commit 0dbb6997353fe36829abd0bfeb3e12c9781cb443
parent bdb3260af991245e3e6f37b044dbdaddac400edb
Author: Silas Brack <s174433@student.dtu.dk>
Date: Tue, 3 Jan 2023 02:09:47 +0100
Define variables.
Diffstat:
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/chapters/sampling.tex b/chapters/sampling.tex
@@ -108,7 +108,7 @@ where \(\sigma\) is obtained from \(\tau\) as a change of variable and \(\Tilde{
We can then solve the \(Q\) systems of equations \(\left( \sigma^2_1 \identity - \bm K \right) \bm c_1 = \bm v, \ldots, \left( \sigma^2_Q \identity - \bm K \right) \bm c_Q = \bm v\) to obtain the solutions \(\bm c_q = \left(\sigma^2_q \identity - \bm K \right)^{-1} \bm v\).
Note that, since the integrand is symmetric with respect to the real axis, only the imaginary component of \(\Gamma_\sigma\) is required.\sidenote{This is due to the fact that the circle with its centre on the \(x\)-axis is symmetric with respect to said \(x\)-axis.}
Since \(\sigma\) is then imaginary (its real part is zero), \(\tau := \sigma^2\) is real such that \(\tau = - \Im(\sigma)^2\).
-As such, since \(\Im(\sigma)^2\) must be positive, \(\tau\) must then be real-valued and negative.\marginnote{For simplicity, since \(w_q\) happens to also be real-valued and negative, from now on we will redefine the variables \(w_q \gets -\Tilde{w}_q\) and \(t_q \gets -\sigma^2_q\).}
+As such, since \(\Im(\sigma)^2\) must be positive, \(\tau\) must then be real-valued and negative.\marginnote{For simplicity, since \(w_q\) happens to also be real-valued and negative, from now on we will define the variables \(w_q = -\Tilde{w}_q\) and \(t_q = -\sigma^2_q\).}
In practice, we incorporate the \(1 / (\pi i) \) term into the quadrature weights and, with the quadrature points \(t_1, \ldots, t_Q\), compute \(\bm K^{-1/2} \bm v \approx \sum^Q_{q=1} w_q \bm c_q\).
Since we will be sampling a fixed number of quadrature points from the circle, to improve the closeness of the approximation of the integral, it would be beneficial to choose a closed circle which intersects the real axis \emph{at} the minimum and maximum eigenvalues, since this will avoid sampling quadrature points which lie outside the spectrum of \(\bm K\).
