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
