Convergence Properties of Certain Minres Algorithms.
Levin (1988) has challenged the convergence properties of the Harman and Jones (1966) method of Minres factor analysis. Levin claimed that convergence of the Harman and Jones method is not guaranteed and that a modified version of this method, with proven convergence, is to be preferred. In the present article it is shown that both claims are invalid. Monotone convergence of the Harman and Jones method is guaranteed whereas the modified version, proposed by Levin, may converge to an incorrect solution. Levin has also claimed that the rank-one version of the Harman and Jones method, as implemented in a method by Zegers and ten Berge (1983) lacks a valid convergence proof, and that a method suggested by Comrey and Ahumada (1964, 1965) should be used instead. It is shown that these claims, too, should be reversed.
An alternating least squares method for the weighted approximation of a symmetric matrix
Bailey and Gower examined the least squares approximationC to a symmetric matrixB, when the squared discrepancies for diagonal elements receive specific nonunit weights. They focussed on mathematical properties of the optimalC, in constrained and unconstrained cases, rather than on how to obtainC for any givenB. In the present paper a computational solution is given for the case whereC is constrained to be positive semidefinite and of a fixed rankr or less. The solution is based on weakly constrained linear regression analysis.
On oblique procrustes rotation
The equations involved in the rotation of an arbitrary factor matrix to a least squares fit to a specified factor structure have been known for many years. These equations, in general, cannot be solved by purely algebraic means, and an approximate solution has previously been used in practical applications.In this paper an effective iterative method for obtaining the exact solution is developed. By algebraic manipulation the set of equations is expressed in the form of one polynomial equation in one unknown. Newton's method is suggested for solving this equation. Practical applications of the procedure indicate that convergence within small tolerance limits is generally attained after few iterations.