Lecture 3 - CE 30125 Lecture 3 LECTURE 3 DIRECT SOLUTIONS TO LINEAR ALGEBRAIC SYSTEMS CONTINUED Ill-conditioning of Matrices There is no clear cut

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CE 30125 - Lecture 3 p. 3.1 LECTURE 3 DIRECT SOLUTIONS TO LINEAR ALGEBRAIC SYSTEMS - CONTINUED Ill-conditioning of Matrices • There is no clear cut or precise definition of an ill-conditioned matrix. Effects of ill-conditioning • Roundoff error accrues in the calculations • Can potentially result in very inaccurate solutions • Small variation in matrix coefficients causes large variations in the solution Detection of ill-conditioning in a matrix • An inaccurate solution for can satisfy an ill-conditioned matrix quite well! • Apply back substitution to check for ill-conditioning • Solve through Gauss or other direct method • Back substitute • Comparing we find that X A XB = X poor A X B B B
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CE 30125 - Lecture 3 p. 3.2 • Back substitution is not a good detection technique. • The effects of ill-conditioning are very subtle! • Examine the inverse of matrix • If there are elements of which are many orders of magnitude larger than the orig- inal matrix, , then is probably ill-conditioned • It is always best to normalize the rows of the original matrix such that the maximum magnitude is of order 1 • Evaluate using the same method with which you are solving the system of equa- tions. Now compute and compare the results to . If there’s a significant devi- ation, then the presence of serious roundoff exists! • Compute using the same method with which you are solving the system of equations. This is a more severe test of roundoff since it is accumulated both in the original inversion and the re-inversion. A A 1 A A A 1 A 1 A I A 1  1
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CE 30125 - Lecture 3 p. 3.3 • Can also evaluate ill-conditioning by examining the normalized determinant . The matrix may be ill-conditioned when: where Euclidean Norm of • If the matrix is diagonally dominant , i.e. the absolute values of the diagonal terms the sum of the off-diagonal terms for each row, then the matrix is not ill-condi- tioned det A a ij 2 j 1 = N i 1 = N ------------------------------- < 1 A a 2 j 1 = N i 1 = N A a ii a j 1 = N i 12 N   =
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CE 30125 - Lecture 3 p. 3.4 • Effects of ill-conditioning are most serious in large dense matrices (e.g. especially those obtained in such problems as curve fitting by least squares)
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This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.

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Lecture 3 - CE 30125 Lecture 3 LECTURE 3 DIRECT SOLUTIONS TO LINEAR ALGEBRAIC SYSTEMS CONTINUED Ill-conditioning of Matrices There is no clear cut

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