1
Contents
1
ntroducti on
2
Spectral
3
An
3
ro ectors and the Separati on of
pper Bound on
k k
f or
ondi ti oni ng of
i genval ues
ondi ti oni ng of
i ght
l obal
o
atri ces
rror Bounds
4. 1 Condi t i oni ng of Si mpl e Ei genval ues : : : : : : : : : : : : : : : : : : : : : : :
4. 2 Condi t i oni ng of Cl us t er ed Ei genval ues : : : : : : : : : : : : : : : : : : : :
4. 3 St abi l i t y : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
i genvectors and
i ght
nvari ant Subspaces
9
9
10
1
5. 1 Angl es Between Subs paces : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5. 2 Condi t i oni ng of Ri ght Ei genvect or s and Ri ght I nvari ant Subspaces: : : : :
5. 3 ( Bl ock) di agonal i zi ng a Matri x wi th a Si mi l ari ty
: : :: : : : : : : : : : : : :
Sum
m
ary:
erturbati on
abl e
13
STRSNA
sti m
ati ng the ondi ti on of ndi vi dual
7. 1 Usage : : : : : : : : : : : : : : : : : : : : : : :
7. 2 Exampl e : : : : : : : : : : : : : : : : : : : : : :
7. 3 Out l i ne of t he Al gori thm: : : : : : : : : : : :
::::::::::::::::
::::::::::::::::
::::::::::::::::
STRSEN sti mati ng the ondi ti on of a l uster of
8. 1 Usage : : : : : : : : : : : : : : : : : : : : : : :
8. 2 Exampl e : : : : : : : : : : : : : : : : : : : : : :
8. 3 Out l i ne of t he Al gori thm: : : : : : : : : : : :
::::::::::::::::
::::::::::::::::
::::::::::::::::
A Sol uti on of the Syl vester
B S appi ng
Li st of
i genpai rs
1
i genval ues
2
uati on
2
i agonal Bl oc s
LAPACK
10
11
12
outi nes f or the
2
onsym
m
etri c
2
i genprobl em
3
14
17
19
21
23
24
ntro ucti on
W
e r evi ewt he t heory of condi t i on number s f or t he nonsymmetri c ei genprobl em, and descri be
al gor i thms f or es t i mati ng them. W
e provi de a manual f or the LAPACK subrouti nes STRSNA
and STRSEN, whi ch comput e t hese condi ti on numbers f or matri ces i n Schur canoni cal f orm.
W
e ass ume the r eader i s f ami l i ar wi th the basi c theory of the nonsymmetri c ei genprobl em:
ei genval ues, ri ght and l ef t ei genvect or s , mul ti pl e ei genval ues and ri ght and l ef t i nvar i ant
s ubs paces.
The condi t i on number of a probl emmeas ures the sensi ti vi ty of the sol uti on to smal l
changes i n the i nput . W
e cal l the probl emi l l - condi ti oned i f i ts condi ti on number i s l arge,
and i l l - pos ed i f i t s condi t i on number i s i nni te. W
e may use condi ti on number s t o bound
err or s i n computed sol ut i ons of numeri cal probl ems.
W
e i l l us t r at e t hi s wi t h a s i mpl e exampl e. I t i s wel l known t hat the condi ti on number f or
s ol vi ng a s ys t emof l i near equati ons i s ( AA
) k1 kk 01
A k , wher e k 1 k i s any matri x oper at or
nor m( we wi l l be more speci c about norms l ater). Suppose that l i near s ystem Ax = i s
s ol ved vi a Gaus s i an el i mi nati on wi th parti al pi voti ng, or some other stabl e schem
xe. Let be the computed sol ut i on. Then one may bound the error by:
kx0 x k = ( ma
kxk
p s ) 1 (A)
wher e ma
p s i s t he machi ne pr eci s i on. The si ze of the constant i mpl i ci t i n the ( 1 )
not at i on depends on the si ze of the matri x, pi vot growth, etc.
Condi t i on numbers may be expensi ve to compute exactl y. For exampl e, comput i ng
( A ) f or even the s i mpl es t mat ri x norms i s t hree ti mes as expensi ve as sol vi ng Ax = i n
t he r st pl ace. Theref ore, one usual l y uses an i nexpensi ve esti mate i n pl ace of the exact
( A ). For exampl e, a method f or esti mati ng ( A ) i s i ncl uded i n LI NPACK, whi ch costs
j us t ( n2 ) ext r a beyond the ( n3) cost of sol vi ng Ax = . The pri ce one pays f or usi ng an
est i mate i s occasi onal (but hopef ul l y rare) mi sesti mates of ( A ). Year s of experi ence wi th
t he LI NPACKes t i mat or at t es t t o i ts rel i abi l i ty, al t hough exampl es do exi st wher e i t can
under esti mate ( A ) badl y.
The codes we di s cus s f or t he nonsymmetri c ei genprobl emwi l l al so use such condi ti on
3) an ( n
es ti mators. Here, the savi ngs wi l l be even gr eat er t han f or l i near equati on sol vi ng:
est i mator us i ng ( n ) works pace i n pl ace of an ) (exact
n
sol uti on usi ng ()nworkspace
i n s ome cases.
Our condi t i on es t i mators wi l l compute two quanti ti es, the r eci procal of a condi ti on
number f or an ei genval ue (or cl uster of ei genval ues), and t he r eci procal of a condi ti on
number f or an ei genvector (or i nvari ant s ubspace) . W
e compute reci pr ocal s of condi ti on
number s to avoi d overow an i nni te or over owed condi ti on number i s i ndi cated by a
zer o r eci pr ocal . By combi ni ng thes e t wo val ues i n s i mpl e al gebrai c f ormul as, a great deal
of detai l ed i nf ormati on about the ei genprobl emcan be obtai ned. Thi s report wi l l descri be
bot h t hese basi c condi t i on number s and these f ormul as.
Our condi t i on numbers wi l l meas ure the changes i n the ei genval ues , r i ght ei genvectors,
means of cl us t er s of ei genval ues, and r i ght i nvari ant s ubspaces of a matri x A when a perturbat i on i s added to i t our bounds wi l l be f unct i ons (usual l y mul ti pl es) of k k . Thi s may
be us ed t o es t i mate the er r or i n sol uti ons computed by LAPACK routi nes because they ar e
3
backward st abl e , i . e. they compute the exact ei gendecomposi ti on of a matri x A + where
A i s t he i nput mat r i x, and k
k = ( ma p s ) k A k . We meas ure changes i n ei genvectors
and i nvari ant subs paces by thei r change i n angl e we di scuss the angl e bet ween s ubspaces
i n mor e detai l i n secti on 5. Our condi ti on number s yi el d both asympt ot i c bounds , whi ch are
accur at e onl y when the normk k i s smal l , and gl obal bounds , whi ch work f or al l k k up t o
a cer tai n upper bound, whose si ze depends on t he probl emand may be l arge or smal l . W
e
s howhowto obt ai n t hese upper bounds on k k as wel l .
W
e i l l us t r at e t he reason f or provi di ng such a vari ety of bounds wi th an exampl e. Let
A be 11 by 11 of t he f ol l owi ng f orm:
0
A =
1
0
..
... ...
.
.
...
1 ..
0 0
: 5
Her e, bl ank entri es are al so zero. Thus,i sA a bl ock di agonal matri x wi th a 10 by 10 bl ock
at t he upper l ef t and a 1 by 1 bl ock at the l ower r i ght. When =0, the upper l ef t bl ock i s
a 10 by 10 Jordan bl ock wi t h a s i ngl e mul ti pl e ei genval ue at 0 and a si ngl e ri ght ei genvector
= [ 1; 0; .; . 0].
. Such a mat r i x i s cal l ed def ect i ve . For smal l nonzer o the ei genval ues
become di s t i nct numbers al l wi t h absol ute val ue1, and ei genvect or s whi ch have rot at ed
away f rom by about 1 r adi ans. When = 10 01 , 1 = : 1, a much l ar ger change. I n
t hi s case we cal l the ei genval ue at 0 and i t s as s oci at ed ei genvector i l l - posed, because thei r
s ens i t i vi t y i s not pr opor t i onal to the normof the perturbati on , but a root of .
The pr act i cal s ol ut i on t o t hi s probl em i s to consi der t hi s matri x as havi ng a cl uster
of 10 ei genval ues near zero wi t h a si ngl e i nvari ant subspace whi ch i s spanned by al l thei r
ei genvectors, as wel l as a si ngl e ei genval ue near . 5 wi th i ts ei genvect or . The mean of thi s
cl us ter of 10 ei genval ues wi l l be much l ess sensi ti ve to smal l perturbati ons than t he i ndi vi dual ei genval ues (i n f act i t wi l l be i ndependent of i n thi s exampl e). For smal l enough k k ,
our asympt ot i c er r or bounds wi l l showthat the mean of t he cl uster of 10 ei genval ues near
0 of A + i s bounded by k k ( s ee Bound 4 bel ow) i . e. the mean i s ver y wel l - condi ti oned.
Si mi l arl y, the ei genval ue near . 5 can al so onl y change by k k f or s mal l enough k k (Bound
2) . The i nvari ant subs paces wi l l al so be much l ess sensi ti ve t han the i ndi vi dual ei genvectors.
I n t hi s exampl e, t he r i ght i nvari ant s ubspace bel ongi ng t o t he cl uster of 10 ei genval ues near
0 i s s panned by t he r s t 10 col umns of the 11 by 11 i denti ty matri x i ndependent of mor e
general l y our bounds wi l l s ay that f or smal l k k the r i ght i nvari ant s ubspace can rotate
by at most 2731k k r adi ans (Bound 6) . The ei genvect or bel ongi ng to the ei genval ue . 5 i s
equal l y i ns ens i t i ve i n thi s exampl e.
Thi s i l l us t r at es our asympt oti c error bounds, val i d f or suci entl y smal l k k . I n contrast,
our gl obal bounds gi ve bounds val i d f or al l k k up to an upper bound whi ch we al s o
est i mate. The mat r i x A i l l us t r ates the source of thes e upper bounds on k k . Suppos e we
make =2 01
: 001 t hen one of the ei genval ues ori gi nal l y at 0 nowequal s . 5, t he same
as t he ei genval ue i n t he l ower r i ght corner . Thus, we can no l onger say that thi s matri x
has a cl us t er of ei genval ues near 0 and one near . 5, and so we can no l onger tal k about the
s ensi ti vi t y of t he mean of the cl uster. W
e can al so no l onger i dent i f y a uni que ei genvector
4
as s oci ated wi t h an ei genval ue near . 5 the ei genvect or s have become i l l - posed. I ndeed, wi th
addi ti onal arbi t r ar i l y s mal l perturbati ons the two ei genvectors f or the ei genval ues at . 5 can
be made to rotate arbi t r ar i l y wi thi n a two di mensi onal subspace, or one of themcan even
01 2can we hope to
di sappear. Thus , onl y i f we bound k k to be s ome val ue l es s t han
have er ror bounds . For thi s exampl e, the upper bound computed by our s of t ware wi l l be
0 ( Bound 1) . For k k < 2 1 010, our upper bound on t he change i n
appr oxi matel y 2 1 10
t he mean of the ei genval ue cl us t er wi l l be 2k k (Bound 5), and our bound on howmuch the
r i ght i nvari ant s ubs pace can r otate wi l l be arctan
(2731k k (1 0 5462k k )) radi ans (Bound
7) , both cl os e t o t he as ympt ot i c bounds.
I n t hi s exampl e, i t i s eas y to i denti f y the cl usters by i nspect i .on Thi
of sA i s not
al ways the cas e i n practi ce, when t he user i s conf r onted wi th a matri x whose ei genval ues
f or ma cl oud rather than wel l separated cl usters. Unf ortunatel y, ther e i s as yet no rel i abl e,
aut omated procedur e f or cl us t er i ng ei genval ues see
, 9,[ 828] f or di scussi on. Our s of twar e
mer el y pr ovi des the tool s f or eval uati ng a parti cul ar cl usteri ng. Agood cl uster wi l l have a
much l es s s ens i t i ve mean and i nvari ant subspace than any s ubcl uster, and must be made
par t of a much l ar ger ( or t r i vi al ) cl uster bef or e i t becomes s i gni cant l y l es s s ensi ti ve. The
10 zer o ei genval ues of As at i s f y thi s cri teri on.
There i s a very l arge l i terature on perturbati on theor y f or t he ei genprobl em. See
[ 3, 8, 9, 11, 15, 16, 22, 24, 25, 27, 28] f or vari ous theoreti cal bounds. Chan, Fel dman
and Par l et t [ ]6 pr ovi ded a For t r an r outi ne t o compute the condi ti on number of si mpl e ei genval ue i n conj uncti on wi t h EI SPACK r outi nes ORTHES and HQR, but i t does not pr ovi de
any i nf ormati on about condi t i oni ng f or ei genvect or s and subspaces . Ruhe[
] suggested
20
usi ng t he Gol ub- Rei ns ch SVDal gori thmt o cal cul ate the condi ti on number f or ei genvectors,
but thi s r equi r es 3()n ops per ei genval ue- ei genvect or pai r, whi ch i s too expensi ve. Van
Loan[ 26] devel oped an eci ent al gori thmf or esti mati ng condi ti on number s of al l ei genval ueei genvector pai rs of a Hes s enberg matri x. I t onl y costs2 ) ops
( n per ei genpai r, assumi ng
t hat t he ei genval ues are known.
W
e have devel oped newal gori thms, whi ch assume t he matri x has been r educed t o Schur
canoni cal f or m(real or compl ex). Reduct i on t o Schur canoni cal f ormi s done by LAPACK
s ubr out i nes SGEHRD and SHSEQR i n the r eal cas e, and CGEHRD and CHSEQR i n the compl ex
cas e. Si nce thi s r educti on i s done vi a orthogonal (or uni tary) si mi l ari ti es, the condi ti on
number s are i dent i cal t o t hose of the ori gi nal matri x. As we wi l l see, starti ng wi th the
matr i x i n Schur f or ms i mpl i es many of t he al gori thms and l et s us use exi sti ng condi ti on
est i mati on sof tware f or (quas i ) tri angul ar matri ces
, 18[, 14
19] .
The r es t of t hi s r eport i s organi zed as f ol l ows. Secti on 2 di scusses spectral proj ectors and
t he s eparati on of mat r i ces , quanti ti es on whi ch l ater bounds ar e based. Sect i on 3 di scusses
t he upper bound on k k f or our gl obal error bounds. Sect i on 4 di scusses asymptoti c and
gl obal bounds f or ei genval ues and means of cl usters of ei genval ues . I n s ect i on 5, we rst
dene t he angl e between two s ubs paces , t he quanti ty bounded by our error bounds. Second,
we present as ympt ot i c and gl obal perturbati on bounds f or both ri ght ei genvect or s and ri ght
i nvar i ant subs paces. Thi r d, we di scuss (bl ock)di agonal i zi ng a matri x by a s i mi l ari ty. The
r es ul ts i n s ect i ons 2 t hr ough 5 are stated wi thout proof ref erences t o proof s i n the l i terature
ar e gi ven. At abul ar s ummary of al l bounds i s gi ven i n secti on 6. Sect i ons 7 and 8 descri be
t he us age of the LAPACK r out i nes STRSNA and STRSEN f or esti mati ng the des i r ed condi ti on
number s (actual l y t hei r reci pr ocal s). STRSNA computes the r eci procal condi ti on numbers
5
of user- s peci ed ei genval ues and/or ei genvect or s of t he i nput matri x. STRSEN computes
t he r eci pr ocal condi t i on number s of t he mean and/or i nvari ant subspace of a si ngl e users peci ed cl us t er of ei genval ues. Two exampl es are provi ded to showhowto use thes e codes.
Outl i nes of the al gori thms are al so gi ven.
The r s t t wo appendi ces descr i be detai l s of the sol uti on of the Syl vester matri x equat i on
and s wappi ng di agonal bl ocks of a quasi tri angul ar matri x. The thi rd appendi x l i sts the
names and basi c f uncti ons of LAPACK routi nes needed f or t he nonsymmetri c ei genval ue
probl em.
W
e end wi t h s ome notati on we wi l l need l ater. Capi tal l etters are used to denot e
mat ri ces , t he cor r es pondi ng l owercase l etter wi th the subscri pt i ref erri ng to the ( i ; )
component (e. g. , a i s t he ( i ; ) component of A ). Asubmatri x of a matri x A i s wri tten as
A . Vectors are al so denoted by l owercase l etters and wi l l be cl earl y i ndi cated i n the text.
Lower case Gr eek l et t er s wi l l denote scal ars.
k x 1k; k x2 kand k x k denote the one- norm, the Eucl i dean norm, and the i nni ty- nor m,
r es pecti vel y, of the n - vector x :
ij
ij
n
k x 1k =
=1
n
k x2 k=
x ;
i
=1
i
x 2
12
=
i
ax x :
k x k= m
1
;
i
i
n
i
k T k1; k T2;k k T k; k T kdenote the matri x norms:
k T k1 =max
ij
j
kT k
i
=max
i
k T2 k=sup kkTxxk2k;
2
=
;
ij
k T k=
;
j
n
=1
ij
2
12
=
:
i; j
Not e t hat k k2 and k k are i nvari ant wi th respect t o uni tary transf ormati on.
W
e wi l l t hr oughout l et 2 denote k k2, and denote k k , the norms of our pert urbat i on mat r i x.
The condi t i on number of T i s ( T ) =k 2Tk kT01 k2 . As ubspace spanned by the col umns
of matr i x A i s denoted as ( A ) (the range of matri x A ).
( A ) denotes the s et of al l
ei genval ues of mat r i x A . A B denotes the Kronecker product of t wo matri ces: A B =
(a B).
The Schur matri x (or Schur f orm) of a real matri x i s an orthogonal l y si mi l ar upper
quas i - tri angul ar mat r i x whose 2 by 2 di agonal bl ocks (i f any exi st) are of the f or m
ij
Such a bl ock has compl ex conj ugate ei genval ues wher2e=0 . The Schur f ormof
a compl ex mat r i x i s a uni t ar i l y si mi l ar upper tri angul ar matri x.
pectr l
ro ectors
n
the
ep r ti on of
o
tri ces
To expl ai n t he bounds i n l ater s ect i ons, we need t o i ntroduce t wo quanti ti es, the spect ral
pro ect or [ 22
, 15] , and t he separat i on of t wo mat ri ces A and B , ( sep
A; B ) [ ]22
.
6
Suppose our cl us t er cont ai ns m 1 ei genval ues , counti ng mul ti pl i ci ti es. Assume the n
by n mat r i x A i s i n Schur canoni cal f orm
12
A = A011 A
A22
(2: 1)
wher e t he ei genval ues of the m by m matri x11Aare exactl y those i n whi ch we are i nterested.
I n pr acti ce, i f the ei genval ues on t he di agonal of A are i n the wrong or der , t hey ar e s or t ed
t o put the desi red ones i n the upper l ef t cor ner as s hown by usi ng the subrouti ne STREXC
i n Appendi x B.
W
e dene t he spect ral pro ect or , or si mpl y pr oj ect or bel ongi ng to the ei genval ues of
A11 as
=
wher e
I
0
(2: 2)
0
s at i s es the sys t emof l i near equati ons
A11
0 A22 =A 12
(2: 3)
Equati on ( 2. 3) i s cal l ed a Syl vester equati on. Gi ven t he Schur canoni cal f orm(2. 1), we
s ol ve t he Syl vester (2. 3) f or usi ng s ubrouti ne STRSYL i n Appendi x A.
has several i mpor t ant pr operti es. Fi rst, the s pace spanned by i ts col umns i s the same
as t he r i ght i nvari ant s ubs pace of A bel ongi ng 11
to. ASecond, t he s pace spanned by i ts
r ows i s t he s ame as the l ef t i nvari ant s ubspace of A bel ongi ng11t. oThus,
A
des cr i bes the
s paces spanned by both the l ef t and r i ght ei genvect or s bel ongi ng11to
. Thi
A rd, i t s nor m
k k2 =(1 +k k 22)1 2 pl ays an i mportant rol e i n our error bounds, as we wi l l see.
I n practi ce, we do not us e k 2 kf or m 1 si nce t hi s i s expensi ve to compute, but r at her
t he cheaper overesti mate
(2: 4)
k k (1 +k k2 )1 2
=
=
The separat i on s ep(A11; A22) of the matri ces 1A1 and A 22 i s dened as t he smal l est
s i ngul ar val ue of the l i near map i n (2. 3) whi ch takes X 11toX A0 XA22, i . e.
s ep( A11; A
22) =mi n
=
k A11X 0 XA22k
k Xk
(2: 5)
Thi s f or mul at i on l et s us es t i mate sep
usi ng the condi ti on esti mator SLACON ,[ 14
18, 19] ,
whi ch es t i mates the nor mof a l i near operator k1 Tgikven t he abi l i ty to compute T x and
T qui ckl y f or ar bi t r ar y x and . I n our case, mul ti pl yi ng an ar bi trary vect or by T means
s ol vi ng t he Syl ves t er equati on (2. 3) wi th an arbi trary ri ght hand s i de, and mul ti pl yi ng by
T means sol vi ng the s ame equati on wi th A
11 repl aced by A11 and A 22 repl aced by A22.
3
Sol vi ng ei t her equati on costs at most )( noperati ons, or as f ewas 2()ni f m n .
Another f ormul ati on whi ch i n pri nci pl e per mi ts an exact eval uati on of
( A11
sep
; A22) i s
s ep( A11; A22) =
(I 0
n
A11 0 A22 I )
(2: 6)
wher e
i s t he s mal l es t s i ngul ar val ue. Thi s met hod i s general l y i mpracti cal , however,
becaus e t he mat r i x whose smal l est si ngul ar val ue we need i s m( n 0 m) di mensi onal , whi ch
7
can be as l arge as 2n 4. Thus we woul d requi re as much as ( n ) extra workspace and
( n ) operati ons , much more than the esti mati on met hod of the l ast paragraph.
sep( A11; A22) measur es t he \separati on" of the s pectrumof11Aand A 22 i n the f ol l owi ng
s ens e. I t i s zer o i f and onl y 11i f and
A A 22 have a common ei genval ue, and smal l i f ther e i s
a s mal l per t ur bat i on of ei t her one that makes t hemhave a common ei genval ue. I11f and
A
A22 ar e both normal mat r i ces , t hen s ep
( A11; A22) i s j ust the mi ni mumdi stance bet ween an
ei genval ue of 1A1 and an ei genval ue of A
22.
STRSNA comput es 1 k k2 ( whi ch i s al ways 1, avoi di ng the possi bi l i ty of over ow) and
s ep f or us er - s el ect ed i ndi vi dual ei genval ues (11i . ie.s 1Aby 1) . STRSEN computes 1 k k
and sep f or a s i ngl e us er - s peci ed cl uster wi th m 1 ei genval ues .
n
pper
oun
on
f or
l ob l Error
oun s
W
e di scus s t he upper bound on k k whi ch l i mi ts the appl i cabi l i ty of our gl obal bounds
i n the next t wo secti ons . As s t ated i n the i ntroduct i on, thi s upper bound occurs because i f
k k i s l ar ge enough that the ei genval ue bei ng consi der ed ( or one of the ei genval ues i n the
cl us ter bei ng cons i dered) moves and coal es ces wi th another ei genval ue ( outsi de t he cl uster),
t hen we can no l onger uni quel y i denti f y the cl uster f or whi ch we want bounds. Thus, i n
t hi s s ect i on we pr es ent l ower bounds on t he smal l est k such
k t hat A + has a mul ti pl e
ei genval ue (or a mul t i pl e ei genval ue i nvol vi ng at l east one ei genval ue wi thi n the or i gi nal
cl us ter and one out s i de).
Bound 1:
[ 22 , Theorem4. 14] Let A ,
2 5 Then as l ong as
and sep
( A11; A22) be de ned as i n 2
k k < sep(4 1A11k; Ak22)
2
, 2 2 and
(3: 7)
t he ei genval ues i n t he cl ust er bel ongi ng11t owiAl l remai n di s oi nt f romt he ei genval ues out si de t he cl ust er In part i cul ar, t he gl obal error bounds of sect i ons and 5 wi l l be guarant eed
e may repl ace2kby kk k as de ned i n 2 t o get
val i d onl y f or sat i sf yi ng 3
a sl i ght l y smal l er upper bound
Bound 1 can be qui t e cons er vati ve, gr eat l y under es t i mati ng the smal l est perturbat i on
needed to make ei genval ues coal es ce. However, i t i s near l y exact i n some cases ( e. g. f or
2 by 2 mat r i ces and normal mat r i ces), and a good esti mate i n many ot her s see, [9]8 f or
di scus si on.
1 k k ( or 1 k k2 i f m =1) and sep( A11; A22) are computed by STRSNA and STRSEN as
des cr i bed i n secti on 2.
Con i ti oni ng of Ei gen
l ues
I n thi s s ect i on, we r evi ew how to meas ure the s ensi ti vi ty of both si mpl e ei genval ues and
cl us ters of ei genval ues.
8
.
Let be a si mpl e ei genval ue of the n by n matri x A , wi th uni t ri ght ei genvect or x and uni t
, and k x 2k = k k2 =1. Let be
l ef t ei genvector . I n other words Ax = xA, =
t he spect r al proj ector f or one may wri te =( x )1 ( 1 x ). Note that k 2 =1
k
x ,
t he s ecant of the angl e between x and .
Let
be a per t ur bati on of A , and2 = k k2. Let be t he perturbed ei genval ue of
A+ .
Bound 2:
[ 27 , p. 68]
0
k2 + ( 22)
2 k
The ( 22) t er mi ndi cat es t hi s i s an asymptoti c bound, appl i cabl e onl y f or suci entl y
s mal l 2 . Thi s bound i s at t ai nabl e, i n the sense that 2f or
suci entl y smal l , ther e exi sts
2
an s uch that 0 = 2 k k2 + ( 2 ).
There i s al s o a gl obal ver s i on of thi s bound:
[ 3 ] Suppose A has al l si mpl e ei genval ues Let
be t he correspondi ng spect ral
pro ect ors Then any ei genval ue of A + must l i e i n one of t he di sks
Bound 3:
i
0
:
n 2 k
i
i
i
k2
There i s no l i mi t on t he si ze of
2 i n Bound 3. Note that the s i zes of t he di sks are j ust
n t i mes l arger than i n Bound 1, wher e2 must be smal l . Bound 3 i s an stronger versi on of
what i s of ten cal l ed the Bauer- Fi ke Theor em, al though Bauer and Fi ke proved t hi s stronger
ver s i on as wel l . I n t he weaker Bauer - Fi ke Theor em, al l of the di sks have the same radi us,
appr oxi matel y equal t o t he l argest radi us max
n 2 k k2 i n Bound 3. Note that Bound 3
i s onl y us ef ul when al l the r adi 2i k n k2 are of modes t s i ze, s i nce i f one or more di sks i s
s o l arge t hat i t i nt er s ect s al l the other di sks, there i s l i ttl e i nf or mati on about l ocat i ons of
i ndi vi dual ei genval ues we onl y knowthey l i e i n the uni on of al l the di sks.
The s ubr out i ne STRSNA can compute 1 k k2 f or a user- speci ed s ubset of the ei genval ues
of A .
i
i
i
.
I t i s easi est to thi nk of A i n Schur f orm(2. 1), wi th the ei genval 11
uesbei
ofngA t he cl uster
of i nt er es t . W
e ar e i nt er es t ed i n boundi ng the perturbati on i n the aver age of the ei genval ues
of t he cl us t er , whi ch may be wr i ttenAtr
11 m, t he trace of 1A1 di vi ded by m. Let be
a pert ur bati on of A , and2 = k k2. Let = tr A11 m be t he mean of t he unperturbed
ei genval ues, and be the mean of t he perturbed ei genval ues .
Bound :
[ 15 , s ec. I I . 2. 2]
0
2 k
9
k2 + ( 22)
e may subst i t ut e
k kof
equat i on 2
f or
k 2 kt o get
a sl i ght l y weaker bound
The ( 22 ) i ndi cat es t hi s i s an asymptoti c bound, appl i cabl e onl y f or suci entl y smal l
2 . Thi s bound i s near l y at t ai nabl e, i n the sense that2 fsu
or ci
entl y smal l , ther e exi sts
an such that 0 k k22 m + ( 22). When m =1, i t i s of course i denti cal to the
bound i n t he pr evi ous s ubsecti on.
k sf yi ng Bound 1 of s ecti on
Our gl obal bound on 0 wi l l onl y be val i d f or k sati
2:
Bound :
[ 22 , page 748] Suppose
k k sat i s es ound
0
22 k k2
Then
Thus, t he gl obal bound i s ust t wi ce as l arge as t he asympt ot i c bound Agai n we may subst i t ut e k kof equat i on 2 f or k 2 kt o get a sl i ght l y weaker bound
STRSNA comput es 1 k k2 f or a user- speci ed s et of i ndi vi dual ei genval ues . STRSEN can
compute 1 k k f or a si ngl e us er- speci ed cl uster of m 1 ei genval ues .
.
When t he ei genval ues of a f ul l matri x A have been f ound f romi ts computed Schur f or
A,m
t he comput ed s wi l l be t hose appropri ate to
A. Thes e s can di er s ubstanti al l y f romthe
t rue s . I ndeed, when A i s def ect iAve,
wi l l usual l y not be, and hence zer o s wi l l become
nonzer o s. The r ever s e s i t uati on coul d occur though thi s i s much l es s probabl e. Si nce some
of t he smay not even be t he correct order of magni tude, i t mi ght be f el t t hat our heavy
r el i ance on themi s unj us t i ed. Si nce our al gori thms f or computi ng the Schur f ormand s ( )
ar e backwar d s t abl e, s ( ) i s the cor r ect val ue f or a matri x A + very cl ose to the or i gi nal
matr i x A . Thi s j us t i es thei r use. The same comments appl y to the computati on and use
of s epdescri bed i n the next secti on.
Con i ti oni ng of R
i ght Ei gen ectors
n
R
i ght
n
ri nt
ubsp ces
I n thi s s ect i on, we r evi ew how to meas ure the sensi ti vi ty of ei genvect or s and i nvar i ant
s ubs paces. W
e begi n by deni ng of the angl e bet ween t wo s ubspaces , and t hen use i t to
des cr i be t he condi t i oni ng of ei genvect or s and i nvari ant subspaces .
.
Let ( x ; ) denote the acut e angl e bet ween t wo nonzer o n - vect or s x and . W
e may wri te
cos ( x ; ) = x ( k x 2kk k2) . W
e wi sh to general i ze thi s to the ( maxi mum) angl e bet ween
t wo m 1 di mens i onal subs paces, whi ch we denote and . One way to dene thi s angl e
i s as
( ; ) = max
mi n ( x ; ) (= max mi n
( x ; ))
(5: 8)
x
x =0
=0
=0
10
x
x =0
A mor e comput at i onal deni t i on of
( ; ) i s the f ol l owi ng. Suppose
i s spanned
by t he col umns of t he n by m orthonormal matri x X , and s i mi l arl y i s spanned by the
col umns of t he n by m or t honormal matri x . Then
( ;
) =ar ccos
(
X)
(5: 9)
Our bounds of t he next t wo secti ons wi l l bound ( ; ) wher e
i nvar i ant subs pace, and i s a perturbed i nvari ant subspace.
W
e may al so dene t he mi ni mumangl e bet ween and as
( ;
) = mi n
(x ;
mi n
x
x =0
) (= mi n
=0
=0
mi n
x
x =0
i s an unperturbed
(x ;
))
Thi s may be reexpr es s ed comput ati onal l y as
( ;
) =ar ccos
(
X)
The norms of t he spect r al proj ect or s k2 i kntroduced i n s ect i on 2 have a si mpl e i nterpr et at i on i n t er ms of angl es bet ween s ubspaces . Let
be the s pectral pr oj ect or f or the
ei genval ue cl us t er wi t h r i ght i nvari ant s ubspace and l ef t i nvari ant s ubspace .beLet
t he compl ement ar y r i ght i nvari ant s ubspace (the s ubspace f or the ot her ei genval ues) and
be t he compl ement ar y l ef t i nvari ant subspace. Then
k k2 = cs c ( ; ) =csc
k k2 = sec ( ; ) =s ec
as k 2kgrows and the cl uster becomes
( ; )
( ; )
more i l l - condi ti oned, t he mi ni mum
I n ot her words ,
angl e between compl ement ar y r i ght (or compl ementary l ef t) subspaces s hri nks. Al so, the
maxi mumangl e between correspondi ng l ef t and ri gth i nvari ant subspaces grows.
.
W
e ass ume A i s i n Schur canoni cal f orm(2. 1), wi th the ei genval ues 11ofbei
A ng t he cl uster
whos e r i ght i nvari ant subs pace we are i nterested i n. Let be a perturbati on of A , and
=k k . Let be the per t ur bed r i ght i nvari ant subspace of A + .
Bound :
[ 8 , Lemma 7. 8]
( ;
)
2
+ ( 2 )
sep( A11; A22)
The ( 2 ) i ndi cat es t hat thi s i s an asymptoti c bound, appl i cabl e onl y f or suci entl y
smal l . I t i s near l y at t ai nabl e f or suci entl y smal
. l Bound :
[ 8 , Lemma 7. 8] Suppose
( ;
)
k k
sat i s es ound
arctan
2
sep( A11; A22)
11
Then
0 4k k2
k k2 may be repl aced by k kof
equat i on 2
t o obt ai n a sl i ght l y weaker bound
Bounds 6 and 7 i mpl y t hat sep i s the r eci procal of the condi ti on number f or ei genvect or s and i nvari ant subs paces. Routi nes STRSNA and STRSEN compute sep
f or i ndi vi dual
ei genvectors and a gi ven i nvari ant subspace, respect i vel y.
.
Occas i onal l y one wi s hes to nd a matri x whi ch di agonal i zes A01: A = , wher e i s
a di agonal mat r i x wi t h t he ei genval ues on t he di agonal . Thi s may be usef ul f or computi ng
f unct i ons of mat r i ces . For exampl e, to exponenti ate a matri x one may use the i denti ty
exp( A ) = exp( ) 01 = i a ( 1 ; . .; . ) 01 . The accuracy of such an al gori t hm
depends on t he condi t i on number ( ) of . The col umns of must be ei genvect or s of A ,
but t hei r norms are arbi t r ar y we woul d l i ke to choose thes e norms to mi ni mi ze ( ). The
next bound gi ves a nearl y opt i mal choi ce of the norms of the col umns of , and bounds the
r es ul ti ng ( ) .
[ 7 ] Suppose A has di st i nct ei genval ueswi t h correspondi ng ri ght ei genvect ors
Let = [ 1; . .; . ] be t he mat ri x of
, where we assume k k2 = 1, and pro ect ors
t hese ei genvect ors Let =[ 1 1; . .; . ] be anot her mat ri x where t he are arbi t rary
Bound :
i
i
i
i
n
nonzero scal ars Then
max
i
Al so
max
i
k k2
i
n
n
i
k k2
( )
i
( )
n
1
maxk
i
i
k2
In ot her words choosi ng t he col umns of t o have norm nearl y mi ni mi zes ( ) over al l
mat ri ces whose col umns are ei genvect ors
Avari ati on on di agonal i zat i on i s bl ock- di agonal i zati on, wher e we ask onl y01that
A =
B be bl ock di agonal , where the di agonal bl ocks of
B B have speci ed ei genval ues ( whi ch
ar e al l di s j oi nt subs et s of t he ei genval ues of A ). Suppose
i s l ocated
B
i n rows and col umns
t hrough of B . Then col umns through of
must span the ri ght i nvari ant subspace
of A cor r es pondi ng t o t he ei genval ues of B . Let
denote these col umns of . Just as we
coul d choos e t he normof each col umn of when we wanted to di agonal i ze A , her e we have
ii
ii
i
t he f r eedomt o choos e t o be any basi s of the ri ght i nvari ant subspace we l i ke. Agai n, we
woul d l i ke t o choos e t he basi s whi ch mi ni mi zes ( ). The next bound says howto do thi s.
i
Bound :
[ 7 ] Let t he set ( A ) of ei genval ues of A be wri t t en as a di s oi nt uni
=1 onof
i
i
set s of ei genval ues Let n be t he number of ei genval ues i n , count i ng mul t i pl i ci t i es Let
be t he pro ect or correspondi ng t o, and t he correspondi ng ri ght i nvari ant subspace
Let be any mat ri x whose n col umns f orman ort honormal basi s of , and wri t e =
[ 1; . .; . ] Then 01 A = B i s bl ock di agonal where di agonal bl ock has
B ei genval ues
Let
be an arbi t rary mat ri x whose ncol umns f orm a basi s of , and wri t e =
01 A =B i s al so bl ock di agonal where di agonal bl ockhas
[ 1 ; . .; . ]
B ei genval ues
Then
max k k2
( )
i
i
i
i
i
i
i
i
i
ii
i
i
i
i
ii
i
i
i
12
Al so
max
i
k k2
1
( )
i
maxk
i
i
k2
In ot her words choosi ng t he col umns of whi ch span t o be ort honormal nearl y mi ni mi zes
( ) over al l bl ock- di agonal i zi ng si mi l ari t i es whi ch reduce A t o di agonal bl ocks wi t h t he same
ei genval ues i n each bl ock
i
um
mry
erturb ti on
i
bl e
For conveni ence, the bounds pr esented i n the pr ecedi ng s ect i ons are summar i zed i n the
f ol l owi ng t abl e. The notati on i s as f ol l ows. W
e assume the matri x A i s i n Schur canoni cal
f or m (2. 1) .
denotes the spectral pr oj ect or as s oci at ed wi th wi th ei genval ue(s)
11 of A
dened i n ( 2. 2) . s ep
wi l l be shor thand f or sep
( A11; A
22), dened i n (2. 5). wi l l denot e t he
unpert ur bed ei genval ue i f 11Ai s 1 by 1, and i f 1A1 i s l arger wi l l denote the unpertur bed
, respect i vel y,
mean of i ts ei genval ues. and wi l l denote the perturbed val ues of and
f or A + .
denotes the unper turbed r i ght i nvari ant subspace correspondi ng11to
, and
A
denotes i ts per t ur bed count erpart of A + .
wi l l denote ( ; ), the angul ar
pert ur bati on of the r i ght i nvari ant subspace as dened i n (5. 8) or (5.
9).l l denot e
2 wi
k k2 and wi l l denote k k, norms of the perturbati on . I n t he tabl e, each asymptoti c
bound has a + ( 2 ) termwhi ch i s not wri tten. Superscri pts i n parenthes es on each bound
i ndi cate whi ch Bound i n t he body of text they ar e. The s uperscri pt
i ndi cates that t he
bound appl i es onl y i f < s ep ( 4k k2) (Bound 1).
As ympt oti c Bounds
Si mpl e ei genval ue
2 k k2 2
0
Ei genval ue Cl us t er 0 I nvari ant subs pace
Gl obal Bounds
0
2 k k2
0
2
arctan
n 2 k k2 3
22 k
0
k2
2
2
01 A
I n addi t i on, Bound 8 s ays t hat the near l y bes t condi ti oned matri x such t hat
i s di agonal has as i ts col umns the ei genvect or s of A al l wi th normequal to 1. The condi ti on
number ( ) of t hi s
s at i s es maxk k2
( )
n 1 maxk k2, wher e i s the
pr oj ector cor r es pondi ng t o ei genval ue.
01 A = B i s bl ock
Bound 9 descri bes a nearl y bes t condi ti oned matri x such t hat
di agonal , s uch that the di agonal bl ocks of B have speci ed ei genval ues . Thi s nearl y
bes t
may be wr i t t en
= [ 1; . . .] wher e
i s any orthonormal basi s of the r i ght
i nvar i ant s ubs pace of A cor r es pondi ng to the ei genval ues i n t he i - th di agonal bl ock of B .
i
i
i
i
i
13
i
i
Est i mati ng t he one- normof k 01 k1 can been done by cal l i ng SLACON vi a a reverse communi cati on i nt er f ace. Thi s means one must provi de t he sol uti on vect or s x and of t he
quas i - tri angul ar s ys t ems :
x =z ; T =z
wher e z i s determi ned by SLACON. Thi s i s the f unct i on of t he subrouti ne SLAQTR. Not e t hat
i s a compl ex mat r i x i f i s a compl ex ei genval ue, but i t i s of t he f orm
=C +i
wher e t he real part C i s a r eal upper quasi - tri angul ar matri x, and the i magi nary par t i s
BB x xx
=B
B
...x
...
:
x
Hence we can eas i l y s ol ve t he compl ex systems
( C +i
) ( p +i q ) =( +i ) ;
i n r eal ari thmeti c, and us e 2( n
( C +(i +)i ) =( +i )
0 1) l ength vect or s xpq =
and =
as the i nput
vect or s of SLACON. W
e al s o us e t he f act that f or any compl ex matri x C +i
1
k
2
C
0
C
k1 k C +i 1k k C 0C
,
k1:
The cos t of the al gori thmdepends on t he l ocati on of the s el ect ed ei genval ues al ong the
di agonal of the i nput mat r i x. Swappi ng adj acent di agonal bl ocks costs ( n ), so movi ng
an ei genval ue at di agonal pos i ti on to the upper l ef t cos t s ( n ) operati ons. Si nce i t
r equi res about (2n) to sol ve a quasi - tri angul ar system, esti mati ng the condi ti on number
of an ei genvector costs 2()n operati ons once t he ei genval ue i s i n t he upper l ef t corner.
Ther ef ore the total cost i s 2 )( nper ei genvect or condi ti on number .
The vari abl e SEP
cont ai ns t he esti mat ed r eci procal condi ti on number s of t he sel ected
ei genvectors.
Esti mti ng the C
on i ti on of
Cl uster of Ei gen-
l ues
I n t hi s s ect i on, we r s t s how the usage of LAPACK subrouti ne STRSEN f or esti mat i ng
t he r eci pr ocal condi t i on number of a speci ed mul ti pl e (or cl ustered) ei genval ue and i ts
cor respondi ng i nvari ant subs pace, and then outl i ne the al gori thm.
20
N
T
LDT
.
- REAL
O
S
2
N
2
.
,
- INTEGER
O
, LDRE
I
S
RE
.
RE
DIMENSION LDLE,MM .
, LE
RE
. I
,
STREVC
,
. LDLE
1, N .
.
- REAL
DIMENSION MM .
15
SHSEIN.
LE
.
.
- INTEGER
O
, LDLE
N
SHSEIN.
. LDRE
1, N .
.
N
LDLE
. I
STREVC
,
.
- REAL
O
T
DIMENSION LDRE,MM .
RE
N
LE
. LDT
, RE
N
.
1, N .
.
- REAL
O
I
LDRE
. T
- INTEGER
O
, LDT
N
RE
DIMENSION LDT,N .
, T
LE
O
S
SEP
, S
1
. I
S
- REAL
O
1
MM
1
SEP
- INTEGER
O
, MM
S
S
SEP. T
N
M
DIMENSION MM .
, SEP
I
.
SEP
1
- REAL
S
.
V
- REAL
B
- REAL
ISGN
- INTEGER
,
. N
, RE
S
.
,
LE
SEP
DIMENSION LD ORK,N
.
ORK
. LD ORK
1, N .
.
N
- REAL
,
, SEP
.
LD ORK - INTEGER
O
, LD ORK
X
.
SEP
- INTEGER
O
, M
ORK
,
.
.
.
DIMENSION 2
N-1
.
DIMENSION 2
N-1
.
DIMENSION N
.
DIMENSION 2
.
16
N-1
INFO
- INTEGER
O
, INFO
,
-K
N 1
K
S
SEP
.
.
The cal l i ng s equence of t he doubl e preci si on routi ne DTRSNA i s the
s ame as t hat of the cor r es pondi ng si ngl e pr eci s i on \S" s ubrouti ne except that al l t he real
var i abl es ar e doubl e preci si on.
om
pl ex. The cal l i ng s equence of t he si ngl e pr eci s i on compl ex i s essenti al l y the same
as STRSNA, except that the T, RE, ORK, X, V arrays are compl ex, and the i nteger array
I SGN i s r eal .
oubl e preci si on com
pl ex. The cal l i ng s equence of t he doubl e pr eci s i on compl ex
r out i ne ZTRSNA i s t he same as that of the correspondi ng si ngl e pr eci s i on \C" subrouti ne
except t hat al l t he r eal vari abl es are doubl e preci si on and al l the compl ex var i abl es are
doubl e preci si on compl ex.
oubl e preci si on.
.
The f ol l owi ng program segment i l l ustrates the use of the si ngl e preci si on subr outi ne to
est i mate sel ected reci pr ocal condi ti on number s of t he ei genval ues and ei genvectors of a
gener al mat r i x. The programrst reduces a gener al matri x to upper Hessenberg f or m
by SGEHRD, and then comput es t he Schur decomposi ti on by the mul ti shi f t QR i terati on
( SHSEQR) . Af ter that the us er s houl d i nput the l ogi cal array SELECT to speci f y t he ei genpai r s
whos e condi t i on numbers wi l l be esti mated, and STREVC i s cal l ed to compute the sel ected
r i ght and l ef t ei genvectors and compactl y store themi n array RE and LE. Fi nal l y STRSNA i s
cal l ed to retur n t he desi red reci procal condi ti on numbers.
PROGRAM
INTEGER
LOGICAL
REAL
REAL
REAL B
INTEGER
INTEGER
REAL
PARAMETER
PARAMETER
TEST
ISGN 1
SELECT
A ,
, R
, I
, RE
,
, LE ,
S
, SEP
, R ORK
, X 1
, V 1 , ORK
,1
, U ,1
N, LDA, LDRE, LDLE, LD ORK, M, I, , MAXN, INFO
NBLCK1, NSHIFT, NBLCK2
DUMMY
LDA =
, LDRE =
, LDLE =
, LDU =
LD ORK =
, MAXN = 1
I
:
N:
A: N
N
NBLCK1:
NSHIFT:
A.
H
17
.
QR
.
.
NBLCK2:
1
READ , N
DO 1 I = 1,N
READ ,
A I,
CONTINUE
READ , NBLCK1
READ , NSHIFT
READ , NBLCK2
C
2
, = 1,N
S
.
CALL XENVIR 'BLOCK',NBLCK1
CALL SGEHRD 'H', N, A, LDA, DUMMY, DUMMY, R,
MAXN, INFO
CALL XENVIR 'SHIFT', NSHIFT
CALL XENVIR 'BLOCK', NBLCK2
CALL SHSEQR 'S', N, A, LDA, DUMMY, DUMMY, R,
MAXN, ORK, LD ORK, MAXN, INFO
DO 2 I = 1, N
RITE ,
R I , I I
CONTINUE
I
3
QR
SELECT
ORK, LD ORK,
I, U, LDU,
.
DO 3 I = 1,N
READ , SELECT I
CONTINUE
C
CALL STREVC 'B', SELECT, N, A, LDA, RE, LDRE, LE, LDLE,
N, M, R ORK, INFO
E
CALL STRSNA SELECT, N, A, LDA, RE, LDRE, LE, LDLE, S, SEP,
N, M, ORK, LD ORK, X, V, B, ISGN, INFO
P
DO
I = 1,M
RITE , S I ,SEP I
CONTINUE
18
.
STOP
END
.
The STRSNA r out i ne i s desi gned to esti mate the r eci pr ocal s of condi ti on number s of the
s el ect ed ei genval ue- ei genvector pai rs of a Schur canoni cal matri x T .
Logi cal ar r ar y SELECT s peci es t he condi ti on number s t o be esti mated. The arrays
RE and LE are us ed i n STREVC t o compactl y store the s el ect ed r i ght and l ef t ei genvectors
r es pecti vel y. Then RE and LE are used i n STRSNA to compute the r eci procal condi ti on
number f or i ndi vi dual ei genval ues :
s( )=
k k2 k k2
wher e and are the r i ght and l ef t ei genvect or s of T correspondi ng to the ei genval ue .
Not e t hat f or compl ex ei genval ues , t he next two col umns of RE ( LE) store the r eal and
i magi nary par t s of t he ri ght ( l ef t) ei genvectors, respect i vel y. W
e s ee t hat computi ng the
r eci pr ocal s of condi t i on number of an ei genval ue costs ( n ) operati ons.
Var i abl e S
cont ai ns the reci pr ocal s of condi ti on number s of t he s el ect ed ei genval ues.
For es t i mati ng t he reci procal s of condi ti on number s of t he associ ated ri ght ei genvectors,
STRSNA r s t cal l s s ubr out i ne STREXC to swap the di agonal bl ocks of matri x T by ort hogonal
t rans f ormati on to the f orm:
T11 T12
0 T22
T T=
wher e t he n1 by n 1 mat r i x T11 i s 1 by 1 or 2 by 2 dependi ng on whet her t he ei genval ue i s
r eal or compl ex. I f 11T i s a 1 by 1 bl ock , we have
s ep( ) = mi n
2
=1
k ( I 022T) x k2
I f T11 i s a 2 by 2 bl ock, then we use a uni tary rotati on to tri angul ari ze the 2 by 2 bl ock to
get
12
0 T~22
whence
s ep( ) = mi n
2
=1
k ( 0T~22) x k2:
I n bot h cases sepcan be es t i mated by esti mati ng t he one- normof
01 =( T
0
)01
becaus e of t he rel ati ons hi p
1
k
n 01
01 k1
1
=k
sep( )
19
01 k2
n
0 1k
01 k1
Est i mati ng t he one- normof k 01 k1 can been done by cal l i ng SLACON vi a a reverse communi cati on i nt er f ace. Thi s means one must provi de t he sol uti on vect or s x and of t he
quas i - tri angul ar s ys t ems :
x =z ; T =z
wher e z i s determi ned by SLACON. Thi s i s the f unct i on of t he subrouti ne SLAQTR. Not e t hat
i s a compl ex mat r i x i f i s a compl ex ei genval ue, but i t i s of t he f orm
=C +i
wher e t he real part C i s a r eal upper quasi - tri angul ar matri x, and the i magi nary par t i s
BB x xx
=B
B
...x
...
:
x
Hence we can eas i l y s ol ve t he compl ex systems
( C +i
) ( p +i q ) =( +i ) ;
i n r eal ari thmeti c, and us e 2( n
( C +(i +)i ) =( +i )
0 1) l ength vect or s xpq =
and =
as the i nput
vect or s of SLACON. W
e al s o us e t he f act that f or any compl ex matri x C +i
1
k
2
C
0
C
k1 k C +i 1k k C 0C
,
k1:
The cos t of the al gori thmdepends on t he l ocati on of the s el ect ed ei genval ues al ong the
di agonal of the i nput mat r i x. Swappi ng adj acent di agonal bl ocks costs ( n ), so movi ng
an ei genval ue at di agonal pos i ti on to the upper l ef t cos t s ( n ) operati ons. Si nce i t
r equi res about (2n) to sol ve a quasi - tri angul ar system, esti mati ng the condi ti on number
of an ei genvector costs 2()n operati ons once t he ei genval ue i s i n t he upper l ef t corner.
Ther ef ore the total cost i s 2 )( nper ei genvect or condi ti on number .
The vari abl e SEP
cont ai ns t he esti mat ed r eci procal condi ti on number s of t he sel ected
ei genvectors.
Esti mti ng the C
on i ti on of
Cl uster of Ei gen-
l ues
I n t hi s s ect i on, we r s t s how the usage of LAPACK subrouti ne STRSEN f or esti mat i ng
t he r eci pr ocal condi t i on number of a speci ed mul ti pl e (or cl ustered) ei genval ue and i ts
cor respondi ng i nvari ant subs pace, and then outl i ne the al gori thm.
20
.
Si ngl e preci si on
CALL STRSEN SELECT,N,T,LDT,S,SEP, ORK,LD ORK,N ORK,
X,V,ISGN,INFO
.. S
A
INTEGER
REAL
..
N, LDT, LD ORK, N ORK, INFO
S, SEP
.. A
A
LOGICAL
INTEGER
REAL
..
SELECT
ISGN
T LDT,
, ORK LD ORK,
, X
, V
A
=========
SELECT - LOGICAL
O
, SELECT
DIMENSION N
1
1
2
2
. F 2
2
.TRUE.
SELECT
. C
O
, SELECT
SELECT
.TRUE.,
N
- INTEGER
O
, N
N
T
LDT
- REAL
O
S
2
N
2
.
. I
.FALSE..
T. N
DIMENSION LDT,N
-
. T
.
.
- INTEGER
O
, LDT
N
2
.
.
, T
2
,
,
. LDT
1, N .
.
21
T
S
- REAL
O
, S
.
S
SEP
ORK
N .
- REAL
O
, SEP
- REAL
,
,
.
DIMENSION LD ORK, N2
N2
.
LD ORK - INTEGER
O
, LD ORK
1, N-N2 .
.
N
N ORK - INTEGER
O
, N ORK
U. N ORK
N
.
N2.
X
- REAL
DIMENSION N1 N2
V
- REAL
DIMENSION N1 N2
ISGN
- INTEGER
DIMENSION N1 N2
INFO
- INTEGER
O
, INFO
-K
N 1
ORK
. LD ORK
.
K
.
ORK.
oubl e preci si on. The cal l i ng s equence of t he doubl e preci si on routi ne DTRSEN i s the
s ame as t hat of the cor r es pondi ng si ngl e pr eci s i on \S" s ubrouti ne except that al l t he real
var i abl es ar e doubl e preci si on.
22
The cal l i ng sequence of t he s i ngl e pr eci s i on compl ex routi ne i s essenti al l y
t he same as STRSNA, except that the T, RE, ORK, X, V arrays ar e compl ex. I nteger array
ISGN i s not needed.
oubl e preci si on com
pl ex. The cal l i ng s equence of t he doubl e pr eci s i on compl ex
r out i ne ZTRSEN i s t he same as that of the correspondi ng s i ngl e pr eci s i on compl ex \C" subr out i ne except t hat al l the real vari abl es are doubl e preci si on and al l the compl ex var i abl es
ar e doubl e preci si on compl ex.
om
pl ex.
.
The f ol l owi ng program segment i l l ustrates the use of the si ngl e preci si on subr outi ne to
est i mate the reci pr ocal condi t i on number s of a s peci ed cl uster of ei genval ues and i ts corr es pondi ng i nvari ant s ubs pace f or a general matri x. The programrst reduces a general
mat ri x t o upper Hes s enberg f ormby SGEHRD, and then computes the Schur decomposi ti on
by the mul t i s hi f t QRal gori thm( SHSEQR). Af t er t hat the user shoul d i nput the l ogi cal ar r ay
SELECT t o s peci f y the ei genval ues i n t he cl uster. Then STRSEN i s cal l ed to es ti mat e t he
r eci pr ocal condi t i on number s .
1
PROGRAM
INTEGER
REAL
REAL
LOGICAL
INTEGER
INTEGER
REAL
PARAMETER
TEST
ISGN
A ,
X 1
,
SELECT
N, LDA,
NBLCK1,
S, SEP,
LDA =
I
:
, R
V 1
, ORK
,1
LD ORK, M, I, , MAXN, INFO
NSHIFT, NBLCK2
DUMMY
, LDU =
, LD ORK = , MAXN = 1
N:
A: N
N
NBLCK1:
NSHIFT:
NBLCK2:
READ , N
DO 1 I = 1,N
READ ,
A I,
CONTINUE
READ , NBLCK1
READ , NSHIFT
READ , NBLCK2
C
, I
, U
,1
A.
H
.
.
QR
, = 1,N
S
.
CALL XENVIR 'BLOCK', NBLCK1
23
QR
.
.
CALL SGEHRD 'H', N, A, LDA, DUM, DUM, R, ORK, LD ORK,
MAXN, INFO
CALL XENVIR 'SHIFT', NSHIFT
CALL XENVIR 'BLOCK', NBLCK2
CALL SHSEQR 'S', N, A, LDA, DUM, DUM, R, I, U, LDU,
MAXN, ORK, LD ORK, MAXN, INFO
DO 2 I = 1, N
RITE ,
R I , I I
CONTINUE
2
I
,
SELECT
.
DO 3 I = 1,N
READ , SELECT I
CONTINUE
3
CALL STRSEN SELECT, N, A, LDA, S, SEP,
ISGN, X, V, INFO
ORK, LD ORK,
P
RITE
STOP
END
,
S,SEP
.
STRSEN r out i ne es t i mates the reci procal condi ti on number s of s peci ed mul ti pl e (or cl ust er ed) ei genval ues and thei r correspondi ng i nvari ant subspace f or a real Schur canoni cal
mat ri x T .
Logi cal array SELECT s peci es t he 1 by 1 (f or real ei genval ues ) or 2 by 2 (f or compl ex
conj ugate ei genval ues) di agonal bl ocks that are to be col l ect ed t oget her to f ormthe desi red
cl us ter. Note that f or 2 by 2 di agonal bl ocks, the rst i ndex of SELECT must be s et t o
.TRUE. i f t he bl ock t o be col l ect ed. For real matri ces, compl ex conj ugate ei genpai r s wi l l
bot h be sel ected i f one i s sel ect ed.
STRSEN r s t cal l s s ubr out i ne STREX2 t o col l ect t he s el ect ed di agonal bl ocks by orthogonal
t rans f ormati on to the top- l ef t corner of T such t hat
T T=
T11 T12
0 T22
wher e t he sel ected bl ocks have been col l ect ed1i by
n nn 1 matri x T11. Then STRSEN computes
24
t he pr oj ector
on the i nvari ant subspace associ ated wi11th
. ITt i s known that
wher e
0
I
=
0
0
;
i s t he sol uti on of the Syl vester equati on
0 T22 =T 12:
T11
Thi s i s done by s ubr out i ne STRSYL . The programtests to avoi d over ow i f k k i s very
l ar ge, ret ur ni ng zero as the reci procal condi ti on number .
The r et ur n val ue S of STRSEN i s the l ower bound (1 +k 2 )k01 2 on the r eci procal of
12
k k2. I t cannot underesti mate k 01
2 k by more than a f actor of n .
Fi nal l y, STRSEN es t i mates the separati on 11
of and
T T 22. We know that thi s can be
est i mated by t he one- normof
=
=
01 =( I
n2
T11 0 T22 I 1 )01 ;
n
n =n1 +n 2
becaus e of t he rel ati ons hi p
1
k
n1 n2
01 k1
1
=k
s ep( T11; T22)
01 k2
n1 n2 k 01 k1:
Thi s i s done by cal l i ng SLACON vi a a reverse communi cati on i nterf ace, provi di ng the s ol uti on
vect or s x and of the equati ons :
T =z
x =z;
wher e z i s determi ned by SLACON. Thi s means we must sol ve the Syl vester equati ons:
T11X 0 XT22 =
T
T 11
0 T22T =
Thi s i s agai n i s done by t he subrouti ne STRSYL.
The retur n val ue SEP of STRSEN i s the esti mated (upper bound) of (sep
T11; T22).
Swappi ng adj acent di agonal bl ocks on t he di agonal of the i nput matri x costs ( n ), so
s wappi ng n1 sel ected ei genval ues to the top l ef t corner cos t s at most
n as l i ttl e
1n2 ) ((and
as not hi ng i f they are al ready at the t op l ef t cor ner). Once t he des i r ed ei genval ues ar e at t he
21 n2 +n( n1 n22 ) operati ons. Ther ef ore
t op l ef t, s ol vi ng ei t her above Syl vester equati on costs
t he condi t i on number esti mati on of a cl uster of ei genval ues and thei r correspondi ng i nvar i ant
s ubs pace costs at most (3n) operati ons, or as f ewas 2()ni f n1 n2 .
25
ol uti on of the
yl ester E u ti on
Cons i derabl e at t ent i on i n t he l i terature has been pai d to sol vi ng the Syl vester equat i on.
Among pr oposed sol ut i ons , Bar t el s and Stewart' s met hod[
] ,2 and Gol ub, Nash and Van
Loan' s method[ 12] are di rect matri x f actori zati on met hods. I n thi s appendi x, we di scuss
t he met hod or i gi nal l y pr es ent ed by Bartel s and Stewart, and the associ ated routi ne STRSYL.
The Syl ves t er mat r i x equati on i s of the f orm
op( A ) X 0 X op( B ) =s C
(A: 10)
wher e A , B and C are real m 2 m, n 2 n and m 2 n matri ces respect i vel( A
y.) =
opA
or AT i s a t r ans pose opt i on. s i s a scal i ng f actor ( 1) whi ch i s so chosen so t hat X can
be comput ed wi t hout overow. W
e wi l l al so suppose that A and B are i n upper quas i t ri angul ar f orm, ot herwi s e we shoul d compute the Schur decomposi ti on of A and B (by
SGEHRD and SHSEQR) ,
TA = ;
TB =
(A: 11)
wher e and are upper quasi - tri angul ar, and and are orthogonal . The r educti ons
( A. 11) l ead to the sys t em( A. 10) .
I t i s wel l known t hat (A. 10) has a uni que sol uti on i f and onl y i f ther e ar e no common
ei genval ues of A and B .
Let p be t he number of 1 by 1 and 2 by 2 bl ocks al ong t he di agonal of A , and l et q be
t he number of the 1 by 1 and 2 by 2 bl ocks al ong the di agonal of B . Parti ti on the A , B , X
and C conf ormal l y. I f op
( A ) =A and op( B ) =B then t he i th bl ock Xsati ses
ij
A X
ii
ij
0X B
ij
wher e
p
C =
0
ij
k
= +1
jj
=s 1 ( C
ij
A X
ik
kj
01
j
0
=1
i
0C
0
ij
)
(A: 12)
X B
il
lj
l
Not e t hat si nce A and B are each 1 by 1 or 2 by 2, the sol uti on of (3) can be obtai ned by
s ol vi ng a l i near sys t emof order at most f our. That can be sol ved eas i l y. Once cal cul ated, the
s ol ut i on X can be s t or ed i n t he l ocati ons occupi ed by, Cwhi ch i s no l onger needed. The
s ol ut i on mat r i x X can be s uccess i vel y sol ved col umn by col umn starti ng f rombottom- l ef t
cor ner of X , i . e. , i n or der
1 , XX 01 1 , . . ,. X11, X 2 , . . ,. X1 .
For sol vi ng equati on (A. 12), we note that i f and/or
A
B are bal anced 2 by 2 bl ocks,
i . e. , t hey are of the f orm
ii
jj
ij
ij
p
p
;
p
ii
q
jj
2 . Equati on (A. 12) can then be expressed as
t hen are the ei genval ues wher e =0
a 2 by 2 l i near sys t em
( 0 w) =s 1
Her e
i s an n a by n a real matri x ( n a =1; 2), w i s real or compl ex,
and ar e n a by 1
matr i ces whi ch ar e r eal i f w i s real and compl ex i f w i s compl ex,1 iand
s asl ocal scal i ng
26
f act or ( 1) whi ch i s s o chosen that can be computed wi thout over ow(see SLALN2 and
SLASY2) . I n par t i cul ar , i f and
A B have t he same ei genval ues , 1si s set to 0.
Si mi l ar l y, i f (op
A ) = AT and op( B ) = B , t he i th bl ock Xof the sol uti on X can be
s ucces si vel y sol ved col umn by col umn starti ng f romtop- l ef t corner of X , i . e. , i 11
n ,order X
X21, . . ,. X 1 , X12, . . ,. X .
I f op( A ) =A and op( B ) =BT , the i th bl ock Xof the sol uti on X can be succes s i vel y
s ol ved col umn by col umn s t ar t i ng f rombottom- ri ght corner of X , i . e. , i n or der
, XX
01 ,
. . ,. X1 , X 01 , . . ,. X11.
I f op( A ) =AT and op( B ) =BT , t he i th bl ock Xof the sol uti on X can be s ucces si vel y
s ol ved col umn by col umn s t ar t i ng f r omt op- ri ght corner of X , i . e. , i n or
, X2X, . . ,.
1 der
X , X1 01 , . . ,. X 1 .
Us i ng t he above di erent s ubsti tuti on order i ngs enabl es one to work on the i nput mat ri ces di r ect l y r at her than t o transpose the i nput matri ces.
The overal l number of ops f or the above substi tuti on sol uti on i s
ii
jj
ij
p
pq
ij
pq
q
p
;q
p;q
ij
q
pq
;q
q
p
2n +mn 2 )
2: 5( m
wher e we have as s umed that A and B are al ready i n Schur f orm.
The programmay be us ed t o i t erati vel y r ene of t he computed sol uti on1 of
X (A. 10): l et
t he r esi dual mat r i x 1 =C 0 AX 1 +X 1B be computed i n doubl e preci si on and rerounded
to s i ngl e preci si on. Us e t he s ame programto sol ve the s ystem2 AX
0 X2B = 1. Then
X1 +X 2 wi l l i n general be a mor e accurate approxi mati on. Thi s pr oces s may be i t erated.
Thi s i t er at i on i s anal ogous t o the i terati ve r enement of approxi mate sol uti ons of l i near
s ys t emas descri bed by Wil ki ns on[ 27
, p. 255] . (Thi s i s not done i n STRSEN and STRSNA. )
ppi ng
i gon l
l oc s
The cr ux of s wappi ng a sel ected bl ock of a r eal Schur f ormto a speci ed posi ti on al ong the
di agonal ( s ubr out i ne STREXC), or col l ect i ng s el ect ed bl ocks together (subrouti ne STREX2)
i s the swappi ng of adj acent bl ocks by an orthogonal si mi l ari ty transf ormati on (subrouti ne
SLAEXC) . St ewar t [ 23
] devel oped an adj acent bl ock swappi ng al gori thmusi ng one or t wo QR
s teps wi t h a pr e- determi ned shi f t to f orce the or der i ng of t he ei genval ues of t he newbl ocks.
Mor e r ecent l y, Ng and Parl ett[ 17
] present a more strai ghtf orward al gori thmf or t he same
t as k. The pr es ent at i on i n t hi s appendi x i s based on Ng and Parl ett's work. W
e di scuss i n
mor e detai l the t r eat ment of pat hol ogi cal cases.
Cons i der a submat r i x of t he f orm
T11 T12
0 T22
wher e T11 i s a p by p mat r i x, and22Ti s a q by q matri x, p ; q =1 or 2, and assume that
T11 and T 22 have no ei genval ue i n common. Moreover , we assume that i f ei ther i s a 2 by 2
mat ri x, i t has been standardi zed ( i . e. , i t has i denti cal di agonal entri es. ) Now, we want to
nd an or t hogonal mat r i x whi ch swaps T11 and T 22, i . e. ,
T11 T12
0 T22
T=
27
T~22 T~12
0 T~11
wher e T~ i s s i mi l ar t o T, i =1; 2.
Si nce T11 and T 22 have no ei genval ue i n common, i t f ol l ows t hat ther e exi sts a uni que
p 2 q mat r i x X s uch that
T11X 0 XT22 =T 12:
ii
ii
Hence
T11 T12
0 T22
0X
I
=
p
T11
I
0X I
I
0
0
=
T22
p
q
0
q
T11
0
q
X
0 I
0 I
I X
p
T22
0
q
I
0
p
W
e s ee that i t i s eas y t o nd an orthogonal ( p +q ) 2 ( p +q ) matri x
0X
2
=
I
q
such t hat
(B: 13)
0
wi th s ome i nver t i bl e q 2 q2, e. g. , usi ng Househol der matri ces to do the QRdecomposi ti on.
Let pr emul t i pl y and pos t mul ti pl y the ori gi nal matri x, yi el di ng
T11 T12
T22
0X I
T =
=
=
=
0
q
0
0
2
2
T22
p
I
0
T22
1
1
2T22 201
0
0
T11
I
0
2
0
T11
T22
0
p
0
T11
0
T12
T
1 11 101
0
01
2
0
I
X
q
1
0
T
01
01
01
2 01 1
1
:
T11 and T 22 have been swapped.
The above cons i derati ons ar e summed up i n the f ol l owi ng steps.
1. Sol ve T11X 0 XT22 =s T12. s i s a scal e f actor i ntroduced t o avoi d over ow.
2. Check i f the magni t ude of X
k k exceeds t he square root of the over owt hreshol d. I n
t hi s cas e 1T1 and T 22 are too cl ose to swap, so we exi t.
3. Us e a Hous ehol der mat r i x
by : T T ,
to do the QR decomposi ti on of ( X IT )and update T
4. Accumul at e t he or t hogonal transf ormati ons i f des i r ed.
5. To pr es er ve t he s t andard Schur f orm, make the di agonal el ements equal i n each 2 by
2 bl ock us i ng or t hogonal t ransf ormati ons.
6. Accumul at e t he or t hogonal transf ormati ons i f des i r ed.
28
Several comment s s houl d be made. Fi rst, the sol uti on of the matri x equati11on
X T0
XT22 = sT 12 has been di s cussed i n det ai l i n Appendi x A, the routi ne SLASY2. Second,
t her e i s no danger i n worki ng wi th X of l arge normprovi ded that
X k2k does not over ow.
2
Mor eover i f kX k does overow t hen t he bl ocks shoul d not be swapped because a ti ny
per tur bati on wi l l caus e11Tand T 22 to have at l east one common ei genval ue[] 9. Hence i n
s tep 2, we check the normof X , and i f X sati ses
s 1 max( kT11k ; kT22k)
< ;
kX k +s
wher e i s t he machi ne preci si on, then t he two bl ocks are regarded as too cl ose to swap.
29
C
List of
LAPAC
R
outi nes f or the
onsymm
etri c Ei gen-
problem
LAPACK mai n routi nes f or the nonsymmetri c ei genprobl em:
SGEBAL Bal ance an i nput general matri x and i sol ate ei genval ues whenever possi bl e.
SGEBAK Formt he ei genvectors f or a general matri x by back transf ormi ng t hose of t he cor r es pondi ng bal anced mat r i x det er mi ned by SGEBAL.
SGEHRD Reduce a general mat r i x to an upper Hessenber g matri x.
SHSEQR Comput e t he ei genval ues of an upper Hessenber g matri x by the mul ti shi f t QR
al gori thm, and retur n t he Schur f orm, accumul ati ng the orthogonal matri x i f desi red.
STREVC Comput e s el ect ed r i ght and/or l ef t ei genvect or s of a Schur matri x.
SHSEIN Comput e s el ect ed r i ght and/or l ef t ei genvect or s of a Hessenber g matri x by i nverse
i ter at i on.
SORGC3 Overwr i t e a mat r i x cont ai ni ng Househol der vectors stored i n the stri ctl y l ower par t
by t he or t hogonal mat r i x t hey r epresent.
STRSNA Es t i mate sel ected reci procal condi ti on number s of i ndi vi dual ei genpai rs of Schur
mat r i x.
STRSEN Es t i mate sel ected reci procal condi ti on number s of a mul ti pl e (or cl uster of ) ei genval ues and t hei r cor r es pondi ng i nvari ant subspace of a Schur matri x.
STRSYL Sol ve the Syl ves t er equati on wi th coeci ent matri ces i n Schur f orm.
STREXC Swap a sel ected di agonal 1 by 1 or 2 by 2 bl ock of a Schur matri x to a speci ed
pos i t i on.
STREX2 Col l ect several sel ected di agonal 1 by 1 or 2 by 2 bl ocks of a Schur mat r i x to the
t op- l ef t or bot t omr i ght corner .
LAPACK auxi l i ary routi nes f or the nonsymmetri c ei genprobl em:
SLAHRD Chase a k by k bul ge of an upper Hessenber g matri x one bl ock down f roma speci ed
col umn number.
SLAHQR BLAS 1 based QR rout i ne to compute the ei genval ues of an upper Hessenberg
mat r i x, and r et ur n t he Schur f orm, accumul ati ng t he orthogonal matri x i f desi red.
SLATRS Mi xed subr out i ne of BLAS 1 and BLAS 2 to sol ve tri angul ar equati ons whi l e avoi di ng overow.
SLAQTR Sol vi ng real or compl ex quasi - tri angul ar systems wher e t he real part i s quas i t ri angul ar , and t he i magi nary part i s of a speci al f or m.
30
SLALN2 Solve a2by2linear equation.
SLAE2 Com
pute the eigenvalues of a2by2nonsymmetric m
atrix.
SLAEXC Swapadjacent diagonal 1 by1or 2by2 blocks of aSchur m
atrix.
SLASY2 Solve the Sylvester equationwithcoecient m
atrices upto2 by2.
SLAEQU Equalize the diagonal elem
ents of a2by2 blockwithanorthogonal similarity.
LAPACK
Other
routines, auxi li ary routi nes, functi ons cal l ed by ei gensystem
subrouti nes (except Level 1, 2 or 3 BLAS routi nes).
XERBLA, LSAME, R1MACH, ENVIR
SLACON, SLACPY, SLAZRO, SLARFG, SLARF, SLANHS, SLAPY2, SLAPY3
31
References
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[2] R. S. Bartels and G. W
. Stewart, Sol ut i on of t he mat ri x equat i on AX + XB = C ,
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[3] F. Bauer and C. Fike, Norms and Excl usi on Theorems , Num. M
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(1960)
[4] A. Bjorck and G. H. Golub, Numeri cal Met hods f or comput i ng angl es bet ween l i near
subspaces , M
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IERRTRANS. Automat. Control AC-29: 926-928(1984).
0 XB T =C ,
[6] S. P. Chan, R. Feldman and B. N. Parlett, A programf or comput i ng t he condi t i on
numbers of mat ri x ei genval ues wi t hout comput i ng ei genvect ors ACMTOM
S, 3: 186203(1977).
[7] J. W
. Demmel, The condi t i on number of equi val ence t ransf ormat i ons t hat bl ock di agonal i ze mat ri x penci l s , SIAMJ. Num. Anal. 20, no. 3, June 1983, pp599{610
[8] J. W
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.W
ilkinson, Il l - condi t i oned ei gensyst emand comput at i on of t he
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[12] G. Golub, S. Nash and C. Van Loan, A Hessenberg- Schur Met hod f or t he Probl em
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[13] G. GolubandC. VanLoan, Mat ri x Comput at i ons (2ndEdition) , Johns Hopkins U.P.
Baltimore, 1988
[14] W
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[16] W
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Universityof California, Berkeley1972.
[17] K. C. NgandB. N. Parlett, Programs t o swap di agonal bl ocks (1988)
32
[18] N. J. Higham, A survey of conditi on number est i mat i on f or t ri angular mat ri ces , SIAM
Rev. 29: 575-596(1987).
[19] N. J. Higham, FORTRAN codes f or est i mat i ng t he one-norm of a real or compl ex
mat ri x, wi t h appl i cat i ons t o condi t i on est i mat i on, ACMTOM
S, 14: 381-396(1988).
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BIT10: 196-216, 1970.
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ei genval ue probl ems , SIAMRev. 15: 727-764(1973).
[23] G. W
. Stewart, Al gori t hm506 HQR3 and EXCHANG: Fort ran subrout i ne f or cal cul at i ng and orderi ng t he ei genval ues of a real upper Hessenberg mat ri x [F2], ACMTOM
S
2:275-280(1976).
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33