Computing the Polar Decomposition in Matrix Groups - University of ...

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Some Automorphism Groups. Space. M. A⋆. Automorphism group, G. Groups corresponding to a bilinear form. R n. I. A. T. Real orthogonals. C n.
Computing the Polar Decomposition in Matrix Groups Nick Higham Department of Mathematics University of Manchester [email protected] http://www.ma.man.ac.uk/~higham/

Joint work with Niloufer Mackey, D. Steven Mackey, and Françoise Tisseur.

Polar Decomp in Group – p. 1/11

Group Background Given nonsingular M and K = R or C, ½ xT M y, real or complex bilinear forms, hx, yiM = x∗ M y, sesquilinear forms. Define automorphism group G = { A ∈ Kn×n : hAx, AyiM = hx, yiM , ∀x, y ∈ Kn }.

Recall adjoint A? of A ∈ Kn×n wrt h·, ·iM defined by hAx, yiM = hx, A? yiM ∀x, y ∈ Kn×n . ½ M −1 AT M , for bilinear forms, ? Can show: A = M −1 A∗ M , for sesquilinear forms, G = { A ∈ Kn×n : A? = A−1 } . Polar Decomp in Group – p. 2/11

Some Automorphism Groups Space

A?

M

Automorphism group, G

Groups corresponding to a bilinear form Rn

I

AT

Real orthogonals

Cn

I

AT

Complex orthogonals

Rn

Σp,q

Σp,q AT Σp,q

Pseudo-orthogonals

Rn

R

RAT R

Real perplectics

R2n

J

−JAT J

Real symplectics

C2n

J

−JAT J

Complex symplectics

Groups corresponding to a sesquilinear form Cn

I

A∗

Unitaries

Cn

Σp,q

Σp,q A∗ Σp,q

Pseudo-unitaries

C2n

J

−JA∗ J

Conjugate symplectics

R=

·

... 1

1

¸

,



 J=

0

−In



In  , 0



 Ip Σp,q = 0

0 −Iq

  

Polar Decomp in Group – p. 3/11

Questions Recall polar decomposition of A ∈ Cn×n : A = U H,

U ∗ U = I,

H = H ∗ ≥ 0.

If A ∈ G I When do its polar factors lie in the group? I How can we exploit any group structure when computing U and H?

Polar Decomp in Group – p. 4/11

Structure of Polar Factors Let U denote set of autom. groups for which M is unitary. Two results of Mackey, Mackey & Tisseur, 2003: Theorem 1 Let G ∈ U and A ∈ G. Then in A = U H the polar factors U and H also belong to G . Theorem 2 Let G ∈ U and A ∈ G. The singular values of A occur in reciprocal pairs σ and 1/σ , with the same multiplicity.

Polar Decomp in Group – p. 5/11

Structure-Preserving Iterations Theorem 3 Consider Zk+1 = Zk Pmm (I − Zk∗ Zk ) Qmm (I − Zk∗ Zk )−1 ,

Z0 = A,

where Pmm (t)/Qmm (t) is the [m/m] Padé approximant to (1 − t)−1/2 and m ≥ 1. If G ∈ U and A ∈ G then Zk ∈ G for all k , Zk → U at order 2m + 1.

Polar Decomp in Group – p. 6/11

Structure-Preserving Iterations Theorem 4 Consider Zk+1 = Zk Pmm (I − Zk∗ Zk ) Qmm (I − Zk∗ Zk )−1 ,

Z0 = A,

where Pmm (t)/Qmm (t) is the [m/m] Padé approximant to (1 − t)−1/2 and m ≥ 1. If G ∈ U and A ∈ G then Zk ∈ G for all k , Zk → U at order 2m + 1.

Iterations zk+1 = f (zk ):

m

f (x)

1

x(3 + x2 ) 1 + 3x2

2

x(5 + 10x2 + x4 ) 1 + 10x2 + 5x4 Polar Decomp in Group – p. 6/11

Iterations (all with X0 = A) Cubic (structure-preserving): 1 Xk+1 = Xk [I + 8(I + 3Xk∗ Xk )−1 ]. 3 Quintic (structure-preserving): · 1 8 xk+1 = xk + 5 5x2k + 7 −

16 5x2k + 3

¸ .

Scaled Newton iteration (not structure-preserving): !1/2 Ã · ¸ −1 kXk kF 1 (k) 1 −∗ (k) . Xk+1 = γ Xk + (k) Xk , γ = 2 kXk kF γ X0 ∈ G ∈ U



γ (0) = 1. Polar Decomp in Group – p. 7/11

Experiment Random symplectic A ∈ R12×12 , kAk2 = 310 = kA−1 k2 . kA∗ A − Ik2 , µO (A) = 2 kAk2 k

0 1 2 3 4 5 6 7 8

Newton (scaled) µO (Xk ) µG (Xk ) 1.0e+0 7.0e-18 1.0e+0 1.0e+0 8.6e-01 8.6e-01 2.0e-01 2.0e-01 3.2e-03 3.2e-03 9.0e-07 9.0e-07 6.0e-14 1.3e-13 4.3e-16 1.1e-13

kA? A − Ik2 . µG (A) = 2 kAk2

Cubic µO (Xk ) µG (Xk ) 1.0e+0 7.0e-18 1.0e+0 8.9e-17 1.0e+0 8.1e-16 9.9e-01 6.3e-15 9.4e-01 5.0e-14 5.7e-01 2.8e-13 3.6e-02 5.2e-13 3.2e-06 5.3e-13 3.8e-16 5.3e-13 Polar Decomp in Group – p. 8/11

Newton Behaviour Theorem 5 Let G ∈ U, A ∈ G, and Xk be the Newton iterates, either unscaled or with Frobenius scaling. Then Xk? = Xk∗ , k ≥ 1. Moreover, M Xk = Xk M ,

real bilinear, complex sesquilinear forms,

M Xk = Xk M ,

complex bilinear forms.

Implications: F Tethering. F Structure in Xk : A pseudo-orthogonal ⇒ Xk block-diagonal, ¸ · Ek Fk . A symplectic ⇒ Xk = −Fk Ek Polar Decomp in Group – p. 9/11

Convergence Tests Padé iteration function fmm satisfies fmm (σ −1 ) = fmm (σ)−1 , fmm (1) = 1, 1 < σ ⇒ 1 < fmm (σ) < σ, 1 ≤ µ < σ ⇒ fmm (µ) < fmm (σ).

Can show that for A ∈ G and G ∈ U: (k)

kU − Zk k2 = fmm (σ1 ) − 1 .

Moreover, kZk kF decreases monotonically. Stop when kZbk+1 kF ≥ 1−δ. kZbk kF

Scaled Newton: latter test applicable for k ≥ 1. Polar Decomp in Group – p. 10/11

Conclusions Structured iteration (cubic, quintic) versus scaled Newton F Newton has slightly better observed numerical stability. F Newton usually requires the fewest flops. F Convergence prediction possible with structured iterations. F Which is best depends on matrix A, group G, and user’s accuracy requirements.

Have analogous results for matrix sign function, with no restrictions on G. I http://www.ma.man.ac.uk/~nareports/narep426.pdf

Polar Decomp in Group – p. 11/11