By C.; Hemery, C. Lebosse
Manuel scolaire de mathématiques, niveau seconde C, programmes de 1965. Algèbre. Cet ouvrage fait partie de l. a. assortment Lebossé-Hémery dont les manuels furent à l’enseignement des mathématiques ce que le Bled et le Bescherelle furent à celui du français.
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Extra resources for Algèbre. Classe de Seconde C
N o t e t h a t in b o t h t h e s e formulae t h e r e a r e d i v i s i o n s by s c a l a r p r o d u c t s , and t h i s i s worrying b e c a u s e of t h e p o s s i b i l i t y t h a t two v e c t o r s in a s c a l a r p r o d u c t may b e n e a r l y o r t h o g o n a l . Therefore in t h i s p a p e r w e offer a n e w formula for r e v i s i n g s e c o n d d e r i v a t i v e a p p r o x i m a t i o n s , t h a t i s a t t r a c t i v e b e c a u s e i t d o e s not i n v o l v e d i v i s i o n s b y s c a l a r p r o d u c t s of different v e c t o r s .
C 2 p " 2 ) , from which we deduce the inequality lim ||G ( ' -G*|| < 4 . 8 9 ( K + 1)LTI/(1-C ) Now 7) is any positive number, so this statement implies that U G ^ ) - Cftl tends to zero. Theorem 5 is proved. We now use this theorem to prove that usually the rate of convergence of the algorithm is super-linear. Theorem 6. If the algorithm is applied with z = 0, if the calculated sequence of points x , ^ = *> 2> • • • > converges to x*> if the derivatives of F(x) satisfy conditions (30) and (31), and if the second derivative matrix at x*, namely GT, is strictly positive definite, then the rate of convergence of the points x ^ ) is super-linear.
Therefore every value of $( k )(x( k ) + _6' k ') is l e s s than or equal to the value that would have been obtained if were equal to r\ . Thus we deduce from expressions (44), (46) and (50) that the condition (51) ^ 3 is satisfied by every ordinary iteration of the algorithm. It follows that condition (28) holds only if the reduction in 47 M. J. D. POWELL F(x) obtained by the kth iteration is bounded by the i n equality F(x ( k ) + _6(k)) - F(x ( k ) ) < - 0 . 075 r\ e . (52) Now if F(x) is bounded below the condition (52) is satisfied only a finite number of times.