Feedbak Gain
Kazufumi Ito and Jari Toivanen
July 18, 2008
Abstrat
A fast algorithm to ompute the optimal feedbak gain for the
linear quadrati regulator problem is developed and analyzed. The
algorithm utilizes the relation between an invariant subspae of the
orresponding Hamiltonian operator and the solution to the Riati
equationand thereduedordermethods. It isbasedon theinverseof
theHamiltonianoperatoronthereduedordersubspae. Largesale
ontrol systems that arise from a disretization of a lass of ontrol
problemsgoverned bypartialdierentialequationsisused to
demon-strate the feasibility and appliability of Algorithm. A sparsity and
strutural propertyof systemmatriesareinorporatedinAlgorithm
and it enables us to ompute a stabilizing feedbak law for a large
lass ofdistributedontrolsystems.
1 Introdution
Consider the Linear Quadrati Regulator(LQR) problem;
min Z
1
0
((x(t);Qx(t))
X
+ju(t)j 2
)dt over u2L 2
(0;1;U); (1.1)
Center for Researh in Sienti Computation, Department of Mathematis, North
CarolinaStateUniversity;researhpartiallysupportedbytheArmyResearhOÆeunder
d
dt
x(t)=Ax(t)+Bu(t); x(0)=x
0
2X: (1.2)
Here, x(t) 2 X and u(t) 2 U is the state and ontrol funtions,
respe-tively. The optimal ontrol to (1.1){(1.2) is given in the feedbak form
u(t) = B
x(t) where the bounded, self-adjoint (operator) on X is
the unique nonnegative solution tothe algebraiRiati equation (ARE)
A
x+Ax BB
x+Qx=0 (1.3)
for everyx2dom(A). Here A
; B
are the Hilbert spae adjoints of A and
B, respetively and x 2 dom(A
) for x 2 dom(A), and we assume that
(A;B)is(exponentially)stabilizableand(A;Q)is(exponentially)detetable.
This result on the LQR problem is stated for the ase when X and U are
Hilbertspaes andA isaninnitesimalgenerator ofC
0
semigroup onX [18℄
and is standard, we refer e.g. to,[5,8℄.
In order to onstrut the optimal feedbak gain K op
= B
via ARE
(1.3) it is normally done that we onsider a sequene of LQR problems
(A N
;B N
;Q N
) on R N
based on approximation methods. It follows from
the Gibson's approximation theory for ARE that if a sequene of
approx-imating nite dimensional ontrol problem (A N
;B N
;Q N
) satises the
sta-bility and onsisteny ondition, e.g., see [6, 8℄, then the feedbak law by
K N
= (B N
)
N
for (A N
;B N
;Q N
) onverges toK op
in norm. Certain lass
of ontrol systems, inluding the uid ontrol and vibration ontrol
prob-lems resultsinalargesaleontrolsystem(A N
;B N
;Q N
)viadisretizations.
So, it is essential to have an eÆient method for the Riati equation that
overomes problems assoiated with large dimensionality N.
In this paper we develop and analyze a fast method toompute the
op-timalfeedbak gain K op
based on the orresponding Hamiltonian,i.e., if we
let p(t)=x(t), then we have
d
dt
(x(t);p(t)) T
=H (x(t);p(t)) T
H= 0
A BB
Q A
1
A
:
If is a losed loop eigenvalue of A BB
, then is aneigenvalue of H .
In fat, if x 2 dom(A) satises (A BB
)x = x, then x 2 dom(A
)
and
(A BB
)x=x
Qx A
x=(A BB
)x=x:
(1.4)
Thus, is an eigenvalue of H and the orresponding eigenfuntion is given
by (x;x). That is, if X =R N
is nite dimensional,then =YV 1
where
(V;Y)iseigenvetors(ingeneralShurvetors)orrespondingtoeigenvalues
(H )withR e<0,e.g.,[19,15,20℄. Whilethese"eigenvetor"methodsan
beusedsatisfatorily(foradisussionofrealadvantagesoeredbythe
Laub-Shur approah over Potter method, see [15℄) for systems with N relatively
small, the omputational eort (and time) grows like N 3
and the storage
requirement is of orderN 2
and they beome prohibitive forlarge systems.
Our proposed algorithms overome this diÆulty by the redued order
approah in [9℄. Algorithm I omputes a sub-invariant subspae of H that
orresponds to eigenvalues 2 (H ) whose real part R e < 0 is large and
then forms a feedbak gain on the invariant subspae. Algorithm II uses
the inverse of H on the redued subspae and approximates the invariant
subspae of H . Both Algorithms employ an impliitly Re-Started Arnold
iteration algorithm [21, 22℄ for nding sub-invariant subspae H (whih is
a generalization of the inverse power iterate for H ) and use the subspae
method (2.3){(2.4) for omputation of H 1
(f;g) whih requires the basi
operations A 1
f and A
g. Algorithm I omputes the exat eigenvalues
and assoiated invariant subspae X
1
of A BB
and thus K op
x = K L
x
for x 2 X
1
. Algorithm II yields approximations to suh eigenvalues and
invariant subspaes. However, for Algorithm II these basi operations are
L. The operation ount and storage requirements are of O(L ). As a result
it does not have a storage limitation as Algorithm I may have (see Setion
2) and it is very eÆient and aurate as well. We strongly believe that by
Algorithms the myth of "solving Riati equation to determine the optimal
feedbak gain is very impratial" is no longer true. We will demonstrate
it through some spei ontrolproblems in Setion 2. It is of our plans to
apply Algorithms to onrete ontrol problems inluding 3D Navier Stokes
ontrolow problems.
In[2℄ahybridmethodombiningtheChandrasekharalgorithm[3℄,
Newton-Kleiman method [13℄ along with an innovative use of the Smith algorithm
[23℄ for solution of Lyapunov equations is developed. It requires the basi
operation(I rA) 1
x; x2X and avery eÆientmethod whenthe rankof
Q issmall. In[17℄animprovementofthe hybridmethodusing ADImethod
[16℄for Lyapunov equations is reported.
Theoutlineofthepaperisasfollows. AlgorithmsI andII areintrodued
andexplainedandtheomputationalissuesareaddressedinSetion2. Also,
numerialtests forontrolproblemsof theheat equationare presented. The
orrespondingformulato(2.3){(2.4)forthegeneralizedontrolsystemofthe
form (2.5) and the orresponding seond order ontrol system is developed
in Setion 3. Setion 4 presents a onvergene analysis of Algorithm I. In
Setion 5 the redued order approahes to address the storage problem of
Algorithm I are desribed. Setion6 validatesAlgorithm II.
2 Algorithms
Algorithm I Find the (generalized) eigenpairs (
i ;(x
i ;p
i
)) with R e
i < 0
of H , where the eigenvalues
i
are ordered with respet to their real part.
Form the matries V and Y onsisting of the rst L vetors of x
i
and p
i ,
respetively. Dene K L
= B
Y(V
V) 1
(V;)
X
; 2 X, where !
(V
V) 1
(V;)
X
is the orthogonal projetion of X onto X
1
= range(V).
Moreover, thespan X
1 =fx
i
losed loopsystem A BK and K x=K x for x2X
1 .
If we use the approximation system (A N
;B N
;Q N
) and approximate H
by H N
;
H N
= 0
A
N
B N
(B N
) T
Q N
(A N
) T
1
A
:
in Algorithm I, then L an be muh smaller than N and thus Algorithm
oers a redued order method for onstrutionof the optimalfeedbak gain
[9℄. If X =R N
is equipped with the norm p
x T
Mx , then K L
is the matrix
representation
K L
=B
Y(V
MV) 1
V
(2.1)
where (V;Y) isonsisting of the rst L eigen (Shur)-vetors of H N
.
Algorithm I requires to nd sub-invariant subspaes of the Hamiltonian
H . We employ animpliitly Re-Started Arnold iterationalgorithm [21, 22℄
for nding the sub-invariant subspae X
1
for alinear system
H (x;p)=(f;g) in XX: (2.2)
Thus,theeÆienyoftheombinedalgorithmdependsonaneÆientmethod
tosolve(2.2). Tothis end, we apply the subspae method[12℄ (see, Remark
1.1) for solving (2.2);i.e.,
(x;p)=H 1
(f;g)=(A 1
(Bu+f); A
(g+QA 1
(Bu+f)): (2.3)
where u=B
p2R m
satises
u+B
A
QA 1
Bu= B
A
(g+QA 1
f) on R m
: (2.4)
In fat, if u=B
p,then
0=u B
p=u+B
A
QA 1
Bu+B
A
(g+QA 1
f):
Equation(2.4)isonR m
foruwithsymmetripositivematrix(I+B
A
QA 1
B).
evaluationof A f; r = B A (g+QA f),
solution u to(2.4) with righthand side r
x = A 1
f +A 1
Bu and p = A
(QA 1
(g + Qx)) to omplete the
solution to(2.3).
Thus, the subspae method for solving (2.2) needs only the one solution
for Ax = f, the one solution A
p = g~ and the solution to (2.4) in R m
. In
summaryitreduesdramatiallystorageandomputationalrequirementsfor
solving (2.2). Theproposedombinedalgorithmisvery eÆientandenables
usto ompute the Riati-basedfeedbak gain for a(super) largesystem. If
m is very large, thenwe use the onjugategradient methodfor solving (2.4)
whih onlyneeds the vetor operationB
A
QA 1
Bu; u2R m
. Otherwise,
we form A 1
B and solve (2.4) using the Cholesky fatorization on R m
.
Animplementation ofthe proposed algorithmusing matlabis asfollows.
[u;e℄=eigs(ri;2N 2
;2L;0;a;b;q); e=diag(e);
j =find(real(e)<0); u=v(:;j); [t;i℄=sort( real(e(j)));
Y =u([N +1:2N℄;i(1:L)); V =u(1:N;i(1:L));
K =V (V 0
M V)n(Y 0
b);
funtion [y℄=ri(xx)
n=size(xx;1)=2; x=reshape(xx;n;2); f =x(:;1); g =x(:;2);
f0=anf; b0=anb; u= (eye(m)+b0 0
qb0)n(b0 0
(g+qf0));
x=f0+b0u; p= (a 0
)n(g+qx); y=[x;p℄;
where eigsis anmatlabroutine and ri.mis anmatlabm-le. A user an
provide the routine toevaluate anf and (a 0
)ng.
Remark 1.1
Formula(2.3){(2.4)diretlyapplytoHfortheoriginalontrolproblem
(1.1){(1.2). In pratieweuse anapproximatingsystem(A N
;B N
;Q N
)
to approximate H by H N
and then a matlab routine eigs, whih
im-plements the Arnold method among with (2.3){(2.4) is used to nd
invariant subspaes of H N
Auray of sub-invariantsubspaes based onH dependsonthose of
(A N
;B N
;Q N
).
In general, anapproximating system has the form
M d
dt
x(t)=Hx(t)+B
0
u(t) (2.5)
where M is the mass (symmetri, positive) matrix on R N
, H is the
stiness operator onR N
and B
0
is the input matrix. Thus,
A=M 1
H; B =M 1
B
0
and Q N
= M if Q =the identity operator in (1.1). It will be shown
in Setion 2 that for system of the form (2.5) we have an equivalent
formulationof (2.3){(2.4) in whih one an avoidforming A=M 1
H
and B =M 1
B
0
and performingM 1
f ompletely. This is espeially
eÆient for the seond order ontrol (3.3) asdesribed inSetion 2.
Formula(2.3){(2.4) is the rightpreondition system [12℄ of (2.2);
H R 1
z =(f;g); (x;p)=R 1
z
with preonditioner
R=
A 0
Q A
:
In fat, note that if z^=z (f;g),then
H R 1
^
z =(f;g) R 1
(f;g)= (H R)R 1
(f;g)=^r2Y
where H R 1
=(H R)R 1
+I and Y =range(H R). Thusz^2Y
satises
^
z+(H R)R 1
^
z =r^in Y (2.6)
This is the redued equationin the sparse subspae Y =R m
with
H R =
0 BB
0 0
and Y =range(B)f0g
boundaryontrolof2and3Dheatequationsondomain=(0;1) d
; d=2;3;
t
y(t;x)=y(t;x); x2
n
y(t;x)= m
X
`=1 u
` (t)b
`
(x); x2
where > 0 is a diusion onstant and b
i
(x) are the distribution funtions
on the boundary . Suppose we use the entral dierene approximation
based on the uniform Cartesian gridpoints with mesh size h = 1
n
in eah
diretion. ThenwehaveN =(n+1) d
and forthetwodimensionalase(2.5)
is with
H =(H
1 Q
1 +Q
1 H
1
); M =h(Q
1 Q
1 )
(B
0 )
ijk
=b(x
i ;y
j ;z
k
)2;
where denotes the Kroneker produt and H
1
is the tridiagonalmatrix of
the form
H
1 =
h 0
B
B
B
B
B
1 1
1 2 1
.
.
. .
.
. .
.
.
1 2 1
1 1: 1
C
C
C
C
C
A
and Q
1
is the diagonal matrix of the form Q
1
= diag([:5;1;;1;:5℄). In
this ase H 1
x an arriedout byFFT with orderNlog(N) operationsand
O(N)storage. Inthe followingswesummarize theresultsforthe aseQ=I
and = :1. In Figure 1 the resulting feedbak gains for L = 30;50;100;N
are shown for the two dimensional ase with the ontrol at the x = 0 side
with b(0;y)=exp( 50(y :5) 2
)and N =21 2
. Table 1 shows Error dened
by
Error =jK L
K N
j
2 =jK
N
j
2
and CPU times forthe above matlabimplementationand for the ase using
FFT solver. The full dimension K N
usingFFT,i.e., even forfulldimension aseN =LAlgorithmIan bemore
eÆient due to Formula(2.3){(2.4).
0
10
20
30
0
20
40
0.01
0.012
0.014
L=30
0
10
20
30
0
20
40
0.01
0.012
0.014
L=50
0
10
20
30
0
20
40
0.01
0.012
0.014
L=100
0
10
20
30
0
20
40
0.01
0.012
0.014
L=N=441
Figure1: Feedbak Gains based oneign-subspae with dimension L
Forthe three dimensional ase in Table 2we summarize our results.
With2N =281 3
=1;062;882itwasnotpossibletouse AlgorithmIwith
L=20due tolakofmemoryspae(>1GB). CPUtimeglowslinearlyinL
for this ase. The storagebeomes aproblemwith inreasing L beforeCPU
time does for alarge sale problem.
ThisstorageproblemisresolvedbythereduedorderapproahinSetion
30 0.0124 4.00 0.50
50 0.0086 5.97 0.91
100 0.0053 14.58 4.45
Table 1: Feedbak Gains based on eign-subspae with dimension L
L N CPU(se) iter.
50 21 3
33.82 7
100 21 3
69.68 4
200 21 3
206.15 2
50 41 3
234.75 7
100 41 3
601.96 4
10 81 3
421.36 7
Table 2: 3D boundary ontrolfor heat equation with DimensionL of
eigen-subspae, iteration ounts, and CPU times
It isbased on the fat that invariantsubspaes of H 1
are same as those of
H and that H 1
is Hamiltonian. Ituses the restrition of H 1
on ^
X ^
X by
^
H 1
=
W
0
0 W
H 1
W 0
0 W
where ^
X =W
X and W isthe redued order orthonormalbasis of X. Suh
a basis an be generated by applying the proper orthogonal deomposition
approah [1, 14℄ and the redued-basis method [10, 11℄. Let L = dim( ^
X),
^
H 1
has the (2L)(2L) matrixrepresentation. Thus, we just need tostore
(2L)(2L)Shurbasis (omparedto(2N)(2L) ShurbasisforAlgorithm
I).
Algorithm IIFind the (generalized) eigenpairs (
i ;(^x
i ;p^
i
)) with R e
i <0
of ^
H 1
. Form the matries ^
V and ^
Y onsisting of vetors of x
i
and p
i ,
respetively. Dene ^
K L
=B
W
^
Y( ^
V
^
V) 1
^
VW
; 2X.
tation of ^
H 1
( ^
f;g)^ an be arried out exatly as for H 1
(see, (6.2){(6.3))
and is restrited to the redued basis. The other advantage of Algorithm
II is that matlab routine eigs is applied for (2L)(2L) matrix (ompared
to (2N) (2N) for Algorithm I) and results in a signiant redution of
storage requirement and omputationalomplexity. On the other hand,
Al-gorithm I leads to the exat eigenvalues and assoiated invariant subspae
X
1
of A BB
and thus K op
x = K L
x for x 2 X
1
. Algorithm II yields
approximationsto suheigenvalues and invariant subspaes.
Ournumerialtestsof AlgorithmII forour heatequationexamplesshow
that it performs as well as Algorithm I. We are able to solve 3D problem
with L=10 3
=1000 and the resulting feedbak is very aurate.
3 Seond order System
For system of the form (2.5) A = M 1
H; B = M 1
B
0
and A
= H
M 1
and (2.3){(2.4) an beequivalently writtenas
(x;p)=H 1
(f;g)=(H 1
(B
0
u+Mf); MH
(g+QH 1
(B
0
u+Mf)):
(3.1)
where u=B
0 M
1
p2R m
satises
u+B
0 H
QH 1
B
0
u= B
0 H
(g+QH 1
Mf) on R m
: (3.2)
In fat,
0=u B
0 (M
1
p)=u+B
0 H
QH 1
B
0 u+B
0 H
(g+QH 1
Mf):
Thus, we an avoid forming A; B and performing M 1
f ompletely for
(2.3){(2.4).
Furthermore,for the seond order ontrolsystem;
M
0 d
dt
y(t)+D
0 d
dt
y(t)+H
0
y(t)=B
0
0 0 0
we an take an advantage of the struture property as follows. If we let
x(t)=(y(t); d
dt
y(t))2X, then
2 4 H 0 0 0 M 0 3 5 d dt x(t)= 2 4 0 H 0 H 0 D 0 3 5 x(t)+ 2 4 0 B 0 3 5 u(t) (3.4)
and weequip X by normj(u;v)j 2
X =(M
0
v;v)+(H
0
u;u). Notethat for(3.4)
A 1
f =H 1
M(f
1 ;f
2
)=( H 1 0 (M 0 f 2 +D 0 f 1 );f 1 ) A
g =MH
(g
1 ;g
2 )=(g
2 D 0 (H 1 0 g 1 ); M 0 H 1 0 g 1 ) and H 1 B 0
u=( H 1
0 B
0
u;0); B 0 H (g 1 ;g 2
)= B
0 H 1 0 g 1
Thus, itfollows from(3.1){(3.2) we havefor (3.3){(3.4)
(x;p)=H 1
(f;g) with
x=( H 1
0 (B
0 u+M
0 f 2 +D 0 f 1 );f 1 )
p=((g+Qx)
2 D 0 H 1 0
(g+Qx)
1 ;M 0 H 1 0
(g+Qx)
1 );
(3.5)
where u2R m
satises
u+B 0 H 0 Q 11 H 1 0 B 0 u=B
0 H
0
(g+QH 1
Mf)
1
on R m
: (3.6)
All required operations are arried out with the original system matries
(M 0 ;H 0 ;D 0
) of (3.3).
4 Convergene Analysis of Algorithm I
Let us assume that A has a ompat resolvent. Let P be the projetion
operator dened by
loopoperator ~
A=A BK op
. We letX
1
=PX and X
2
=(I P)X. Then
we have the deomposition onX
1 X
2
A BK
L
=A BK op
+ 2
4
0 PB(K op
K L
)
0 (I P)B(K op K L ) 3 5
Note that K L =K op ~ P where ~
P is the orthogonal projetion of X onto X
1 .
Thus,
Iftheeigenspaespannedbyfx
i
gofA BK op
isomplete,then ~
P !I
and thus
jK op
K L
j=jK
op (I
~
P)j!0 asL!1:
If j(I P)BK op
(I ~
P)jis suÆientlysmall, then A BK L
generates
an exponentially stable semigroup onX.
It follows from (1.4) that for 2X
1
(A BB
)+A
+Q=0;
or equivalently
( ;(A BB
))+((A BB
) ;)+((Q+(K op
)
K op
) ;)) (4.1)
for all 2 X
1
and 2 X. Let ^
= ~
P ~
P. Then, ^
has a matrix
represen-tation onX
1 as
MV(V
MV) 1
Y(V
MV) 1
V
M:
It follows from(4.1) that
( ^
;(A BB
))+((A BB
) ; ^
)+((Q+(K op ) K op ) ;))
for all; 2X
1
. That is,
^ = Z 1 0 ~ PS (t) ~
P((Q+(K op ) K op ) ~ PS(t) ~ P dt
where S(t) isthe semigroup onX generated by A BK op
order Control
In this setion we desribe the redued order approahes. As pointed out
in Introdutionthe storage requirement for AlgorithmI beomes a problem
with inreasing L. It is neessary to redue the dimension of X for large
sale systems.
Let W 2 R N
^
N
be the redued order orthonormal basis of X. Suh
a basis an be generated by applying the proper orthogonal deomposition
approah [1, 14℄ and the redued-basis method [10, 11℄. If Q has a small
rank, the basis an be generated by the Krylov subspae
spanf[B;Q℄;A 1
[B;Q℄;:::;A n 1
[B;Q℄g: (5.1)
ThissubspaeisbasedonthefatthatthesolutiontotheRiatiequation
satises a Liyapunov equation;
(A BK op
)
+(A BK
op
)+(K op
)
K op
+Q=0:
The redued order ontrolsystem on ^
X =W
X is
min Z
1
0
((QWx;^ Wx)^
X
+ju(t)j 2
)dt
subjet to
W
MW d
dt ^
x (t)=W
HW^x(t)+W
B
0
u(t); x (t)^ 2 ^
X:
Thus, we an apply our algorithm (3.1){(3.2) for the redued order system
with (W
MW;W
HW;W
B
0 ;W
QW).
6 Algorithm II and Inverse Hamiltonian
Let ^
X = M 1
2
W
X with W
MW = I be the redued order subspae and
^
H 1
bethe redued restrition of H 1
on ^
X ^
X dened by
^
H 1
= W
M 1
2
0
0 W
M 1
2 !
H 1
M 1
2
W 0
0 M
1
2
W !
(x;p)= ^
H 1
(f;g)
=(W
MH 1
(B
0
u+MWf); W
MH
(MWg+QH 1
(B
0
u+MWf)):
(6.2)
where u=B
0
Wp2R m
satises
u+B
0 H
QH 1
B
0
u= B
0 H
(Wg+QH 1
MWf) on R m
: (6.3)
In fat let =M 1
2
x(t), then (2.5) isequivalently writtenas
d
dt
(t)=A(t)+Bu(t)
with
A=M 1
2
HM 1
2
; B =M 1
2
B
0 ;
~
Q=M 1
2
QM 1
2
:
Then, the orresponding HamiltonianH satises
H 1
= 0
M
1
2
0
0 M
1
2 1
A 0
H B
0 B
0
Q H
1
A 1
0
M
1
2
0
0 M
1
2 1
A
and thus from(6.1)
^
H 1
= 0
W
M 0
0 W
M 1
A 0
H B
0 B
0
Q H
1
A 1
0
MW 0
0 MW
1
A
whih results in (6.2){(6.3). Note that Formula (6.2){(6.3) uses the same
Next,welaim that ^
H is Hamiltonian. In fat,
0 I I 0 ^ H 1 0 I I 0 = 0 I I 0 W M 1 2 0 0 W M 1 2 ! H 1 M 1 2 W 0 0 M 1 2 W ! 0 I I 0 = W M 1 2 0 0 W M 1 2 ! 0 I I 0 H 1 0 I I 0 W 0 0 W = W M 1 2 0 0 W M 1 2 ! (H 1 ) M 1 2 W 0 0 M 1 2 W ! = ( ^ H 1 )
Thus, the Potter theory [19, 20℄ validates AlgorithmII.
6.1 Spetral Approximations
Let be an eigenvalue of H 1
and =fz 2 C : jz j = Æg is a
ounter-lokwise oriented urve in (H 1
) and isolates and dene the spetral
projetion P:
P = 1
2i Z
(zI H 1 ) 1 dz (6.4) Let H 1 n
be a(nite rank) approximatingsequene of H 1
. For example,
H 1 n = ~ P n (H Nn ) 1 ~ P n ; where ~ P n
is the orthogonal projetion on to X
n X n and H N n : (X N n X N n
) ! (X
N
n X
N
n
) approximates H . Let be an eigenvalue of H 1
and is the domain enlosed by . Assume the strong stability: for eah
z 2nfg
H 1
n
!H 1
forall 2XX
and there exists M =M(z) suh that
j(zI H 1
n )
1
n
H 1
n
is dened forsuÆiently large n and
j(P P
n
)jÆM( ) max
z2 j(H
1
H 1
n
)(zI H 1
) 1
j (6.5)
for 2XX. It follows from[4℄ that
lim
n [(H
1
n
)\℄ =fg
lim
n
dimP
n
(X X)dimP(XX):
(6.6)
Moreover, assume the uniform radial onvergene: for all > 0 and every
ompat subsetK of (H 1
)there exists anN suhthat forn N
sup
z2K r
([(H
1
H 1
n
)(zI H 1
) 1
℄)
where r
(S) denotes the spetral radius of a bounded operator S. It then
follows from[4℄
lim
n
dimP
n
(X X)=dimP(XX): (6.7)
For example, if H 1
n
is dened by (6.1) with W = W
n
inreasing family of
the redued orderbasis, thenitfollowsfrom [7℄that the strongstabilityand
uniform radial onvergene hold and the auray estimate of Algorithm II
is redued from (6.5).
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