LER =

U 1 U 2 Σ 1 0

0 Σ 2

V ₁ ^T V ₂ ^T

,

U 1 , V 1 ∈ R ^n×r

U 2 , V 2 ∈ R ^n×(n−r)

having orthogonal olumns and

Σ 1 =

diag

(ς 1 , . . . ς d )

Σ 2 =

^diag

(ς d+1 , . . . ς n )

5.4.3. Bilinear Krylov Subspae Methods. Model Order Redution

forbilinearsystemsviaKrylovsubspaeshasbeen examinedbyseveral

re-searherssuhasPhilipps[54℄,CondonandIvanov[23℄,BreitenandDamm

[17℄, BaiandSkoogh[8℄, andLin andoworkers[45℄. Momentmathing

anbeahievedbyseriesexpansionsofthemultivariatetransferfuntions

asgivenin(2.37). Foreaseofpresentation,weassume

E = I n

^throughout

thefollowingsetion. A multimomentanbedenedas:

Denition5.4.1([45℄,[34℄). Let

Σ bil

^be^a^bilinear^system^as^givenⁱⁿ^(2.30).

Fornonnegativeintegers

m 1 , . . . , m i ,

^amultimoment

H ^(m _i ¹ ^,...,m ⁱ ⁾ (s 1 , . . . , s i )

ofthetransferfuntion

H i (s 1 , . . . , s i )

^as^givenⁱⁿ^(2.37)^is^dened^as

H ^(m _i ¹ ^,...,m ⁱ ⁾ (s 1 , . . . , s i ) =(−1) ⁱ C(s i I n − A) ^−m ⁱ N [I m ⊗ (s _i−1 I n − A) ^−m ⁱ ⁻¹ N ] . . .

· [I | m ⊗ · · · ⊗ {z I m }

i−2

^times

⊗(s 2 I n − A) ^−m ² N ]

· [I | m ⊗ · · · ⊗ {z I m }

i−1

^times

⊗(s 1 I n − A) ^−m ¹ B],

(5.41)

where

N = [N 1 . . . N m ]

Toensuremoment mathing,Krylovsubspaes(f. 5.2.2)needtobe

built. Often(seef.e. [8,45,17℄),thefollowingKrylovsubspaesareused

formomentmathingaround

s = 0

span(V ⁽¹⁾ ) = K q (A ⁻¹ , A ⁻¹ B), span(V ⁽ⁱ⁾ ) =

[ m k=1

K q (A ⁻¹ , A ⁻¹ N k V ⁽ⁱ⁻¹⁾ ),

span(V ) = span [ r i=1

span(V ⁽ⁱ⁾ )

! .

Momentmathinginpointsotherthantheoriginanbeguaranteedbythe

followingresultgivenbyFlagg[34℄:

Theorem5.4.2 ([34℄, SubsystemInterpolation). Let

{ξ j } ^k _j=1 , {ζ j } ^k _j=1 ⊂ C

andvetors

c ^T ∈ C ^p

^and

b ∈ C ^m

^be^given. ^Dene

b _j = 1 _j ⊗ b

^and

N ^⊕T = N 1 ^T , . . . , N m ^T

where

1 _j

isaolumnof

m ^j−1

^ones. ^To^onstrut^a^redued

ordersystemthatmathesallthemultimoments

H ^(l _j ¹ ^,...,l ^j ⁾ (ξ 1 , . . . , ξ j ) b _j

and

c H ^(l _j ¹ ^,...,l ^j ⁾ (ζ j , . . . , ζ 1 )

^for

j = 1, . . . , k

^and

l 1 , . . . , l j = 1, . . . , q,

^onstrut^the

matries

V

^and

W

^as^follows:

span(V ⁽¹⁾ ) = K q {(ξ 1 I − A) ⁻¹ , (ξ 1 I − A) ⁻¹ Bb}, span(W ⁽¹⁾ ) = K q {(ζ 1 I − A) ^−∗ , (ζ 1 I − A) ^−∗ C ^∗ c ^∗ },

span(V ^(j) ) = K q {(ξ j I − A) ⁻¹ , (ξ j I − A) ⁻¹ N (I m ⊗ V ^(j ⁻¹⁾ )}

^for

j = 2, . . . , k, span(W ^(j) ) = K q {(ζ j I − A) ^−∗ , (ζ j I − A) ^−∗ N ^⊕T (I m ⊗ W ^(j−1) )}

^for

j = 2, . . . , k, span(V ) = span{

[ k j=1

span(V ^(j) )},

span(W ) = span{

[ k j=1

span(W ^(j) )}.

Provided

W ˜ ^T = (W ^T V ) ⁻¹ W ^T

^is^dened,^the^redued^system

A ˆ = ˜ W ^T AV

N ˆ k = ˜ W ^T N k V

C ˆ = CV

^and

B ˆ = ˜ W ^T B

^satises:

H ^(l _j ¹ ^,...,l ^j ⁾ (ξ 1 , . . . , ξ j ) b _j = ˆ H ^(l ¹ ^,...,l ^j ⁾ (ξ 1 , . . . , ξ j ) b _j

and

c H ^(l _j ¹ ^,...,l ^j ⁾ (ζ 1 , . . . , ζ j ) = c H ˆ ^(l ¹ ^,...,l ^j ⁾ (ζ 1 , . . . , ζ j )

for

j = 1, . . . , k

^and

l 1 , . . . , l k = 1, . . . , q

5.5.

H ₂

-OPTIMALBILINEARMODELORDERREDUCTION 73

Usingthismomentmathingofmultimomentswouldinvolveastrategy

forndingpoints

{ξ j } ^k _j=1 , {ζ j } ^k _j=1 ⊂ C

^and ^vetors

c ^T ∈ C ^p

^and

b ∈ C ^m

suhthatthereduedmodeldeliversagoodapproximation totheoriginal

model. Theadvantageofthisapproahisthatit doesnotdependonthe

onvergeneof theunderlyingVolterraseries, whihmight notbe known

apriori (f. the denitionof BIBO stability and the onvergene of the

VolterraseriesgiveninSetion2.3.2).Inadditiontothemomentmathing

approah, one might thinkof the interpolation of the multivariate

trans-fer funtions

H i (s 1 , . . . , s i )

^, ^or ⁱⁿ ^other ^words ^the interpolation of theVolterraseries. ThisapproahhasbeenexaminedbyFlagg[34℄inhis

dissertationandresultedin aderivationofinterpolationonditionsforthe

Volterraseriesrepresentationof abilinearsystem. Flaggwas ableto

es-tablishaonnetion betweenVolterra series interpolationand the results

onerningthe

H ₂

-optimalonditionsforbilinearsystems reentlyderived byZhangandLam[72℄andBennerandBreiten[12℄.

5.5.

H ₂

-optimalbilinearModelOrderRedution

As in the linear ase, one is interested in

H ₂

-optimalbilinear MOR.

Withinthissetion,neessary

H ₂

-optimalityonditionsforbilinearsystems are obtainedby deriving the

H ₂

-norm (5.39) ofthe error system (5.35).

First,thebilinearWilsononditionsoriginallyobtainedby ZhangandLam

[72℄ will bederived. Usingadierent approah,Bennerand Breiten[12℄

obtainedtheBilinearInterpolatory RationalKrylovAlgorithm(BIRKA),a

generalizationtobilinearsystemsofthelinearIRKA(Algorithm1). In

addi-tion,wewillderiveanew

H ₂

-optimalalgorithmrelyingonoptimizationon Grassmannmanifolds,whihisageneralizationofthemethodsgiveninthe

linearasebyYanandLam[69℄andXuandZeng[68℄.

AstheFiniteElement Disretisationofindustrialmodelsleads tosystems

with

E 6= I n

^,^we^need^to^inorporate

E

ⁱⁿ^ourderivation.Weannotsimply invertthematrix

E

^as^due^to^their^large^dimension,^the^inversion^would^be

numeriallyexpensive or evenimpossible. Hene, we willderiveoptimality

onditionsfor systems with

E 6= I n

E

nonsingular, whihhavenot been statedelsewhere. Allsystemswillbeassumedto bereahable,observable,

BIBOstableandtheGramiansshallexist.

5.5.1. Wilsononditionsforbilinearsystems. Dening

C = _C T

− C ˆ ^T

[ C − C ˆ ]

^,^the^norm^of^the^error^system^an^be^given^as:

J = ||Σ ^err _bil || ² H ₂ = tr [ C − C ˆ ] P ^err _C T

− C ˆ ^T

= tr (P ^err C) .

^(5.42)

Bydierentiatingthenorm(5.42)andusingtheLyapunovequations(5.36)

and(5.38)weobtainthefollowingonditions(foradetailedderivationsee

AppendixA.1):

E ˆ = −Y ₂₂ ⁻¹ Y 12 ^T EP 12 P ₂₂ ⁻¹ ,

^(5.43)

A ˆ = −Y ₂₂ ⁻¹ Y ₁₂ ^T AP 12 P ₂₂ ⁻¹ ,

^(5.44)

N ˆ k = −Y ₂₂ ⁻¹ Y ₁₂ ^T N k P 12 P ₂₂ ⁻¹ ,

^for

k = 1, . . . , m,

^(5.45)

B ˆ = −Y ₂₂ ⁻¹ Y ₁₂ ^T B,

^(5.46)

C ˆ = CP 12 P ₂₂ ⁻¹ ,

^(5.47)

with

Y i j

^as^givenⁱⁿ^(5.37)^and

P i j

^asⁱⁿ^(5.36). ^This^leads^to^the^following

theorem:

Theorem5.5.1 ([72℄). If thereduedsystem

Σ ˆ bil

^,^whih^is^reahable^and

observable, isan

H ₂

-optimalreduedorder modelforthe system

Σ bil

^and

the reahabilityand observabilityGramians

P ^err

^and

Q ^err

^exist,^then ^there

existmatries

W, V ∈ R ^n×r

suhthat

E ˆ = W ^T EV, A ˆ = W ^T AV, N ˆ k = W ^T N k V, B ˆ = W ^T B, C ˆ = CV.

^(5.48)

Theyanbeobtainedbyequations (5.43)to (5.44)as

W := −Y 12 Y ₂₂ ⁻¹

^and

V := P 12 P ₂₂ ⁻¹ .

Remark 5.5.2. Inserting the observability Gramian

Q ^err

ⁱⁿ ^the ^equations

leadstotheprojetionsforthesystemmultipliedby

E ⁻¹

E ˆ = −Y ₂₂ ⁻¹ Y ₁₂ ^T EP 12 P ₂₂ ⁻¹

= − EQ ˆ ⁻¹ ₂₂ E ˆ ^T E ˆ ^−T Q ^T ₁₂ E ⁻¹ EP 12 P ₂₂ ⁻¹ ,

⇒ I r = −Q ⁻¹ ₂₂ Q ^T ₁₂ P 12 P ₂₂ ⁻¹ , A ˆ = −Y ₂₂ ⁻¹ Y ₁₂ ^T AP 12 P ₂₂ ⁻¹

= − EQ ˆ ⁻¹ ₂₂ E ˆ ^T E ˆ ^−T Q ^T ₁₂ E ⁻¹ AP 12 P ₂₂ ⁻¹ ,

⇒ E ˆ ⁻¹ A ˆ = −Q ⁻¹ ₂₂ Q ^T ₁₂ E ⁻¹ AP 12 P ₂₂ ⁻¹ ,

withanaloguealulationsfor

N k

B

^and

C

5.5.

H ₂

-OPTIMALBILINEARMODELORDERREDUCTION 75

5.5.2. TheoptimalityonditionsderivedbyBennerandBreiten. As

intheaseoftheWilsononditions,BennerandBreitendeduethe

opti-malityonditionsbydierentiatingthe

H ₂

-normoftheerrorsystem(5.40).

Inontrasttotheirderivation,weneedtoonsider

E 6= I n

E

nonsingular.

Theobtainedreduedsystem anbewrittenas

( ˆ A, N ˆ k , B, ˆ C) ˆ

^after

multi-plyingwith

E ˆ ⁻¹

^from^the^left,^and^hene^we^will^assume

E ˆ = I r

^. ^In^addition,

weassumethat

A ˆ

^isdiagonalizable.

Itispossibletorewritetherepresentationofthe

H ₂

-normasgivenin(5.40) byusing:

A ˆ = SΛS ⁻¹ , B ˜ ^T = S ⁻¹ B, ˆ C ˜ = ˆ CS, N ˜ _k ^T = S ⁻¹ ( ˆ N) k S,

whihleadsto:

J = ||Σ ^err bil || ² H ₂

= vec(I 2p ) ^T ([ C − C ˜ ] ⊗ [ C − C ˜ ])

× − _A

Λ

⊗ _E

I r

− _E

I r

⊗ _A

Λ

− X m

k=1

h _N

k N ˜ _k ^T

i

⊗ h _N

k N ˜ _k ^T

i ! −1

× _B

B ˜ ^T

⊗ _B

B ˜ ^T

vec(I 2m ).

(5.49)

Derivations with respet to the eigenvalues of the redued system

Λ = diag(ˆ λ 1 , . . . , ˆ λ r )

^and ^the ^matries

N ˜ k , B, ˜

^and

C ˜

^lead^to ^the

follow-ingoptimalityonditions(theirderivationanbefoundinAppendixA.2):

vec(I p ) ^T ( ˜ C ⊗ C) −I r ⊗ A − Λ ⊗ E − X m

k=1

N ˜ k T ⊗ N k

! −1

(e i e i ^T ⊗ E) −I r ⊗ A − Λ ⊗ E − X m

k=1

N ˜ k T ⊗ N k

! −1

( ˜ B ^T ⊗ B)vec(I m )

= vec(I p ) ^T ( ˜ C ⊗ C) ˆ −I r ⊗ A ˆ − Λ ⊗ I r − X m

k=1

N ˜ k T ⊗ N ˆ k

! −1

(e i e _i ^T ⊗ I r ) −I r ⊗ A ˆ − Λ ⊗ I r − X m

k=1

N ˜ k T ⊗ N ˆ k

! −1

( ˜ B ^T ⊗ B)vec(I ˆ m ),

(5.50)

vec(I p ) ^T ( ˜ C ⊗ C) −I r ⊗ A − Λ ⊗ E − X m

k=1

N ˜ k T ⊗ N k

! −1

(e i e j ^T ⊗ N) −I r ⊗ A − Λ ⊗ E − X m

k=1

N ˜ k T ⊗ N k

! −1

( ˜ B ^T ⊗ B)vec(I m )

= vec(I p ) ^T ( ˜ C ⊗ C) ˆ −I r ⊗ A ˆ − Λ ⊗ I r − X m

k=1

N ˜ k T ⊗ N ˆ k

! −1

(e _i e j ^T ⊗ N) ˆ −I r ⊗ A ˆ − Λ ⊗ I r − X m

k=1

N ˜ k T ⊗ N ˆ k

! −1

( ˜ B ^T ⊗ B)vec(I ˆ m ),

(5.51)

vec(I p ) ^T ( ˜ C ⊗ C) −I r ⊗ A − Λ ⊗ E − X m

k=1

N ˜ k T ⊗ N k

! −1

· (e j e i ^T ⊗ B)vec(I m )

= vec(I p ) ^T ( ˜ C ⊗ C) ˆ −I r ⊗ A ˆ − Λ ⊗ I r − X m

k=1

N ˜ k T ⊗ N ˆ k

! −1

· (e j e i ^T ⊗ B)vec(I ˆ m ),

^(5.52)

vec(I p ) ^T (e i e ^T _j ⊗ C) −I r ⊗ A − Λ ⊗ E − X m

k=1

N ˜ k T ⊗ N k

! −1

· ( ˜ B ^T ⊗ B)vec(I m )

= vec(I p ) ^T (e i e _j ^T ⊗ C) ˆ −I r ⊗ A ˆ − Λ ⊗ I r − X m

k=1

N ˜ k T ⊗ N ˆ k

! −1

· ( ˜ B ^T ⊗ B)vec(I ˆ m ).

^(5.53)

Thefollowingtheoremshowsthe onnetionbetween anoptimalredued

ordermodelandtheonditions(5.50)(5.53).

5.5.

H ₂

-OPTIMALBILINEARMODELORDERREDUCTION 77

Theorem5.5.3([12℄). Let

Σ bil

^denote^a^BIBO^stable^bilinear^system.

As-sume that

Σ ˆ bil

^is^a ^redued^bilinear^system ^of ^order

r

^that ^minimizes^the

H ₂

-normoftheerrorsystemamongallotherbilinearsystemsofdimension

r

^. ^Then,

Σ ˆ bil

^fullls^the^onditions^(5.50)^(5.53).

5.5.3. Algorithmsresultingfromthe

H ₂

-optimalityonditions. Now itispossibletoobtaintwodierentalgorithmsforthealulationofbilinear

optimalreduedordermodels. First,asseenin theontext oftheWilson

onditions,optimal models anbe obtainedby using

W = −Y 12 Y ₂₂ ⁻¹

^and

V = P 12 P ₂₂ ⁻¹

^(f. ^Theorem ^5.5.1). ^Hene ^it ^holds

span(Y 12 ) ⊂ W

^and

span(P 12 ) ⊂ V

^. ^It^is^suient^to^determine

Y 12

^and

P 12

^whih^an^be^done

bysolvingSylvesterequationsobtainedbysplittingtheequations(5.36)and

(5.38). Thisleads to thefollowingalgorithm(fora more detailedinsight

werefertothederivationofBennerandBreiten[12℄):

Algorithm2GeneralizedSylvesteriteration(f.[12℄).

Input:

E, A, N k , B, C, E, ˆ A, ˆ N ˆ k , B, ˆ C ˆ

Output:

E ˆ ^opt , A ˆ ^opt , N ˆ _k ^opt , B ˆ ^opt C ˆ ^opt

1: whilenotonvergeddo

2: Solve

AX E ˆ ^T + EX A ˆ ^T + X m

k=1

N k X N ˆ k + B B ˆ ^T = 0

^(5.54)

3: Solve

A ^T Y E ˆ + E ^T Y A ˆ + X m

k=1

N k Y N ˆ k − C ^T C ˆ = 0

^(5.55)

V = orth(X)

W = orth(Y )

^%^orth^omputes^anorthonormalbasis 5:

E ˆ = W ^T EV

A ˆ = W ^T AV

N ˆ k = W ^T N k V

B ˆ = W ^T B

6: endwhile

E ˆ ^opt = ˆ E, A ˆ ^opt = ˆ A, N ˆ _k ^opt = ˆ N k , B ˆ ^opt = ˆ B, C ˆ ^opt = ˆ C

Theorem5.5.4([12℄). IfAlgorithm2onverges,then

E ˆ ^opt , A ˆ ^opt , N ˆ _k ^opt , B ˆ ^opt

and

C ˆ ^opt

^fulll^the^Wilson^optimality^onditions(5.43)-(5.47).

Proof. TheproofofthisTheoremanbefoundintheAppendixA.3.

AswederivedtheoptimalityonditionsaordingtoBreitenandBenner

[12℄byusingreduedsystemsassuming

E ˆ = I r

^,^we^obtain^for^the^solution

ofthebilinearSylvesterequations(5.54)and(5.55):

vec(X) = −I r ⊗ A − A ˆ ⊗ E − X m k=1

N ˆ k ⊗ N k

! − 1

vec(B B ˆ ^T )

= −SS ⁻ ¹ ⊗ A − SΛS ⁻ ¹ ⊗ E − X m k=1

S N ˜ ^T k S ⁻ ¹ ⊗ N k

! − 1

( ˆ B ⊗ B)vec(I m )

= (S ⊗ I n ) −I r ⊗ A − Λ ⊗ E − X m k=1

N ˜ k ^T ⊗ N k

! S ⁻ ¹ ⊗ I n

! − 1

( ˆ B ⊗ B)vec(I m )

= (S ⊗ I n ) −I r ⊗ A − Λ ⊗ E − X m k=1

N ˜ k ^T ⊗ N k

! − 1

( ˜ B ^T ⊗ B)vec(I m )

| {z }

vec(V )

,

and

vec(Y ) = I r ^T ⊗ A ^T + ˆ A ^T ⊗ E ^T + X m

k=1

N ˆ k ^T ⊗ N ^T k

! −1

( ˆ C ^T ⊗ C ^T )vec(I p )

= S ^−T S ^T ⊗ A ^T + S ^−T ΛS ^T ⊗ E ^T + X m

k=1

S ^−T N ˜ k S ^T ⊗ N k ^T

! −1

( ˆ C ^T ⊗ C ^T )vec(I p )

= −S ^−T ⊗ I n

−I r ⊗ A ^T − Λ ⊗ E ^T − X m k=1

N ˜ k ⊗ N k ^T

! −1

S ^T ⊗ I n

( ˆ C ^T ⊗ C ^T )vec(I p )

= −S ^−T ⊗ I n

−I r ⊗ A ^T − Λ ⊗ E ^T − X m k=1

N ˜ k ⊗ N k ^T

! −1

( ˜ C ^T ⊗ C ^T )vec(I p )

| {z }

vec(W)

,

Thisleadstothefatthat

span(X) ⊂ V

^and

span(Y ) ⊂ W

^. ^Instead^of

solvingtheSylvesterequationsasgivenin(5.54)and(5.55) ,weanusethe

vetorizedformoftheSylvesterequationstoalulateanoptimalredued

model,whihleadstoAlgorithm3.

5.5.

H ₂

-OPTIMALBILINEARMODELORDERREDUCTION 79

Algorithm3BilinearIRKAforsystemswith

E 6= I

E

nonsingular(f.[12℄).

Input:

E, A, N k , B, C, A, ˆ N ˆ k , B, ˆ C ˆ

Output:

A ˆ ^opt , N ˆ _k ^opt , B ˆ ^opt , C ˆ ^opt

1: whilenotonvergeddo

A ˆ = SΛS ⁻¹

B ˜ ^T = S ⁻¹ B ˆ

C ˜ = ˆ CS N ˜ _k ^T = S ⁻¹ N ˆ k S

vec(V ) =

−I r ⊗ A − Λ ⊗ E − P m k=1 N ˜ k

T ⊗ N k

−1

( ˜ B ^T ⊗ B)vec(I m )

vec(W ) = −I r ⊗ A ^T − Λ ⊗ E ^T − P m

k=1 N ˜ k ⊗ N _k ^T −1

( ˜ C ^T ⊗ C ^T )vec(I p )

V = orth(V )

W = orth(W )

^%^orth^omputes^anorthonormalbasis 6:

A ˆ = (W ^T EV ) ⁻¹ W ^T AV

N ˆ k = (W ^T EV ) ⁻¹ W ^T N k V

B ˆ =

(W ^T EV ) ⁻¹ W ^T B

C ˆ = CV

7: endwhile

A ˆ ^opt = ˆ A

N ˆ _k ^opt = ˆ N k

B ˆ ^opt = ˆ B

C ˆ ^opt = ˆ C

TheonvergeneofAlgorithm3willbemeasuredintermsofthehange

intheeigenvaluesofthereduedsystem. Ineveryiterationthe hangein

theeigenvaluesbetweenthelasttwoiterationsisheked. Ifitissuiently

small,thealgorithmstopsandreturnsthenalreduedordermodel.

5.5.4.

H ₂

-optimalMORbyusingmethodsfromdierential geome-try. Wewillestablishanewresultforthederivationof

H ₂

-optimalbilinear reduedordermodels.Foreaseofpresentationwewillassume

E = I n

^. ^As^a

systemwith

E

^invertible^is^equivalent^to^the^system^multiplied^by

E ⁻¹

^,^this

ispossible. Inaddition,ageneralizationtosystemswith

E 6= I n

^should^be

possible.

5.5.4.1. The minimizationproblem. As in the preedingsetions we

aregoingto minimizethe

H ₂

-normoftheerrorsystem. However,we use adierent approah, whihwas originallygiven for linear systemsby Yan

andLam in1999 [69℄. Itis basedon minimizingthenorm on theStiefel

manifold. Thisapproahwasreentlytransferredto Grassmannmanifolds

by Xuand Zeng [68℄. We will nowdevelop the methods for the bilinear

ase. In ontrastto the methodsinthe previoussetions, thesemethods

diretlypreservetheBIBOstabilityofthemodel. Hene thereisno need

forstabilizationmethodsthat anbeusedforexampletostabilizeredued

ordermodelsobtainedbyBIRKAseeSetion6.2.

First,theobjetivefuntionfortheminimizationhastobefound. We

denethefollowingfuntion:

J (W, V ) = J (W ^T AV, W ^T N 1 V, . . . , W ^T N m V, W ^T B, CV ) := ||Σ ^err _bil || ² H ₂

= tr(

C − C ˆ

Im Dokument Bilinear H 2-optimal Model Order-Reduction with applications to thermal parametric systems (Seite 92-101)

U 1 U 2 Σ 1 0

0 Σ 2

V 1 T V 2 T

,

U 1 , V 1 ∈ R n×r

U 2 , V 2 ∈ R n×(n−r)

Σ 1 =

(ς 1 , . . . ς d )

Σ 2 =

(ς d+1 , . . . ς n )

E = I n

Σ bil

m 1 , . . . , m i ,

H (m i 1 ,...,m i ) (s 1 , . . . , s i )

H i (s 1 , . . . , s i )

H (m i 1 ,...,m i ) (s 1 , . . . , s i ) =(−1) i C(s i I n − A) −m i N [I m ⊗ (s i−1 I n − A) −m i −1 N ] . . .

· [I | m ⊗ · · · ⊗ {z I m }

i−2

⊗(s 2 I n − A) −m 2 N ]

· [I | m ⊗ · · · ⊗ {z I m }

i−1

⊗(s 1 I n − A) −m 1 B],

N = [N 1 . . . N m ]

s = 0

span(V (1) ) = K q (A −1 , A −1 B), span(V (i) ) =

[ m k=1

K q (A −1 , A −1 N k V (i−1) ),

span(V ) = span [ r i=1

span(V (i) )

! .

{ξ j } k j=1 , {ζ j } k j=1 ⊂ C

c T ∈ C p

b ∈ C m

b j = 1 j ⊗ b

N ⊕T = N 1 T , . . . , N m T

1 j

m j−1

H (l j 1 ,...,l j ) (ξ 1 , . . . , ξ j ) b j

c H (l j 1 ,...,l j ) (ζ j , . . . , ζ 1 )

j = 1, . . . , k

l 1 , . . . , l j = 1, . . . , q,

V

W

span(V (1) ) = K q {(ξ 1 I − A) −1 , (ξ 1 I − A) −1 Bb}, span(W (1) ) = K q {(ζ 1 I − A) −∗ , (ζ 1 I − A) −∗ C ∗ c ∗ },

span(V (j) ) = K q {(ξ j I − A) −1 , (ξ j I − A) −1 N (I m ⊗ V (j −1) )}

j = 2, . . . , k, span(W (j) ) = K q {(ζ j I − A) −∗ , (ζ j I − A) −∗ N ⊕T (I m ⊗ W (j−1) )}

j = 2, . . . , k, span(V ) = span{

[ k j=1

span(V (j) )},

span(W ) = span{

[ k j=1

span(W (j) )}.

W ˜ T = (W T V ) −1 W T

A ˆ = ˜ W T AV

N ˆ k = ˜ W T N k V

C ˆ = CV

B ˆ = ˜ W T B

H (l j 1 ,...,l j ) (ξ 1 , . . . , ξ j ) b j = ˆ H (l 1 ,...,l j ) (ξ 1 , . . . , ξ j ) b j

c H (l j 1 ,...,l j ) (ζ 1 , . . . , ζ j ) = c H ˆ (l 1 ,...,l j ) (ζ 1 , . . . , ζ j )

j = 1, . . . , k

l 1 , . . . , l k = 1, . . . , q

H 2

{ξ j } k j=1 , {ζ j } k j=1 ⊂ C

c T ∈ C p

b ∈ C m

H i (s 1 , . . . , s i )

H 2

H 2

H 2

H 2

H 2

H 2

E 6= I n

E

E

E 6= I n

E

C = C T

− C ˆ T

V ₁ ^T V ₂ ^T

U 1 , V 1 ∈ R ^n×r

U 2 , V 2 ∈ R ^n×(n−r)

H ^(m _i ¹ ^,...,m ⁱ ⁾ (s 1 , . . . , s i )

H ^(m _i ¹ ^,...,m ⁱ ⁾ (s 1 , . . . , s i ) =(−1) ⁱ C(s i I n − A) ^−m ⁱ N [I m ⊗ (s _i−1 I n − A) ^−m ⁱ ⁻¹ N ] . . .

⊗(s 2 I n − A) ^−m ² N ]

⊗(s 1 I n − A) ^−m ¹ B],

span(V ⁽¹⁾ ) = K q (A ⁻¹ , A ⁻¹ B), span(V ⁽ⁱ⁾ ) =

K q (A ⁻¹ , A ⁻¹ N k V ⁽ⁱ⁻¹⁾ ),

span(V ⁽ⁱ⁾ )

{ξ j } ^k _j=1 , {ζ j } ^k _j=1 ⊂ C

c ^T ∈ C ^p

b ∈ C ^m

b _j = 1 _j ⊗ b

N ^⊕T = N 1 ^T , . . . , N m ^T

1 _j

m ^j−1

H ^(l _j ¹ ^,...,l ^j ⁾ (ξ 1 , . . . , ξ j ) b _j

c H ^(l _j ¹ ^,...,l ^j ⁾ (ζ j , . . . , ζ 1 )

span(V ⁽¹⁾ ) = K q {(ξ 1 I − A) ⁻¹ , (ξ 1 I − A) ⁻¹ Bb}, span(W ⁽¹⁾ ) = K q {(ζ 1 I − A) ^−∗ , (ζ 1 I − A) ^−∗ C ^∗ c ^∗ },

span(V ^(j) ) = K q {(ξ j I − A) ⁻¹ , (ξ j I − A) ⁻¹ N (I m ⊗ V ^(j ⁻¹⁾ )}

j = 2, . . . , k, span(W ^(j) ) = K q {(ζ j I − A) ^−∗ , (ζ j I − A) ^−∗ N ^⊕T (I m ⊗ W ^(j−1) )}

span(V ^(j) )},

span(W ^(j) )}.

W ˜ ^T = (W ^T V ) ⁻¹ W ^T

A ˆ = ˜ W ^T AV

N ˆ k = ˜ W ^T N k V

B ˆ = ˜ W ^T B

H ^(l _j ¹ ^,...,l ^j ⁾ (ξ 1 , . . . , ξ j ) b _j = ˆ H ^(l ¹ ^,...,l ^j ⁾ (ξ 1 , . . . , ξ j ) b _j

c H ^(l _j ¹ ^,...,l ^j ⁾ (ζ 1 , . . . , ζ j ) = c H ˆ ^(l ¹ ^,...,l ^j ⁾ (ζ 1 , . . . , ζ j )

H ₂

{ξ j } ^k _j=1 , {ζ j } ^k _j=1 ⊂ C

c ^T ∈ C ^p

b ∈ C ^m

H ₂

H ₂

H ₂

H ₂

H ₂

H ₂

C = _C T

− C ˆ ^T

J = ||Σ ^err _bil || ² H ₂ = tr [ C − C ˆ ] P ^err _C T

− C ˆ ^T

= tr (P ^err C) .

E ˆ = −Y ₂₂ ⁻¹ Y 12 ^T EP 12 P ₂₂ ⁻¹ ,

A ˆ = −Y ₂₂ ⁻¹ Y ₁₂ ^T AP 12 P ₂₂ ⁻¹ ,

N ˆ k = −Y ₂₂ ⁻¹ Y ₁₂ ^T N k P 12 P ₂₂ ⁻¹ ,

B ˆ = −Y ₂₂ ⁻¹ Y ₁₂ ^T B,

C ˆ = CP 12 P ₂₂ ⁻¹ ,

H ₂

P ^err

Q ^err

W, V ∈ R ^n×r

E ˆ = W ^T EV, A ˆ = W ^T AV, N ˆ k = W ^T N k V, B ˆ = W ^T B, C ˆ = CV.

W := −Y 12 Y ₂₂ ⁻¹

V := P 12 P ₂₂ ⁻¹ .

Q ^err

E ⁻¹

E ˆ = −Y ₂₂ ⁻¹ Y ₁₂ ^T EP 12 P ₂₂ ⁻¹

= − EQ ˆ ⁻¹ ₂₂ E ˆ ^T E ˆ ^−T Q ^T ₁₂ E ⁻¹ EP 12 P ₂₂ ⁻¹ ,

⇒ I r = −Q ⁻¹ ₂₂ Q ^T ₁₂ P 12 P ₂₂ ⁻¹ , A ˆ = −Y ₂₂ ⁻¹ Y ₁₂ ^T AP 12 P ₂₂ ⁻¹

= − EQ ˆ ⁻¹ ₂₂ E ˆ ^T E ˆ ^−T Q ^T ₁₂ E ⁻¹ AP 12 P ₂₂ ⁻¹ ,

⇒ E ˆ ⁻¹ A ˆ = −Q ⁻¹ ₂₂ Q ^T ₁₂ E ⁻¹ AP 12 P ₂₂ ⁻¹ ,

H ₂

H ₂

E ˆ ⁻¹

H ₂

A ˆ = SΛS ⁻¹ , B ˜ ^T = S ⁻¹ B, ˆ C ˜ = ˆ CS, N ˜ _k ^T = S ⁻¹ ( ˆ N) k S,

J = ||Σ ^err bil || ² H ₂

= vec(I 2p ) ^T ([ C − C ˜ ] ⊗ [ C − C ˜ ])

× − _A

⊗ _E

− _E

⊗ _A

h _N

k N ˜ _k ^T

⊗ h _N

k N ˜ _k ^T

× _B