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Showing papers on "QR decomposition published in 1997"


Journal ArticleDOI
TL;DR: An inverse-free, highly parallel, spectral divide and conquer algorithm that can compute either an invariant subspace of a nonsymmetric matrix, or a pair of left and right deflating subspaces of a regular matrix pencil.
Abstract: We discuss an inverse-free, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix \(A\), or a pair of left and right deflating subspaces of a regular matrix pencil \(A - \lambda B\). This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm.

145 citations


Book
01 Jan 1997
TL;DR: In this paper, the singular value decomposition of a square matrix has been shown to be unitarily invariant in terms of the scalar product length of a vector isometric matricies.
Abstract: Lecture 1: metric space some useful definitions nested balls normed space popular vector norms matrix norms equivalent norms operator norms. Lecture 2: scalar product length of a vector isometric matricies preservation of length and unitary matricies Schur theorum normal matricies positive definite matricies the singular value decomposition unitarily invariant norms a short way to the SVD approximations of a lower rank smoothness and ranks. Lecture 3: perturbation theory condition of a matrix convergent matricies and series the simplest iteration method inverses and series condition of a linear system consistency of matrix and right-hand side eigenvalue perturbations continuity of the polynomial roots. Lecture 4: diagonal dominance Gerschgorin disks small perturbations of eigen values and vectors condition of a simple eigenvalue analitic perturbations. Lecture 5: spectral distances "symmetric" theorums Hoffman-Wielandt theorum permutation vector of a matrix "unnormal" extension eigenvalues of Hermitian matrices interlacing properties what are clusters? singular value clusters eigenvalue clusters. Lecture 6: floating-point numbers computer arithmetic axioms round-off errors for the scalar product forward and backward analysis some philosophy an example of "bad" operation one more example ideal and machine tests up or down solving the triangular systems. Lecture 7: direct methods for linear systems theory of the LU decomposition round-off errors for the LU decomposition growth of matrix entries and pivoting complete pivoting the Cholesky method triangular decompositions and linear systems solution how to refine the solution. Lecture 8: the QR decomposition of a square matrix the QR decomposition of a rectangular matrix householder matrices elimination of elements by reflections Givens matricies elimination of elements by rotations computer realizations of reflections and rotations orthgonalization method loss of orthogonality modified Gram-Schmidt algorithm bidiagonalization unitary similarity reduction to the Hessenberg form. Lecture 9: the eigenvalue problem the power method subspace iterations distances between subspaces subspaces and orthoprojectors distances and orthoprojectors subspaces of equal dimension the CS decomposition convergence of subspace iterations for the block diagonal matrix convergance of subspace iterations in the general case. Lecture 10: the QR algorithm generalised QR algorithm basic formulas the QR iteration lemma convergance of the QR iterations pessimistric and optimistic Bruhat decomposition what if the inverse matrix is not strongly regular the QR iterations and the subspace iterations. Lecture 11: quadratic convergence cubic convergence what makes the QR algorithm efficient implicit QR iterations arrangement of computations how to find the singular value decomposition. Lecture 12: function approximation (Part contents)

125 citations


Journal ArticleDOI
TL;DR: In this paper, first-order perturbation analysis for QR matrix factorization is presented for a given real matrix of rank n and general perturbations in rank n which are sufficiently small in norm.
Abstract: This paper gives perturbation analyses for $Q_1$ and $R$ in the QR factorization $A=Q_1R$, $Q_1^TQ_1=I$ for a given real $m\times n$ matrix $A$ of rank $n$ and general perturbations in $A$ which are sufficiently small in norm. The analyses more accurately reflect the sensitivity of the problem than previous such results. The condition numbers here are altered by any column pivoting used in $AP=Q_1R$, and the condition number for $R$ is bounded for a fixed $n$ when the standard column pivoting strategy is used. This strategy also tends to improve the condition of $Q_1$, so the computed $Q_1$ and $R$ will probably both have greatest accuracy when we use the standard column pivoting strategy. First-order perturbation analyses are given for both $Q_1$ and $R$. It is seen that the analysis for $R$ may be approached in two ways---a detailed "matrix--vector equation" analysis which provides a tight bound and corresponding condition number, which unfortunately is costly to compute and not very intuitive, and a simpler "matrix equation" analysis which provides results that are usually weaker but easier to interpret and which allows the efficient computation of satisfactory estimates for the actual condition number. These approaches are powerful general tools and appear to be applicable to the perturbation analysis of any matrix factorization.

64 citations


Journal ArticleDOI
01 Sep 1997
TL;DR: A fast backward elimination algorithm is introduced based on a QR decomposition and Givens transformations to prune radial-basis-function networks and provides a hybrid supervised centre selection approach.
Abstract: A fast backward elimination algorithm is introduced based on a QR decomposition and Givens transformations to prune radial-basis-function networks. Nodes are sequentially removed using an increment of error variance criterion. The procedure is terminated by using a prediction risk criterion so as to obtain a model structure with good generalisation properties. The algorithm can be used to postprocess radial basis centres selected using a k-means routine and, in this mode, it provides a hybrid supervised centre selection approach.

35 citations


Journal ArticleDOI
TL;DR: It is argued that perturbing towards the orthonormal polar factor is an attractive choice, and it is shown that the perturbations improve the departure from orthonormality without significantly degrading the finite-time global error bound for the ODE solution.
Abstract: Certain applications produce initial value ODEs whose solutions, regarded as time-dependent matrices, preserve orthonormality. Such systems arise in the computation of Lyapunov exponents and the construction of smooth singular value decompositions of parametrized matrices. For some special problem classes, there exist time-stepping methods that automatically inherit the orthonormality preservation. However, a more widely applicable approach is to apply a standard integrator and regularly replace the approximate solution by an orthonormal matrix. Typically, the approximate solution is replaced by the factorQ from its QR decomposition (computed, for example, by the modified Gram-Schmidt method). However, the optimal replacement—the one that is closest in the Frobenius norm—is given by the orthonormal polar factor. Quadratically convergent iteration schemes can be used to compute this factor. In particular, there is a matrix multiplication based iteration that is ideally suited to modern computer architectures. Hence, we argue that perturbing towards the orthonormal polar factor is an attractive choice, and we consider performing a fixed number of iterations. Using the optimality property we show that the perturbations improve the departure from orthonormality without significantly degrading the finite-time global error bound for the ODE solution. Our analysis allows for adaptive time-stepping, where a local error control process is driven by a user-supplied tolerance. Finally, using a recent result of Sun, we show how the global error bound carries through to the case where the orthonormal QR factor is used instead of the orthonormal polar factor.

32 citations


Journal ArticleDOI
TL;DR: An algorithm to compute an approximate rank revealing sparse QR factorization that achieves an exponential bound like methods that use exact singular vectors and presents a theoretical analysis that shows that the use of approximate singular vectors does not degrade the quality of the rank-revealing factorization.
Abstract: We describe an algorithm to compute an approximate rank revealing sparse QR factorization. We use a two phase algorithm to provide especially high accuracy in the labeling of some columns as ``redundant,'' which ensures robustness in the use of our factorization in computing explicit bases of the nullspace. Our first phase is similar in outline to other proposed sparse RRQR factorizations, in that we couple a standard sparse QR factorization scheme with a condition estimator to develop a factorization with a well-conditioned leading block. There are important details in our implementation of the condition estimator and pivoting that enhance efficiency and reliability. However, the exceptional characteristic of our algorithm is its second phase, which ensures that columns labeled as redundant lead to highly accurate nullvectors. The second phase requires that we compute all columns of R explicitly in the first phase; we cannot discard ``redundant'' columns as is often done elsewhere. This condition, in the presence of pivoting to reveal the rank, requires dynamic data structures and necessarily degrades sparsity. But the additional work fits naturally into the multifrontal factorization's use of efficient dense vector kernels, minimizing overall cost. We present a theoretical analysis that shows that our use of approximate singular vectors does not degrade the quality of our rank-revealing factorization; we achieve an exponential bound like methods that use exact singular vectors. We provide results of numerical experiments and close with a discussion of limitations of this approach.

27 citations


Journal ArticleDOI
TL;DR: Fast, efficient and reliable algorithms for discrete least-squares approximation of a real-valued function given at arbitrary distinct nodes in [0,2π) by trigonometric polynomials are presented.
Abstract: Fast, efficient and reliable algorithms for discrete least-squares approximation of a real-valued function given at arbitrary distinct nodes in [0,2π) by trigonometric polynomials are presented. The algorithms are based on schemes for the solution of inverse unitary eigenproblems and require only O(mn) arithmetic operations as compared to O(mn 2 ) operations needed for algorithms that ignore the structure of the problem. An algorithm which solves this problem with real-valued data and real-valued solution using only real arithmetic is given. Numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general QR decomposition methods for the least-squares problem.

27 citations


Dissertation
01 Jul 1997
TL;DR: The design of an application-specific integrated circuit of a parallel array processor is considered for recursive least squares by QR decomposition using Givens rotations, applicable in adaptive filtering and beamforming applications and a novel algorithm, based on the Squared Given Rotation algorithm, is developed.
Abstract: The design of an application-specific integrated circuit of a parallel array processor is considered for recursive least squares by QR decomposition using Givens rotations, applicable in adaptive filtering and beamforming applications. Emphasis is on high sample-rate operation, which, for this recursive algorithm, means that the time to perform arithmetic operations is critical. The algorithm, architecture and arithmetic are considered in a single integrated design procedure to achieve optimum results. A realisation approach using standard arithmetic operators, add, multiply and divide is adopted. The design of high-throughput operators with low delay is addressed for fixed- and floating-point number formats, and the application of redundant arithmetic considered. New redundant multiplier architectures are presented enabling reductions in area of up to 25%, whilst maintaining low delay. A technique is presented enabling the use of a conventional tree multiplier in recursive applications, allowing savings in area and delay. Two new divider architectures are presented showing benefits compared with the radix-2 modified SRT algorithm. Givens rotation algorithms are examined to determine their suitability for VLSI implementation. A novel algorithm, based on the Squared Givens Rotation (SGR) algorithm, is developed enabling the sample-rate to be increased by a factor of approximately 6 and offering area reductions up to a factor of 2 over previous approaches. An estimated sample-rate of 136 MHz could be achieved using a standard cell approach and O.35pm CMOS technology. The enhanced SGR algorithm has been compared with a CORDIC approach and shown to benefit by a factor of 3 in area and over 11 in sample-rate. When compared with a recent implementation on a parallel array of general purpose (GP) DSP chips, it is estimated that a single application specific chip could offer up to 1,500 times the computation obtained from a single OP DSP chip.

25 citations


Journal ArticleDOI
TL;DR: The authors determine expressions for the degradation in the performance of these algorithms due to finite precision by exploiting the steady-state properties of thesegorithms, and show that the three algorithms have about the same finite-precision performance, with PSTAR-RLS performing better than STAR-Rls, which does better than QRD- RLS.
Abstract: The QR decomposition-based recursive least-squares (RLS) adaptive filtering (QRD-RLS) algorithm is suitable for VLSI implementation since it has good numerical properties and can be mapped onto a systolic array. A new fine-grain pipelinable STAR-RLS algorithm was developed. The pipelined STAR-RLS algorithm (PSTAR-RLS) is useful for high-speed applications. The stability of QRD-RLS, STAR-RLS, and PSTAR-RLS has been proved, but the performance of these algorithms in finite-precision arithmetic has not yet been analyzed. The authors determine expressions for the degradation in the performance of these algorithms due to finite precision. By exploiting the steady-state properties of these algorithms, simple expressions are obtained that depend only on known parameters. This analysis can be used to compare the algorithms and to decide the wordlength to be used in an implementation. Since floating- or fixed-point arithmetic representations may be used in practice, both representations are considered. The results show that the three algorithms have about the same finite-precision performance, with PSTAR-RLS performing better than STAR-RLS, which does better than QRD-RLS. These algorithms can be implemented with as few as 8 bits for the fractional part, depending on the filter size and the forgetting factor used. The theoretical expressions are found to be in good agreement with the simulation results.

23 citations


Journal ArticleDOI
TL;DR: This letter presents a new fast QR algorithm based on Givens rotations using a priori errors using the principles behind the triangularization of the weighted input data matrix via QR decomposition and the type of errors used in the updating process.
Abstract: This letter presents a new fast QR algorithm based on Givens rotations using a priori errors. The principles behind the triangularization of the weighted input data matrix via QR decomposition and the type of errors used in the updating process are exploited in order to investigate the relationships among different fast algorithms of the QR family. These algorithms are classified according to a general framework and a detailed description of the new algorithm is presented.

22 citations


Journal ArticleDOI
TL;DR: A procedure for predicting software reliability, using an Auto Regressive (AR) model, using computationally efficient numerical methods like Singular Value Decomposition ( SVD) and QR factorization is proposed.
Abstract: This paper proposes a procedure for predicting software reliability, using an Auto Regressive (AR) model. The parameters of the models are selected using computationally efficient numerical methods like Singular Value Decomposition ( SVD) and QR factorization. For better prediction of time between failures, the AR models have been selected using Akaike Information Criterion (AIC) and Schwarz's Information Criterion ( SIC). A comparative study with the Jelinski-Moranda (JM) and Schick-Wolverton ( SW) models has been performed. Some real life data has been used for illustration purposes.

Journal ArticleDOI
TL;DR: Three methods have been described for solving SEMs subject to separable linear equalities constraints, considers the constraints as additional precise observations while the other two methods reparameterized the constraints to solve reduced unconstrained SEMs.
Abstract: Algorithms for computing the three-stage least squares (3SLS) estimator usually require the disturbance convariance matrix to be non-singular. However, the solution of a reformulated simultaneous equation model (SEM) results into the redundancy of this condition. Having as a basic tool the QR decomposition, the 3SLS estimator, its dispersion matrix and methods for estimating the singular disturbance covariance matrix and derived. Expressions revealing linear combinations between the observations which become redundant have also been presented. Algorithms for computing the 3SLS estimator after the SEM have been modified by deleting or adding new observations or variables are found not to be very efficient, due to the necessity of removing the endogeneity of the new data or by re-estimating the disturbance covariance matrix. Three methods have been described for solving SEMs subject to separable linear equalities constraints. The first method considers the constraints as additional precise observations while the other two methods reparameterized the constraints to solve reduced unconstrained SEMs. Method for computing the main matrix factorizations illustrate the basic principles to be adopted for solving SEMs on serial or parallel computers.

Journal ArticleDOI
TL;DR: Near perfect load balancing is achieved by exploiting a ‘commutativity’ property of the Kronecker product, and communication requirements are minimized by employing a binary exchange algorithm for matrix transposition.

Journal ArticleDOI
TL;DR: The singular value and QR decompositions of the projection matrix give new types of series expansions of the plasma image and present stable and high speed imaging which might be useful for monitoring the time evolution of plasma in a fusion device with angular-limited, sparse and noisy data in projection observation.

Journal ArticleDOI
TL;DR: It is shown that it is the modified Bruhat decomposition that governs the eigenvalue disorder in the QR (GR) algorithm.

Book
31 Aug 1997
TL;DR: This paper presents a review of Solution Techniques for Forward-Looking Models, a method for simulating model simulation on Parallel Computers using the Newton Method, and some examples of such models.
Abstract: Preface. 1: Introduction. 2: A Review of Solution Techniques. 2.1. LU Factorization. 2.2. QR Factorization. 2.3. Direct Methods for Sparse Matrices. 2.4. Stationary Iterative Methods. 2.5. Nonstationary Iterative Methods. 2.6. Newton Methods. 2.7. Finite Difference Newton Method. 2.8. Simplified Newton Method. 2.9. Quasi-Newton Methods. 2.10. Nonlinear First-Order Methods. 2.11. Solution by Minimization. 2.12. Globally Convergent Methods. 2.13. Stopping Criteria and Scaling. 3: Solution of Large-Scale Macroeconometric Models. 3.1. Block Triangular Decomposition of the Jacobian Matrix. 3.2. Orderings of the Jacobian Matrix. 3.3. Point Methods versus Block Methods. 3.4. Essential Feedback Vertex Sets and the Newton Method. 4: Model Simulation on Parallel Computers. 4.1. Introduction to Parallel Computing. 4.2. Model Simulation Experiences. 5: Rational Expectations Models. 5.1. Introduction. 5.2. The Model MULTIMOD. 5.3. Solution Techniques for Forward-Looking Models. A: Appendix. A.1. Finite Precision Arithmetic. A.2. Condition of a Problem. A.3. Complexity of Algorithms. Index.

01 Nov 1997
TL;DR: The BR algorithm as discussed by the authors is a new method for calculating the eigenvalues of an upper Hessenberg matrix, which is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR, it is well adapted to computing the eigvalues of the narrowband, nearly tridiagonal matrices generated by the look-ahead Lanczos process.
Abstract: The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrowband, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the QR algorithm by a factor of 30--60 in computing time and a factor of over 100 in matrix storage space.


Journal ArticleDOI
Chunguang Sun1
15 Dec 1997
TL;DR: New block-oriented parallel algorithms for sparse triangular solution for large-scale sparse linear least squares problems on distributed-memory multiprocessors are proposed and the arithmetic and communication complexities of the new algorithms applied to regular grid problems are analyzed.
Abstract: This paper studies the parallel solution of large-scale sparse linear least squares problems on distributed-memory multiprocessors. The key components required for solving a sparse linear least squares problem are sparse QR factorization and sparse triangular solution. A block-oriented parallel algorithm for sparse QR factorization has already been described in the literature. In this paper, new block-oriented parallel algorithms for sparse triangular solution are proposed. The arithmetic and communication complexities of the new algorithms applied to regular grid problems are analyzed. The proposed parallel sparse triangular solution algorithms together with the block-oriented parallel sparse QR factorization algorithm result in a highly efficient approach to the parallel solution of sparse linear least squares problems. Performance results obtained on an IBM Scalable POWERparallel system SP2 are presented. The largest least squares problem solved has over two million rows and more than a quarter million columns. The execution speed for the numerical factorization of this problem achieves over 3.7 gigaflops per second on an IBM SP2 machine with 128 processors.

Journal ArticleDOI
TL;DR: Numerical and computational aspects of direct methods for large and sparse least squares problems are considered, and a Householder multifrontalscheme and its implementation on sequential and parallel computers are described.
Abstract: Numerical and computational aspects of direct methods for large and sparse least squares problems are considered After a brief survey of the most often used methods, we summarize the important conclusions made from a numerical comparison in matlab Significantly improved algorithms have during the last 10-15 years made sparse QR factorization attractive, and competitive to previously recommended alternatives Of particular importance is the multifrontal approach, characterized by low fill-in, dense subproblems and naturally implemented parallelism We describe a Householder multifrontal scheme and its implementation on sequential and parallel computers Available software has in practice a great influence on the choice of numerical algorithms Less appropriate algorithms are thus often used solely because of existing software packages We briefly survey software packages for the solution of sparse linear least squares problems Finally, we focus on various applications from optimization, leading to the solution of large and sparse linear least squares problems In particular, we concentrate on the important case where the coefficient matrix is a fixed general sparse matrix with a variable diagonal matrix below Inner point methods for constrained linear least squares problems give, for example, rise to such subproblems Important gains can be made by taking advantage of structure Closely related is also the choice of numerical method for these subproblems We discuss why the less accurate normal equations tend to be sufficient in many applications

Journal ArticleDOI
TL;DR: A bound on the performance of QR-factorization with column pivoting with column pivototing is derived and two classes of matrices are constructed for which the bound is sharp or asymptotically sharp.

Journal ArticleDOI
TL;DR: An upper bound for the spectral condition number of a diagonalizable matrix is obtained and can be used in the QR algorithm to estimate the accuracy of the computed eigenvalues.

Journal ArticleDOI
TL;DR: The order-recursive solutions of the least-squares problem, in particular the QR decomposition, are used to obtain the order recursive URLS algorithms, which provide the flexibility to alter the order of the URLS algorithm without adding extra variables.

Journal ArticleDOI
TL;DR: The Modified Covariance method of spectral estimation, proposed as a replacement of the conventional short term Fourier transform for use with pulsed Doppler ultrasound blood flow detectors in order to improve time/frequency resolution, is computationally far more demanding.
Abstract: The Modified Covariance method of spectral estimation, proposed as a replacement of the conventional short term Fourier transform for use with pulsed Doppler ultrasound blood flow detectors in order to improve time/frequency resolution, is computationally far more demanding. The use of an application specific architecture for real-time implementation is of interest. A cost/benefit selection of systolic arrays, firstly for a matrix solution method with suitable bus width and secondly for calculation of covariance matrix elements, led to the choice of Cholesky matrix decomposition, 12 bit bus width and a bi-linear systolic array designed using a data dependence graph method.

Book ChapterDOI
01 Jan 1997
TL;DR: New parallel algorithms for computing rank-revealing QR (RRQR) factorizations of dense matrices on multicomputers, based on a serial approach developed by C. H. Bischof and G. Quintana-Orti are presented.
Abstract: The solution to many scientific and engineering problems requires the determination of the numerical rank of matrices We present new parallel algorithms for computing rank-revealing QR (RRQR) factorizations of dense matrices on multicomputers, based on a serial approach developed by C H Bischof and G Quintana-Orti The parallel implementations include the usual QR factorization with column pivoting, and a new faster approach that consists of two stages: a QR factorization with local column pivoting and a reliable rank-revealing algorithm appropriate for triangular matrices Our parallel implementations include the BLAS-2 and BLAS-3 QR factorizations without pivoting since they are a good reference point, though they are not appropriate for rank-revealing purposes

Book ChapterDOI
01 Jan 1997
TL;DR: In this paper, a QR factorization of the Universal Kriging matrix (M) is proposed to reduce the condition number of M (cond(M)) by using a set of functions derived from the eigenvectors of the variogram matrix (Γ).
Abstract: Many hydrological variables usually show the presence of spatial drifts Most often they are accounted for by either universal or residual kriging and usually assuming low order polynomials The Universal Kriging matrix (M), which includes in it the values of the polynomials at data locations (matrix F), in some cases may have a too large condition number and can even be nearly singular due to the fact that some columns are close to be linearly dependent These problems are usually caused by a combination of pathological data locations and an inadequate choice of the coordinate system As suggested by others, we have found that an appropriate scaling can alleviate the problem by significantly reducing the condition number of M (cond(M)) This scaling, however, does not affect the linear independence of this matrix We show that a QR factorization of matrix F leads to a significant improvement on cond(M) An alternative to drift polynomials consists on using a set of functions derived from the eigenvectors of the variogram matrix (Γ) Although their potential usefulness as interpolating functions remains to be ascertained, they are optimal from the point of view of optimizing cond(M) In fact we are able to provide a rigorous proof for an upper bound of cond(M) Applications of the theoretical developments to hydraulic head data from an alluvial aquifer are also presented

Proceedings ArticleDOI
02 Nov 1997
TL;DR: A modular parallel architecture for a MMSE-DFE coefficient computation processor is presented that is based on the QR factorization of a channel-and-noise-dependent data matrix and is implemented using CORDIC processors within a systolic array architecture.
Abstract: A modular parallel architecture for a MMSE-DFE coefficient computation processor is presented. The architecture is based on the QR factorization of a channel-and-noise-dependent data matrix and is implemented using CORDIC processors within a systolic array architecture. Implementation issues including the number of CORDIC stages and the bit precision required in a fixed-point implementation are investigated through computer simulations.


01 Jan 1997
TL;DR: A new model error indicator function based on singular value decomposition (SVD) and QR permutation decomposition techniques is proposed and is compared with two other popular error indicator functions.
Abstract: A new model error indicator function based on singular value decomposition (SVD) and QR permutation decomposition techniques is proposed. Since an updating problem including large numbers of updating parameters is usuallyill-conditioned, a singular value decomposition technique is first used to determine the meaningful sub-matrix of the system data matrix. A QR permutation decomposition with column pivoting is then performed on this sub-matrix to find the permuted parameters. The equation error norms corresponding to sequential reduced systems are calculated and an error termination criterion is applied to determine the number of meaningful updating parameters. The proposed method is compared with two other popular error indicator functions. Test case includes a free-free supported beam structure.

Proceedings ArticleDOI
02 Jul 1997
TL;DR: Two new fast multichannel QR decomposition (QRD) least squares (LS) adaptive algorithms are presented, which are based on numerically robust orthogonal Givens rotations and offers substantially reduced computational complexity compared to previously derived multich channel fast QRD schemes.
Abstract: Two new fast multichannel QR decomposition (QRD) least squares (LS) adaptive algorithms are presented. Both algorithms deal with the general case of channels with different orders, comprise scalar operations only and are based on numerically robust orthogonal Givens rotations. The first algorithm is a block type scheme which processes all channels jointly. The second algorithm processes each channel separately and offers substantially reduced computational complexity compared to previously derived multichannel fast QRD schemes. This is demonstrated in the context of Volterra filtering.