solution ◽  
Differentiable Functions ◽  
The Matrix

Separable nonlinear equations have the form where the matrix and the vector are continuously differentiable functions of and . We assume that and has full rank. We present a numerical method to compute the solution for fully determined systems () and compatible overdetermined systems (). Our method reduces the original system to a smaller system of equations in alone. The iterative process to solve the smaller system only requires the LU factorization of one matrix per step, and the convergence is quadratic. Once has been obtained, is computed by direct solution of a linear system. Details of the numerical implementation are provided and several examples are presented.

scholarly journals On some Hermite–Hadamard inequalities involving k-fractional operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shanhe Wu ◽  
Sajid Iqbal ◽  
Muhammad Aamir ◽  
Muhammad Samraiz ◽  
Awais Younus
Keyword(s):  
Fractional Integrals ◽  
Quadrature Formulas ◽  
Fractional Operators ◽  
Differentiable Functions ◽  
Error Estimations

AbstractThe main objective of this paper is to establish some new Hermite–Hadamard type inequalities involving k-Riemann–Liouville fractional integrals. Using the convexity of differentiable functions some related inequalities have been proved, which have deep connection with some known results. At the end, some applications of the obtained results in error estimations of quadrature formulas are also considered.

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