lec04_09132006 - 10.34, Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #4: When Algorithms Run Into Problems: Numerical Error; Ill- conditioning, and Tolerances Homework find balance between concise and detailed Æ manager is audience include solutions in word document include any issues you had Factorization All NxN real matrices can be written: A Æ P T LU (L : lower triangular, U : upper, P : permutation) Linear algebra is a WELL-DEVELOPED area of study MATLAB is an offshoot of LAPACK (linear algebra package) Can download LAPACK from netlib A x = b in MATLAB: x = A \ b ‘\’ is an “amazing function” – very powerful * WARNING: do not always know what it’s doing Uniqueness and Existence of Solution if rank(A ) = N NxN –- solution exists if rank(A ) < N, det(A ) = 0 -- singular, rank deficient A z = 0 , where z 0 *find additional equation* at least one eigenvalue = 0 n o i n v e r s e : z A -1 *0 A x = b does rank(A ) = rank([A b ])? if yes, there is a solution if yes and rank(A )<N, there are an infinite number of solutions Symmetry A T = A Also called Hermitian Matrix if all coefficients are real General Hermitian matrix: transpose and complex-conjugate. positive definite: x
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lec04_09132006 - 10.34, Numerical Methods Applied to...

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