CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019

Lecture 17Finish defining the condition number.* The condition number of subtraction shows when subtraction is hard!* The condition number of Ax=b The condition number of Ax = b: kappa(A) = ||A|| ||A^{-1}||The point of this lecture is to establishthe condition number of a matrix as a fundamental quantity that arises in many algorithms that explains how hard a linear system is to solve. — We need to work out the matrix 2-norm,which involves the singular values, so we show- The SVD exists.- The SVD is the eigenvalue decomp. for sym. pos. def.- ||A||_2 is orthogonal invariant. - ||A||_2 = max sing. val.- ||A^{-1}||_2 = 1/min sing. val - 2-norm for sym. pos. def.Convergence of Richardsonat rate 1 - 1/kappa(A).Convergence of Steep. Descent at rate (1-1/kappa(A))^2)

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CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019 CS 515 - Purdue University - Computer Science - David F. Gleich - 2019

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