On Deflations in Extended QR Algorithms
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×××××××××××××××××××××××××××××××××××××××××××
×
On Deflations in Extended QRAlgorithms
Thomas Mach Raf Vandebril
Dept. Computer Science, KU Leuven, Belgium
GAMM Workshop Appl. Numer. Linear Algebra09 September 2013Wuppertal, Germany
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 1/27
![Page 2: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/2.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
QR factorization of the Hessenberg matrixRotation of the form:
[c s−s c
], |c |2 + |s|2 = 1
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 2/27
![Page 3: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/3.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
0×××××
��
QR factorization of the Hessenberg matrixRotation of the form:
[c s−s c
], |c |2 + |s|2 = 1
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 2/27
![Page 4: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/4.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
0×××××
��
QR factorization of the Hessenberg matrix
Rotation of the form:
[c s−s c
], |c |2 + |s|2 = 1
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 2/27
![Page 5: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/5.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
0××××
����
QR factorization of the Hessenberg matrixRotation of the form:
[c s−s c
], |c |2 + |s|2 = 1
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 2/27
![Page 6: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/6.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
0×××
����
��
QR factorization of the Hessenberg matrixRotation of the form:
[c s−s c
], |c |2 + |s|2 = 1
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 2/27
![Page 7: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/7.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
0××
����
����
QR factorization of the Hessenberg matrixRotation of the form:
[c s−s c
], |c |2 + |s|2 = 1
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 2/27
![Page 8: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/8.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
0×
����
����
��
QR factorization of the Hessenberg matrixRotation of the form:
[c s−s c
], |c |2 + |s|2 = 1
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 2/27
![Page 9: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/9.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××0
����
����
����
QR factorization of the Hessenberg matrixRotation of the form:
[c s−s c
], |c |2 + |s|2 = 1
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 2/27
![Page 10: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/10.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
QR factorization of the Hessenberg matrix
Rotation of the form:
[c s−s c
], |c |2 + |s|2 = 1
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 2/27
![Page 11: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/11.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
There is a relation to extended Krylov subspaces.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 3/27
![Page 12: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/12.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Rotators
Fusion:
�� �� = c1−s1
s1c1
c2−s2
s2c2
= c3−s3
s3c3
= ��
Turnover:
����
��=
Ö
Ö
Ö
Ö
Ö
Ö
Ö
Ö
Ö
= ����
��
Transfer Through Upper Triangular:
����
��Ö
Ö Ö Ö Ö
Ö Ö Ö
Ö Ö=
��
��Ö
Ö Ö Ö Ö
Ö Ö Ö
Ö Ö Ö=
��
��
Ö
Ö
Ö
Ö
Ö
Ö
Ö
Ö
Ö
Ö
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 4/27
![Page 13: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/13.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
1) Compute bulge.
A) Eigenvalues of trailing matrix provide shift.B) x = p(A)e1 = (A− µmI ) · · · (A− µ1I )e1.C) Q0x = γe1.D) Apply Q0.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 14: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/14.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
1) Compute bulge.
A) Eigenvalues of trailing matrix provide shift.B) x = p(A)e1 = (A− µ1I )e1.C) Q0x = γe1.D) Apply Q0.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 15: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/15.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
��
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 16: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/16.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
×
2) Bulge chasing.
A) Compute zeroing similarity transformation.B) Apply transformation.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 17: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/17.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
��
� �
2) Bulge chasing.
A) Compute zeroing similarity transformation.B) Apply transformation.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 18: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/18.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
×
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 19: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/19.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
��
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 20: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/20.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
×
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 21: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/21.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
��
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 22: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/22.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
×
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 23: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/23.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
��
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 24: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/24.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××××××
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 25: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/25.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××××××
��� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 26: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/26.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
Why Francis’s Algorithm Works [Watkins 1982, ’07, ’08, ’11]
Power method.
Krylov subspaces.
Shifts accelerate convergence.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 5/27
![Page 27: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/27.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
QR Factorization
QR Factorization
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/27
![Page 28: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/28.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 29: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/29.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
��
Fusion
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 30: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/30.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 31: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/31.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 32: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/32.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 33: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/33.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 34: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/34.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 35: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/35.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 36: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/36.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 37: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/37.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 38: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/38.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 39: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/39.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 40: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/40.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 41: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/41.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 42: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/42.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
![Page 43: On Deflations in Extended QR Algorithms](https://reader030.fdocuments.us/reader030/viewer/2022032715/55ae49141a28abf3178b465c/html5/thumbnails/43.jpg)
Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
������
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Transfer through up. triang.
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
���� ��
Fusion
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Francis’s Algorithm on the QR Factorization[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
��
Fusion
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Move pattern
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Similarity transformation
Move pattern
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
��
��
������
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
��
��
����
� �
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
��
��
���� ��
Turnover
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
��
��
����
��
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
��
��
����
� �
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
��
��
����
��
Fusion
Move pattern
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
��
��
����
����
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
��
��
����
����
Why the Extended QR Algorithm Works
Power method.
Rational Krylov subspaces.
Shifts accelerate convergence.
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Standard Deflation
Standard Deflation
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Deflation Criterion
× × × × ×× × × ×× × ×× ××
××××
=
����
����
× × × × ×× × × ×× × ×× ××
A) |H(k + 1, k)| ≤ ε ‖H‖F
B) |H(k + 1, k)| ≤ ε (|H(k , k)|+ |H(k + 1, k + 1)|)
A = Gτ(1)Gτ(2) · · ·Gτ(n−1)R
∥∥I − Gτ(i)
∥∥2≤ ε
A = Gτ(1)Gτ(2) · · ·Gτ(i−1)Gτ(i+1) · · ·Gτ(n−1)R
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Why is that a Useful Deflation Criterion?
Why is that a useful deflation criterion?
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Bauer-Fike Theorem
Bauer-Fike Theorem [Bauer, Fike 1960]
minµ∈Λ(A)
|λ− µ| ≤ κ2(X )‖A− A‖2
‖A− A‖ ≤ ‖Gτ(1) · · ·Gτ(i−1)‖ ‖Gτ(i) − I‖ ‖Gτ(i+1) · · ·Gτ(n−1)‖ ‖R‖≤ ‖Gτ(i) − I‖ ‖R‖≤ ‖Gτ(i) − I‖ ‖A‖
Lemma
Let A be diagonalizable, A = XΛX−1, and λ an eigenvalue of A.Then there exists an eigenvalue µ of A, so that
|λ− µ| ≤ κ2(X )‖Gτ(i) − I‖2‖A‖2.
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
But we can do better:
Theorem [Eisenstat, Ipsen 1995]
A = A1A2, E = A− A
mini
|λ−µi ||µi | ≤ κ2(X )
∥∥A−11 EA−1
2
∥∥2.
A1 = Gτ(1) · · ·Gτ(i) and A2 = Gτ(i+1) · · ·Gτ(n−1)R
⇒ A−11 EA−1
2 = I − G−1τ(i)
‖I − G−1τ(i)‖2 = ‖G−1
τ(i)(Gτ(i) − I )‖2 ≤ ‖Gτ(i) − I‖2
Lemma
Let A be diagonalizable, A = XΛX−1, and λ an eigenvalue of A.Then there exists an eigenvalue µ of A, so that
|λ−µ||µ| ≤ κ2(X )‖Gτ(i) − I‖2.
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Conditioning of One Eigenpair
Theorem [Wilkinson 1965]
Let
µ ∈ Λ(A), Ax = µx , yHA = µyH , be a simple eigenvalue,
E = εF , with ‖F‖2 = 1.
For µ = µ(ε) ∈ Λ(A + E ) with µ(0) = λ, we have
|µ− λ| ≤ ‖E‖2
yH x+O(‖E‖2
2).
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Relative Perturbation Estimate
Following the proofs of [Eisenstat, Ipsen 1995] we can proof thefollowing theorem.
Theorem [M., Vandebril ’13]
Let
A be invertible,
µ ∈ Λ(A), Ax = µx , yHA = µyH , be a simple eigenvalue,
E = εF , with ‖F‖2 = 1.
For µ = µ(ε) ∈ Λ(A + E ) with µ(0) = λ, we have
|µ−λ||λ| ≤
‖A−1E‖2
yH x+O(
∥∥A−1E∥∥2
2).
If A = A1A2 and A1 and A2 invertible, then
|µ−λ||λ| ≤
‖A−11 EA−1
2 ‖2
yH x+O
(‖A−1
1 EA−12 ‖2
2
).
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Very Good Relative Perturbation Estimate
Corollary [M., Vandebril ’13]
Let A = Gτ(1)Gτ(2) · · ·Gτ(n−1)R and
A = Gτ(1)Gτ(2) · · ·Gτ(i−1)Gτ(i+1) · · ·Gτ(n−1)R. Let further A be
nonsingular, and let λ be a single eigenvalue of A. Then thereexists an eigenvalue µ of A, so that
|µ−λ||λ| ≤
‖I−Gτ(i)‖2
xH y+O
(‖I − Gτ(i)‖2
2
),
with x the right and y the left eigenvector to λ.
The deflation criterion∥∥I − Gτ(i)
∥∥2≤ ε is very good.
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Aggressive Early Deflation (AED)[Braman, Byers, Mathias ’02]
Aggressive Early Deflation
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Aggressive Early Deflation (AED)[Braman, Byers, Mathias ’02]
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××××××××
A
⇒
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××⊗⊗⊗
A′ = QHAQ
VH
× × × ×× × ×× ××
V
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 18/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
AED for Extended QR Algorithms
����
����
����
����
��
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
AED for Extended QR Algorithms
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
AED for Extended QR Algorithms
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ×××××
×××
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
AED for Extended QR Algorithms
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
VH V
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
AED for Extended QR Algorithms
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
���� ��
�� �� ��
���� ��
�� �� ��
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 19/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
AED for Extended QR Algorithms
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
���� ��
�� �� ��
������
������
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 19/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
AED for Extended QR Algorithms
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
���� ��×
�� �� ��× ×
������ ×
���� ××��
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
AED for Extended QR Algorithms
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Wedge-to-Broadhead Transformation
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Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 20/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Restore Original Pattern
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
The effect of aggressive early deflation for n = 500
1020
3040
50
020
4060
80100
0
200
400
*
*
# shifts#
deflation window
run
tim
ein
s
aggressive early deflationstandard deflation only
best without early deflation best with early deflation savingsn sh t #it·sh sh m t #it·sh
100 4 6.00 s 215 10 18 5.62 s 223 6% time100 2 6.42 s 198 2 28 8.60 s 163 18% iterations500 6 148.49 s 803 20 20 100.45 s 727 32% time500 2 161.38 s 732 2 70 314.85 s 391 46% iterations
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
The effect of aggressive early deflation for n = 500
1020
3040
50
020
4060
80100
0
200
400
*
*
# shifts#
deflation window
run
tim
ein
s
aggressive early deflationstandard deflation only
best without early deflation best with early deflation savingsn sh t #it·sh sh m t #it·sh
100 4 6.00 s 215 10 18 5.62 s 223 6% time100 2 6.42 s 198 2 28 8.60 s 163 18% iterations500 6 148.49 s 803 20 20 100.45 s 727 32% time500 2 161.38 s 732 2 70 314.85 s 391 46% iterations
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Middle Deflation
Middle Deflation
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Middle Deflation
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Middle Deflation
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Middle Deflation
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Middle Deflation
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Middle Deflation
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Middle Deflation
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Middle Deflation
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Predicting Middle Deflations
One needs to define a deflation window that most likely leads to amiddle deflation.
g∏i=f
|si | < ε2
For random matrices a successful middle deflation window of size0.75 · n × 0.75 · n is found after a few iterations.
A ∈ C1500×1500, random, three multishift steps (150 shifts),deflation window 363:1026, leads to middle deflation
The corresponding deflation windows for other test-matrices(tols2000, pde2961) have not been successful.
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Predicting Middle Deflations
One needs to define a deflation window that most likely leads to amiddle deflation.
g∏i=f
|si | < ε2
For random matrices a successful middle deflation window of size0.75 · n × 0.75 · n is found after a few iterations.
A ∈ C1500×1500, random, three multishift steps (150 shifts),deflation window 363:1026, leads to middle deflation
The corresponding deflation windows for other test-matrices(tols2000, pde2961) have not been successful.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 25/27
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Conclusions
We have investigated deflations for extended QR algorithms.
Deflating almost diagonal rotations preserves absolute andrelative accuracy also with respect to the conditioning ofsingle eigenvalues.
We have seen aggressive early deflation in extended QRalgorithms.
These ideas can be used for middle deflations in special cases.
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Basics Standard Deflation Aggressive Early Deflation Middle Deflation
Acknowledgments
The research was partially supported by
the Research Council KU Leuven
CREA-13-012 Can Unconventional Eigenvalue AlgorithmsSupersede the State of the Art (CUASSA),F+/13/020 Exploiting Unconventional QR-Algorithms for Fastand Accurate Computations of Roots of Polynomials,OT/11/055 Spectral Properties of Perturbed Normal Matricesand their Applications,CoE EF/05/006 Optimization in Engineering (OPTEC),
DFG research stipend MA 5852/1-1,
Fund for Scientific Research–Flanders (Belgium) project G034212N,Reestablishing Smoothness for Matrix Manifold Optimization viaResolution of Singularities,
Interuniversity Attraction Poles Programme, initiated by the BelgianState, Science Policy Office, Belgian Network DYSCO – DynamicalSystems, Control, and Optimization.
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