Lanczos 方法 Louis Komzsik 著张伟廖本善译 演变与应用 清华大学出版社 北京
内容简介 Lanczos 方法是 20 世纪计算数学方向最有影响的方法之一, 并且已经在工程中得到了广泛应用. 本书兼顾了 Lanczos 方法的理论演变和工程中的实际应用, 其内容分为两部分 : 第一部分阐述了方法的演变, 并提供了具体算法 ; 第二部分讨论了工业中的实际应用, 包括常用的模态分析 复特征值分析 频率响应分析以及线性系统问题的求解. 对于应用数学和工业工程专业的研究人员, 以及工程计算领域的工程师, 本书是一本很有价值的参考书. The Lanczos Method: Evolution and Application copyright 2003 Society for Industrial and Applied Mathematics. Published by Tsinghua University Press with permission. Chinese edition copyright 2011 by Tsinghua University Press. 版权所有, 侵权必究 侵权举报电话 :010 62782989 13701121933 图书在版编目 (CIP) 数据 Lanczos 方法 : 演变与应用 /( 美 ) 科姆日克 (Komzsik, L.) 著 ; 张伟, 廖本善译. 北京 : 清华大学出版社,2011.3 书名原文 :The Lanczos Method: Evolution and Application ISBN 978 7 302 25271 9 Ⅰ. 1 L Ⅱ. 1 科 2 张 3 廖 Ⅲ.1 电子计算机 - 计算方法 Ⅳ. 1TP301.6 中国版本图书馆 CIP 数据核字 (2011) 第 060555 号 责任编辑 : 佟丽霞责任校对 : 刘玉霞责任印制 : 王秀菊 出版发行 : 清华大学出版社 地 址 : 北京清华大学学研大厦 A 座 http://www.tup.com.cn 邮 编 :100084 社总机 :010 62770175 邮 购 :010 62786544 投稿与读者服务 :010 62776969, c-service@tup.tsinghua.edu.cn 质量反馈 :010 62772015, zhiliang@tup.tsinghua.edu.cn 印刷者 : 北京清华园胶印厂 装订者 : 经 销 : 全国新华书店 开 本 :170 230 印张 :6 字数 :106 千字 版 次 :2011 年 3 月第 1 版 印次 :2011 年 3 月第 1 次印刷 印 数 :1 4 000 定 价 :14.00 元 产品编号 :032543 01
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,..,.. Louis Komzsik 2009
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iv, QR Givens., MATLAB. 25, Lanczos. Beresford Parlett Gene Golub,.,. Horst Simon, John Lewis Zhaojun Bai. Lanczos., Tom Kowalski, NASTRAN 1,,. Louis Komzsik 2002 1 NASTRAN.
1 Lanczos................................................. 3 1.1.............................................................3 1.2.............................................................4 1.3.............................................. 6 1.4............................................................... 8 2 Lanczos........................................... 10 2.1.............................................................. 10 2.2.................................................... 12 2.3......................................................... 14 2.4........................................................... 14 3 Lanczos........................................... 20 3.1 Lanczos................................................... 20 3.2.................................................... 21 3.3............................................. 22 3.4......................................................... 23 4 Lanczos......................................... 25 4.1 Lanczos.................................................. 25 4.2............................................. 27 4.3............................................... 28 4.4.................................................... 29 5 Lanczos......................................... 30 5.1 Lanczos........................................... 30 5.2............................................... 31 5.3.............................................................. 31 5.4.......................................................32 5.5......................................................... 33
vi 5.6.............................................................. 34 5.7.......................................................35 6 Lanczos............................................ 41 6.1................................................................ 41 6.2........................................................... 43 6.3........................................................... 43 6.3.1......................................................... 44 6.3.2................................................... 45 6.3.3....................................................... 45 6.3.4 Lanczos.............................................. 46 6.4............................................... 47 7...................................................... 49 7.1.......................................................49 7.2.................................................. 51 7.3.................................................... 52 8...................................................... 54 8.1.................................................. 54 8.2.......................................................55 8.3........................................................... 57 8.4......................................................... 58 8.5......................................................... 59 8.6............................................... 61 9.........................................................63 9.1......................................................... 63 9.2................................................... 64 9.3.................................................... 66 9.4 Lanczos Padé....................................... 66 9.4.1.................................................... 67 9.4.2.................................................... 67 9.5.......................................................69
vii 10 Lanczos.......................................... 71 10.1............................................................... 71 10.2............................................................... 71 10.3.......................................................... 73 10.4 Lanczos............................................... 74 10.5.................................................75.......................................................................... 77 Cornelius Lanczos........................................... 78.............................................................................79.............................................................................80........................................................................81
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1 Lanczos, Lanczos.,. Lanczos [2],, (The method of minimized iterations).,. 1.1 n A, n n f(x) = x T Ax = a kl x k x l (1.1) k=1 l=1. x = (x 1, x 2,, x n ), Lanczos. A, x T Ax = 1 (1.2) R n n. n. x i, i = 1, 2,, n. λ i, Ax i = λ i x i. (1.3),, x i, n ( ), x i = cn, (1.4) c., n = (x T Ax 1) = Ax i, (1.5) c = 1/λ i.
4 1 Lanczos 1.2 Lanczos (A T = A), G(µ) = det(a µi) = 0, (1.6) Au = µu, (1.7) u, µ. Lanczos, Lanczos., b 0 b T 0 b 0, b 2 0,. Lanczos,. b, b 1 b 0 Ab 0 b 1 = Ab 0 α 0 b 0. (1.8) α 0 b 1 b 2 1 = (Ab 0 α 0 b 0 ) 2. (1.9). α 0 = (Ab 0)b 0 b 2. (1.10) 0 b 1 b 0.,, b 1 b 0 = 0. (1.11) b 2 = Ab 1 α 1 b 1 β 0 b 0, (1.12) b 2., b 2 2. α 1 = (Ab 1)b 1 b 2 1, β 0 = (Ab 1)b 0 b 2. (1.13) 0
1.2 5 (Ab 1 )b 0 = b 1 (Ab 0 ) = b 2 1, (1.14) b 2 b 1 b 0., b 3 = Ab 2 α 2 b 2 β 1 b 1 γ 0 b 0. (1.15), b 2 (orthogonality), γ 0 = (Ab 2)b 0 b 2 0 = b 2(Ab 0 ) b 2 0 = 0. (1.16) Lanczos,,, b 0, b 1 = (A α 0 I)b 0, b 2 = (A α 1 I)b 1 β 0 b 0, b 3 = (A α 2 I)b 2 β 1 b 1,. b m = (A α m 1 I)b m 1 β m 2 b m 2 = 0. m,. Lanczos (order), m n, n A.,,, β k, k < m. 20 60,, Lanczos. Lanczos, A A T,, (bi-orthogonal process). b 0 b 0 (b 0, b 0 b H 0 ), α 0 b 1 = Ab 0 α 0 b 0, (1.17) b 1 = A T b 0 α 0 b 0, (1.18) α 0 = (Ab 0)b 0 b 0 b 0 = (AT b 0)b 0 b 0b 0. (1.19)
6 1 Lanczos b 2 = Ab 1 α 1 b 1 β 0 b 0, (1.20) b 2 = A T b 1 α 1 b 1 β 0 b 0, (1.21) α 1 β 0 α 1 = (Ab 1)b 1 b 1 b 1 = (AT b 1)b 1 b 1b 1, (1.22) β 0 = (Ab 1)b 0 b 1 b 0, = (AT b 1)b 0 b 0b 0 = b 1b 1 b 0b 0. (1.23) p 0 = 1, p 1 (µ) = µ α 0, p 2 (µ) = (µ α 1 )p 1 (µ) β 0 p 0 (µ),. p n (µ) = (µ α n 1 )p n 1 (µ) β n 2 p n 2., n ( β i ), p n A, µ i, i = 1, 2,, n. 1.3 Lanczos {b i }, {b i }. A, b i b i = p i (µ 1 )u 1 + p i (µ 2 )u 2 + + p i (µ n )u n. (1.24) b i u k, u k u i, b i u k = p i (µ k )u k u k. (1.25). u k uh k, k., u i b i, u i = α i,0 b 0 + α i,1 b 1 + + α i,n 1 b n 1. (1.26)
1.3 7 u i b k = α i,k b k b k, (1.27) α i,k = u ib k b k b. (1.28) k u i = b 0 b 0 b + p 1 (µ i ) b 1 0 b 1 b 1 u i = b 0 b 0 b + p 1 (µ i ) b 1 0 b 1 b 1 b n 1 + + p n 1 (µ i ) b n 1 b. (1.29) n 1 b n 1 + + p n 1 (µ i ) b n 1 b. (1.30) n 1, m n,..,, A (B) * T, B T = (B ) T AB, (1.31) [ B = b 0 b 1 b n 1 ], (1.32) [ B = b 0 b 1 b n 1 ], (1.33) α 0 β 0 β 0 α 1 β 1 T =.......... (1.34) β n 2 α n 1 β n 1 β n 1 α n., (1.7) (B ) T AB(B ) T u = µ(b ) T u, (1.35) b ( bi-orthogonality) B(B ) T = I, (1.36)
8 1 Lanczos T v = µv, (1.37) v = (B ) T u, u = Bv. (1.38) : T v, Lanczos. 1.4,. Lanczos b 0, b 1, S, S = span(b 0, b 1 ). (1.39) Lanczos. Ab 1 (1.13) proj S (Ab 1 ) = (Ab 1)b 0 b 0 b 0 b 0 + (Ab 1)b 1 b 1 b 1 b 1, (1.40) proj S (Ab 1 ) = (Ab 1)b 0 b 2 0 b 0 + (Ab 1)b 1 b 2 b 1. (1.41) 1 proj S (Ab 1 ) = β 0 b 0 + α 1 b 1. (1.42) Ab 1,, Lanczos b 2 = Ab 1 proj S (Ab 1 ) = Ab 1 α 1 b 1 β 0 b 0 (1.43) (1.12), Lanczos., Lanczos, Ab j Lanczos, Ab j,, b j+1, Lanczos b j+1., [ B j = b 0 b 1 b j ], (1.44) S j = span(b j ), (1.45)
1.4 9 a = Ab j, (1.46) p = proj Sj a, (1.47) b j+1 = a p. (1.48), Lanczos., Lanczos,,.
2 Lanczos, Lanczos,,. 2.1 Ax = λx, y T A = λy T, (2.1) A., x, y, Lanczos. Lanczos X n Y n ( b i b i ), Y T nx n = I, (2.2) I n, Y T nax n = T n, (2.3) A, T n., : AX n = X n T n, (2.4) Y T na = T n Y T n. (2.5), Ax k = γ k 1 x k 1 + α k x k + β k x k+1, (2.6) y T ka = β k 1 y T k 1 + α k y T k + γ k y T k+1. (2.7) y k x k Y n X n k, k = 1, 2,, n 1. k (k < n), AX k = X k T k + β k x k+1 e T k, (2.8)