... 50 CHAPTER 4. FUNCTION OPTIMIZATIONso that orthogonalityfor qbecomespporp p (4. 19)This condition is called -conjugacy,or -orthogonality: if equation (4. 19) holds, then p and p are ... the procedure, due to Hestenes and Stiefel (Methods of conjugate gradients for solving linear systems,J.Res. Bureau National Standards, section B, Vol 49 , pp. 40 9 -4 36, 1952), which also incorporates ... equations (4. 3) and (4. 6) and theKantorovich inequality we obtainx x yg gg g g gy y (4. 11)x x (4. 12)Since the ratio in the last term is smaller than one, it follows immediately that x x and hence,...