GNU Octave, version 2.1.36 (i686-pc-linux-gnu). Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002 John W. Eaton. This is free software; see the source code for copying conditions. There is ABSOLUTELY NO WARRANTY; not even for MERCHANTIBILITY or FITNESS FOR A PARTICULAR PURPOSE. For details, type `warranty'. Report bugs to . octave> A=[2 1 2; 3 1 2; 1 2 3] A = 2 1 2 3 1 2 1 2 3 octave> b=[1;2;3] b = 1 2 3 octave> b b = 1 2 3 octave> A A = 2 1 2 3 1 2 1 2 3 octave> b=[1;2;3] b = 1 2 3 octave> x=A\b x = 1.0000 7.0000 -4.0000 octave> A*x ans = 1.0000 2.0000 3.0000 octave> lu(A) ans = 0.66667 0.20000 1.00000 1.00000 0.00000 0.00000 0.33333 1.00000 0.00000 octave> [L,U,P]=lu(A) L = 1.00000 0.00000 0.00000 0.33333 1.00000 0.00000 0.66667 0.20000 1.00000 U = 3.00000 1.00000 2.00000 0.00000 1.66667 2.33333 0.00000 0.00000 0.20000 P = 0 1 0 0 0 1 1 0 0 octave> P*A-L*U ans = 0 0 0 0 0 0 0 0 0 octave> A=[.985 .002 .001; .005 .991 .001; .01 .007 .998] A = 0.9940000 0.0020000 0.0010000 0.0050000 0.9910000 0.0010000 0.0100000 0.0070000 0.9980000 octave> [1 1 1] * A ans = 1.0090 1.0000 1.0000 octave> A=[.985 .002 .001; .005 .991 .001; .01 .007 .998] A = 0.9850000 0.0020000 0.0010000 0.0050000 0.9910000 0.0010000 0.0100000 0.0070000 0.9980000 octave> [1 1 1] * A ans = 1 1 1 octave> A^3 ans = 0.9702450 0.0039590 0.0019850 0.0098900 0.9820980 0.0019940 0.0198650 0.0139430 0.9960210 octave> A^30 ans = 0.9557310 0.0058778 0.0029552 0.0146721 0.9732929 0.0029820 0.0295970 0.0208294 0.9940628 octave> A^35 ans = 0.641891 0.044894 0.024331 0.109668 0.768355 0.027412 0.248442 0.186751 0.948257 octave> A^100 ans = 0.597554 0.049916 0.027429 0.121420 0.736520 0.031472 0.281026 0.213564 0.941099 octave> A^1000 ans = 0.256859 0.080490 0.052518 0.184490 0.444775 0.072601 0.558651 0.474735 0.874881 octave> A^10000 ans = 0.070518 0.070522 0.070511 0.128237 0.128266 0.128193 0.801246 0.801212 0.801296 octave> A^100001 ans = 0.070513 0.070513 0.070513 0.128205 0.128205 0.128205 0.801282 0.801282 0.801282 octave> A^100001 ans = 0.070513 0.070513 0.070513 0.128205 0.128205 0.128205 0.801282 0.801282 0.801282 octave>