Search Results (14)

Traditional time-frequency diagnostics for high-speed bearings face an entrenched trade-off between resolution and real-time feasibility. We present a fast Dual-Phase Short-Time Root-MUSIC pipeline that exploits Hankel structure via FFT-accelerated Lanczos bidiagonalization and Sliding-window Singular Value Decomposition to deliver sub-Hz super-resolution under millisecond budgets. Validated on the Politecnico di Torino aerospace dataset (seven fault classes, three severities), fDSTrM detects 150 μm inner-race and rolling-element defects with 98% and 95% probability, respectively, at signal-to-noise ratio down to −3 dB (78% detection), while Short-Time Fourier Transform and Wavelet Packet Decomposition fail under identical settings. Against classical Root-MUSIC, the approach sustains approximately 200 times speedup with less than ${10}^{-11}$ relative frequency error in offline scaling, and achieves 1.85 milliseconds per 4096-sample frame on embedded-class hardware in streaming tests. Subspace order pre-estimation with adaptive correction preserves closely spaced components; Kalman tracking formalizes uncertainty and yields 95% confidence bands. The resulting early warning margin extends maintenance lead-time by 24–72 h under industrial interferences (Gaussian, impulsive, and Variable Frequency Drive harmonics), enabling field-deployable super-resolution previously constrained to offline analysis. Full article

This paper presents a fast and robust approach to evaluate the singular values of small (e.g., $4\times 4$, $5\times 5$) matrices on single- and multi-Graphics Processing Unit (GPU) systems, enabling the modulation of the accuracy–speed trade-off. Targeting applications that require only computations of the SVs in electromagnetics (e.g., Multiple Input Multiple Output—MIMO link capacity optimization) and emerging deep-learning kernels, our method contrasts with existing GPU singular value decomposition (SVD) routines by computing singular values only, thereby reducing overhead compared to full-SVD libraries such as cuSOLVER’s gesvd and MKL’s desvg. The method uses four steps: interlaced storage of the matrices in GPU global memory, bidiagonalization via Householder transformations, symmetric tridiagonalization, and root finding by bisection using Sturm sequences. We implemented the algorithm in CUDA and evaluated it on different single- and multi-GPU systems. The approach is particularly suited for the analysis and design of multiple-input/multiple-output (MIMO) communication links, where thousands of tiny SVDs must be computed rapidly. As an example of the satisfactory performance of our approach, the speed-up reached for large matrix batches against cuSOLVER’s gesvd has been around 20 for $4\times 4$ matrices. Furthermore, near-linear scaling across multi-GPUs systems has been reached, while maintaining root mean square errors below $2.3\times {10}^{-7}$ in single precision and below $2.3\times {10}^{-13}$ in double precision. Tightening the tolerance from $\delta ={10}^{-7}$ to $\delta ={10}^{-9}$ increased the total runtime by only about 10%. Full article

The presence of strong clutter remains a critical challenge for radar system target detection. Traditional clutter suppression techniques such as Doppler-based filters often fail to extract low-velocity targets from clutter. To address this limitation, this paper proposes an adaptive singular value decomposition (A-SVD) method utilizing support vector machines (SVM). The proposed approach leverages the augmented implicitly restarted Lanczos bidiagonalization (AIRLB) algorithm to decompose echo matrices into different subspaces, which are then characterized in relation to Doppler frequency, energy, and correlation. These features are employed to classify the clutter subspaces using an SVM classifier, which solves the problem of selecting the SVD threshold. The clutter subspaces are suppressed by zeroing out corresponding singular values, and the matrix is then recomposed by the rest of the subspaces to recover the echo. Experiments on simulated and real datasets show that the proposed method achieves an average improvement factor (IF) above 40 dB and reduces runtime by over 85% in most scenarios. Full article

Min and max matrices are structured matrices that appear in diverse mathematical and computational applications. Their inherent structures facilitate highly accurate numerical solutions to algebraic problems. In this research, the total positivity of generalized Min and Max matrices is characterized, and their bidiagonal factorizations are derived. It is also demonstrated that these decompositions can be computed with high relative accuracy (HRA), enabling the precise computations of eigenvalues and singular values and the solution of linear systems. Notably, the discussed approach achieves relative errors on the order of the unit roundoff, even for large and ill-conditioned matrices. To illustrate the exceptional accuracy of this method, numerical experiments on quantum extensions of Min and L-Hilbert matrices are presented, showcasing their superior precisions compared to those of standard computational techniques. Full article

Corner cutting algorithms are important in computer-aided geometric design and they are associated to stochastic non-singular totally positive matrices. Non-singular totally positive matrices admit a bidiagonal decomposition. For many important examples, this factorization can be obtained with high relative accuracy. From this factorization, a corner cutting algorithm can be obtained with high relative accuracy. Illustrative examples are included. Full article

This paper presents an accurate method to obtain the bidiagonal decomposition of some generalized Pascal matrices, including Pascal k-eliminated functional matrices and Pascal symmetric functional matrices. Sufficient conditions to assure that these matrices are either totally positive or inverse of totally positive matrices are provided. In these cases, the presented method can be used to compute their eigenvalues, singular values and inverses with high relative accuracy. Numerical examples illustrate the high accuracy of our approach. Full article

The elements of the bidiagonal decomposition (BD) of a totally positive (TP) collocation matrix can be expressed in terms of symmetric functions of the nodes. Making use of this result, and studying the relation between Wronskian and collocation matrices of a given TP basis of functions, we can express the entries of the BD of Wronskian matrices as the values of certain symmetric functions evaluated at a single node. Moreover, in the case of polynomial bases, we obtain compact formulae for the entries of the BD of their Wronskian matrices. Interesting examples illustrate the applications of the obtained formulae. Full article

We consider generalized Green matrices that, in contrast to Green matrices, are not necessarily symmetric. In spite of the loss of symmetry, we show that they can preserve some nice properties of Green matrices. In particular, they admit a bidiagonal decomposition. Moreover, for convenient parameters, the bidiagonal decomposition can be obtained efficiently and with high relative accuracy and it can also be used to compute all eigenvalues, all singular values, the inverse, and the solution of some linear system of equations with high relative accuracy. Numerical examples illustrate the high accuracy of the performed computations using the bidiagonal decompositions. Finally, nonsingular and totally positive generalized Green matrices are characterized. Full article

In order to achieve an effective approximation solution for solving discrete ill-conditioned problems, Golub, Hansen, and O’Leary used Tikhonov regularization and the total least squares (TRTLS) method, where the bidiagonal technique is considered to deal with computational aspects. In this paper, the generalized singular value decomposition (GSVD) technique is used for computational aspects, and then Tikhonov regularized total least squares based on the generalized singular value decomposition (GTRTLS) algorithm is proposed, whose time complexity is better than TRTLS. For medium- and large-scale problems, the randomized GSVD method is adopted to establish the randomized GTRTLS (RGTRTLS) algorithm, which reduced the storage requirement, and accelerated the convergence speed of the GTRTLS algorithm. Full article

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included. Full article

Extensions of Filbert and Lilbert matrices are addressed in this work. They are reciprocal Hankel matrices based on Fibonacci and Lucas numbers, respectively, and both are related to Hilbert matrices. The Neville elimination is applied to provide explicit expressions for their bidiagonal factorization. As a byproduct, formulae for the determinants of these matrices are obtained. Finally, numerical experiments show that several algebraic problems involving these matrices can be solved with outstanding accuracy, in contrast with traditional approaches. Full article

In this work, block checkerboard sign pattern matrices are introduced and analyzed. They satisfy the generalized Perron–Frobenius theorem. We study the case related to total positive matrices in order to guarantee bidiagonal decompositions and some linear algebra computations with high relative accuracy. A result on intervals of checkerboard matrices is included. Some numerical examples illustrate the theoretical results. Full article

A new class of matrices defined in terms of r-Stirling numbers is introduced. These r-Stirling matrices are totally positive and determine the linear transformation between monomial and r-Bell polynomial bases. An efficient algorithm for the computation to high relative accuracy of the bidiagonal factorization of r-Stirling matrices is provided and used to achieve computations to high relative accuracy for the resolution of relevant algebraic problems with collocation, Wronskian, and Gramian matrices of r-Bell bases. The numerical experimentation confirms the accuracy of the proposed procedure. Full article

Green matrices are interpreted as discrete version of Green functions and are used when working with inhomogeneous linear system of differential equations. This paper discusses accurate algebraic computations using a recent procedure to achieve an important factorization of these matrices with high relative accuracy and using alternative accurate methods. An algorithm to compute any minor of a Green matrix with high relative accuracy is also presented. The bidiagonal decomposition of the Hadamard product of Green matrices is obtained. Illustrative numerical examples are included. Full article