Y.Shen.J.Zhao Appl.Math.Comput.144 (2003)441-455 447 i.e., 0>arctany Proposition 2.2 shows that if over some interval of iterations residual norms generated by Lanczos are increasing at least as a specified rate,then the angle (K,AK)cannot decrease at a rate faster than the bound on 0 given in (2.9),i.e.,the corresponding MINRES residual norms cannot decrease at a rate faster than the bound on sin 0.For example,if y>2 then 0> arctan2≈63.4349，sin0≈0.89442.▣ 3.Peaks and some related factors The reason why the residual norm is not always minimized is more inter- esting,for it touches a deeper issue and is a topic of current concern and im- perfect understanding [3].In this section we consider four factors which are related to the Lanczos peaks. I.Numerical Instabilities.What role do numerical instabilities play in the generation of the peak formations observed in the Lanczos residual norm plots?In [1],we know that if the linear system (1.1)is sufficiently well condi- tioned [1,Definition 3.3],then numerical instabilities have no role in any ob- served peak formations. II.Finite precision arithmetic.Are the peaks and plateaus artifacts of the finite precision arithmetic?See [1],we know that peaks and plateaus are not artifacts of the finite precision arithmetic.Peaks and plateaus can also occur when the arithmetic is exact.However,more peaks or plateaus will occur in finite precision arithmetic than would occur if the computations were exact. Moreover,the effect of finite precision arithmetic is an open problem. III.Angle between subspaces.Based on properties of angle between sub- spaces and use the relationship between the orthogonal residual norm and the minimal residual norm,we obtain a sufficient and necessary condition for occurring of peaks. Theorem 3.1.In the exact arithmetic,for c0,k 1,2,...,L-1 the condi- tion 序+答<序+山，B≥ 1 (3.1) is satisfied if and only if >B (3.2) where F=sin∠(K,AK)i.e., h > arctan c Proposition 2.2 shows that if over some interval of iterations residual norms generated by Lanczos are increasing at least as a specified rate c, then the angle \ðKk ; AKk Þ cannot decrease at a rate faster than the bound on h given in (2.9), i.e., the corresponding MINRES residual norms cannot decrease at a rate faster than the bound on sin h. For example, if c > 2 then h > arctan 2  63:4349, sin h  0:89442. 3. Peaks and some related factors The reason why the residual norm is not always minimized is more inter￾esting, for it touches a deeper issue and is a topic of current concern and im￾perfect understanding [3]. In this section we consider four factors which are related to the Lanczos peaks. I. Numerical Instabilities. What role do numerical instabilities play in the generation of the peak formations observed in the Lanczos residual norm plots? In [1], we know that if the linear system (1.1) is sufficiently well condi￾tioned [1, Definition 3.3], then numerical instabilities have no role in any ob￾served peak formations. II. Finite precision arithmetic. Are the peaks and plateaus artifacts of the finite precision arithmetic? See [1], we know that peaks and plateaus are not artifacts of the finite precision arithmetic. Peaks and plateaus can also occur when the arithmetic is exact. However, more peaks or plateaus will occur in finite precision arithmetic than would occur if the computations were exact. Moreover, the effect of finite precision arithmetic is an open problem. III. Angle between subspaces. Based on properties of angle between sub￾spaces and use the relationship between the orthogonal residual norm and the minimal residual norm, we obtain a sufficient and necessary condition for occurring of peaks. Theorem 3.1. In the exact arithmetic, for ck 6¼ 0; k ¼ 1; 2; ... ; L  1 the condi￾tion 1 F 2 k þ F 2 k1 b2 < 1 b2 þ 1; b P1 ð3:1Þ is satisfied if and only if krLR k k krLR k1k > b ð3:2Þ where Fk ¼ sin \ðKk ; AKk Þ Y. Shen, J. Zhao / Appl. Math. Comput. 144 (2003) 441–455 447