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We describe an algorithm that can be used to select a set of independent columns of a matrix when numerical rank can be determined, and we associate the uncertainty of the data with rank selection. Mathematical software written in Fortran which implements this algorithm is available for a variety of computing machines.

This paper discusses the robustness and the computational stability of the singular value decomposition algorithm used at the NBER Computer Research Center. The effect of perturbations on input data is explored. Suggestions are made for using the algorithm to get information about the rank of a real square or rectangular matrix. The algorithm can also be used to compute the best approximate solution of linear system of equations in the least squares sense, to solve linear systems of equations with equality constraints, and to determine dependencies or near dependencies among the rows or columns of a matrix. A copy of the subroutine that is used and some examples on which it has been tested are included in the appendixes.

In this paper we shall be concerned with the following problem. Let A be an m x n matrix with m being greater than or equal to n, and suppose that A is near (in a sense to be made precise later) a matrix B whose rank is less than n. Can one find a set of linearly independent columns of A that span a good approximation to the column space of B? The solution of this problem is important in a number of applications. In this paper we shall be chiefly interested in the case where the columns of A represent factors or carriers in a linear model which is to be fit to a vector of observations b. In some such applications, where the elements of A can be specified exactly (e.g. the analysis of variance), the presence of rank degeneracy in A can be dealt with by explicit mathematical formulas and causes no essential difficulties. In other applications, however, the presence of degeneracy is not at all obvious, and the failure to detect it can result in meaningless results or even the catastrophic failure of the numerical algorithms being used to solve the problem. The organization of this paper is the following. In the next section we shall give a precise definition of approximate degeneracy in terms of the singular value decomposition of A. In Section 3 we shall show that under certain conditions there is associated with A a subspace that is insensitive to how it is approximated by various choices of the columns of A, and in Section 4 we shall apply this result to the solution of the least squares problem. Sections 5, 6, and 7 will be concerned with algorithms for selecting a basis for the stable subspace from among the columns of A.

This paper presents a means for detecting the presence of multicollinearity and for assessing the damage that such collinearity may cause estimated coefficients in the standard linear regression model. The means of analysis is the singular value decomposition, a numerical analytic device that directly exposes both the conditioning of the data matrix X and the linear dependencies that may exist among its columns. The same information is employed in the second part of the paper to determine the extent to which each regression coefficient is being adversely affected by each linear relation among the columns of X that lead to its ill conditioning.

Transferring Fortran subroutines from one manufacturer' s machine to another, or from one operating system to another, puts certain constraints on the construction of the Fortran statements that are used in the subroutines. The reliable performance of mathematical software should be unaffected by the host environment in which the software is used or by the compiler from which the cod eis generated. In short, the reliable performance of the algorithm is to be independent of the computing environment in which it is run. The subroutines of ROSEPACK (Robust Statistics Estimation Package) are Fortran IV source code designed to be semi-portable where semi-portable is defined to mean transportable with minimum change. *This paper described the guidelines by which ROSEPACK subroutines were written.

A description of a system of subroutines to compute solutions to the iteratively reweighted least squares problem is presented. The weights are determined from the data and linear fit and are computed as functions of the scaled residuals. Iteratively reweighted least squares is a part of robust statistics where "robustness" means relative insensitivity to moderate departures from assumptions. The software for iteratively reweighted least squares is cast as semi-portable Fortran code whose performance is unaffected (in the sense that performance will not be degraded) by the computer or operating-system environment in which it is used. An [ell sub1] start and an [ell sub2] start are provided. Eight weight functions, a numerical rank determination, convergence criterion, and a stem-and-leaf display are included.

In this paper we shall be concerned with the following problem. Let A be an m x n matrix with m being greater than or equal to n, and suppose that A is near (in a sense to be made precise later) a matrix B whose rank is less than n. Can one find a set of linearly independent columns of A that span a good approximation to the column space of B? The solution of this problem is important in a number of applications. In this paper we shall be chiefly interested in the case where the columns of A represent factors or carriers in a linear model which is to be fit to a vector of observations b. In some such applications, where the elements of A can be specified exactly (e.g. the analysis of variance), the presence of rank degeneracy in A can be dealt with by explicit mathematical formulas and causes no essential difficulties. In other applications, however, the presence of degeneracy is not at all obvious, and the failure to detect it can result in meaningless results or even the catastrophic failure of the numerical algorithms being used to solve the problem. The organization of this paper is the following. In the next section we shall give a precise definition of approximate degeneracy in terms of the singular value decomposition of A. In Section 3 we shall show that under certain conditions there is associated with A a subspace that is insensitive to how it is approximated by various choices of the columns of A, and in Section 4 we shall apply this result to the solution of the least squares problem. Sections 5, 6, and 7 will be concerned with algorithms for selecting a basis for the stable subspace from among the columns of A.

Transferring Fortran subroutines from one manufacturer's machine to another or from one operating system to another puts certain constrains on the construction of the Fortran statements that are used in the subroutines. The reliable performance of this mathematical software should be unaffected by the host environment in which the software is used or by the compiler from which the code is generated. In short, the algorithm is to he independent of the computing environment in which it is run. The subroutines of ROSEPACK (Robust Statistics Estimation Package) are Fortran IV source code designed to be semi-portable where semi-portable is defined to mean transportable with minimum change.*

This paper discusses the robustness and the computational stability of the singular value decomposition algorithm used at the NBER Computer Research Center. The effect of perturbations on input data is explored. Suggestions are made for using the algorithm to get information about the rank of a real square or rectangular matrix. The algorithm can also be used to compute the best approximate solution of linear system of equations in the least squares sense, to solve linear systems of equations with equality constraints, and to determine dependencies or near dependencies among the rows or columns of a matrix. A copy of the subroutine that is used and some examples on which it has been tested are included in the appendixes.

Transferring Fortran subroutines from one manufacturer' s machine to another, or from one operating system to another, puts certain constraints on the construction of the Fortran statements that are used in the subroutines. The reliable performance of mathematical software should be unaffected by the host environment in which the software is used or by the compiler from which the cod eis generated. In short, the reliable performance of the algorithm is to be independent of the computing environment in which it is run. The subroutines of ROSEPACK (Robust Statistics Estimation Package) are Fortran IV source code designed to be semi-portable where semi-portable is defined to mean transportable with minimum change. *This paper described the guidelines by which ROSEPACK subroutines were written.

This grant, under the URIP program, provided instrumentation equipment used to enhance research in concurrent computing. The equipment provided a network of concurrent computing machines. Keywords: Numerical algorithms; computations. (Author)

The final report for the project, Mathematical Software for Linear Control and Estimation Theory, includes a list of publications, conference proceedings, and invited talks by the principal investigators. Emphasis throughout the project was placed on interdisciplinary work in control theory, numerical methods suitable for certain control problems and mathematical software to do the frequency response problem. Numerically stable methods were used, and portability of the mathematical software was stressed. (Author)

ROSEPACK (RObust Statistics Estimation PACKage) is a system of Fortran subroutines and an interactive driver to compute solutions to the iteratively reweighted least square problem. The weights are determined from the data and the linear fit and are computed as functions of the scaled residuals.

We provide further discussion of the use of least-squares techniques for this and G. W. Stewart, LINPACK Users' Guide (Society of Industrial and Applied. User's Guide for Intel® Math Kernel Library 11.3 for Linux* OS. Revision: Benchmark your cluster with Intel® Optimized MP LINPACK Benchmark for Clusters. Running Linpack on Linux High performance computer. No problem. I need a working guide for hpl if someone can do help me at this. The errors are.