RJS Software

The following products are developed by RJS Software - a third party company, for use with GAUSS. Technical support is provided directly through the developer.

LAPACK for GAUSS

The LALIB package is an implementation of LAPACK as an extension of the GAUSS Run-time Library. The LAPACK routines for real and complex general, real symmetric, complex symmetric, and complex Hermitian matrices are implemented.

LAPACK ­ Linear Algebra PACKage ­ is the long awaited update to the well known LINPACK and EISPACK software packages. For more than 20 years LINPACK and EISPACK have been the standard for numerical computation. Currently used by GAUSS and other numerical and statistical software as their core routines, LINPACK and EISPACK have now been upgraded under the direction of many of the same people who created the original software. LAPACK, not only contains the latest, state-of-the-art numerical algorithms, it also provides many new features for the serious numerical analyst. These features emphasize the most important numerical analysis issue, the accuracy and precision of the ill-conditioned problem.

An important addition is the "expert" routine. The linear equation, least squares, and eigenvalue functions have both regular and expert versions. The expert versions, in addition to returning the usual results, also provide extensive information about the problem. For example, the expert version of the linear equation solver for the real or complex square matrices equilibrates and scales the input matrices, and returns the LU factorization, the pivoting information, scaling vectors, condition estimate, and forward error bounds and relative backward error estimates.

LALIB contains routines for solving linear equations, least squares problems, eigensystems, and factorizations. The following routines are included:

Linear Equations

Ordinary Least Squares

Eigen systems

LALIB contains a full complement of eigen system functions in both regular and expert versions. Subsets of eigenvalues/vectors may be computed by specifying a range of either values or indices. For square input matrices either left or right eigenvectors, or both, may be computed. There are also functions for computing the singular value decomposition and Schur form and vectors.

Solves

LALIB contains solve functions for real or complex general matrices, real or complex, symmetric or Hermitian, positivdefinite or indefinite matrices, as well as triangular matrices, and Sylvester's equation. The expert versions return appropriatfactorizations, pivot vectors, scaling vectors, condition numbers, and forward and backward error bounds.

Factorizations

LALIB implements real and complex versions of the QR, RQ, LDL, LU, and Cholesky factorizations.

LQR, LQRE, LQREP, LQQR, LQQRE, LQQREP, LQYR, LQYRE, LQYREP, LQYTR, LQYTRE, LQYTREP	QR factorization for real or complex rectangular matrices, with and without pivoting, with and without Q, QY, and Q'Y
LLU, LINV, LLUCOND, LLUDET	For real or complex rectangular matrices LU factorization with pivoting, inverse (for square matrices), condition number, determinant
LCHOL, LINVPD, LCHCOND, LCHDET	For real symmetric or complex Hermitian positive definite matrices, Cholesky factorization, inverse, condition number, determinant
LDL, LDLINV, LDLCOND, LDLDET	For real or complex symmetric, complex Hermitian indefinite matrices, LDL factorization, inverse, condition number, determinant

Platform: Windows

Requires: GAUSS Mathematical & Statistical System v3.2 and above.

QP v1.0- Quadratic Programming

QP solves the standard quadratic programming problem: min{1/2x'Qx - x'R}, subject to constraints: Ax = B and Cx >= D, with bounds: Xl <= x <= Xu, where x is a vector of unknown coefficients, and Q, R, A, B, C, D, Xl, and Xu are known matrices.

CLSQ

Constrained least squares is a special case of the the quadratic programming problem. CLSQ is a procedure included in the QP module for computing constrained least squares regression estimates. The ability to specify inequality constraints and to place bounds on the coefficients is unique to this procedure and not available in other GAUSS applications. CLSQ also computes the correct standard errors of the constrained coefficients.

Most regression models contain coefficients that can be bounded or constrained in some way. For example, it is often known that one or more coefficients are positive or are in some range. Incorporating this information into the estimation using CLSQ always improves the t-statistics of the estimates over the unconstrained estimation. Even specifying very broad ranges for the coefficients can improve the efficiency of the estimates, and for that reason the use of CLSQ could be recommended for all least squares problems.

Portfolio Management

The "Mean-Variance", "Mean-SemiVariance" and "Effective Mix" models are important applications of the QP problem in investment portfolio management. The Effective Mix model is a constrained least squares problem for which CLSQ is suited. The Mean-Variance and Mean-Semivariance models are quadratic programming problems where Q is the covariance matrix of a portfolio of stocks, bonds, options, etc., and R is a vector of their mean values. The QP solution yields estimates of the ideal distribution of the portfolio among the securities.

Parametric Quadratic Programming

PQP is a procedure included in the QP module for simulating portfolio distribution under various assumptions about investment strategies. Mean values, risk tolerance and structural constraints can all be varied, and the implications for the portfolio distribution can be explored.

Platform: Windows, LINUX and UNIX

Requires: GAUSS Mathematical & Statistical System v3.2 or above.