Numerical aspects of computing the Moore-Penrose inverse of full column rank matrices

Abstract

This paper presents a comparison of certain direct algorithms for computing the Moore-Penrose inverse, for matrices of full column rank, from the point of view of numerical stability. It is proved that the algorithm using Householder QR decomposition, implemented in floating point arithmetic, is forward stable but only conditionally mixed forward-backward stable. A similar result holds also for the Classical Gram-Schmidt algorithm with reorthogonalization (CGS2). This algorithm was developed and analyzed by Abdelmalek (BIT, 11 (4) We incorporate the recent result of Byers and Xu (SIAM J. Matrix Anal. Appl. 30:822-843, 2008) to show that the algorithm based on the Golub-Kahan bidiagonalization is mixed forward-backward stable under natural assumptions.

Stability of Two Direct Methods for Bidiagonalization and Partial Least Squares

“…Their proof that this algorithm is mixed forward-backward stable applies also to the Householder PLS algorithm. Similar results are shown in Smoktunowicz and Wróbel [23].…”

Application of the partitioning method to specific Toeplitz matrices

“…In order to confirm the efficiency of Algorithm 2, we compared the block partitioning method with three recently announced methods for computing the Moore-Penrose inverse (Chountasis et al, 2009a;2009b;Courrieu, 2005;Katsikis and Pappas, 2008). Therefore, the following algorithms for computing the Moore-Penrose inverse are compared: A comparison of several direct algorithms for computing the Moore-Penrose inverse of full column rank matrices was presented by Smoktunowicz and Wróbel (2012). Also, the computational cost of these methods for computing the Moore-Penrose inverse of a full column rank matrix A ∈ R m×n is given in Table 1 of Smoktunowicz and Wróbel (2012).…”

“…Therefore, the following algorithms for computing the Moore-Penrose inverse are compared: A comparison of several direct algorithms for computing the Moore-Penrose inverse of full column rank matrices was presented by Smoktunowicz and Wróbel (2012). Also, the computational cost of these methods for computing the Moore-Penrose inverse of a full column rank matrix A ∈ R m×n is given in Table 1 of Smoktunowicz and Wróbel (2012). In our case, we have the situation A = H T ∈ R m+l−1×m .…”

“…In our case, we have the situation A = H T ∈ R m+l−1×m . According to the computational complexities presented by Smoktunowicz and Wróbel (2012), the complexity of the Courrieu method is equal to E Chol = 3(m + l − 1)m 2 + m 3 /3, and the complexity of Qrginv method is…”

Convergence of Rump’s method for computing the Moore-Penrose inverse

“…A number of numerical and symbolic algorithms, see [5], [6], [19], [20], [36] for computing the Moore-Penrose inverse of the structured and block matrices have been presented. Rump et al develop the numerically verified methods for the matrix inversion, see [24], [26], [29], [33], the matrix equations in [25], the linear least squares problem and the under-determined linear system in [28].…”