LSEQIEQ: a FORTRAN IV subroutine package for the analysis of multiple linear regression problems with possibly deficient pseudorank and linear equality and inequality constraints

Abstract

An algorithm and FORTRAN IV subroutine package are presented which solve the general linear least-squares problem: minimize∥AX−B∥ minimize∥X∥ where A is a p by n data matrix of values preceding the n unknown coefficients symbolized by the vector X and B is a p-vector of dependent variables corresponding to the p rows of the data matrix A, subject to an arbitrary set of linear equality constraints and inequality bounds on the coefficients of the vector X. The solution is effected by orthogonal decomposition of the data matrix and the code includes a provision for estimating the true rank (actual number of independent variables) of the problem and extracting a solution vector which incorporates this intervariable correlation. The computer storage requirements of the subroutines are kept to a minimum by implementing a sequential accumulation algorithm which alleviates the need to store the data matrix in memory at any time. Thus, an effectively infinite number of data points can be processed in a small fixed storage space. A detailed description of subroutine usage is given and three applications of the code are presented which deal with rank deficient data matrices which are: (1) large and overdetermined, (2) small, but whose unknown coefficients are bounded by inequality constraints, and (3) underdetermined, with bounded coefficients subject to linear equality constraints. The first problem is concerned with the application of thermodynamics to the solution properties of silicate liquids where a strongly rank deficient data matrix is analyzed. The second application of the algorithm involves a rapid method of estimating the precious metal content of mineral phases in an ore body. Lastly, a complex underdetermined problem, from the solution chemistry of clay minerals, is investigated. A solution is obtained only after careful and judicious construction of the inequality constraint matrix.These applications stem from typical least-squares problems encountered in igneous petrology, economic geology, and geochemistry and emphasize the advantages of the present approach over standard regression techniques.