The condition of gram matrices and related problems

1978 ◽  
Vol 80 (1-2) ◽  
pp. 45-56 ◽  
Author(s):  
J. M. Taylor
Keyword(s):  
Least Squares ◽  
Product Space ◽  
Linear Equations ◽  
Accurate Solution ◽  
Inner Product ◽  
Condition Numbers ◽  
Least Squares Problems ◽  
Normal Equations ◽  
Gram Matrices ◽  
Ill Conditioning

SynopsisIt has been known for some time that certain least-squares problems are “ill-conditioned”, and that it is therefore difficult to compute an accurate solution. The degree of ill-conditioning depends on the basis chosen for the subspace in which it is desired to find an approximation. This paper characterizes the degree of ill-conditioning, for a general inner-product space, in terms of the basis.The results are applied to least-squares polynomial approximation. It is shown, for example, that the powers {1, z, z2,…} are a universally bad choice of basis. In this case, the condition numbers of the associated matrices of the normal equations grow at least as fast as 4n, where n is the degree of the approximating polynomial.Analogous results are given for the problem of finite interpolation, which is closely related to the least-squares problem.Applications of the results are given to two algorithms—the Method of Moments for solving linear equations and Krylov's Method for computing the characteristic polynomial of a matrix.

scholarly journals Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
K. Appi Reddy ◽  
T. Kurmayya
Keyword(s):  
Product Space ◽  
Matrix Multiplication ◽  
Convex Cones ◽  
Inner Product Space ◽  
Inner Product ◽  
Indefinite Inner Product ◽  
Indefinite Matrix ◽  
Gram Matrices ◽  
Indefinite Inner Product Space

AbstractIn this paper we characterize Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space using the indefinite matrix multiplication. This characterization includes the acuteness (or obtuseness) of certain closed convex cones.

scholarly journals Accurate Solution of Structured Least Squares Problems via Rank-Revealing Decompositions

2013 ◽  
Vol 34 (3) ◽  
pp. 1112-1128 ◽  
Author(s):  
Nieves Castro-González ◽  
Johan Ceballos ◽  
Froilán M. Dopico ◽  
Juan M. Molera
Keyword(s):  
Least Squares ◽  
Accurate Solution ◽  
Least Squares Problems ◽  
Rank Revealing

scholarly journals Condition numbers of the least squares problems with multiple right-hand sides

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1667-1676
Author(s):  
Lingsheng Meng ◽  
Bing Zheng
Keyword(s):  
Least Squares ◽  
Numerical Experiments ◽  
Upper Bounds ◽  
Condition Numbers ◽  
Least Squares Problems ◽  
Least Squares Problem ◽  
Right Hand

In this paper, we investigate the normwise, mixed and componentwise condition numbers of the least squares problem min X?Rnxd ||X - B||F, where A ? Rmxn is a rank-deficient matrix and B ? Rmxd. The closed formulas or upper bounds for these condition numbers are presented, which extend the earlier work for the least squares problem with single right-hand side (i.e. B ? b is an m-vector) of several authors. Numerical experiments are given to confirm our results.

scholarly journals On the Evaluation of Determinants, the Formation of their Adjugates, and the Practical Solution of Simultaneous Linear Equations

1933 ◽  
Vol 3 (3) ◽  
pp. 207-219 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Form ◽  
Least Squares ◽  
Linear Equations ◽  
Statistical Correlation ◽  
Simultaneous Equations ◽  
Normal Equations ◽  
Practical Solution ◽  
Simultaneous Linear Equations ◽  
Reciprocal Matrix ◽  
Image Position

There are various methods in existence for the practical solution of a set of simultaneous equationsSome of these methods are appropriate to special systems, as for example to the axisymmetric “normal equations” of Least Squares. In many applications, however, as in problems of statistical correlation of many variables, it may be desired not merely to solve a given set of equations but to obtain as much knowledge as possible about the system or matrix of coefficients; perhaps to evaluate its determinant and various minors, such as principal minors, possibly also to determine the elements of the adjugate matrix, or the reciprocal matrix. The examination of the sign of successive principal minors of an axisymmetric determinant, in order to find the signature of the corresponding quadratic form, is a case in point; and there are many such applications.

Condition Numbers for Structured Least Squares Problems

2006 ◽  
Vol 46 (1) ◽  
pp. 203-225 ◽  
Author(s):  
W. Xu ◽  
Y. Wei ◽  
S. Qiao
Keyword(s):  
Least Squares ◽  
Condition Numbers ◽  
Least Squares Problems

Perturbation analysis and condition numbers of scaled total least squares problems

2009 ◽  
Vol 51 (3) ◽  
pp. 381-399 ◽  
Author(s):  
Liangmin Zhou ◽  
Lijing Lin ◽  
Yimin Wei ◽  
Sanzheng Qiao
Keyword(s):  
Least Squares ◽  
Perturbation Analysis ◽  
Total Least Squares ◽  
Condition Numbers ◽  
Least Squares Problems

scholarly journals On the Condition Number Theory of the Equality Constrained Indefinite Least Squares Problem

2018 ◽  
Vol 34 ◽  
pp. 619-638
Author(s):  
Shaoxin Wang ◽  
Hanyu Li ◽  
Hu Yang
Keyword(s):  
Number Theory ◽  
Least Squares ◽  
Condition Number ◽  
Condition Numbers ◽  
Least Squares Problems ◽  
Unified Framework ◽  
Least Squares Problem ◽  
Practical Applications ◽  
New Findings ◽  
Special Cases

In this paper, within a unified framework of the condition number theory, the explicit expression of the \emph{projected} condition number of the equality constrained indefinite least squares problem is presented. By setting specific norms and parameters, some widely used condition numbers, like the normwise, mixed and componentwise condition numbers follow as its special cases. Considering practical applications and computation, some new compact forms or upper bounds of the projected condition numbers are given to improve the computational efficiency. The new compact forms are of particular interest in calculating the exact value of the 2-norm projected condition numbers. When the equality constrained indefinite least squares problem degenerates into some specific least squares problems, our results give some new findings on the condition number theory of these specific least squares problems. Numerical experiments are given to illustrate our theoretical results.

scholarly journals Mixed and component wise condition numbers for weighted Moore-Penrose inverse and weighted least squares problems

Filomat ◽  
2009 ◽  
Vol 23 (1) ◽  
pp. 43-59 ◽  
Author(s):  
Li Zhao ◽  
Jie Sun
Keyword(s):  
Numerical Analysis ◽  
Least Squares ◽  
Weighted Least Squares ◽  
Full Column Rank ◽  
Condition Numbers ◽  
Linear Least Squares ◽  
Least Squares Problems ◽  
Linear Least Squares Problem ◽  
Classical Condition ◽  
The Matrix

Condition numbers play an important role in numerical analysis. Classical condition numbers are norm-wise: they measure both input perturbations and output errors with norms. To take into account the relative scaling of data components or a possible sparseness, component-wise condition numbers have been increasingly considered. In this paper, we give explicit expressions for the mixed and component-wise condition numbers for the weighted Moore-Penrose inverse of a matrix A, as well as for the solution and residue of a weighted linear least squares problem ||W 1 2 (Ax-b) ||2 = minv2Rn ||W 1 2 (Av-b) ||2, where the matrix A with full column rank. .

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