A breakdown-free Lanczos type algorithm for solving linear systems.
A novel parallel algorithm based on the Gram-Schmidt method for tridiagonal linear systems of equations.
A Numerical Procedure for Computing the Moore-Penrose Inverse.
A parallel projection method for linear algebraic systems
A direct projection method for solving systems of linear algebraic equations is described. The algorithm is equivalent to the algorithm for minimization of the corresponding quadratic function and can be generalized for the minimization of a strictly convex function.
A robust and efficient parallel SVD solver based on restarted Lanczos bidiagonalization.
A simple recursive estimator with on-line orthogonalization of the input sequence
A unified approach to some strategies for the treatment of breakdown in Lanczos-type algorithms
The Lanczos method for solving systems of linear equations is implemented by using some recurrence relationships between polynomials of a family of formal orthogonal polynomials or between those of two adjacent families of formal orthogonal polynomials. A division by zero can occur in these relations, thus producing a breakdown in the algorithm which has to be stopped. In this paper, three strategies to avoid this drawback are discussed: the MRZ and its variants, the normalized and unnormalized...
Algorithms 80-81. Calculation of projections
An algebraic construction of discrete wavelet transforms
Discrete wavelets are viewed as linear algebraic transforms given by banded orthogonal matrices which can be built up from small matrix blocks satisfying certain conditions. A generalization of the finite support Daubechies wavelets is discussed and some special cases promising more rapid signal reduction are derived.
An Extension of 3D Zernike Moments for Shape Description and Retrieval of Maps Defined in Rectangular Solids
Zernike polynomials have been widely used in the description and shape retrieval of 3D objects. These orthonormal polynomials allow for efficient description and reconstruction of objects that can be scaled to fit within the unit ball. However, maps defined within box-shaped regions ¶ for example, rectangular prisms or cubes ¶ are not well suited to representation by Zernike polynomials, because these functions are not orthogonal over such regions. In particular, the representations require many...
Computing the CS Decomposition of a Partitioned Orthonormal Matrix
Decomposition of an updated correlation matrix via hyperbolic transformations
An algorithm for hyperbolic singular value decomposition of a given complex matrix based on hyperbolic Householder and Givens transformation matrices is described in detail. The main application of this algorithm is the decomposition of an updated correlation matrix.
Derivation of BiCG from the conditions defining Lanczos' method for solving a system of linear equations
Lanczos’ method for solving the system of linear algebraic equations consists in constructing a sequence of vectors in such a way that and . This sequence of vectors can be computed by the BiCG (BiOMin) algorithm. In this paper is shown how to obtain the recurrences of BiCG (BiOMin) directly from this conditions.
Error Analysis of an Algorithm for Solving an Undetermined Linear System.
Fast block Toeplitz orthogonalization.
Fast Givens transformation for quaternion valued matrices applied to Hessenberg reductions.
Fast Toeplitz Orthogonalization.
Filter factors of truncated TLS regularization with multiple observations
The total least squares (TLS) and truncated TLS (T-TLS) methods are widely known linear data fitting approaches, often used also in the context of very ill-conditioned, rank-deficient, or ill-posed problems. Regularization properties of T-TLS applied to linear approximation problems were analyzed by Fierro, Golub, Hansen, and O’Leary (1997) through the so-called filter factors allowing to represent the solution in terms of a filtered pseudoinverse of applied to . This paper focuses on the situation...
Hybrid Algorithm for Fast Toeplitz Orthogonalization.
Modified Gram-Schmidt for solving linear least squares problems is equivalent to Gaussian elimination for the normal equations