[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-326248-105":59,"doc-detail-326248-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","math-6610-lecture-20-the-qr-algorithm-with-shifts","MATH 6610 Lecture 20 - The QR Algorithm with Shifts","","Lecture 20 focuses on eigenvalue computation using the QR algorithm, starting from the basic QR iteration for Hermitian matrices. It explains the QR step (A=QR then A←RQ), connects QR to simultaneous power iteration, and discusses convergence speed relative to power iteration. The lecture then develops links between QR iteration and inverse iteration, including an unshifted inverse iteration viewpoint. Finally, it motivates how shifting can enable a Rayleigh-iteration-like method and presents the shifted QR update rule.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/math-6610-lecture-20-the-qr-algorithm-with-shifts/326248/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/math-6610-lecture-20-the-qr-algorithm-with-shifts/326248.png","ImageObject",300,407,{"name":92,"@type":93},"Jiven","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-20","2026-09-21",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What are the basic steps of the QR algorithm for computing eigenvalues of a Hermitian matrix?","Question",{"text":112,"@type":113},"Compute the QR decomposition A=QR, then replace A by the product RQ and repeat the process on the updated matrix.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the QR algorithm relate to power iteration and convergence behavior?",{"text":117,"@type":113},"For Hermitian A, the QR algorithm can be viewed as a form of simultaneous power iteration, so its convergence is comparable to power iteration in speed.",{"name":119,"@type":110,"acceptedAnswer":120},"How is the QR algorithm connected to inverse iteration, and how can shifting be incorporated?",{"text":121,"@type":113},"The QR algorithm also performs an unshifted inverse iteration viewpoint, and the lecture discusses how shifting is needed to achieve a Rayleigh-iteration-like algorithm via a shifted QR procedure.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},326248,1791971873,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},1099513958607,"https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399","L20-S00  \nEigenvalue algorithms: The QR algorithm with shifts  \nMATH 6610 Lecture 20  \nOctober 26, 2020  \nTrefethen & Bau: Lecture 29  \nL20-S01  \nThe QR algorithm  \nAssume A is Hermitian. The QR algorithm for computing eigenvalues:  \n1. Compute A 􀀐 QR, the QR decomposition of A  \n2. Replace A by the procedure A Ð RQ  \n3. Return to step 1  \nWe've seen that this is just simultaneous power iteration.  \nL20-S01  \nThe QR algorithm  \nAssume A is Hermitian. The QR algorithm for computing eigenvalues:  \n1. Compute A 􀀐 QR, the QR decomposition of A  \n2. Replace A by the procedure A Ð RQ  \n3. Return to step 1  \nWe've seen that this is just simultaneous power iteration.  \n which also means that it converges as “quickly” as power iteration.  \nCan we instead develop a Rayleigh iteration type of algorithm?  \nL20-S02  \nQR algorithm and inverse iteration, I  \n(Unshifted) inverse iteration: power iteration on A􀀁1 .  \nThe QR algorithm performs power iteration.  \nL20-S02  \nQR algorithm and inverse iteration, I  \n(Unshifted) inverse iteration: power iteration on A􀀁1 .  \nThe QR algorithm performs power iteration.  \nThe QR algorithm also performs unshifted inverse iteration.  \nL20-S02  \nQR algorithm and inverse iteration, I  \n(Unshifted) inverse iteration: power iteration on A􀀁1 .  \nThe QR algorithm performs power iteration.  \nThe QR algorithm also performs unshifted inverse iteration.  \nRecall: if QQkR is the QR factor computed at the kth iteration, then  \nAk 􀀐 QkRk ; Qk 􀀐 QQ1R QQR2 􀀄 􀀄 􀀄 QQkR ;  \nand Rk is an upper-triangular matrix.  \nL20-S03  \nQR algorithm and inverse iteration, II  \nLet F be a permutation matrix that flips a vector, i.e. , associated to the permutation map:  \nt1; 2; : : : ; n 􀀁 1; nu ÝÑ tn; n 􀀁 1 ; : : : ; 2 ; 1u :  \nThen:  \nL20-S03  \nQR algorithm and inverse iteration, II  \nLet F be a permutation matrix that flips a vector, i.e. , associated to the permutation map:  \nt1; 2; : : : ; n 􀀁 1; nu ÝÑ tn; n 􀀁 1 ; : : : ; 2 ; 1u :  \nThen:  \nA􀀁kF 􀀐 pQkF q 􀀀FRT F􀀈 ;  \nwhich is a QR factorization of A􀀁kF.  \nI.e. , the last column(s) of Qk (computed via the QR algorithm) are inverse iteration with starting vectors given by the first columns of F.  \nThe QR algorithm with shifts L20-S04  \nWe can almost perform Rayleigh iteration with QR procedures.  \nWe can perform inverse iteration; how to accomplish shifting?  \nL20-S04  \nThe QR algorithm with shifts  \nWe can almost perform Rayleigh iteration with QR procedures.  \nWe can perform inverse iteration; how to accomplish shifting?  \nBefore explaining the shift, we show the result: The QR algorithm with shifts.  \nSet AQR0 􀀐 A and 􀀖0 􀀐 0. For k 􀀐 1 ; 2 ; : : : ;  \n QQkRRQRk 􀀐 AQk1 􀀁 􀀖k􀀁1I  AQkR 􀀐 RQRkQQkR 􀀀 􀀖k􀀁1I  “Choose” 􀀖 k","cbCais1ceGcyv9Vw","https://ap.wps.com/l/cbCais1ceGcyv9Vw","pdf",159918,21,"English","# Eigenvalue algorithms: The QR algorithm with shifts\n## The QR algorithm (Hermitian case)\n## QR algorithm and inverse iteration (I)\n## QR algorithm and inverse iteration (II)\n## The QR algorithm with shifts","[{\"question\":\"What are the basic steps of the QR algorithm for computing eigenvalues of a Hermitian matrix?\",\"answer\":\"Compute the QR decomposition A=QR, then replace A by the product RQ and repeat the process on the updated matrix.\"},{\"question\":\"How does the QR algorithm relate to power iteration and convergence behavior?\",\"answer\":\"For Hermitian A, the QR algorithm can be viewed as a form of simultaneous power iteration, so its convergence is comparable to power iteration in speed.\"},{\"question\":\"How is the QR algorithm connected to inverse iteration, and how can shifting be incorporated?\",\"answer\":\"The QR algorithm also performs an unshifted inverse iteration viewpoint, and the lecture discusses how shifting is needed to achieve a Rayleigh-iteration-like algorithm via a shifted QR procedure.\"}]","MATH 6610 Lecture 20 - The QR Algorithm with Shifts | PDF",1789973752,53]