LAPACK is a transportable library of Fortran 77 subroutines for solving the most common problems in numerical linear algebra: systems of linear equations, linear least squares problems, eigenvalue problems and singular value problems. LAPACK is designed to supersede LINPACK and EISPACK, principally by restructuring the software to achieve much greater efficiency on vector processors, high-performance ``superscalar`` workstations, and shared-memory multi-processors. LAPACK also adds extra functionality, uses some new or improved algorithms, and integrates the two sets of algorithms into a unified package. The LAPACK Users` Guide gives an informal introduction to the design of the algorithms and software, summarizes the contents of the package, describes conventions used in the software and documentation, and includes complete specifications for calling the routines. This edition of the Users` guide describes Release 1.0 of LAPACK.