SOLVING LINEAR LEAST SQUARES PROBLEMS BY GRAM-SCHMIDT ORTHOGONALIZATION

1. Introduct ion . L e t A b e a g i v e n m x n r ea l m a t r i x of r a n k n, m > n, a n d b a g i v e n m x 1 r e a l v e c t o r . T h e n t h e r e e x i s t s a u n i q u e v e c t o r x w h i c h so lves t h e l e a s t s q u a r e s p r o b l e m of m i n i m i z i n g ltb A x l t ~ • I t is we l l k n o w n t h a t t h e s o l u t i o n x sa t i s f i e s t h e c o n d i t i o n A ~ ' ( b A x ) = O , (1 . I ) i .e . t h e r e s i d u a l v e c t o r r = b A x is o r t h o g o n a l t o t h e c o l u m n s of A . I t fo l lows t h a t w e c a n c o m p u t e x f r o m t h e n o r m a l e q u a t i o n s A T A x = A r b . (1.2) W e d e f i n e fo l l owing B a u e r [1] t h e ( co lumnw ise ) c o n d i t i o n of t h e r e c t a n g u l a r m a t r i x A t o be c o n d ( A ) = m a x []Axll/min []Ax][. I~1=I II~i=I T h e c o n d i t i o n n u m b e r c o r r e s p o n d i n g t o t h e L 2 n o r m is d e n o t e d b y n(A). I n s e c t i o n 7 we wi l l show t h a t u n d e r s o m e r e s t r i c t i o n s , z (A) c a n b e con-