On Some Structured Inverse Eigenvalue Problems

ISSN 0249-6399 NTTTNTOA ERCECEE NOMTQEE NAUTOMA EN ET INFORMATIQUE EN RECHERCHE DE NATIONAL INSTITUT nsm tutrdIvreEigenvalue Inverse Structured some On oetEr,BradPhilippe Bernard Erra, Robert apport de recherche Problems RGAM 6 PROGRAMME Ê2604 NÊ un1995 Juin TIQUE

On some Structured Inverse Eigenvalue Problems

Rob ert Erra Bernard Philipp e

Programme Calcul scientique mo delisation et logiciel numerique

Pro jet ALADIN

Rapp ort de recherche n Juin pages

Abstract This work deals with various nite algorithms that solve two sp ecial Structured

Inverse Eigenvalue Problem Siep

The rst problem we consider is the Jacobi Inverse Eigenvalue Problem Jiep given

some constraints on two sets of real nd a Jacobi matrix J real symmetric tridiagonal

with p ositive nondiagonal entries that admits as sp ectrum and principal subsp ectrum the

two given sets Two classes of nite algorithms are considered The p olynomial algorithm

is based on a sp ecial EuclidSturm algorithm Householders terminology which has b een

rediscovered several times The matrix algorithm is a symmetric Lanczos algorithm with a

sp ecial initial vector Some characterization of the matrix insures the equivalence of the two

algorithms in exact arithmetic

The results of the symmetric situation are extended to the nonsymmetric case this is

the second Siep which is considered the Tridiagonal Inverse Eigenvalue Problem Tiep

Possible breakdowns may o ccur in the p olynomial algorithm as it may happ en with the

nonsymmetric Lanczos algorithm The connection b etween the two algorithms exhibits a

similarity transformation from the classical Frob enius companion matrix to the tridiagonal

matrix

This result is used to illustrate the fact that when computing the eigenvalues of a matrix

the nonsymmetric Lanczos Algorithm can lead to a slow convergence even for a symmetric

matrix since an outer eigenvalue of the tridiagonal matrix of order n can b e arbitrarily

far from the sp ectrum of the original matrix

Keywords Jacobi matrix Euclid algorithm Sturm sequence Lanczos algorithm inverse

eigenvalue problems

Resume tsvp

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Problemes inverses de valeurs propres

Resume Cet article presente dierents algorithmes directs qui resolvent deux problemes

inverses de valeurs propres structures Le premier probleme est le probleme inverse de Jacobi

etant donne quelques contraintes sur deux ensembles de reels construire une matrice de

Jacobi qui admette comme sp ectre et sp ectre principal dordre immediatement inferieur

les deux ensembles de nombres On etudie deux classes dalgorithmes un algorithme de

p olynomes souvent redecouvert et lalgorithme de Lanczos symetrique avec un vecteur initial

approprie

La situation precedente est ensuite etendue au cas non symetrique elle corresp ond au

deuxieme probleme inverse Dans ce cas lalgorithme de p olynomes p eut sarreter avant

la n comme cela p eut arriver avec lalgorithme de Lanczos non symetrique Dans le cas

sans echec on exhib e une transformation de similitude entre la matrice tridiagonale et la

matrice compagnon de Frob enius qui relie les deux algorithmes Une consequence de ce

dernier resultat p ermet de montrer que lalgorithme de Lanczos non symetrique utilise p our

calculer les valeurs propres dune matrice dordre n p eut mener a une matrice dordre n

dont une valeur propre est arbitrairement loin du sp ectre de la matrice initiale

Motscle Matrice de Jacobi algorithme dEuclide suite de Sturm algorithme de Lanczos

problemes inverses de valeurs propres

Inverse Eigenvalue Problems

Intro duction

In the last centuries the computation or lo calisation of some or all the ro ots of a p oly

nomial has received a considerable attention Another related problem has emerged and

progressively b ecame of fundamental interest for the community of numerical analysis the

computation or lo calisation of some or all of the eigenvalues of a matrix

More precisely for every matrix algorithm we can lo ok at its p olynomial counterpart

For instance Bernoullis metho d for nding the largest ro ot of an algebraic equation

px is equivalent to the Von MisesGeiringer metho d applied on the Frob enius

companion matrix of px which computes its largest ro ot by the p ower metho d However

the translation is not always obvious

Two algorithms have b een the ob ject of a great deal of work the Rouths Algorithm

based on the Sturm theorem which computes the numb er of ro ots of a

p olynomial in a given interval and the Lanczoss Algorithm which tridiagonalises a

matrix The purp ose of this work is to p oint out and make clear the formal connections

existing b etween these two famous algorithms and to give a relevant bibliography

The link b ecomes clear with a mo died version of Rouths Algorithm which we call Routh

Lanczos algorithm and for which we present a brief history Given a p olynomial px with

real and simple ro ots these algorithms build a Jacobi matrix ie a real irreducible symmetric

tridiagonal matrix whose characteristic p olynomial is px For sake of simplicity we will

only consider p olynomials with real co ecients

The fact that we imp ose the structure of the matrix solution of problems and

explains the denomination structured inverse eigenvalue problems due to Boley and Golub

BG that we have declined in Jiep studied in sect Biep and Tiep studied in sect

Notations

In this work J denotes the tridiagonal matrix

k k k +1

1 2

B C

2 2 3

B C

n1 n1 n

n n

When J is real symmetric we always supp ose that we have

i i

RR n

R Erra B Philippe

If px is a p olynomial with real co ecients px a x the classical Frob eniuss

i=0

companion matrix asso cied to p is

C B

C p

C B

a a

1 0

a a a

n n n

For every matrix J of order n we call upper principal submatrix the matrix of order

n obtained by deleting the last row and the last column of J Similarly the matrix of

order n obtained by deleting the rst row and the rst column of J is called the lower

principal submatrix and the sp ectra of these matrices are called principal subspectra I is

the identity matrix of order n

P T

For a matrix A we note A I A I where

n n

C B

I is the reversal matrix of order n and A is the pertransposition of A A nth order

P T

matrix A such that A I A I A is called a p ersymmetric matrix

n n

Jacobi Inverse Eigenvalue Problems

Discrete Inverse Eigenvalue Problems have rst b een prop osed by Downing and Householder

DH see Fri FNO for newer references

We limit our attention in this section to the following situations which we dene as

the upper Jacobi Inverse Eigenvalue Problems an upp er Jiep that has b een raised by

Ho chstadt Ho c

Problem Let f g and f g be two sets of real satisfying the

i i=1n i i=1n1

strict interlacing property

1 1 2 2 n1 n

Find a Jacobi matrix J such that

spJ f g and spJ f g

n i i=1n n1 i i=1n1

where spA denotes the spectrum of A and J the upper principal submatrix of order

n of J

n INRIA

Inverse Eigenvalue Problems

Problem Let p x and p x be two monic polynomials of degree n and n res

n n1

pectively which are assumed to have strictly interlaced real roots

Find a Jacobi matrix J such that

detxI J p x and detxI J p x

n n n n1 n1 n1

where J is the upper principal submatrix of order n of J

n1 n

We can also dene the lower Jiep where J is replaced by the lower principal

submatrix of order n of J We must p oint out that if J is a solution of the upp er

n n

Jiep with data and then J I J I is a solution of the lower Jiep with data

and see also BG

Another related problem is the following persymmetric Jiep

Problem Let f g be a set of real increasing Find a persymmetric Jacobi

i i=1n

matrix J having as spectrum

Problems and come rst from the discretisation of continuous inverse Sturm

Liouville problems The problem of computing Gauss quadrature formula gives also Sieps

for example computing the Gauss quadrature formula of order n knowing the Gauss qua

drature formula of order n can b e interpreted as the following

Problem Let J be a Jacobi matrix of order n and let be a set of n reals

n 2n

f g strictly increasing Find a Jacobi matrix J having as spectrum and whose

i i=12n 2n 2n

principal upper submatrix of order n is J problem DD in BG

References and other exemples of Sieps can b e found in GK Hal BG These

problems can b e seen as particular cases of the more general following problem the Banded

Inverse Eigenvalue Problem Biep ie the problem of the reconstruction of a symmetric

pbanded n n matrix from the sp ectral data with additional interlacing conditions See

BK BG MH for the case p n and Fri for the sp ecial case p n

reconstruction of the complete symmetric matrix with metho ds dierent from those that

we will discuss here

Hald Hal proved that the rst problem is wellp osed

Theorem Let J and J be solutions of Problem with respective data

1 1 2 2 n1 n

and

1 1 2 2 n1 n

There exists a constant K such that

n n

X X

2 12 2

jjJ J jj K

j j i i E

i=1 i=1

RR n

R Erra B Philippe

where jj jj is the Frobenius norm The constant K is bounded in terms of n and K

E 0

O n dened by

0 0

where

minj j j j

j k j k 0

and

minj j j j j j j j

0 j k j k j k j k

j 6=k

with

d max min

n n 1 1

Obviously Problems and are equivalent and therefore this theorem implies

that Problem is wellp osed as well

Existing algorithms for Jiep

For sake of completeness we rst outline two families of algorithms that can b e used to

solve Jiep namely the Lanczos algorithm and the EuclidSturm algorithm with a sp ecial

emphasis on Rouths algorithm We p oint out that we only consider direct or nite algo

rithms ie algorithms that terminate in a nite numb er of steps See FNO for a survey

of iterative algorithms for inverse eigenvalue problems

EuclidSturm Algorithms

The integer Euclids algorithm is one of the oldest nontrivial algorithms For a full des

cription see Knu and for a survey see Bar

Denition Let f and g be two polynomials of degree n and m respectively We cal l a

generalised Euclid algorithm Knu any algorithm based on the fol lowing recursion

p f and p g

n n1

p q p sig n p f or k n n

k k k k 1 k k k 2

where sig n p is the remainder of the polynomial division of p by p

k k k 2 k k k 1

It is often used for computing a GCD of f and g if p then p is a GCD

k 2 k 1

of f and g Traditionnally the sequence fp p g is called a polynomial remainder

n n1

sequence prs We remark that the terms of the remainder sequence are unique up to a

scalar multiplication

Classical choices for sig n and are

k k k INRIA

Inverse Eigenvalue Problems

and which corresp onds to the classical Euclid algorithm

k k k

sig n and for p olynomials over a unique factorisation domain

k k k

p olynomials with integer co ecients for example

and which corresp onds to the classical Sturm algorithm

k k k

Stu If p and p satisfy the assumptions of Problem then the prs

n n1

obtained stops with the constant p olynomial p and is a Sturm sequence Jac

and chosen in order to obtain a monic p olynomial for p

k k k k 2

The latter case has b een prop osed several times with minor mo dications

Schwarz can b e considered as the rst who prop osed an algorithm that builds a tridia

gonal eventually complex matrix admitting f as characteristic p olynomial and g as

the characteristic p olynomial of the upp er principal submatrix His pro cedure can b e

expressed in the formalism of with a sp ecial choice of g Sch called the alter

nant g is dened by g z f z f z where f is a p olynomial whose

co ecients are the conjugate of the co ecients of f This work follows Walls work

who studied connections b etween innite Jacobi matrices submatrices of nite order

of such matrices and the asso ciated continued fractions see chapters IX X XI XII

in Wal

Collins prop osed it as an attempt to keep small the magnitude of the co ecients in

the p olynomial GCD computation Col He chose g f which is not a monic

p olynomial and This algorithm is often used in computer algebra systems

Mig

Householder studied the same recurrence and provided relationships b etween the co

ecients of the prs Hou

Hald seems to b e the rst to have intro duced it for solving Problem He proved

that by taking p and p satisfying the assumptions of Problem the prs

n n1

obtained is still a Sturm sequence but with monic p olynomials Hal

BenOr and al BOFKT BOT following Collinss idea used that algorithm for

computing the ro ots of a given p olynomial which has only real ro ots They also prop ose

a parallel version of their algorithm

More recently Schmeisser has rediscovered Halds algorithm see Sch He prop osed

this algorithm to solve a problem p osed by Fiedler at the International Col loqium on

Applications of Mathematics in Hamburg in july see also Fie Given a monic

polynomial px with only real roots construct a real symmetric matrix for which px

is the caracteristic polynomial

RR n

R Erra B Philippe

Householder calls EuclidSturm algorithms the algorithms based on the fourth recursion

We prop ose to call EuclidSturm algorithms all algorithms based on the third and fourth

recursions

For completeness we must p oint out that Rutishausers QD algorithm prop osed in

can b e used to compute a tridiagonal matrix J of order n which satises

detx I J px

n n

when px is a p olynomial of degree n with distinct ro ots This algorithm can b e viewed as

a sp ecial case and at the origin of the matrix LR algorithm

Wendro Wen has studied a variant of problem the construction of a sequence

of orthogonal polynomials fg g g g g given g p and g p He proved

0 1 n1 n n1 n1 n n

that under the assumptions of problem this problem has a unique solution which is a

Sturm sequence but his demonstration do esnt use explicitly an EuclidSturm scheme and

he do esnt mention problem We can p oint out that problem has b een studied by

Hald Hal and has initially b een studied by GantmacherKrein in the b o ok GK

published in German in but published in russian in

A wellknown EuclidSturm Algorithm is the Rouths Algorithm Routh stated his famous

algorithm to compute the numb er of ro ots of a real p olynomial which lie in the lefthalf

complex plane We follow here the presentation of Barnett and Siljaks extensive survey

BS which was published for the centennial of the algorithm

Given two p olynomials f and g such that

n n1

f x x x

0 1 n

n1 n2

g x x x

0 1 n1

with and

0 0

Then the Routh Array is the set of rows

r r r

01 02 0n+1

r r r

11 12 0n

where r and r for corresp onding j The array is generated by the rule

0j j 1 1j j 1

r r

i21 i2j +1

r r

i11 i1j +1

i11

This is the socalled determinantal form of the Rouths algorithm It avoids the need of

a p olynomial division The relation b etween the Sturm sequence f x f x f x

0 1 n

obtained from which correp onds to the second case for and and the rst column

k k

of the Rouths array is the following INRIA

Inverse Eigenvalue Problems

f x f x

f x g x

Q P

2i2 ni

r r x where sig n f x

j1 2i1mi+j 1 i i i

1 j =0

See also Gan and BS for newer references

From the Rouths array one can construct the Schwarz matrix which is a tridiagonal

real matrix but is generally non symmetric

C B

r r

2 1

where r r r r and r r r for i see Sch A similar result is implicit

1 11 2 21 i i1 i21

in Walls b o ok The tridiagonal matrix R similar to S dened by

12 12

B C

r r

n n1

B C

B 12 C

B C

r A

r r

where the square ro ots can b e complex is called the Rouths matrix see BS The numb er

of p ositive terms in the sequence r r r r r r is equal to the numb er of ro ots of

1 1 2 1 2 n

px with negative real parts BS

The symmetric RouthLanczos algorithm

To solve Problem one can compute the prs by the fourth case of recurrence presented

in subsection Hald proved then that all the quotient p olynomials are monic and of

degree and all the co ecients are p ositive Hal Then the recurrence can b e written

p x x p x p x k n

k k k 1 k k 2

which is the wellknown threeterm recurrence for computing the characteristic p olynomial

of the Jacobi matrix J but computed in a reverse way We note that the

n k k k +1

p olynomials p x p x are called Lanczos Polynomials by Householder in Hou

0 n

RR n

R Erra B Philippe

The obtained tridiagonal matrix J

1 2

B C

2 2 3

B C

n1 n1 n

n n

which is by construction a Jacobi matrix and is solution of Problem We call this

pro cedure the symmetric RouthLanczos Algorithm J is characterized by the following

Theorem Hald Let f g and f g be two sets of real satisfying the

i i=1n i i=1n1

strict interlacing property There exists a unique Jacobi matrix of order n such that its

eigenvalues are f g and the eigenvalues of J are f g

i 1n n1 i 1n1

Pro of See Hal for a complete pro of 2

Ho chstadt Ho c Ho c proved that there exists at most one solution to problem

and GaryWilson GW proved that there exists at least one solution Hald Hal

seems to b e the rst who explicitely proved that problem has a unique solution using

an EuclidSturm scheme

The Lanczos algorithm

The algorithm which was called pq algorithm by Lanczos Lan is well known fre

quently used and has b een thoroughly studied It has b een initially prop osed by Lanczos

circa for computing the minimal p olynomial of a matrix

The use of Lanczos Algorithm to solve Problem has b een rst prop osed in BG

and BG see BG for more information and references The goal is to cho ose the

starting vector from the given eigenvalues data

Prop osition Let f g f g be two sets of strictly interlaced real If

i i=1n i i=1n1

is the diagonal matrix of which diagonal entries are f g and if the vector q

i i=1n

q q is dened by

1 n

j k

j =1

j k

j =1 j 6=k

then the symmetric Lanczos Algorithm applied on with a starting vector q satisfying

generates the Jacobi matrix which is solution of Problem

k k k +1 INRIA

Inverse Eigenvalue Problems

A straight application of Halds Theorem Hal insures that in exact arithmetic the

Jacobi matrix obtained here is equal to the matrix J I J I obtained in the previous

section with the RouthLanczos algorithm The fact that we must use the p ertransp osition

of J comes from the fact that the RouthLanczos Algorithm solves an upp er Jiep while

the symmetric Lanczos Algorithm used in solves a lower Jiep

Such a result can b e used for proving Scotts result ab out the fact that even symmetric

Lanczos can converge slowly in the sense of the KanielPaigeSaads theory Sco

Corollary Let A be a real symmetric matrix of order n with spA f g

1 2 n

For every set of reals f g which strictly interlaces spA there exists a real

1 n1

vector p such that if J is the Jacobi matrix obtained by the symmetric Lanczos algorithm

0 n

with starting vector p and J its principal submatrix then we have

0 n1

spJ f g

n1 1 2

In particular we can choose p such that

i i+1

i n

Other Algorithms for solving a Jiep

We have seen that constructing a Jacobi matrix is equivalent to construct a sequence of

orthogonal p olynomials We must p oint out that the two oldest known pro cedures for ge

nerating orthogonal p olynomials are the Stieltjes pro cedure Sti and the Chebyschev

pro cedure Che Che Both algorithms compute a sequence of orthogonal p olynomials

with a forward process ie they compute p b efore p The Stieltjes pro cedure has b een

k k +1

studied by Gautschi Gau Gau and the Chebyshev pro cedure has b een also advo cated

by Gautschi Gau and by Gutknecht and Gragg GG the two algorithms have a

O n complexity Both algorithms can b e used to solve a Jiep

In Rei Reichel compares the Stieltjes pro cedure and an algorithm prop osed by Gragg

and Harro d GH for solving a Jiep This last algorithm b elongs to another class of

algorithms that can b e used to solve a Jiep they are based on a mo dication of a Ruti

shausers algorithm Rut For example the RutishauserKahanPalWalker algorithm

prop osed by Gragg and Harro d GH BG consists in applying a sequence of carefully

chosen orthogonal plane rotations in a given order onto an arrowheaded matrix also called

diagonal b ordered matrix A similar algorithm has b een rst prop osed by BieglerKonig

BK These algorithms are O n algorithms and evidently they compute in exact

arithmetic the same Jacobi matrix J constructed by the RouthLanczos and the Lanczos

algorithms

Finally we must p oint out that an algorithm of the same class has b een recently prop osed

by Ammar and Gragg AG for solving a Biep It uses an ecient pattern of rotations

which provides a stable O n algorithm It seems to b e the rst numerical ly stable algorithm

that can solve a Jiep in O n arithmetic op erations

RR n

R Erra B Philippe

General p olynomials

Complete and incomplete prs

An interesting question arises when p olynomials are not anymore assumed to have real

interlacing ro ots

Problem b ecomes

Problem Let p x and p x be two real monic polynomials of degree n and n

n n1

respectively

Find a tridiagonal matrix J such that

detxI J p x and detxI J p x

n n n n1 n1 n1

where J is the principal submatrix of order n of J

n1 n

We call this second structured inverse eigenvalue problem a Tiep for Tridiagonal

Inverse Eigenvalue Problem

We can still try to compute the prs fp xg with the RouthLanczos algorithm

k k =n0

which corresp onds to the fourth choice of Section

p x p x p x k n

k k k 1 k k 2

In contrast to Jiep the pro cess computing the prs can stop b efore reaching the

remainder of degree one Before exhibiting such situations let us consider the following

denition

Denition Let p x and p x be the two starting polynomials We say that the prs

n n1

is complete when the degree of polynomial p is equal to k and the last computed polynomial

is of degree

A necessary condition to have a complete prs is to start with two relative prime p o

lymomials p and p since the last remainder of the sequence is their monic GCD If

n n1

the p olynomials are not prime the algorithm stops after having computed a multiple of

the GCDp p This situation is the rst case where an incomplete prs is computed

n n1

instead of the complete prs exp ected

But GC D p p is not a sucient condition to insure a complete prs For

n n1

example the sequence dened by p x x and p x gives p x and the

n n1 n2

pro cess stops immediately for any n This situation is the second case of computation

of an incomplete prs This fact was known during the th century but its not clear

if Sturm knew that his algorithm could suer from breakdown for further references see

KN english translation of a work published in Russian in

The preceeding example shows that Problem contrasts drastically with the Jiep We

can p oint out the fact that the matrix C p when p x x is exactly the example

3 3

chosen in PTL to show that Nonsymmetrix Lanczos Algorithms may breakdown INRIA

Inverse Eigenvalue Problems

Similarly to the nonsymmetric Lanczos Algorithm we call happy breakdown the rst

case and serious breakdown the second It is as hard to overcome this situation by cho osing

another p olynomial p as it is for the selection of a go o d pair of initial vectors in the

Lanczos pro cess

A nonsymmetric RouthLanczos Algorithm

If the recurrence gives a complete prs it allows us to dene the tridiagonal matrix

B C

2 2

B C

n1 n1

n n

which is by construction solution of This is the nonsymmetric RouthLanczos algo

rithm

We establish now the connection b etween this algorithm and the nonsymmetric Lanczos

algorithm

We obtained a result which has b een suggested by a similar relation b etween the Fro

b enius companion matrix and the Comrade Matrix in Bar and a Pury and Weygandts

result which states that a Frob enius companion matrix C is similar to the Routh canonical

tridiagonal matrix R through a tranformation matrix which is triangularBS

If p x and p are p olynomials with real co ecients we denote by T the lower

n n1

triangular matrice T comp osed of the co ecients of the prs p x p x pro duced

0 n1

by the RouthLanczos algorithm when there is no breakdown

[1]

B C

[2] [2]

B C

a a

B C

0 1

B C

[n2]

[n1] [n1] [n1]

a a a

0 1 n2

where

k 1

[k ]

j k

a x k n p x x

j =0

Let J b e the tridiagonal matrix obtained in

k k

We have the following

RR n

R Erra B Philippe

Theorem Let p x and p be polynomials of degree n with real coecients If the

n n1

RouthLanczoss Algorithm produces a complete prs the matrices J and T dened as above

are related by the fol lowing similarity transformation

J TCT

where C is the companion matrix C p

Pro of We follow the pro of given in Bar for a similar result Let fx g b e a set

j j =1n

of n distinct and nonzero real numb ers The matrix X dened by X x is therefore

ij j

a nonsingular Vandermonde matrix

We dene for any nonzero real x the vector

p x

C B

p x

C B

P x

C B

p x

A direct computation gives

B C

J P x xP x

B C

p x

and

P x TX

2 n1 t

where X x x x Similarly

B C

C p X xX

B C

p x

and therefore from

C B

J P x JTX TC p X xP x T

C B

p x

since p x is invariant under T

n INRIA

Inverse Eigenvalue Problems

This is true for any nonzero real x and hence

JT X TC X

which ends the pro of 2

To describ e the connections of the RouthHurwitz algorithm with the Lanczos algorithm

we consider here the version of the nonsymmetric Lanczos Algorithm which pro duces the

real tridiagonal matrix with entries set to one on the upp er diagonal

Corollary The relation shows that the matrix J is also obtained by applying the

Lanczos algorithm on the matrix C when taking as left and right initial vectors respectively

the rst row of T and the rst column of T and for which no breakdown occurs

Ill convergence for the Lanczos algorithm

We can state a result for the nonsymmetric case which is much worse than for its symmetric

version

Theorem Let A be a real matrix with real eigenvalues f g For every real

i i=1n

such that or there exist two initial vectors p and q such that if J is

1 n 0 0 n

the real tridiagonal matrix obtained by the nonsymmetric Lanczos Algorithm with initial

vectors p q we have

0 0

spJ

where J is the principal submatrix of J

n1 n

Let us rst state a lemma

Lemma Let p and q be two real monic polynomials We assume that

p is of degree n and its roots f g are simple and real

i i=1n

q is of degree n and its roots f g are simple and real

i i=1n1

we cal l this property the shifted interlacing

1 1 2 n1 n1 n

assumption

Then if r is dened as the third polynomial in the sequence obtained by the RouthLanczos

algorithm applied on p q ie

px x aq x cr x with degree r n

then we have

i r is of degree n and its roots are simple and real

RR n

R Erra B Philippe

ii the roots f g of r strictly interlace the roots of q

i 1n2

Pro of A computation similar to Hal pp insures that

X X

k k j +1 j

j =1

k =1

which proves that c is negative and cannot vanish Moreover since the p olynomials p and

cr are coincident on f g and since fp g is an alternate sequence the ro ots

i 1n1 j 1j 1

of r strictly interlace the ro ots of q 2

Pro of theorem

Let us dene the p olynomials p x detxI A and p x x where

n n1 i

i=1

f g is any set of reals satisfying the shifted interlacing assumption

i 1n1

1 1 2 n1 n1 n

and where Therefore by applying the previous lemma we can claim that the rst step

of the RouthLanczos algorithm denes a p olynomial p of which ro ots strictly interleave

the set f g We are now in the situation where the RouthLanczos algorithm with the

i 1n1

p olynomials p and p computes a complete prs p p p p of order n

n1 n2 n1 n2 1 0

which is a Sturm sequence We notice that the prs p p p p of order n is also

n n1 1 0

complete but is not a Sturm sequence Let J the obtained tridiagonal matrix and T the

triangular matrix containing the co ecients of the p olynomials of the sequence Theorem

insures that the matrices satisfy J TC p T

n n

Let X b e a matrix such that X AX C p The last two equalities imply that

J TX ATX

Now we can conclude by dening the two vectors p and q equal to resp ectively the rst

0 0

column of TX and the rst row of TX as the starting vectors for the Lanczos pro cess 2

This result illustrates that when computing the eigenvalues of a matrix the nonsymmetric

Lanczos Algorithm can lead to a slow convergence even for a symmetric matrix since a p er

ipherical eigenvalue of the tridiagonal matrix of order n can b e arbitrarily far from the

sp ectrum of the original matrix This p o or b ehavior of the nonsymmetric Lanczos algo

rithm is not shared by the Arnoldi algorithm since for the latter the eld of values of the

Hessenb erg matrices are always included in the eld of values of the original matrix Despite

of that Lanczos algorithms are still attractive for their very cheap computational cost

Parametric Representations of a Jacobi matrix

All the previous results lead to the problem of parametric representations of a Jacobi matrix

and more generally of a tridiagonal matrix This problem is dened and studied in Par

We summarise here some of the results INRIA

Inverse Eigenvalue Problems

Theorem Let J a real Jacobi matrix of order n

n k k k +1

Given the n eigenvalues f g of J and the n eigenvalues f g of

i i=1n n i i=1n1

J principal submatrix of J or those of any principal submatrix or order n

n1 n

if the eigenvalues of J interlace those of J then J is uniquely determinated

n1 n n

Given the polynomials p detx I J and p detxI J if the roots

n n n1 n1 n1

of p interlace strictly the roots of p then J is uniquely determinated

n1 n n

Let B a matrix similar to the Jacobi matrix J dened by

Q BQ J

Then Q and J are determinated to within diagonal scaling by the rst or last column

of Q and the rst or last row of Q

Given the n eigenvalues f g of J and the last row of its orthogonal matrix of

i i=1J n n

eigenvectors then J is uniquely determinated

T k

Given the n eigenvalues f g of J and the n values fe J e g where

i i=1n n 1

1 n

k =1

T T

e then J is uniquely determinated

The rst three determinations are consequences of the previous sections the fourth determi

nation is cited in HP and the last is a sp ecial case of theorem in MH We can p oint

out the fact that the ve results involve n free parameters by considering appropriate

normalisations

Conclusion

This pap er deals with two classes of nite algorithms that solve a sp ecial structured inverse

eigenvalue problemSiep given some constraints on a set of real eigenvalues nd a Jacobi

matrix J real symmetric tridiagonal that admits eigenvalues satisfying the constraints We

have called Jiep Jacobi Inverse Eigenvalue Problem the case where the two sets of given

eigenvalues satisfy the interlacing prop erty and Tiep Tridiagonal Inverse Eigenvalue

Problem the problem obtained when the interlacing prop erty is relaxed

The two classes of nite algorithms that have b een considered are

a p olynomial algorithm based on a sp ecial EuclidSturm algorithm Householders

terminology which has b een rediscovered several times

a matrix algorithm which is a symmetric Lanczos algorithm with a sp ecial choice of

the starting vector

Halds theorem insures the equivalence of the two algorithms in exact arithmetic

We have extended the results of the symmetric situation to the nonsymmetric case

by considering general real p olynomials Possible breakdowns may o ccur in the p olynomial

RR n

R Erra B Philippe

algorithm as it may happ en with the nonsymmetric Lanczos algorithm The connection

b etween the two algorithms exhibits a similarity transformation from the Frob enius matrix

to the tridiagonal matrix

This result has b een used to illustrate the fact that when computing the eigenvalues of

a matrix the nonsymmetric Lanczos Algorithm can lead to a slow convergence even on a

symmetric matrix

Acknowledgement

The authors would like to thank Dr MH Gutknecht for the reading of a preliminary version

of this work and for some references

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