Thisisjointworkwith RafVandebrilKULeuven JaredAurentzWSU DavidSWatkins companioncomradeandrelatedmatrices Zerosofapolynomialmonomialbasis pzznc1zn1c2zn2cn0 companionmatrixA26666 ID: 470639
Download Pdf The PPT/PDF document "Collaborators" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Collaborators Thisisjointworkwith RafVandebril(KULeuven) JaredAurentz(WSU) DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(monomialbasis) p(z)=zn�c1zn�1�c2zn�2��cn=0 companionmatrixA=2666664c1c2cn11...103777775 GetthezerosofpbycomputingtheeigenvaluesofA. Example:MATLAB'srootscommand DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(monomialbasis) p(z)=zn�c1zn�1�c2zn�2��cn=0 companionmatrixA=2666664c1c2cn11...103777775 GetthezerosofpbycomputingtheeigenvaluesofA. Example:MATLAB'srootscommand DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(monomialbasis) p(z)=zn�c1zn�1�c2zn�2��cn=0 companionmatrixA=2666664c1c2cn11...103777775 GetthezerosofpbycomputingtheeigenvaluesofA. Example:MATLAB'srootscommand DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(monomialbasis) p(z)=zn�c1zn�1�c2zn�2��cn=0 companionmatrixA=2666664c1c2cn11...103777775 GetthezerosofpbycomputingtheeigenvaluesofA. Example:MATLAB'srootscommand DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(monomialbasis) p(z)=zn�c1zn�1�c2zn�2��cn=0 companionmatrixA=2666664c1c2cn11...103777775 GetthezerosofpbycomputingtheeigenvaluesofA. Example:MATLAB'srootscommand DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(\arbitrary"basis) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 confederatematrix(Barnett1983)A=2666666437777775 pk(z)bk+1;k=zpk�1(z)�kXj=1pj�1(z)bjk Shortrecurrenceimpliessparsematrix. DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(\arbitrary"basis) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 confederatematrix(Barnett1983)A=2666666437777775 pk(z)bk+1;k=zpk�1(z)�kXj=1pj�1(z)bjk Shortrecurrenceimpliessparsematrix. DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(\arbitrary"basis) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 confederatematrix(Barnett1983)A=2666666437777775 pk(z)bk+1;k=zpk�1(z)�kXj=1pj�1(z)bjk Shortrecurrenceimpliessparsematrix. DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(\arbitrary"basis) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 confederatematrix(Barnett1983)A=2666666437777775 pk(z)bk+1;k=zpk�1(z)�kXj=1pj�1(z)bjk Shortrecurrenceimpliessparsematrix. DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(\arbitrary"basis) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 confederatematrix(Barnett1983)A=2666666437777775 pk(z)bk+1;k=zpk�1(z)�kXj=1pj�1(z)bjk Shortrecurrenceimpliessparsematrix. DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(3-termrecurrence) pk(z) k=(z�k)pk�1(z)� kpk�2(z) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 comradematrixA=2666666437777775 tridiagonalplusspike Examples:Chebyshev,Legendre,Hermite,... Example:Chebfun'srootscommand DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(3-termrecurrence) pk(z) k=(z�k)pk�1(z)� kpk�2(z) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 comradematrixA=2666666437777775 tridiagonalplusspike Examples:Chebyshev,Legendre,Hermite,... Example:Chebfun'srootscommand DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(3-termrecurrence) pk(z) k=(z�k)pk�1(z)� kpk�2(z) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 comradematrixA=2666666437777775 tridiagonalplusspike Examples:Chebyshev,Legendre,Hermite,... Example:Chebfun'srootscommand DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(3-termrecurrence) pk(z) k=(z�k)pk�1(z)� kpk�2(z) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 comradematrixA=2666666437777775 tridiagonalplusspike Examples:Chebyshev,Legendre,Hermite,... Example:Chebfun'srootscommand DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(2-termrecurrence) Newtonbasis pk(z) k=(z�k)pk�1(z) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0A=2666666437777775 bidiagonalplusspike DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(2-termrecurrence) Newtonbasis pk(z) k=(z�k)pk�1(z) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 A=2666666437777775 bidiagonalplusspike DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(2-termrecurrence) Newtonbasis pk(z) k=(z�k)pk�1(z) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0 A=2666666437777775 bidiagonalplusspike DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(1-termrecurrence) monomialbasis pk(z)=zpk�1(z) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0A=2666666437777775 onediagonalplusspike companionmatrix DavidS.Watkins companion,comrade,andrelatedmatrices Zerosofapolynomial(1-termrecurrence) monomialbasis pk(z)=zpk�1(z) p(z)=pn(z)�c1pn�1(z)�c2pn�2(z)��cnp0(z)=0A=2666666437777775 onediagonalplusspike companionmatrix DavidS.Watkins companion,comrade,andrelatedmatrices Howdowecomputetheeigenvalues? narrowbandplusspikeA=2666666437777775 Francis'sbulge-chasingalgorithm(\implicitly-shiftedQR") O(n3)time,O(n2)space Isthereamoreeconomicalapproach? DavidS.Watkins companion,comrade,andrelatedmatrices Howdowecomputetheeigenvalues? narrowbandplusspikeA=2666666437777775 Francis'sbulge-chasingalgorithm(\implicitly-shiftedQR") O(n3)time,O(n2)space Isthereamoreeconomicalapproach? DavidS.Watkins companion,comrade,andrelatedmatrices Howdowecomputetheeigenvalues? narrowbandplusspikeA=2666666437777775 Francis'sbulge-chasingalgorithm(\implicitly-shiftedQR") O(n3)time,O(n2)space Isthereamoreeconomicalapproach? DavidS.Watkins companion,comrade,andrelatedmatrices Howdowecomputetheeigenvalues? narrowbandplusspikeA=2666666437777775 Francis'sbulge-chasingalgorithm(\implicitly-shiftedQR") O(n3)time,O(n2)space Isthereamoreeconomicalapproach? DavidS.Watkins companion,comrade,andrelatedmatrices Howdowecomputetheeigenvalues? bandedplusspikeA=2666666437777775 Exploitrankstructure(generators) Wepursueadi erentapproach. DavidS.Watkins companion,comrade,andrelatedmatrices Howdowecomputetheeigenvalues? bandedplusspikeA=2666666437777775 Exploitrankstructure(generators) Wepursueadi erentapproach. DavidS.Watkins companion,comrade,andrelatedmatrices Howdowecomputetheeigenvalues? bandedplusspikeA=2666666437777775 Exploitrankstructure(generators) Wepursueadi erentapproach. DavidS.Watkins companion,comrade,andrelatedmatrices FastMethods Wehavedevelopedgreatmethodsthatare... lightningfast unstable DavidS.Watkins companion,comrade,andrelatedmatrices FastMethods Wehavedevelopedgreatmethodsthatare... lightningfast unstable DavidS.Watkins companion,comrade,andrelatedmatrices FastMethods Wehavedevelopedgreatmethodsthatare... lightningfast unstable DavidS.Watkins companion,comrade,andrelatedmatrices Ausefulfactorization(byGaussianelimination) 266664377775 andsoon DavidS.Watkins companion,comrade,andrelatedmatrices Ausefulfactorization(byGaussianelimination) 266664377775=266664377775 andsoon DavidS.Watkins companion,comrade,andrelatedmatrices Ausefulfactorization(byGaussianelimination) 266664377775=2666640377775 andsoon DavidS.Watkins companion,comrade,andrelatedmatrices Ausefulfactorization(byGaussianelimination) 266664377775=266664377775 andsoon DavidS.Watkins companion,comrade,andrelatedmatrices Ausefulfactorization(byGaussianelimination) 266664377775=2666640377775 andsoon DavidS.Watkins companion,comrade,andrelatedmatrices Ausefulfactorization(byGaussianelimination) 266664377775=266664377775 andsoon DavidS.Watkins companion,comrade,andrelatedmatrices Ausefulfactorization(byGaussianelimination) 266664377775=2666640377775 andsoon DavidS.Watkins companion,comrade,andrelatedmatrices Ausefulfactorization(byGaussianelimination) 266664377775=266664377775 andsoon...to(banded)triangularform. DavidS.Watkins companion,comrade,andrelatedmatrices Ausefulfactorization(byGaussianelimination) WegetthefactorizationA= 266664377775 coretransformationstimesbandeduppertriangular bandwidth=lengthofrecurrence DavidS.Watkins companion,comrade,andrelatedmatrices Whatcoretransformationslooklike =264113752641137526411375 DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm 266664377775 Wewilldevelopanonunitarybulge-chasingalgorithm thatpreservesthisfactorization. Note:StorageisO(n)(about6nforthiscase). Wehavesingle-shiftanddouble-shiftvariants. DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm 266664377775 Wewilldevelopanonunitarybulge-chasingalgorithm thatpreservesthisfactorization. Note:StorageisO(n)(about6nforthiscase). Wehavesingle-shiftanddouble-shiftvariants. DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm 266664377775 Wewilldevelopanonunitarybulge-chasingalgorithm thatpreservesthisfactorization. Note:StorageisO(n)(about6nforthiscase). Wehavesingle-shiftanddouble-shiftvariants. DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm 266664377775 Wewilldevelopanonunitarybulge-chasingalgorithm thatpreservesthisfactorization. Note:StorageisO(n)(about6nforthiscase). Wehavesingle-shiftanddouble-shiftvariants. DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) 266664377775 DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) 266664377775 DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) 266664377775 DavidS.Watkins companion,comrade,andrelatedmatrices PassingGausstransformthroughtriangularmatrix BluecoretransformationsareGausstransforms."# ="#= "# Wedothisateveryopportunity. Fromthispointon,wesuppressthetriangularmatrix. DavidS.Watkins companion,comrade,andrelatedmatrices PassingGausstransformthroughtriangularmatrix BluecoretransformationsareGausstransforms."# ="#= "# Wedothisateveryopportunity. Fromthispointon,wesuppressthetriangularmatrix. DavidS.Watkins companion,comrade,andrelatedmatrices PassingGausstransformthroughtriangularmatrix BluecoretransformationsareGausstransforms."# ="#= "# Wedothisateveryopportunity. Fromthispointon,wesuppressthetriangularmatrix. DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) 266664377775 DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) 266664377775 DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices Turnoveroperation Canwedothis? , crucialquestion Pessimisticanswer:No!(notalways) Optimisticanswer:Yes!(almostalways) DavidS.Watkins companion,comrade,andrelatedmatrices Turnoveroperation Canwedothis? , crucialquestion Pessimisticanswer:No!(notalways) Optimisticanswer:Yes!(almostalways) DavidS.Watkins companion,comrade,andrelatedmatrices Turnoveroperation Canwedothis? , crucialquestion Pessimisticanswer:No!(notalways) Optimisticanswer:Yes!(almostalways) DavidS.Watkins companion,comrade,andrelatedmatrices Turnoveroperation Canwedothis? , crucialquestion Pessimisticanswer:No!(notalways) Optimisticanswer:Yes!(almostalways) DavidS.Watkins companion,comrade,andrelatedmatrices Turnoveroperation ="0# ="# Redsubmatrixstillhasrankone. DavidS.Watkins companion,comrade,andrelatedmatrices Turnoveroperation ="0# = "# Redsubmatrixstillhasrankone. DavidS.Watkins companion,comrade,andrelatedmatrices Turnoveroperation So..."#= "0#= Gausstransformdoesnotdisturb22submatrix. DavidS.Watkins companion,comrade,andrelatedmatrices Turnoveroperation So... "#= "0# = Gausstransformdoesnotdisturb22submatrix. DavidS.Watkins companion,comrade,andrelatedmatrices Turnoveroperation So... "#= "0# = Gausstransformdoesnotdisturb22submatrix. DavidS.Watkins companion,comrade,andrelatedmatrices Turnoveroperation Completeturnoverlookslikethis: = DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) Done! Nowrepeatagain,again,andagain DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) Done! Nowrepeat again,again,andagain DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) Done! Nowrepeat again, again,andagain DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) Done! Nowrepeat again, again, andagain DavidS.Watkins companion,comrade,andrelatedmatrices BulgeChasingAlgorithm(single-shiftvariant) Done! Nowrepeat again, again, andagain DavidS.Watkins companion,comrade,andrelatedmatrices Computationalcomplexity O(n)storage(everythingincache) O(n) opsperiteration O(n)iterations O(n2) opstotal DavidS.Watkins companion,comrade,andrelatedmatrices Computationalcomplexity O(n)storage(everythingincache) O(n) opsperiteration O(n)iterations O(n2) opstotal DavidS.Watkins companion,comrade,andrelatedmatrices Computationalcomplexity O(n)storage(everythingincache) O(n) opsperiteration O(n)iterations O(n2) opstotal DavidS.Watkins companion,comrade,andrelatedmatrices Computationalcomplexity O(n)storage(everythingincache) O(n) opsperiteration O(n)iterations O(n2) opstotal DavidS.Watkins companion,comrade,andrelatedmatrices Computationalcomplexity O(n)storage(everythingincache) O(n) opsperiteration O(n)iterations O(n2) opstotal DavidS.Watkins companion,comrade,andrelatedmatrices Stabilityissue Thisisastructured,implicitly-shiftedLRalgorithm. Unstable!Canwetrusttheresults?No, butwecandoaposterioritests. k(I�A)vk=jp()j jp() p0()jisestimateoferror. Newtoncorrection Cost:O(n2) opsfornroots. DavidS.Watkins companion,comrade,andrelatedmatrices Stabilityissue Thisisastructured,implicitly-shiftedLRalgorithm. Unstable!Canwetrusttheresults?No, butwecandoaposterioritests. k(I�A)vk=jp()j jp() p0()jisestimateoferror. Newtoncorrection Cost:O(n2) opsfornroots. DavidS.Watkins companion,comrade,andrelatedmatrices Stabilityissue Thisisastructured,implicitly-shiftedLRalgorithm. Unstable! Canwetrusttheresults?No, butwecandoaposterioritests. k(I�A)vk=jp()j jp() p0()jisestimateoferror. Newtoncorrection Cost:O(n2) opsfornroots. DavidS.Watkins companion,comrade,andrelatedmatrices Stabilityissue Thisisastructured,implicitly-shiftedLRalgorithm. Unstable! Canwetrusttheresults? No, butwecandoaposterioritests. k(I�A)vk=jp()j jp() p0()jisestimateoferror. Newtoncorrection Cost:O(n2) opsfornroots. DavidS.Watkins companion,comrade,andrelatedmatrices Stabilityissue Thisisastructured,implicitly-shiftedLRalgorithm. Unstable! Canwetrusttheresults? No, butwecandoaposterioritests. k(I�A)vk=jp()j jp() p0()jisestimateoferror. Newtoncorrection Cost:O(n2) opsfornroots. DavidS.Watkins companion,comrade,andrelatedmatrices Stabilityissue Thisisastructured,implicitly-shiftedLRalgorithm. Unstable! Canwetrusttheresults? No, butwecandoaposterioritests. k(I�A)vk=jp()j jp() p0()jisestimateoferror. Newtoncorrection Cost:O(n2) opsfornroots. DavidS.Watkins companion,comrade,andrelatedmatrices Stabilityissue Thisisastructured,implicitly-shiftedLRalgorithm. Unstable! Canwetrusttheresults? No, butwecandoaposterioritests. k(I�A)vk=jp()j jp() p0()jisestimateoferror. Newtoncorrection Cost:O(n2) opsfornroots. DavidS.Watkins companion,comrade,andrelatedmatrices Stabilityissue Thisisastructured,implicitly-shiftedLRalgorithm. Unstable! Canwetrusttheresults? No, butwecandoaposterioritests. k(I�A)vk=jp()j jp() p0()jisestimateoferror. Newtoncorrection Cost:O(n2) opsfornroots. DavidS.Watkins companion,comrade,andrelatedmatrices Stabilityissue Thisisastructured,implicitly-shiftedLRalgorithm. Unstable! Canwetrusttheresults? No, butwecandoaposterioritests. k(I�A)vk=jp()j jp() p0()jisestimateoferror. Newtoncorrection Cost:O(n2) opsfornroots. DavidS.Watkins companion,comrade,andrelatedmatrices Stabilityissue Thisisastructured,implicitly-shiftedLRalgorithm. Unstable! Canwetrusttheresults? No, butwecandoaposterioritests. k(I�A)vk=jp()j jp() p0()jisestimateoferror. Newtoncorrection Cost:O(n2) opsfornroots. DavidS.Watkins companion,comrade,andrelatedmatrices Performance Firstexample:companionmatrices,complex,single-shiftcodeContestants LAPACKcodeZHSEQR(O(n3)) BBEGG(quasiseparable,unitary) AVWourcode DavidS.Watkins companion,comrade,andrelatedmatrices SpeedComparison,companioncase DavidS.Watkins companion,comrade,andrelatedmatrices