21 lines
1.3 KiB
TeX
21 lines
1.3 KiB
TeX
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\subsection{Problem 7}
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Let $\mathbf{A} = \left[a_{ij}\right] \in\mathbb{R}^{N\times N}$ with, $a_{ij} = \frac{1}{i + j - 1}$ (Hilbert matrix). For $N=5$, perform the LU factorization of the matrix $\mathbf{A}$. Then, compute det($\mathbf{A}$).
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\subsubsection*{Mathematics}
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Hilbert matrices are an example of ill-conditioned matrices, which -- with their high $\kappa$ -- makes the numerical computations highly unstable. For our $\mathbf{H} = \left[h_{ij}\right] \in\mathbb{R}^{5\times 5}$:
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\begin{equation*}
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\matr{H} =
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\begin{bmatrix}
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1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\
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\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\
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\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \\
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\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \\
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\frac{1}{5} & \frac{1}{6} & \frac{1}{7} & \frac{1}{8} & \frac{1}{9}
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\end{bmatrix}
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\end{equation*}
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$\kappa(\matr{H})\approx4.7\cdot10^5$ which is a huge value similar to this in the Problem 4.
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\subsubsection*{Solution}
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\lstinputlisting[style=Matlab-editor]{problems/Problem7.m}
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The above results may look promising, but a determinant calculated both by \MATLAB and our algorithms is of the order of $4\cdot10^{-12}$, which for numerical computations is highly not useful.
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