An evaluation of linear least squares computer programs

Abstract

Two linear least squ a res tes t p roblems, both fifth degree polyno mi a ls, have bee n run on more th an twe nJ y d iffe rent co mput e r progra ms in orde r to assess th e ir num erica l acc uracy. Among the progra ms tes ted were re presentati ves f ro m vari ous sta ti sti ca l pac kages as we ll as some from th e S HA RE libra ry. Essenti a ll y fi ve diffe re nt algorithm s were used in the va ri ous progra ms to obta in the coeffi c ients of the leas t squ a res fit s. The tests were run on severa l diffe rent comput e rs, in doubl e prec i io n as we ll as s ingle precis ion. By co mpa ring the coe ffi c ie nts re port ed , it was found th at those programs us in g orthogona l Householde r transform ations or Gra m-Schmidt orthonorm aliza tion we re much more accura te th an those us ing e liminatio n a lgo rithms. P rogra ms us ing orthogo na l polyno mi als (s uit a bl e onl y for po lynomi a l fit s) a lso pro ved to be superior to those us ing e limin ation a lgo rithm s_ O ne program , us ing congru enti a l me thods and int eger a rithme ti c, obt a ined exac t so lutions. In a number of progra ms , the coeffi cie nts re port ed in one tes t probl e m were sometim es co mple te ly e rroneous, cont aining not even one co rrec t s ignificant digit.