Household Methods Research Articles

A block Cholesky‐LU‐based QR factorization for rectangular matrices

AbstractThe Householder method provides a stable algorithm to compute the full QR factorization of a general matrix. The standard version of the algorithm uses a sequence of orthogonal reflections to transform the matrix into upper triangular form column by column. In order to exploit (level 3 BLAS or structured matrix) computational advantages for block‐partitioned algorithms, we develop a block algorithm for the QR factorization. It is based on a well‐known block version of the Householder method which recursively divides a matrix columnwise into two smaller matrices. However, instead of continuing the recursion down to single matrix columns, we introduce a novel way to compute the QR factors in implicit Householder representation for a larger block of several matrix columns, that is, we start the recursion at a block level instead of a single column. Numerical experiments illustrate to what extent the novel approach trades some of the stability of Householder's method for the computational efficiency of block methods.

THE EIGENVALUES OF SYMMETRIC TRIDIAGONAL MATRIX METHOD WITH ACCELERATION MBY THE QR SHIFT

This research investigates the application of the QR - method for computing all the eigenvalues of the real symmetric tridiagonal matrix. The Householder method will be used for reduction of the real symmetric matrix to symmetric tridiagonal form, and then the so called QR - method with acceleration shift applies a sequence of orthogonal transformations to the symmetric tridiagonal matrix which converges to a similar matrix that is tridiagonal. This tridiagonal matrix possesses an eigenvalues similar to the eigenvalues of the symmetric tridiagonal matrix. Particular attention is paid to the shift technique that accelerates the rate of convergence. Computer algorithms for implementing the Householder's method and QR – method are presented. Computer Matlab programs for performing the Householder algorithm and the QR algorithm (with acceleration shift) are listed in the Appendix.

Householder's method for solving the $p$-adic polynomial equations

This work offers an analogue of Householder's Method for solving a root-finding problem $f(x)=0$ in the $p$-adic setting. We apply this method to calculate the square roots of a $p$-adic number $a\in\mathbb{Q}_{p}$ where $p$ is a prime number, and through the calculation of the approached solution of the $p$-adic polynomial equation $f(x)=x^{2}-a=0$. We establish the rate of convergence of this method. Finally, we also determine how many iterations are needed to obtain a specified number of correct digits in the approximate.

A note on machine method for root extraction

The present investigation deals with the critical study of the works of Lancaster and Traub, who have developed $n$th root extraction methods of a real number. It is found that their developed methods are equivalent and the particular cases of Halley's and Householder's methods. Again the methods presented by them are easily obtained from simple modifications of Newton's method, which is the extension of Heron's square root iteration formula. Further, the rate of convergency of their reported methods are studied.

New Predictor-Corrector Iterative Methods with Twelfth-Order Convergence for Solving Nonlinear Equations

In this paper, we propose and analyze new two efficient iterative methods for finding the simple roots of nonlinear equations. These methods based on a Jarratt's method, Householder's method and Chun&Kim's method by using a predictor-corrector technique. The error equations are given theoretically to show that the proposed methods have twelfth-order convergence. Several numerical examples are given to illustrate the efficiency and robustness of the proposed methods. Comparison with other well-known iterative methods is made.

A new approach for inversion of large random matrices in massive MIMO systems.

We report a novel approach for inversion of large random matrices in massive Multiple-Input Multiple Output (MIMO) systems. It is based on the concept of inverse vectors in which an inverse vector is defined for each column of the principal matrix. Such an inverse vector has to satisfy two constraints. Firstly, it has to be in the null-space of all the remaining columns. We call it the null-space problem. Secondly, it has to form a projection of value equal to one in the direction of selected column. We term it as the normalization problem. The process essentially decomposes the inversion problem and distributes it over columns. Each column can be thought of as a node in the network or a particle in a swarm seeking its own solution, the inverse vector, which lightens the computational load on it. Another benefit of this approach is its applicability to all three cases pertaining to a linear system: the fully-determined, the over-determined, and the under-determined case. It eliminates the need of forming the generalized inverse for the last two cases by providing a new way to solve the least squares problem and the Moore and Penrose's pseudoinverse problem. The approach makes no assumption regarding the size, structure or sparsity of the matrix. This makes it fully applicable to much in vogue large random matrices arising in massive MIMO systems. Also, the null-space problem opens the door for a plethora of methods available in literature for null-space computation to enter the realm of matrix inversion. There is even a flexibility of finding an exact or approximate inverse depending on the null-space method employed. We employ the Householder's null-space method for exact solution and present a complete exposition of the new approach. A detailed comparison with well-established matrix inversion methods in literature is also given.

Householder's approximants and continued fraction expansion of quadratic irrationals

There are numerous methods for rational approximation of real numbers. Continued fraction convergent is one of them and Newton's iterative method is another. Connections between these two approximation methods were discussed by several authors. Householder's methods are generalisation of Newton's method. In this paper, we will show that for these methods analogous connection with continued fractions hold.

Higher-order iterative methods by using Householder's method for solving certain nonlinear equations

In this paper, we suggest and analyze some new higher-order iterative methods by using Householder's method free from second derivative for solving nonlinear equations. Here we use new and different technique for implementation of higher-order derivative of the function and derive new higher- order predictor-corrector iterative methods free from second derivative. The efficiency index equals to

A Useful Form of Unitary Matrix Obtained from Any Sequence of Unit 2-Norm n-Vectors

Charles Sheffield pointed out that the modified Gram–Schmidt (MGS) orthogonalization algorithm for the QR factorization of $B\!\in\!\R^{n\times k}$ is mathematically equivalent to the QR factorization applied to the matrix B augmented with a $k\times k$ matrix of zero elements on top. This is true in theory for any method of QR factorization, but for Householder's method it is true in the presence of rounding errors as well. This knowledge has been the basis for several successful but difficult rounding error analyses of algorithms which in theory produce orthogonal vectors but significantly fail to do so because of rounding errors. Here we show that the same results can be found more directly and easily without recourse to the MGS connection. It is shown that for any sequence of k unit 2-norm n-vectors there is a special $(n\!+\!k)$-square unitary matrix which we call a unitary augmentation of these vectors and that this matrix can be used in the analyses without appealing to the MGS connection. We describe the connection of this unitary matrix to Householder matrices. The new approach is applied to an earlier analysis to illustrate both the improvement in simplicity and advantages for future analyses. Some properties of this unitary matrix are derived. The main theorem on orthogonalization is then extended to cover the case of biorthogonalization.

Computation of Eigenvectors on Concurrent Processors

Suppose a real symmetric general matrix A is reduced into a real symmetric tridiagonal matrix C by either Householder's method or Givens method. Eigenvalues of the tridiagonal matrix are the eigenvalues of the original matrix since they are similar. After eigenvalues are calculated, the eigenvectors of the original matrix may be obtained by the eigenvectors of the tridiagonal matrix and the transformation matrices used in either Householder's or Givens tridiagonalization methods [1]. In these methods, however, transformation matrices usually are not stored due to memory considerations. Only critical information such as rotation angles in Givens method is saved. Generally, the recovery of all transformation matrices involves more computation than the tridiagonalization procedure. Therefore such a scheme is rarely used. In this paper, a serial eigenvector solution procedure, assuming eigenvalues are known, is first described. This procedure applies Gauss elimination with diagonal pivoting on the characteristic equations. A parallel procedure is then developed by using scattered column decomposition. The efficiency and speedup are given in terms of problem and machine parameters. The efficiency of the parallel algorithm increases as the number of scattered columns assigned to each processor increases, and decreases as communication cost between processors increases.

A study of Stark ladders in finite crystals on the basis of the tight-binding approximation

The eigenvalue equation relevant for investigating Stark ladders (SL) in finite crystals on the basis of the tight-binding approximation (TBA) is derived from a difference equation, which is established using (localised) Wannier functions. The treatment makes use of a two-band model, includes the effect of the electric field on both the Coulomb and the resonance integrals and incorporates the finiteness of the crystal via the use of the periodic boundary conditions. Numerical explorations of the eigenvalue equation are carried out with the help of Householder's tridiagonalisation method in combination with the Q-L algorithm. The energy eigenvalues are computed for various values of the electric field and other pertinent parameters, the parameters involving the interband terms being determined on the basis of the (k.p) method. A thorough scanning of the energy eigenvalues reveals the circumstances under which SL would occur in finite crystals conforming with the conditions adopted. The features of the results are discussed critically, both independently and in the light of other similar results. In particular, the treatment predicts well the features expected of SL in finite systems for low fields; for higher fields, which are likely to introduce Zener tunnelling, the application of the TBA to the problem of SL in finite crystals appears to leave room for question.

On the efficient and accurate solution of the skew-symmetric eigenvalue problem: An arrangement of new and already known algorithmic formulations

Algorithms are presented to solve the special eigenvalue problem AZ = Zλ , where A is skew-symmetric. The effective use of Householder's method, the bisection method and inverse iteration for solving the complete eigen-value problem are described in some detail. Simultaneous vector iteration is formulated for skew-symmetric matrices. The amount of work for the skew-symmetric Jacobi algorithm and the simultaneous vector iteration may be reduced by using the solution of a simplified eigenvalue problem. For Hermitian matrices also quadratic eigenvalue bounds for groups of eigenvalues and linear bounds for groups of eigenvectors are derived. The case where the set of calculated eigenvectors is not orthonormal is considered in some detail. In principle, the skew-symmetric eigenvalue problem may be easily transformed into a symmetric eigenvalue problem; but such a procedure has the following disadvantages: first, the results are in general less accurate, and, second, the eigenvectors which belong to well separated eigenvalues are not uniquely determined.

"A conjecture in numerical linear algebra"

In [1] and [2] we proposed a conjecture in numerical linear algebra and here we would like to clarify an implicit assumption in our note. Throughout we meant by a matrix always a <u>dense</u> matrix. It was clear to us that for real symmetric band matrices the "chasing" strategy proposed by Rutishauser [3] is superior to the unmodified form of tridiagonalization. It has also been pointed out [4] that for very sparse matrices the Givens' method can take more advantage of the sparsity than Householder's method. At present intensive research is being done in the field of very large matrix problems [5] and here new approach is necessary. In spite of the development of a fast Givens transformation [6], [7], we still feel that our conjecture may hold true for dense matrices, because we request minimum of operations and optimal numerical properties.

Theory and Applications of Numerical Analysis

(Chapter Heading): Introduction. Basic Analysis. Taylors Polynomial and Series. The Interpolating Polynomial. Best Approximation. Splines and Other Approximations. Numerical Integration and Differentiation. Solution of Algebraic Equations of One Variable. Linear Equations. Matrix Norms and Applications. Matrix Eigenvalues and Eigenvectors. Systems of Non-linear Equations. Ordinary Differential Equations. Boundary Value and Other Methods for Ordinary Differential Equations. Appendices. Solutions to Selected Problems. References. Subject Index. Introduction: What is Numerical Analysis? Numerical Algorithms. Properly Posed and Well-Conditioned Problems. Basic Analysis: Functions. Limits and Derivatives. Sequences and Series. Integration. Logarithmic and Exponential Functions. Taylor's Polynomial and Series: Function Approximation. Taylor's Theorem. Convergence of Taylor Series. Taylor Series in Two Variables. Power Series. The Interpolating Polyomial: Linear Interpolation. Polynomial Interpolation. Accuracy of Interpolation. The Neville Aitken Algorithm. Inverse Interpolation. Divided Differences. Equally Spaced Points. Derivatives and Differences. Effect of Rounding Error. Choice of Interpolation Points. Examples of Bernstein and Runge. BestApproximation: Norms of Functions. Best Approximations. Least Squares Approximations. Orthogonal Functions. Orthogonal Polynomials. Minimax Approximation. Chebyshev Series. Economization of Power Series. The Remez Algorithms. Further Results on Minimax Approximation.Splines and Other Approximations: Introduction. B-Splines. Equally-Spaced Knots. Hermite Interpolation. Pade and Rational Approximation. Numerical Integration and Differentiation: Numerical Integration. Romberg Integration. Gaussian Integration. Indefinite Integrals. Improper Integrals. Multiple Integrals. Numerical Differentiation. Effect of Errors. Solution of Algebraic Equations of One Variable: Introduction. The Bisection Method. Interpolation Methods. One-Point Iterative Methods. Faster Convergence. Higher Order Processes. The Contraction Mapping Theorem. Linear Equations: Introduction. Matrices. Linear Equations. Pivoting. Analysis of Elimination Method. Matrix Factorization. Compact Elimination Methods. Symmetric Matrices. Tridiagonal Matrices. Rounding Errors in Solving Linear Equations. Matrix Norms and Applications: Determinants, Eigenvalues, and Eigenvectors. Vector Norms. Matrix Norms. Conditioning. Iterative Correction from Residual Vectors. Iterative Methods. Matrix Eigenvalues and Eigenvectors: Relations Between Matrix Norms and Eigenvalues Gerschgorin Theorems. Simple and Inverse Iterative Method. Sturm Sequence Method. The QR Algorithm. Reduction to Tridiagonal Form: Householder's Method. Systems ofNon-Linear Equations: Contraction Mapping Theorem. Newton's Method. Ordinary Differential Equations: Introduction. Difference Equations and Inequalities. One-Step Methods. Truncation Errors of One-Step Methods. Convergence of One-Step Methods. Effect of Rounding Errors on One-Step Methods. Methods Based on Numerical Integration Explicit Methods. Methods Based on Numerical Integration Implicit Methods. Iterating with the Corrector. Milne's Method of Estimating Truncation Errors. Numerical Stability. Systems and Higher Order Equations. Comparison of Step-by-Step Methods. Boundary Value and Other Methods for Ordinary Differential Equations: Shooting Method for Boundary Value Problems. Boundary Value Problem. Extrapolation to the Limit. Deferred Correction. Chebyshev Series Method. Appendices. Solutions to Selected Problems. References. Subject Index.

On the transformation of symmetric sparse matrices to the triple diagonal form

The problem of minimizing the number of zero elements that become non-zero during the computation when a sparse symmetric matrix is reduced to a triple diagonal form, either by Givens' or Householder's method, is discussed. Algorithms for minimizing the growth of such non-zero elements are given.

Householder's method for complex matrices and eigensystems of hermitian matrices

Householder's method for complex matrices and eigensystems of hermitian matrices

An error analysis of Householder's method for the symmetric eigenvalue problem

An error analysis of Householder's method for the symmetric eigenvalue problem

Householder's method for symmetric matrices

Householder's method for symmetric matrices

Householder's Method for the Solution of the Algebraic Eigenproblem

Householder's Method for the Solution of the Algebraic Eigenproblem