13 - 2/24/2010 Determinants: If the determinant of A is...

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2/24/2010 1 Determinants: If the determinant of A is zero A is singular ( A 1 does not exist) The system of equations has no unique solution In Matlab the determinant of a matrix can found using function det . >> A = [ 4 2; 2 1 ]; >> det(A) ans = 0 >> A = [ 2 5 6; 4 5 1; 7 6 2 ]; >> A [ 2 5 6; 4 5 1; 7 6 2 ]; >> det(A) ans = 391 Determinants close to zero are an indication of problem cases. Condition Numbers: The condition number of a matrix is defined as cond( A ) = || A || * || A 1 || where || A || is a “norm” of A A “norm” is as measure of the “size” of a matrix. For example: Multiple “norm” definitions exist. The “e” in the above equation indicates a “Euclidian norm”. See text page 254 for more information. Relatively high condition numbers (>> 1) are an indication of problem cases.  2 2 2 2 5 1 2 5 1 e There is no hard and fast line. If the elements of A have t significant digits the elements of x (the solution) will have t – log(cond( A )) significant digits. Note: this rule is conservative.
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2/24/2010 2 Graphical representation of two simultaneous equations (ideal case): 3x 1 + 2 x 2 = 18 x 1 + 2 x 2 = 2 Solution cond(A) = 1.6404, det(A) = 8 Graphical representation of two simultaneous equations (no solution): 0.5x 1 + x 2 = 0.5 0.5x 1 + x 2 = 1 The two equations are contradictory. cond(A) = Inf, det(A) = 0
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2/24/2010 3 Graphical representation of two simultaneous equations (infinite solutions): x 1 + 2x 2 = 2 0.5x 1 + x 2 = 1 The two equations are not linearly independent (they give the same information). cond(A) = 2.5176e+016, det(A) = 0 Graphical representation of two simultaneous equations (problem case): 0.5x 1 + x 2 = 1 (2.3/5)x 1 + x 2 = 1.1 Solution A solution exists but it is very sensitive to small calculation errors. The system is ill conditioned . cond(A) = 61.5237, det(A) = 0.0400
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2/24/2010 4 Tests for ill conditioned systems: Scale A so that the largest element in each row has a magnitude of 1. Then calculate the inverse. If A 1 has elements that are orders of magnitude greater than 1 it is likely that the system is ill conditioned. Multiply A by A 1 . If the result is not close to the identity matrix it is likely that the system is ill conditioned. Invert A 1 . If the result is not close to A it is likely that the system is ill conditioned.
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13 - 2/24/2010 Determinants: If the determinant of A is...

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