\[ A\in \mathbb{R}^{m\times n} \\ x\in \mathbb{R}^n \mapsto Ax \in \mathbb{R}^m \]
Assumption: \(A \in \mathbb{R}^{n \times n}\) is symmetric, no repeated eigenvalues \(\lambda\)s.
Lemma: Every matrix \(A \in \mathbb{R}^{n \times n}\) has at least one (complex) eigenvalue.
很多时候需要把已有问题转化为特征值问题。
\[ \begin{aligned} cond A^T A &= ||A^T A|| \cdot ||(A^T A)^{-1}||\\ &\approx ||A^T|| \cdot ||A|| \cdot ||A^{-1}|| \cdot ||A^{-T}|| &= cond A^2 \end{aligned} \]
为了避免计算\(A^T A\)，我们可以使用QR分解。
Gaussian elimination works in theory, but what about floating point precision?
How much can we trust \(x_0\) if \(0<||Ax_0 - b||_2 \ll 1\)(backwards error)?
Gaussian elimination and/or LU can solve all the example problems above. But these systems can have special properties that make them easier or stabler to solve.
Today's example: Positive definite, sparsity.
\[ \begin{aligned} Ax &= b \\ A &\in \mathbb{R}^{n \times n} \\ x &\in \mathbb{R}^n \\ b &\in \mathbb{R}^n \end{aligned} \]
Mathematically correct != numerically sound. Using Tolerance:
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7double x = 1.0;
double y = x / 3.0;
if(fabs(x-y*3.0) < numeric_limits<double>::epsilon()){
cout << "They are equal" << endl;
}
else
cout << "They are not equal" << endl;