2. 第五章 特征值与特征向量 —— 幂法 /* Power Method */ 计算矩阵的主特征根及对应的特征向量 Wait a second, what does that dominant eigenvalue mean? That is the eigenvalue with the largest magnitude. Why in the earth do I want to know that? Don’t you have to compute the spectral radius from time to time?  原始幂法 /* the original method */ 条件： A 有特征根 | 1 | > | 2 |  …  | n |  0 ，对应 n 个线性无 关的特征向量 思路：从任意 出发，要求 … … … | i / 1 | < 1 当 k 充分大时，有 这是 A 关于 1 的近似 特征向量

3. Ch.5 Power Method – The Original Method 定理 设 A  R n  n 为非亏损矩阵 /* non-derogatory */ ，其主特 征根 /* dominant eigenvalue */ 1 为实根，且 | 1 | > | 2 |  …  | n | 。 则从任意非零向量 ( 满足 ) 出发, 迭代 收敛到主特征向量 ， 收敛到 1 。 每个不同的特征根 只对应一个 Jordan 块 注：  结论对重根 1 = 2 = … = r 成立。  若有 1 =  2 ， 则此法不收敛。  任取初始向量时，因为不知道 ，所以不能保 证  1  0 ，故所求得之 不一定是 ，而是使 得 的第一个 ，同时得到的特征根 是 m 。 HW: p.98 #1

4. Ch.5 Power Method – Normalization  规范化 /* normalization */ 为避免大数出现，需将迭代向量规范化，即每一步先保 证 ，再代入下一步迭代。一般用 。 记：则有： Algorithm: Power Method To approximate the dominant eigenvalue and an associated eigenvector of the n  n matrix A given a nonzero initial vector. Input: dimension n; matrix a[ ][ ]; initial vector V0[ ]; tolerance TOL; maximum number of iterations N max. Output: approximate eigenvalue and approximate eigenvector (normalized) or a message of failure.

5. Algorithm: Power Method (continued) Step 1 Set k = 1; Step 2 Find index such that | V0[ index ] | = || V0 ||  ; Step 3 Set V0[ ] = V0[ ] / V0[ index ]; /* normalize V0 */ Step 4 While ( k  N max ) do steps 5-11 Step 5 V [ ] = A V0[ ]; /* compute V k from U k  1 */ Step 6 = V[ index ]; Step 7 Find index such that | V[ index ] | = || V ||  ; Step 8 If V[ index ] == 0 then Output ( “A has the eigenvalue 0”; V0[ ] ) ; STOP. /* the matrix is singular and user should try a new V0 */ Step 9 err = || V0  V / V[ index ] ||  ; V0[ ] = V[ ] / V[ index ]; /* compute U k */ Step 10 If (err < TOL) then Output ( ; V0[ ] ) ; STOP. /* successful */ Step 11 Set k ++; Step 12 Output (Maximum number of iterations exceeded); STOP. /* unsuccessful */ Ch.5 Power Method – Normalization

6. Ch.5 Power Method – Deflation Technique  原点平移法 /* deflation technique */ 决定收敛的速度，特别 是 | 2 / 1 | 希望 | 2 / 1 | 越小越好。 不妨设 1 > 2  …  n ，且 | 2 | > | n | 。 1 2 n O p = ( 2 + n ) / 2 思路思路 令 B = A  pI ，则有 | I  A | = | I  (B+pI) | = | ( p)I  B |  A  p = B 。 而 ，所以求 B 的特征根收 敛快。 As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality. -- Albert Einstein (1879-1955) How are we supposed to know where p is?

7.  反幂法 /* Inverse Power Method */ Ch.5 Power Method –Inverse Power Method 若 A 有 | 1 |  | 2 |  … > | n | ，则 A  1 有 对应同样一组特征向量。 1 1 1 11  …   nn A  1 的主特征根 A 的绝对值最小的特征根 Q: How must we compute in every step? A: Solve a linear system with A factorized. 若知道某一特征根 i 的大致位置 p ，即对任意 j  i 有 | i  p | << | j  p | ，并且如果 (A  pI)  1 存在，则 可以用反幂法求 (A  pI)  1 的主特征根 1/( i  p ) ，收 敛将非常快。 思路思路

8. Ch.5 Power Method –Inverse Power Method Lab 09. Approximating Eigenvalues Approximate an eigenvalue and an associated eigenvector of a given n  n matrix A near a given value p and a nonzero vector. Input There are several sets of inputs. For each set: The 1 st line contains an integer 100  n  0 which is the size of a matrix. n =  1 signals the end of file. The following n lines contain the matrix entries in the format shown: The next line contains a real number TOL, which is the tolerance for eigenvalues, and an integer N  0 which is the maximum number of iterations. The next line contains an integer n  m > 0 which is the number of eigenvalues to be approximated. Each of the following m lines contains a real number p which is an initial approximation of the eigenvalue, followed by n real number entries of the nonzero vector. The numbers are separated by spaces and new lines. The inputs guarantee that the shifted matrix can be factorized by Doolittle method.

9. Ch.5 Power Method –Inverse Power Method Output (  represents a space) For each p, there must be a set of outputs in the following format: The 1 st line contains the approximation of an eigenvalue printed as in the C printf: fprintf(outfile, "%12.8f\n", lambda ); The 2 nd line contains the n entries of the associated eigenvector. Each entry is printed as in the C fprintf: fprintf(outfile, "%12.8f  ", x ); If the method fails to give a solution after N iterations, print the message “Maximum  number  of  iterations  exceeded.\n”. If p is just the accurate eigenvalue, print the message fprintf(outfile, “%12.8f  is  an  eigenvalue.\n”, p ); The outputs of two test cases must be seperated by a blank line. Sample Input 3 1  2  3 2  3  4 3  4  5 0.0000000001  1000 1 –0.6  1  1  1 2 0  1 1  0 0.0000000001  10 1 1.0  1  1 –1 Sample Output  –0.62347538  1.00000000  0.17206558  –0.65586885   1.00000000  is  an  eigenvalue.