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{SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 55 "Computing the Perron Eige
nvector of a Reciprocal Matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(linalg):\nDigits:=3;\n
" }{TEXT -1 471 "For the following reciprocal matrix M, we find the Pe
rron eigenvalue and eigenvector V.  The procedure is to start with som
e vector X = (1,1,. . . ,1) and compute M^kX.  If M were consistent, t
hen the other n-1 eigenvalues will be zero.  Thus, even if M is not ex
actly consistent, we anticipate that  M^kX ~ lambda^k cV for some cons
tant c.   Thus, normalizing M^kX we can recover the normalized Perron \+
eigenvector V.   Comparing M^\{k+1\} and M^k we can recover lambda.  \+
\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 351 "M:=matrix([[1, 5, 3, 7, 6, 6
, 1/3, 1.0/4],\n          [1/5, 1, 1/3, 5, 3, 3, 1/5, 1/7],\n         \+
 [1/3, 3, 1, 6, 3, 4, 1/2, 1/5],\n          [1/7, 1/5, 1/6, 1, 1/3, 1/
4, 1/7, 1/8],\n          [1/6, 1/3, 1/3, 3, 1, 1/2, 1/5, 1/6],\n      \+
    [1/6, 1/3, 1/4, 4, 2, 1, 1/5, 1/6],\n          [3, 5, 2, 7, 5, 5, \+
1, 1/2],\n          [4, 7, 5, 8, 6, 6, 2, 1]]);  \n" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 48 "Loop to compute MX and then normalize the
 output" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "MainEigenvect := \+
proc(n)\nlocal i,k,X,Y,Z,sum;\nX:=vector([1,1,1,1,1,1,1,1]);\nY:=evalm
(X/8);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "for k from 1 to n\ndo\n \+
 X:=evalm(M&*Y):\n  sum:=0;\n  for i from 1 to 8\n  do \n    sum := su
m+X[i];\n  od;\n  Y:=evalm(X/sum);\nod;\nRETURN(sum, evalm(Y));\nend;
\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Do the computation." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "MainEigenvect(4);\n" }}}
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