Lecture 17 Caehhoinmbs Condition Comet Now Result I By Tf ...* The condition number of subtraction...
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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 Lecture 17 Finish 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 establish the 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 Richardson at rate 1 - 1/kappa(A). Convergence of Steep. Descent at rate (1-1/kappa(A))^2) Caehhoinmbs , If µ , .× . , , snug %TTsTo=←A if the Condition wah Result # I By Comet Tf Oo is dishes , mofHrsb9uk_ One ok note : Letx bears %t%t=zzt Let Uy ,Ry=qcy , ↳tUyTAV×=A , yTAVz= ? HA , Tf we .to , Now , ho - k get Let SUbdB= KCX :D - HTncxHl afamddnx ran the subg , f Q ? -2022 - On vector that attain Zern claim : this .no ! " 2711875JPY11 Sub ? 11 many Sabbah is had ! JNCIJ = A " upper - baton he any pennon c sown obscene oh , i ° 0070J jai . Read 11 Alk = " Atk = HUA Uk the moxy Ux,Ry=q# pro , z tho > o UBEBVRGT Example : mad . * - q IIA 'll IAM Comm # 4 a man " A%yq f- square only qu . 11AM then HE Vx , Rx - great UYTAVX ' I what is A→UyTAv×=A , kcfint-htnckyq.az , Ae the soba sough 9 - aux At > Oi > Ooh where Iaaf , clay ? ! the one go.ae ! gyIAxyTAvz AF Wpf ) " fifth acute , so it :÷÷÷÷÷÷:÷÷.÷ . ¥ . ÷÷÷f÷÷÷÷:÷÷÷÷÷÷÷÷÷ :÷f÷÷÷w÷ : ÷÷÷÷÷:f÷÷÷÷÷÷:÷:÷÷ :÷÷÷÷¥÷¥%l÷÷÷÷÷÷f :* :÷:÷÷÷f÷¥mf÷÷÷÷÷÷ :÷÷÷÷ . - hey Unique ? I - Alsos : z or fwtw 0 a masks - - ix. - ay qq.gg#y= , ,µ , ⇐"*"% V :nxn omg A- o = Uout " Vyll = Some y Yaya - I so E- her Uy Ry - if FIA 't - - o a. ya ,µqq Oc ) Xyy = 2nmaa.ae Simm , diam - Tati A- . I - - neat Teach .is/heagsae.e ME " 4.am#=mafaAIcy=ftaET ) C.zuy.gegqlfyt-UIAx-UEcoyt.ru#f = - fmdbiemmehtn.AM/o%1f8gIfooug;f ( * - Yal E=n[If for any org . u . ok ellipse . " × " ⇒ athlete > oetufo na O
Transcript of Lecture 17 Caehhoinmbs Condition Comet Now Result I By Tf ...* The condition number of subtraction...
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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