By Despres B.
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N − 1, i = j + 1, j + 2, . . 20) L 2jk, k=1 For a nondiagonal term we get j−1 Li j = L ik L jk /L j j , Ai j − k=1 choleski Note that in Eqs. 20) Ai j appears only in the formula for L i j . Therefore, once L i j has been computed, Ai j is no longer needed. This makes it possible to write the elements of L over the lower triangular portion of A as they are computed. The elements above the principal diagonal of A will remain untouched. At the conclusion of decomposition L is extracted with the MATLAB command tril(A).
N where v is a vector. In MATLAB a Vandermode matrix can be generated by the command vander(v). 0 T 1 0 1 0 1 Also evaluate the accuracy of the solution (Vandermode matrices tend to be illconditioned). Solution We used the program shown below. 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.
Anj ··· ··· ··· ··· ··· ··· A1n A2n A3n .. Akn .. Ain .. Ann ⎤ b1 b2 ⎥ ⎥ ⎥ b3 ⎥ ⎥ .. ⎥ ⎥ ⎥ bk ⎥ ← pivot row ⎥ .. ⎥ ⎥ bi ⎥ ⎥ ← row being transformed .. ⎦ bn Let the ith row be a typical row below the pivot equation that is to be transformed, meaning that the element Aik is to be eliminated. We can achieve this by multiplying the pivot row by λ = Aik/Akk and subtracting it from the ith row. The corresponding changes in the ith row are Ai j ← Ai j − λAk j , j = k, k + 1, . . 8b) To transform the entire coefﬁcient matrix to upper triangular form, k and i in Eqs.
Analyse Numerique Matricielle DEA d'analyse numerique by Despres B.