By Maurice Holt
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After constructing A and b, the output format was changed to long so that the solution would be printed to 14 decimal places. 132462079991823e-006 As the determinant is quite small relative to the elements of A (you may want to print A to verify this), we expect detectable roundoff error. Inspection of x leads us to suspect that the exact solution is x = 1250/3 −3125 9250 −13500 29128/3 −2751 T in which case the numerical solution would be accurate to 9 decimal places. 99999999994998 The result seems to conﬁrm our previous conclusion.
002y = 0 and re-solving the equations. 5, y = −1500. 1% change in the coefﬁcient of y produced a 100% change in the solution. 1 Introduction Numerical solutions of ill-conditioned equations are not to be trusted. The reason is that the inevitable roundoff errors during the solution process are equivalent to introducing small changes into the coefﬁcient matrix. This in turn introduces large errors into the solution, the magnitude of which depends on the severity of ill-conditioning. In suspect cases the determinant of the coefﬁcient matrix should be computed so that the degree of ill-conditioning can be estimated.
6 Compute Choleski’s decomposition of the matrix ⎡ ⎤ 4 −2 2 ⎢ ⎥ A = ⎣ −2 2 −4 ⎦ 2 −4 11 Solution First we note that A is symmetric. Therefore, Choleski’s decomposition is applicable, provided that the matrix is also positive deﬁnite. An a priori test for positive deﬁniteness is not needed, since the decomposition algorithm contains its own test: if the square root of a negative number is encountered, the matrix is not positive deﬁnite and the decomposition fails. Substituting the given matrix for A in Eq.