Fred T. Krogh

Basic Linear Algebra Subprograms for Fortran Usage

A package of 38 low level subprograms for many of the basic operations of numerical linear algebra is presented. The package is intended to be used with FORTRAN. The operations in the package are dot products, elementary vector operations, Givens transformations, vector copy and swap, vector norms, vector scaling, and the indices of components of largest magnitude. The subprograms and a test driver are available in portable FORTRAN. Versions of the subprograms are also provided in assembly language for the IBM 360/67, the CDC 6600 and CDC 7600, and the Univac 1108.

Algorithm 539: Basic Linear Algebra Subprograms for Fortran Usage [F1]

Received 13 July 1977. Perm~sion to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. The work of the fast and fourth authors was supported by the National Aeronautics and Space Administration under Contract NAS 7-100. The work of the second author was supported by the U.S Energy Research and Development Administration (ERDA) under Contract AT (29-1)-789 and (at Washington State University) by the Office of Naval Research under Contract NR 044-457. Authors addresses: C.L Lawson, Jet Propulsion Laboratory, M/S 125-128, 4800 Oak Grove Drive, Pasadena, CA 91103; R.J. Hanson, Numerical Mathematics, I :~. 5122, Sandia Laboratories, Albuquerque, NM 87115; D.R. Kincaid, Center for Numerical Analysis, The University of Texas at Austin, Austin, TX 78712; F.T. Krogh, Jet Propulsion Laboratory, M/S 125-128, 4800 Oak Grove Drive, Pasadena, CA 91103.

A quadratic-tensor model algorithm for nonlinear least-squares problems with linear constraints

A new algorithm is presented for solving nonlinear least-squares and nonlinear equation problems. The algorithm is based on approximating the nonlinear functions using the quadratic-tensor model proposed by Schnabel and Frank. The problem statement may include simple bounds or more general linear constraints on the unknowns. The algorithm uses a trust-region defined by a box containing the current values of the unknowns. The objective function (Euclidean length of the functions) is allowed to increase at intermediate steps. These increases are allowed as long as our predictor indicates that a new set of best values exists in the trust-region. There is logic provided to retreat to the current best values, should that be required. The computations for the model-problem require a constrained nonlinear least-squares solver. This is done using a simpler version of the algorithm. In its present form the algorithm is effective for problems with linear constraints and dense Jacobian matrices. Results on standard test problems are presented in the Appendix. The new algorithm appears to be efficient in terms of function and Jacobian evaluations.

The gravitational field of a disk

This note gives the gravitational potential of the disk {(x, y, z):x2+y2≤p2, z=0} and the gravitational field at the point (x, y, z). Formulas for a ring can be obtained as the difference of our results for two different values ofp. Results are obtained in terms of elliptic integrals and we indicate how these functions can be computed efficiently. Formulas necessary for the computation of partial derivatives are also given.

Asymptotic (h\rightarrow\infty) Absolute Stability for BDFs Applied to Stiff Differential Equations

Methods based on backward differentiation formulas (BDFs) for solving stiff differential equations require iterating to approximate the solution of the corrector equation on each step. One hope for reducing the cost of this is to make do w~th iteration matrices that are known to have errors and to do no more iterations than are necessary to maintain the stability of the method. This paper, following work by Klopfenstem, examines the effect of errors m the iteration matrix on the stability of the method. Apphcation of the results to an algorithm is discussed briefly.

Algorithm 653: Translation of algorithm 539: PC-BLAS, basic linear algebra subprograms for FORTRAN usage with the INTEL 8087, 80287 numeric data processor

The Basic Linear Algebra Subprograms (BLAS) are described in [l]. The particular implementation documented here is intended for any of the FORTRAN compilers, [2-41, that run on MS-DOS and PC-DOS operating systems. Source code is provided for an Assembly language implementation of these subprograms, which are designed so that the computation is independent of the interface with the calling program unit. In fact, each of the compilers have different methods of passing pointers to input argument lists and returning results for functions. The independence of the mathematical operations from the particulars of the compiler was achieved by the judicious use of Assembly language macros. We believe that it is now a relatively easy job to write these macros for a FORTRAN compiler that is not on our list. The Assembly language versions of the PC-BLAS are generally more efficient when used in applications than are the FORTRAN versions. (See Appendix B for a brief rationale based on efficiency.) Usage of this code requires that the machine have an 8087 or 80287 Numeric Data Processor. The Assembly code for this translation can be assembled using the product of [5]. That product must be acquired by the reader separately; it is not included here. FORTRAN

Improving the efficiency of portable software for linear algebra

In algorithms for linear algebraic computations there are a fairly small number of basic operations which are generally responsible for a significant percentage of the total execution time. We mention particularly the dot product operation, ω := x^Ty, the elementary vector operation, y :=αx+y, and the euclidean vector norm, η = (x^Tx)^1/2.

Opinions on matters connected with the evaluation of programs and methods for integrating ordinary differential equations

Eighty-seven items which might be considered in the evaluation of programs for solving differential equations are listed, and ten experts in the field give their opinions on the importance of each item.

Algorithms Policy

A contribution should be in the form of an algorithm, certification, remark, or translation. Algortthms Algorithms are published in TOMS to make the fruits of research in mathematical software readily available to as wide an audience as possible. An algorithm must either provide a capability not readily available or perform a task better in some way than has been done before. Better can mean anything from improved reliability or efficiency to more attractive packaging. In all cases, an algorithm must represent a substantial contribution in terms of the amount of work or the originality required for its creation. In most cases the communication of new algorithmic ideas should be done either in a companion TOMS paper or in previously published work. The textual part of an algorithm submission should give a brief description of what the algorithm does and pertinent information on usage and maintenance. It should not duplicate information in another paper or in the algorithm listing. Algorithms which serve only a very narrow application area will be considered only if they require less than a single page of TOMS, and complete documentation is available in other published work or in machine readable form with the algorithm submission. An algorithm submitted to TOMS must be complete, portable, well documented, and well structured. The meaning of these terms is clarified below. Preprocessors or compilers that are required by an algorithm, but not generally available, are treated the same as any other code as far as these items are concerned. 1. C o m p l e t e n e s s : With the exception of code used from a previously published TOMS algorithm, a submission must include all of the code and test data necessary for the effective use and testing of the algorithm by a large segment of its intended audience. To assist those who use the algorithm, a small test driver should be provided that illustrates the use of the algorithm for a simple test problem. For testing purposes, one should provide in a single driver a sufficient variety of test cases to exercise all the main features of the code. All submitted code is subject to the refereeing process. Code subject to more restricted use than specified in the TOMS policy may be used in a supporting role for an algorithm provided it is available from an established source for a nominal fee, and there …

A plea for tolerance in the evaluation of numerical methods and mathematical software

It is a trite observation that experts have difficulty in agreeing on anything, except on matters where they are not expert. My concern here is the effect this characteristic of experts has on the evaluation of numerical methods and mathematical software. I believe that careful evaluations of numerical methods and mathematical software is an important obligation of numerical analysts to the scientific community. My plea is that evaluations be carried out in such a way that people with different points of view have sufficient information to reach conclusions. This tolerance for a diversity of opinion may mean extra work for the evaluator, but without this extra work it will be impossible for some people to interpret the results.