Logarithmic residue based methods for
computing zeros of analytic functions
and related problems
P Kravanja M Van Barel and A Haegemans
Department of Computer Science Katholieke Universiteit Leuven
Celestijnenlaan A B Heverlee Belgium
Abstract
Given an analytic function f and a p ositively oriented Jordan curve we consider the
problem of computing all the zeros of f that lie inside together with their resp ective mul
tiplicities By exploiting the connection b etween logarithmic residue integrals and the theory
of formal orthogonal p olynomials we obtain an accurate algorithm that pro ceeds by solving
generalized eigenvalue problems and a Vandermonde system We show how similar techniques
can b e applied to the following problems computing zeros and p oles of meromorphic functions
and lo cating clusters of zeros of analytic functions
Intro duction
Let W b e a simply connected region in C f W C analytic in W and a p ositively oriented
Jordan curve in W that do es not pass through any zero of f We consider the problem of computing
al l the zeros of f that lie in the interior of together with their resp ective multiplicities
Let N denote the total numb er of zeros of f that lie in the interior of i e the numb er of zeros
where each zero is counted according to its multiplicity Supp ose that N Let the sequence
Z Z consist of all the zeros of f that lie inside Each zero is rep eated according to its
N
0
f has a simple p ole at multiplicity An easy calculation shows that the logarithmic derivative f
each zero of f with residue equal to the multiplicity of the zero It follows that
Z
0
f z
dz N
i f z
and thus N can b e calculated via numerical integration Integrals of this typ e are called logarithmic
residue integrals Metho ds for the determination of zeros of analytic functions that are based on the
numerical evaluation of integrals are called quadrature methods A review of such metho ds is given
in A classical approach is to consider the integrals
Z
0
f z
p
s dz p z
p
i f z
Corresp onding author E mail Peter Kravanja na net ornl gov
The residue theorem implies that the s s are equal to the Newton sums of the unknown zeros
p
p p
p Z s Z
p
N
Delves and Lyness considered the monic p olynomial P z of degree N that has zeros Z Z
N N
and calculated its co e cients via Newton s identities In this way they reduced the problem to the
easier problem of computing the zeros of a p olynomial Unfortunately the map from the Newton
sums to the co e cients in the standard monomial basis of P z is usually ill conditioned Also
N
the p olynomials that arise in practice may b e such that small changes in the co e cients pro duce
much larger changes in some of the zeros The lo cation of the zeros determines their sensitivity to
p erturbations of the co e cients Multiple zeros and very close zeros are extremely sensitive but
even a succession of mo derately close zeros can result in severe ill conditioning The comp osite map
from the Newton sums to the zeros and their resp ective multiplicities is also usually ill conditioned
A similar approach was taken by Li who considered as a system of p olynomial equations
Li used a homotopy continuation metho d to solve this system
What is wrong with these approaches in our opinion is that they consider the wrong set of
unknowns One should consider the mutually distinct zeros and their resp ective multiplicities sep
arately Let n denote the numb er of mutually distinct zeros of f that lie in the interior of Let
z z b e these zeros and their resp ective multiplicities In Section we will show how
n n
these unknowns can b e calculated by solving generalized eigenvalue problems and a Vandermonde
system Our approach exploits the connection b etween logarithmic residue integrals and the theory
of formal orthogonal p olynomials In Section we show how similar techniques can b e applied to
the following problems computing zeros and p oles of meromorphic functions and lo cating clusters
of zeros of analytic functions
Formal orthogonal p olynomials
Let P b e the linear space of p olynomials with complex co e cients We de ne a symmetric bilinear
form h i P P C by setting
Z
n
0
X
f z
z z dz z z h i
k k k
i f z
k
for any two p olynomials P Note that h i can b e evaluated via numerical integration
A monic p olynomial of degree t that satis es
t
k
hz z i k t
t
is called a formal orthogonal polynomial FOP of degree t Observe that condition is void for
t The adjective formal emphasizes the fact that in general the form h i do es not de ne a
true inner pro duct An imp ortant consequence of this fact is that in contrast to p olynomials that
are orthogonal with resp ect to a true inner pro duct FOPs need not exist or need not b e unique
t
for every degree t For details see for example and the references cited therein If is satis ed
and is unique then is called a regular FOP and t a regular index If we set
t t
t t
z u u z u z z
t t t t t
then condition translates into the Yule Walker system
s s s
s u
t
t t
s u
s
t t
s u
t t t s s
t t
t
Hence the regular FOP of degree t exists if and only if the Hankel matrix H s is
t k l
k l
regular Thus the rank pro le of H s determines which regular FOPs exist
k l k l
Theorem n rank H for every integer p In particular n rank H
n p N
Proof Let p b e a nonnegative integer As
n
X
p
p
s h z i z
p k
k
k
for p the matrix H can b e written as
n p
n
h i
X
T
n p
T
H z z where z
z z
n p k k k
k
k
k
k
This implies that rank H n However H is regular Indeed one can easily verify that H can
n p n n
n r T
and D is the where V is the Vandermonde matrix V z b e factorized as H V D V
n n n n n n
r s s n
diagonal matrix D diag Therefore rank H n It follows that rank H n 2
n n n p n p
By assumption s N and thus the regular FOP of degree exists It is given by z z
where s s Theorem implies that the regular FOP of degree n exists and tells us also
n
that regular FOPs of degree larger than n do not exist The p olynomial is easily seen to b e
n
n
Y
z z z
k n
k
It is the monic p olynomial of degree n that has z z as simple zeros
n
If H is strongly regular i e if all its leading principal submatrices are regular then we have a
n
full set f g of regular FOPs
n
What happ ens if H is not strongly regular By lling up the gaps in the sequence of existing
n
1
with a monic p olynomial of degree t regular FOPs it is p ossible to de ne a sequence f g
t t
t
such that if these p olynomials are group ed into blo cks according to the sequence of regular indices
then p olynomials b elonging to di erent blo cks are orthogonal with resp ect to h i More precisely
1
de ne f g as follows If t is a regular index then let b e the regular FOP of degree t Else
t t
t
de ne as where r is the largest regular index less than t and is an arbitrary monic
t r t r t r
t r
p olynomial of degree t r In the latter case is called an inner polynomial If z z then
t t r
1
can b e we say that is de ned by using the standard monomial basis These p olynomials f g
t t
t
group ed into blo cks Each blo ck starts with a regular FOP and the remaining p olynomials are inner
p olynomials Note that the last blo ck has in nite length
The so called block orthogonality prop erty is expressed by the fact that the Gram matrix G
n
n
h i is blo ck diagonal The diagonal blo cks are regular symmetric and zero ab ove the
r s
r s
main antidiagonal If all the inner p olynomials in a certain blo ck are de ned by using the standard
monomial basis then the corresp onding diagonal blo ck has Hankel structure The matrix G
n
n
h i is blo ck tridiagonal The diagonal blo cks are symmetric and lower anti Hessenb erg
r s
r s
i e its entries are equal to zero along all the antidiagonals that lie ab ove the main antidiagonal
except for the antidiagonal that precedes the main antidiagonal Again if all the inner p olynomials
in a certain blo ck are de ned by using the standard monomial basis then the corresp onding diagonal
blo ck is a Hankel matrix The entries of the o diagonal blo cks are all equal to zero except for the
entry in the south east corner For pro ofs and further details we refer to
Theorem The eigenvalues of the pencil G G are z z In other words they
n n
n
are given by z z where s s
n
n n
Proof De ne V as the Vandermonde like matrix V z and let D and D b e
n n r s n
r s
n
the diagonal matrices D diag and D diag z z Then G
n n n n n
n
T T
and G can b e factorized as G V D V and G V D V Let b e an eigenvalue of the
n n n n
n n n n n
p encil G G and x a corresp onding eigenvector Then
n
n
G x G x
n
n
T T
V D V x V D V x
n n n
n n n
T
x y D y if y V D
n
n n
diag z z y y
n
This proves the theorem 2
If n and are known then we can apply the previous theorem to obtain z z
n n
By Theorem the value of n can b e computed as the rank of H but this is not a very practical
N
1
approach Instead we will start to compute the p olynomials f g one by one and determine the
t
t
value of n as the degree of the last existing regular FOP The upp er b ound N will play a crucial role
in the computations
Theorem can b e interpreted as follows the zeros of the nth degree regular FOP can b e calculated
by solving a generalized eigenvalue problem This prop erty holds for al l regular FOPs This will
enable us to compute regular FOPs in their pro duct representation which is numerically very stable
as De ne the matrices G and G
k
k
k k
and G h i G h i
r s k r s
r s k r s
for k
Theorem Let t be a regular index and let z z be the zeros of the regular FOP
t t t t
Then the eigenvalues of the pencil G G are given by z z In other words they
t t t t
t
are given by z z where s s
t t t
t
We will rst show that the zeros of s as H Proof De ne the Hankel matrix H
t k l
k l t t
are given by the eigenvalues of the p encil H H The zeros of are given by the eigenvalues of
t t
t
its companion matrix C Let b e an eigenvalue of C and x a corresp onding eigenvector As H is
t t t
regular we may conclude that C x x H C x H x Using one can easily verify that
t t t t
H C H
t t t
Let A b e the unit upp er triangular matrix that contains the co e cients of
t t
T
Then G can b e factorized as G A H A As z z where s s the matrix G
t t t t
t
t
t t
T
is given by h z i G The matrix h z i can b e written as A H A and thus
r s t r s t
r s r s
t t
T
G A H H A Now let b e an eigenvalue of the p encil H H and x a corresp onding
t t t
t
t t t
eigenvector Then
H x H x
t
t
H H x H x
t t
t
T T
A H H A y A H A y if y A x
t t t t
t
t t t
G y G y
t
t
This proves the theorem 2
Regular FOPs are characterized by the fact that the determinant of a Hankel matrix is di erent
from zero while inner p olynomials corresp ond to singular Hankel matrices By using an explicit
determinant expression for regular FOPs one can show that h i det H det H Therefore
t t t t
if t is a regular index then t is a regular index if and only if h i However from a
t t
numerical p oint of view a test is equal to zero do es not make sense Because of rounding errors in
the evaluation of h i we would encounter only regular FOPs Strictly sp eaking one could say that
inner p olynomials are not needed in numerical calculations However the opp osite is true Let us
call a regular FOP wel l conditioned if its corresp onding Yule Walker system is well conditioned
and il l conditioned otherwise To obtain a numerically stable algorithm it is crucial to generate only
well conditioned regular FOPs and to replace ill conditioned regular FOPs by inner p olynomials
Stable lo ok ahead solvers for linear systems of equations that have Hankel structure are based on
this principle In this approach the diagonal blo cks in G are taken slightly larger than
n
strictly necessary to avoid ill conditioned blo cks A disadvantage is that part of the structure of G
n
and G gets lost and that there will b e some additional ll in
n
We will ask the user for two thresholds and with to decide whether
stop cond stop cond
the algorithm may stop or not and to determine the size of a blo ck Supp ose that the algorithm
has just generated a well conditioned regular FOP If r N then we may stop Else we
r
pro ceed to calculate h i If jh ij then we generate as a regular FOP i e
r r r r cond r
we solve a generalized eigenvalue problem to obtain the zeros of Else we scan the sequence
r
N r
t t
jhz ij If jhz ij for t N r then we conclude that n r
r r r r stop
t
and stop Else we search for the rst element that is larger than The corresp onding value of t
cond
then determines the size of the blo ck of p olynomials If all the elements are less than then we
cond
use the value of t that corresp onds to the maximum to determine the blo ck size and warn the user
that we could not obtain the level of well conditioning that he or she requested
Once approximations for the zeros have b een calculated the multiplicities can b e found by solving
the Vandermonde system
s
s z z
n
n
n
s z z
n n
n
A Fortran implementation of our algorithm is available Numerical tests show that it gives very
accurate results Note that it do es not require initial guesses for the zeros Also as not only
approximations for the zeros are computed but also the corresp onding multiplicities one can use the
quadratically convergent mo di ed Newton s iteration to re ne the approximations for the zeros
Related problems
Computing zeros and p oles of meromorphic functions
Let W and b e de ned as ab ove and let f W C b e meromorphic in W Supp ose that f has
neither zeros nor p oles on We consider the problem of computing al l the zeros and p oles of f that
lie in the interior of together with their resp ective multiplicities and orders Let N denote the
total numb er of zeros of f that lie inside and let P denote the total numb er of p oles of f that lie
inside Supp ose that N P Let n denote the numb er of mutually distinct zeros of f that lie
inside Let z z b e these zeros and their resp ective multiplicities Let p denote the
n n
numb er of mutually distinct p oles of f that lie inside Let y y b e these p oles and
p p
0
their resp ective orders The logarithmic derivative f z f z has a simple p ole at z with residue
k k
for k n and a simple p ole at y with residue for l p It follows that
l l
Z
p
n
0
X X
f z
dz z z z z y y h i
k k k l l l
i f z
k l
for any two p olynomials P The FOP of degree n p with resp ect to this form is given by
p
n
Y Y
z z z y
k l
k l
Our algorithm for computing zeros of analytic functions is essentially an algorithm for computing
zeros of FOPs Provided that an upp er b ound for n p is known one can easily verify that the
algorithm can b e mo di ed to compute the zeros of i e the mutually distinct zeros and p oles
of f that lie inside Once these have b een calculated their corresp onding multiplicities and orders
can b e calculated by solving a Vandermonde system The sign of a comp onent of the solution of this
systems tells us whether the corresp onding zero of the FOP of degree n p is a zero of f or a p ole
As s h i N P can b e computed via numerical integration and n p N P s P
it follows that an upp er b ound for P immediately leads to the required upp er b ound for n p In
case is the unit circle such a b ound for P can b e obtained by using the heuristic approach of
Gleyse and Kaliaguine
Lo cating clusters of zeros of analytic functions
Supp ose that f is analytic and supp ose that the zeros of f that lie inside can b e group ed into m
clusters In this case the problem of computing all the zeros of f that lie inside is notoriously
ill conditioned Let I I b e index sets that de ne these clusters and let
m
X X
z and c
k k j k j
j
k 2I k 2I
j j
for j m In other words is equal to the total numb er of zeros that form cluster j
j
its weight whereas c is equal to the arithmetic mean of the zeros in cluster j its centre of
j
gravity We will show how our algorithm for computing zeros of analytic functions can b e used
to obtain approximations for the centres c c of the clusters together with the corresp onding
m
weights This information enables one to zo om into a certain cluster its zeros can b e
m
calculated separately from the other zeros of f By shifting the origin in the complex plane to
the centre of a certain cluster its zeros b ecome b etter relatively separated which is appropriate in
oating p oint arithmetic and reduces the ill conditioning
For k n we de ne z c if k I From the de nition of and c it follows that
k k j j j j
X
j m
k k
k 2I
j
De ne the symmetric bilinear form h i by
m
m
X
c c h i
j j j m
j
for any two p olynomials P This form is related to the form h i in an obvious way instead
of the zeros z z and their multiplicities we now use the centres of gravity c c
n n m
and the weights of the clusters Let
m
max j j
k
k n
The following theorem tells us that h i approximates h i and vice versa
m
Theorem Let P Then h i h i O
m
Proof This follows immediately by expanding the terms in
n m
X X X
h i z z c c
k k k k j k j k
j
k k 2I
j
into a Taylor series and by taking into account equation 2
In other words if is su ciently small then we can exp ect our algorithm for computing zeros of
analytic functions to b ehave as if the underlying symmetric bilinear form is h i instead of h i
m
The FOP of degree m will b e a regular FOP and its zeros will b e go o d approximations for the centres
c c The FOPs of degree larger than m will b e ill conditioned The following result con rms
m
this intuition
Corollary The matrix H is regular if Let t m Then det H O
m t
Let z be the FOP of degree m with respect to the form h i Then c O
m m j
p
for j m Also hz z i O for al l p m
m
Once approximations for the centres have b een found the corresp onding weights can b e calculated
by solving a Vandermonde system For more details including a pro of of the previous corollary
numerical examples as well as a di erent approach based on rational interp olation at ro ots of unity
we refer to
Acknowledgements
This research was supp orted by the Fund for Scienti c Research Flanders FWO V pro jects Or
thogonal Systems and their Applications grant G and Counting and computing all
isolated solutions of systems of nonlinear equations grant G and by the Flemish Insti
tute for the Promotion of Scienti c and Technological Research in the Industry IWT
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