Burton S. Garbow

Testing Unconstrained Optimization Software

Much of the testing of optimization software is inadequate because the number of test functmns is small or the starting points are close to the solution. In addition, there has been too much emphasm on measurmg the efficmncy of the software and not enough on testing reliability and robustness. To address this need, we have produced a relatwely large but easy-to-use collection of test functions and designed gmdelines for testing the reliability and robustness of unconstrained optimization software.

Software for estimating sparse Jacobian matrices

In many nonlinear problems it is necessary to estimate the Jacobian matrix of a nonlinear mapping F. In large scale problems the Jacobian of F is usually sparse, and then estimation by differences is attractive because the number of differences can be small compared to the dimension of the problem. For example, if the Jacobian matrix is banded then the number of differences needed to estimate the Jacobian matrix is, at most, the width of the band. In this paper we describe a set of subroutines whose purpose is to estimate the Jacobian matrix of a mapping F with the least possible number of function evaluations.

Software for estimating sparse Hessian matrices

The solution of a nonlinear optimization problem often requires an estimate of the Hessian matrix for a function f. In large scale problems, the Hessian matrix is usually sparse, and then estimation by differences of gradients is attractive because the number of differences can be small compared to the dimension of the problem. In this paper we describe a set of subroutines whose purpose is to estimate the Hessian matrix with the least possible number of gradient evaluations.

Algorithm 566: FORTRAN Subroutines for Testing Unconstrained Optimization Software [C5], [E4]

A partial listing of the FORTRAN package of subroutines for testing unconstrained optimazation software is given with a brief description of the subroutines. The following three problem areas are considered: (1) zeros of systems of N nonlinear functions in N variables; (2) least square minimization of M nonlinear functions in N variables; (3) unconstrained minimization of an objective function with N variables. To test a code in any of the three problem areas, the user must provide a driver and interface routine. The package includes example drivers and interface routines for each of the problem areas. Sample data are also provided. (SC)

Remark on “Algorithm 535: The QZ Algorithm to Solve the Generalized Eigenvalue Problem for Complex Matrices [F2]”

Three FORTRAN subroutines are provided that implement a complex form of the QZ algorithm for finding lambda and z such that Az = lambda Bz, where A and B are complex N by N matrices. The complex QZ algorithm is unaffected by singularity or near singularity of B. Subroutie CQZHES implements the first step of the algorithm wherein A and B are simultaneously reduced by unitary transformations to upper Hessenberg and upper triangular form, respectively. Subroutine CQZVAL implements an iterative process that reduces A to upper triangular form while maintaining the trianglar form of B. The eigenvalues are derivable from the corresponding diagonal elements of the reduced A and B. Subroutine CQZVEC applies the accumulated transformations from the two earlier steps onto the eigenvectors of the triangular problem. No facility is provided for obtaining just a few eigenvectors or, for balancing A and B. A long-precision IBM version of the subroutines was tested on a 370/195. There are no machine-dependent constants in the subroutines, so the standard version should run directly on different machines. (RWR)

Algorithm 662: A Fortran software package for the numerical inversion of the Laplace transform based on Weeks' method

Here we present a brief documentation of the software package WEEKS written in FORTRAN 77. The mathematical background and general information about its performance are described in the accompanying paper on pages 163-170 of this issue. Further information of a theoretical nature may be found in [3]. Some of the design characteristics of this package are described in detail in a preliminary report [2]. However, there are minor differences between the implementation described there and that described here.

EISPACK — A package of matrix eigensystem routines

Abstract EISPACK is a package of Fortran subroutines that find eigenvalues and eigenvectors of matrices. This paper introduces each of the current members of EISPACK and describes the capability it contributes to the package. The final section describes the planned further extensions of the package.

FORTRAN subroutines for estimating sparse Hessian matrices (Algorithm 649).

This is the FORTRAN package of subroutines described in [1] for estimating the Hessian matrix of a mapping f: Rn ~ R. Given the sparsity pattern of the lower triangular part of the (symmetric) Hessian matrix, the package partitions the columns of the Hessian into p groups such that the entire matrix can be estimated from p gradient evaluations. The package contains two principal subroutines, DSSM and FDHS. DSSM calls the remaining subroutines of the package plus the subroutines of the earlier Jacobian package [2] to determine an appropriate partition of the columns of the Hessian matrix. Given the partition, FDHS then computes an approximation to the Hessian matrix. The package includes an example program that calls both DSSM and FDHS. No data are required. The example is described in Section 4 of [1].

Remark on algorithm 662

There are several errors in two subroutines, MODULl and MODUL2, of the package that this Remark corrects. MODULl.