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

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= ��


Hessenberg Matrix

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=

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0×××××

��

QR factorization of the Hessenberg matrix

Rotation of the form:

[c s−s c

], |c |2 + |s|2 = 1


Hessenberg Matrix

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××××××

=

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0××××

��

QR factorization of the Hessenberg matrixRotation of the form:

[c s−s c

], |c |2 + |s|2 = 1


Hessenberg Matrix

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[c s−s c

], |c |2 + |s|2 = 1


Hessenberg Matrix

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[c s−s c

], |c |2 + |s|2 = 1


Hessenberg Matrix

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0×

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[c s−s c

], |c |2 + |s|2 = 1


Hessenberg Matrix

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[c s−s c

], |c |2 + |s|2 = 1


Hessenberg Matrix

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=

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QR factorization of the Hessenberg matrix

Rotation of the form:

[c s−s c

], |c |2 + |s|2 = 1


Extended Hessenberg Matrix

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There is a relation to extended Krylov subspaces.


Rotators

Fusion:

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s1c1

c2−s2

s2c2

= c3−s3

s3c3

= ��

Turnover:

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Transfer Through Upper Triangular:

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Ö Ö Ö=

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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.



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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.



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2) Bulge chasing.

A) Compute zeroing similarity transformation.B) Apply transformation.



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2) Bulge chasing.

A) Compute zeroing similarity transformation.B) Apply transformation.



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Why Francis’s Algorithm Works [Watkins 1982, ’07, ’08, ’11]

Power method.

Krylov subspaces.

Shifts accelerate convergence.


Extended QR Algorithm[Vandebril ’11, Watkins, Vandebril ’12]

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Fusion

Transfer through up. triang.



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Turnover



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Similarity transformation



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Move pattern



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Why the Extended QR Algorithm Works

Power method.

Rational Krylov subspaces.

Shifts accelerate convergence.


Deflation Criterion

× × × × ×× × × ×× × ×× ××

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=

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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


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.


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.


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).


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

).


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.


Aggressive Early Deflation (AED)[Braman, Byers, Mathias ’02]

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×××××××××

A

⇒

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××××××⊗⊗⊗

A′ = QHAQ

VH

× × × ×× × ×× ××

V


AED for Extended QR Algorithms

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VH V



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Wedge-to-Broadhead Transformation

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Test for deflations.


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


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


Middle Deflation

× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××


Middle Deflation

× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××


Middle Deflation

× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××


Middle Deflation

× × × × × × × × × ×× × × × × × × × ×× × × × × × × ×× × × × × × ×× × × × × ×× × × × ×× × × ×× × ×× ××

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Middle Deflation

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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.


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.


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
