on deflations in extended qr algorithms
TRANSCRIPT
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On Deflations in Extended QRAlgorithms
Thomas Mach Raf Vandebril
Dept. Computer Science, KU Leuven, Belgium
IWASEP 10Dubrovnik, 3 June 2014
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 1/20
Rotation
11
1
11
c
c
s
−s= ��
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 2/20
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 3/20
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 3/20
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 3/20
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 3/20
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 3/20
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 3/20
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 3/20
Extended Hessenberg Matrix
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××××× ×
××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
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����
There is a relation to extended Krylov subspaces.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 4/20
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:
����
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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 5/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
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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 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
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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 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
��
� �
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
��
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
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 6/20
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 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
��
� �
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
× =
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
× =
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
× =
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
��
� �
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
��
� �
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
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��
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
×
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
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� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
×
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
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��
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
×
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
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Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
��
� �
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
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��
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
������
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××××××
��
� �
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
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� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
Francis’s Implicit Hessenberg QR Algorithm[Francis 1961]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
���� ��
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 6/20
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 6/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
��
� �
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
��
Fusion
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
��
� �
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��� �
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Move pattern
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
��
� �
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
��
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
×=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
��
� �
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
� �
��
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Move pattern
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
×××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
����
����
��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
××× ×××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
��
��
������
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×× × ×
×
××× ×××
��
� �
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
��
��
������
� �
Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×
××
××
×
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
��
��
����
� �
Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×
××
××
×
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
����
��
��
���� ��
Turnover
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×
××
××
×
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
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Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
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=
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Similarity transformation
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×
××
××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
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Transfer through up. triang.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×
××
××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
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Fusion
Move pattern
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××× ×
××
××
=
× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
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Why the Extended QR Algorithm Works
Power method.
Rational Krylov subspaces.
Shifts accelerate convergence.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 7/20
Deflation Criterion
× × × × ×× × × ×× × ×× ××
××××
=
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× × × × ×× × × ×× × ×× ××
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
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 8/20
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.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 9/20
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.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 10/20
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).
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 11/20
Relative Perturbation Estimate (1)
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
).
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 12/20
Relative Perturbation Estimate (2)
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 good.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 13/20
Aggressive Early Deflation (AED)[Braman, Byers, Mathias ’02]
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
×××××××××
A
⇒
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
××××××⊗⊗⊗
A′ = QHAQ
VH
× × × ×× × ×× ××
V
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 14/20
AED for Extended QR Algorithms
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Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 15/20
AED for Extended QR Algorithms
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Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 15/20
AED for Extended QR Algorithms
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×××
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 15/20
AED for Extended QR Algorithms
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VH V
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 15/20
AED for Extended QR Algorithms
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Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 15/20
AED for Extended QR Algorithms
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Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 15/20
AED for Extended QR Algorithms
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Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 15/20
AED for Extended QR Algorithms
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Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 15/20
Wedge-to-Broadhead Transformation
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Test for deflations.
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 16/20
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
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 17/20
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
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 17/20
Middle Deflation
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 18/20
Middle Deflation
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 18/20
Middle Deflation
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 18/20
Middle Deflation
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
V VH V VH
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 18/20
Middle Deflation
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Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 18/20
Middle Deflation
× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 18/20
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 19/20
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 19/20
Conclusions & Acknowledgments
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.
The research was partially supported by the Research Council KULeuven (CREA-13-012, F+/13/020, OT/11/055, CoEEF/05/006), by DFG research stipend MA 5852/1-1, by Fund forScientific Research–Flanders (Belgium) project G034212N, and byBelgian Network DYSCO – Dynamical Systems, Control, andOptimization
Thomas Mach, Raf Vandebril, On Deflations in Extended QR Algorithms 20/20