Lecture+24--Iterative+method

Lecture+24--Iterative+method - Example of ill-conditioned...

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4 3 2 1 0 -1 -2 -2 -1 0 1 2 3 4 1.01x + 0.99y = 1.98 0.99x + 1.01y = 2.02 x y 4 3 2 1 0 -1 -2 -2 -1 0 1 2 3 4 1.01x + 0.99y = 2.02 0.99x + 1.01y = 1.98 x y Example of ill-conditioned system
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Matrix Condition Evaluation Hilbert matrix – notoriously ill-conditioned Row-sum norm (last row has the largest sum) Normalization - + + + + = - + + + + = 1 n 2 n 2 n n 1 n n 1 2 n 3 5 3 4 3 1 1 n 2 4 2 3 2 1 n 1 3 1 2 1 1 A 1 n 2 1 2 n 1 1 n 1 n 1 2 n 1 5 1 4 1 3 1 1 n 1 4 1 3 1 2 1 n 1 3 1 2 1 1 A ] [ ] [ 1 n 2 n 2 n n 1 n n 1 A - + + + + + + =
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Matrix Condition Evaluation 5  5 Hilbert matrix: A=hilb(5) Condition number = 9 5 8 5 7 5 6 5 1 8 4 7 4 6 4 5 4 1 7 3 6 3 5 3 4 3 1 6 2 5 2 4 2 3 2 1 5 1 4 1 3 1 2 1 1 A / / / / / / / / / / / / / / / / / / / / ] [ 736 434260 116480 7282 3 A A A Cond 116480 17640 44800 39200 13440 1400 A 7282 3 9 5 8 5 7 5 6 5 1 A 1 1 . ) ( ) . ( ] [ . = = = = - + + - + + - = = + + + + = - - - - - - - - - - - - - - = - 8820 22050 18900 6300 630 17640 44800 39200 13440 1400 11340 29400 26460 9450 1050 2520 6720 6300 2400 300 126 350 350 150 25 A 1 ] [
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MATLAB Norms and Condition Numbers >> A=[1 1/2 1/3 1/4 1/5; 1 2/3 2/4 2/5 2/6; 1 3/4 3/5 3/6 3/7; 1 4/5 4/6 4/7 4/8; 1 5/6 5/7 5/8 5/9];
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Lecture+24--Iterative+method - Example of ill-conditioned...

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