C. Reinsch · Mathematics of Computation · 1971
The most efficient program for finding all the eigenvalues of a symmetric matrix is a combination of the Householder tridiagonalization and the <italic>QR</italic> algorithm. The latter, if carried out in a natural way, requires 4<italic>n</italic> additions, 10<italic>n</italic> multiplications, 2<italic>n</italic> divisions, and <italic>n</italic> square roots per iteration (<italic>n</italic> the order of the matrix). In 1963, Ortega and Kaiser showed that the process can be carried out using no square roots (and saving 7<italic>n</italic> multiplications). However, their algorithm is unstable and several modifications were suggested to increase its accuracy. We, too, want to give such a modification together with some examples demonstrating the achieved accuracy.
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Joseph Thomas Parker, P. Hill, David Dickinson et al. · Zenodo (CERN European Organization for Nuclear Research) · 2020
This archive contains the files required to reproduce the results and figures presented in <em>Parallel tridiagonal matrix inversion with a hybrid multigrid--Thomas algorithm method</em>, J. T. Parker, P. A. Hill, D. Dickinson and B. D. Dudson. Also available at the repository: https://gitlab.com/JosephThomasParker/files-and-plotting-scripts-for-parallel-tridiagonal-matrix-inversion-with-a-hybrid-multigrid-thomas-algorithm-method/ Questions to joseph.parker@ukaea.uk. This version is before submission to journal.
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Sho Araki, Tanaka Hiroki, Kimura Kinji et al. · 2013
The orthogonal qd algorithm with shifts (oqds algorithm), proposed by von Matt, is an algorithm for computing the singular values of bidiagonal matrices. This algorithm is accurate in terms of relative error, and it is also applicable to general triangular matrices. In particular, for lower tridiagonal matrices, BLAS Level 2.5 routines are available in preprocessing stage for this algorithm. BLAS Level 2.5 routines are faster than BLAS Level 2 routines widely used in preprocessing for bidiagonalization. Generally, it takes O(n3) operations to reduce a full n-by-n matrix to a band matrix such as bidiagonal or lower tridiagonal matrix. On the other hand, computing the singular values of a bidiagonal or lower tridiagonal matrices takes only O(n2) operations. Consequently, if we have an algorithm for computing the singular values of lower tridiagonal matrices, we can expect that the total computation time including preprocessing to obtain the singular values is reduced. In this paper, we consider the oqds algorithm for lower tridiagonal matrices. We propose a shift strategy for lower tridiagonal matrices to accelerate convergence and derive criteria for deflation or splitting. (This paper is submitted to PDPTA’13)
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Nanda , Dwi Saputra · Digilib Repository Unila (Lampung University) · 2025
Penelitian ini membahas penentuan invers dari matriks k-tridiagonal menggunakan algoritma dekomposisi Doolittle. Matriks k-tridiagonal merupakan generalisasi dari matriks tridiagonal yang memiliki elemen tidak nol hanya pada diagonal utama serta k diagonal di atas dan di bawahnya. Dalam penelitian ini, digunakan pendekatan analitik melalui dekomposisi LU dengan metode Doolittle untuk membagi matriks A menjadi matriks segitiga bawah L dan matriks segitiga atas U. Invers dari matriks A kemudian diperoleh dengan menyelesaikan sistem persamaan linear dua tahap menggunakan substitusi maju dan substitusi mundur. Selain itu, disusun algoritma numerik berbasis Python untuk mempermudah proses komputasi invers matriks k-tridiagonal. Hasil penelitian menunjukkan bahwa metode Doolittle efektif digunakan dalam menentukan invers matriks k-tridiagonal secara sistematis dan terstruktur. Kata-kata kunci: matriks k-tridiagonal, dekomposisi Doolittle, invers matriks, dekomposisi LU, Python. This research discusses the determination of the inverse of a k-tridiagonal matrix using the Doolittle decomposition algorithm. A k-tridiagonal matrix is a generalization of a tridiagonal matrix with non-zero entries only on the main diagonal and k diagonals above and below it. An analytical approach is applied using LU decomposition via the Doolittle method to factor the matrix A into a lower triangular matrix L and an upper triangular matrix U. The inverse of matrix A is then computed by solving the linear system equations in two stages using forward and backward substitution. Additionally, a numerical algorithm based on Python programming is developed to facilitate the computation of the inverse of a k-tridiagonal matrix. The results show that the Doolittle method is effective in systematically and structurally determining the inverse of k-tridiagonal matrices. Keywords: k-tridiagonal matrix, Doolittle decomposition, matrix inverse, LU decomposition, Python.
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M. M. Chawla, Kalpdrum Passi, R.A Zalik · International Journal of Computer Mathematics · 1990
Evans [2, 3] introduced the method of recursive point partitioning algorithm for the solution of sparse banded matrix systems and investigated the “one-line at a time” strategy for the solution of tridiagonal linear systems. Recursive block partitioning schemes resulting from variation in the size of the block structure using “two-lines at a time” have been investigated for both the tridiagonal and the quindiagonal matrix systems in Okolie [6]. The case of partitioning strategy for an nth order system has been considered by Evans and Okolie [4] resulting in a recursive decoupling algorithm for tridiagonal linear systems. Following the recursive point partitioning algorithm of Evans [2, 3], Chawla et al [1] developed a recursive partitioning algorithm for inverting tridiagonal matrices. In the present paper we present a method for inverting tridiagonal matrices by adopting the strategy resulting in a recursive doubling algorithm; the present algorithm has a highly parallel structure.
Cited by 2
Shuotian Chen · 2015
Many eigenvalue and eigenvector algorithms begin with reducing the input matrix into a tridiagonal form. A tridiagonal matrix is a matrix that has non-zero elements only on its main diagonal, and the two diagonals directly adjacent to it. Reducing a matrix to a tridiagonal form is an iterative process which uses Jacobi rotations to reduce matrix elements to zero. The purpose of this research project is to implement an existing algorithm for tridiagonal reduction using CUDA, thus leveraging the parallelism present in GPUs to accelerate the process. In a serial implementation of the algorithm, at each step only the elements in 2 rows/ columns are modified. Therefore, the CUDA implementation takes the form of a parallel reduction algorithm which will simultaneously apply multiple Jacobi rotations to the matrix, thus zeroing out multiple elements at the same time. The CUDA implementation of this algorithm was measured to have a performance improvement of roughly an order of magnitude for sufficiently large matrices as compared to the reference serial CPU implementation. This result shows that the parallel algorithm is able to successfully exploit the GPU’s parallel architecture and provide a significant improvement to the performance of the original algorithm. Subject
Cited by 1
Akiko Fukuda · Kyushu University Institutional Repository (QIR) (Kyushu University) · 2011
The target matrix of the dhLV algorithm is already shown to be a class of nonsymmetric band matrix with complex eigenvalues. In the case where the band width M = 1 in the dhLV algorithm, it is applicable to a tridiagonal matrix, with real eigenvalues, whose upper and lower subdiagonal entries are restricted to be positive and 1, respectively. In this paper, we first clarify that the dhLV algorithm is also applicable to the eigenvalue computation of nonsymmetric tridiagonal matrix with relaxing the restrictions for subdiagonal entries. We also demonstrate that the wellknown packages are not always desirable for computing nonsymmetric eigenvalues with respect to numerical accuracy. Through some numerical examples, it is shown that the tridiagonal eigenvalues computed by the dhLV algorithm are to high relative accuracy.
Cited by 1Open access
Peter Bennery, Heike Fassbendery, David S. Watkinsz · 1997
i, i = 1; : : : ; k. (2) An iteration of the SR algorithm on a 2n \\Theta 2n J-tridiagonal Hamiltonian matrix using shifts _i, \\Gamma _i, i = 1; : : : ; k, is equivalent to an iteration of the HR algorithm on an n \\Theta n tridiagonal sign-symmetric matrix using shifts _2i, i = 1; : : : ; k.
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Rosendo Monroy‐Loperena · Industrial & Engineering Chemistry Research · 2002
It is shown that the well-known and extensively used bubble-point methods, such as the one proposed by Wang and Henke for simulating vapor−liquid multistage distillation problems, can be used efficiently for the simulation of other types of vapor−liquid multistage separation problems, such as absorption and reboiled-absorption processes, even when the thermodynamic properties are a strong function of the composition. Via numerical simulations, it is shown that a robust bubble-point temperature iteration scheme provides convergence stability in the whole bubble-point method. The proposed robust bubble-point temperature iteration scheme is a combination of an equation decoupling method and a simultaneous method. This hybrid bubble-point temperature iteration scheme exploits the reliability of the equation decoupling method, which makes good convergence progress from a poor starting point. However, if the initial point in the bubble-point temperature calculation is good enough, only the equation decoupling method is used, to avoid excessive successive substitution steps when the separation factors are updated. As a consequence of the extension of the bubble-point method to simulate absorption cases, a didiagonal matrix with an off column is generated, when the interstage vapor rates are calculated. An efficient and simple algorithm derived from the Gaussian elimination method to solve the didiagonal matrix with an off column is presented. Results are reported for distillation, absorption, and reboiled-absorption processes, using the Soave−Redlich−Kwong equation of state.
Cited by 8
Hongwei Liu · ASME Press eBooks · 2011
An ergodic algorithm was proposed for solving a class of nonlinear equations and a new algorithm was proposed for solving tridiagonal linear equations. By traversing a single variable, the ergodic algorithm for nonlinear equations can find all real solutions in the specified interval and avoid the iteration divergence. In the algorithm for tridiagonal linear equations, the equations were turned into a linear equation with only one variable. Compared to the computation time by the chasing algorithm for tridiagonal linear equations, the time by this new algorithm is reduced by 40%1¼50% according to simulation results when the tridiagonal matrix dimension is less than 30000.
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Ming Gu, Stanley C. Eisenstat · SIAM Journal on Matrix Analysis and Applications · 1995
The authors present a stable and efficient divide-and-conquer algorithm for computing the spectral decomposition of an $N \times N$ symmetric tridiagonal matrix. The key elements are a new, stable method for finding the spectral decomposition of a symmetric arrowhead matrix and a new implementation of deflation. Numerical results show that this algorithm is competitive with bisection with inverse iteration, Cuppen’s divide-and-conquer algorithm, and the QR algorithm for solving the symmetric tridiagonal eigenproblem.
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Calvetti Golub, D. Calvetti, G. H. Golub et al. · 2007
. Recently Laurie presented a new algorithm for the computation of (2n + 1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n + 1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation phase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible. Numerical examples illustrate the performance of the algorithm. Key words. Jacobi matrix, inverse eigenvalue problem, divide-and...
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C.R. Dun, M. Hegland, M. R. Osborne · 1996
this paper, the wrap-around partitioning methodology, originally proposed by Hegland [1], is considered for the solution of a tridiagonal matrix. This approach can also be formulated as a generalisation of the cyclic reduction algorithm. However, the wrap-around partitioning algorithm described here can be used in conjunction with partial pivoting or QR factorisation. The algorithm presented here is a recursive implementation of the wrap-around partitioning scheme. This has the added advantage of removing the restriction, common to many cyclic reduction algorithms, that the order of the matrix must be a power of 2. In section 2, the wrap around partitioning scheme is presented in terms of simple bidiagonal matrices. It is shown in Section 5 that tridiagonal matrices can be transformed into block-bidiagonal matrices. Thus, the algorithm for the simple bidiagonal matrix forms a prototype for the tridiagonal algorithm. 2 C.R. Dun, M. Hegland &amp; M.R. Osborne The recursive implementation of the wrap-around partitioning scheme is detailed in Section 3. Numerical results of the simple bidiagonal algorithm are presented in Section 4. In Section 5, issues associated with extending the bidiagonal algorithm to the tridiagonal algorithm are discussed.
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Carla Ferreira, Beresford Parlett · 2015
Abstract. We prove convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one-point spectrum- the Jordan form is one big Jordan block. First we develope properties of eigenvector matrices. Key words. unsymmetric tridiagonal matrices, multiple eigenvalues, LR algorithm AMS subject classifications. 65F15 1. Introduction. This paper presents a rigorous proof that the LR algorithm, without shifts, applied to an unreduced tridiagonal matrix with a one-point spectrum converges to an upper bidi-agonal matrix. The rate of convergence is very slow, like 1/k after k steps, but what is remarkable is that the algorithm actually converges. We hasten to say that this result is not exactly new. In the middle of the 1960’s J.H. Wilkinson sketched out the underlying reason for this surprising result,
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Ermuhammad Dushanov · Physics of Particles and Nuclei Letters · 2008
In case of block-tridiagonal matrix, the problem of calculation of a generalized double-point matrix sequence is examined. The general form of inverse matrix of the bordered matrix is obtained when the initial matrix is singular. The criterion of existence of the generalized matrix sequence is found, and the algorithm of calculation of the sequence and the structure elements of the block-tridiagonal matrices are given.
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T. Y. Li, Zhonggang Zeng · 1992
. In this paper we present an algorithm for the eigenvalue problem of symmetric tridiagonal matrices. Our algorithm employs the determinant evaluation, split-and-merge strategy and Laguerre&apos;s iteration. The method directly evaluates eigenvalues and uses inverse iteration as an option when eigenvectors are needed. This algorithm combines the advantages of existing algorithms such as QR, bisection/multisection and Cuppen&apos;s divide-and-conquer method. It is fully parallel, and competitive in speed with the most efficient QR algorithm in serial mode. On the other hand, our algorithm is as accurate as any standard algorithm for the symmetric tridiagonal eigenproblem and enjoys the flexibility in evaluating partial spectrum. Key words. eigenvalue, Laguerre&apos;s iteration, symmetric tridiagonal matrix 1. Introduction. For a symmetric tridiagonal matrix T with nonzero subdiagonal entries, the eigenvalues of T or the zeros of its characteristic polynomial f() = det[T \\Gamma I ] (1.1) are all rea...
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I. BAR ON, B. CODENOTTI, LEONCINI, Mauro · IRIS UNIMORE (University of Modena and Reggio Emilia) · 1996
In this paper we present a linear time algorithm for checking whether a tridiagonal matrix will become singular under certain perturbations of its coefficients. The problem is known to be NP-hard for general matrices. Our algorithm can be used to get perturbation bounds on the solutions to tridiagonal systems.
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Hcini Fahd · Results in Applied Mathematics · 2025
This paper presents two fast numerical algorithms for solving opposite-bordered tridiagonal Toeplitz linear systems. Both algorithms are designed to solve a system of n equations in linear time. The first algorithm uses a block 2×2-LU factorization combined with a fast approach for solving upper quasi-triangular Toeplitz systems. The second algorithm applies a splitting technique to the opposite-bordered tridiagonal Toeplitz matrix, along with a fast algorithm for solving tridiagonal Toeplitz systems. The effectiveness of the proposed algorithms is demonstrated through numerical experiments.
Cited by 1Open access
Daniela Calvetti, Gene H. Golub, William B. Gragg et al. · Mathematics of Computation · 2000
Recently Laurie presented a new algorithm for the computation of $(2n+1)$-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order $2n+1$ from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation phase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible. Numerical examples illustrate the performance of the algorithm.
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Kang, Ji-Hoon · Zenodo (CERN European Organization for Nuclear Research) · 2020
Parallel tridiagonal matrix solver using cyclic reduction / parallel cyclic reduction algorithm in Fortran
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