The Matrix Sign Function Metho d and the
Computation of Invariant Subspaces
y y
Ralph Byers Chunyang He Volker Mehrmann
November
Abstract
A p erturbation analysis shows that if a numerically stable pro ce
dure is used to compute the matrix sign function then it is comp etitive
with conventional metho ds for computing invariant subspaces Stabil
ity analysis of the Newton iteration improves an earlier result of Byers
and con rms that ill conditioned iterates may cause numerical insta
bility Numerical examples demonstrate the theoretical results
Introduction
n n
If A R has no eigenvalue on the imaginary axis then the matrix sign
function sign A may b e de ned as
Z
sign A z I A dz I
i
where is any simple closed curve in the complex plane enclosing all eigen
values of A with p ositive real part The sign function is used to compute
eigenvalues and invariant subspaces and to solve Riccati and
Sylvester equations The matrix sign function is attractive for
machine computation b ecause it can b e e ciently evaluated by relatively
University of Kansas Dept of Mathematics Lawrence Kansas USA Partial
supp ort received from National Science Foundation grant INT Some supp ort
was also received from University of Kansas General Research Allo cation
y
TU Chemnitz Zwickau Fak f Mathematik D Chemnitz FRG Partial sup
p ort was received from Deutsche Forschungsgemeinschaft Pro jekt La
simple numerical metho ds Some of these are surveyed in It is partic
ularly attractive for large dense problems to b e solved on computers with
advanced architectures
Beavers and Denman use the following equivalent de nition Let
A XJX b e the Jordan canonical decomp osition of a matrix A having
no eigenvalues on the imaginary axis Let the diagonal part of J b e given
by the matrix D diag d d If S diag s s where
n n
if d
i
s
i
if d
i
then sign A is given by sign A XSX
Let V V A b e the invariant subspace of A corresp onding to eigen
values with p ositive real part let V V A b e the invariant subspace of
A corresp onding to eigenvalues with negative real part let P A P b e
the skew pro jection onto V parallel to V and let P P A b e the
skew pro jection onto V parallel to V Using the same contour as in
the pro jection P has the resolvent integral representation Page
Z
P z I A dz
i
It follows from and that sign A P P P I I P
The matrix sign function was introduced using de nition by Rob erts
in a technical rep ort which was not published until Kato
Page rep orts that the resolvent integral go es back to
and
There is some concern ab out the numerical stability of numerical meth
o ds based up on the matrix sign function In this pap er we demon
strate that evaluating the matrix sign function is a more ill conditioned
computational problem than the problem of nding bases of the invariant
subspaces V and V Sometimes it is tremendously more ill conditione d
See Example in Section Never the less we also give p erturbation and
error analyses which show that at least for Newton s metho d for the compu
tation of the matrix sign function in most circumstances the accuracy
is comp etitive with conventional metho ds for computing invariant subspaces
Our analysis improves some of the p erturbation b ounds in
In Section we establish some notation and clarify the relationship b e
tween the matrix sign function and the Schur decomp osition The next
two sections give a p erturbation analysis of the matrix sign function and
its invariant subspaces Section gives a posteriori b ounds on the forward
and backward error asso ciated with a corrupted value of sign S Section
is a stability analysis of the Newton iteration Section demonstrates the
results with some numerical examples
Throughout the pap er k k denotes the sp ectral norm k k the norm
q
P
ja j The or column sum norm and k k the Frobenius norm k k
ij F F
set of eigenvalues of a matrix A is denoted by A The op en left half plane
is denoted by C and the op en right half plane is denoted by C Borrowing
some terminology from engineering we refer to the invariant subspace V
n n
V A of a matrix A R corresp onding to eigenvalues in C as the
stable invariant subspace and the subspace V V A corresp onding to
eigenvalues in C as the unstable invariant subspace We use P P A
for the the skew pro jection onto V parallel to V and P P A for the
skew pro jection onto V parallel to V
Relationship with the Schur Decomp osition
Supp ose that A has the Schur form
k n k
k A A
H
Q AQ
n k A
where A C and A C If Y is a solution of the Sylvester
equation
YA A Y A
then
k n k
k I Y
H
Q sign A Q
n k I
k n k
k I Y
H
Q P Q
n k
and
k n k
Y k
H
Q P Q
n k I
The solution of has the integral representation
Z
z I A A z I A dz Y
i
where is a closed contour containing all eigenvalues of A with p ositive real
part
The stable invariant subspace of A is the range or column space of
sign A I P If
i h
R R
sign A I QR Q Q
is a QR factorization with column pivoting then the columns of Q
form an orthonormal basis of this subspace Here Q is orthogonal is a
p ermutation matrix R is upp er triangular and R is nonsingular
It is not di cult to use the singular value decomp osition of Y to show
that
r
k sign A k kY k kY k
It follows from that
kA k
kY k
sep A A
kA Z ZA k
11 22
F
where sep is de ned as in by sep A A min
Z
kZ k
F
The E ect of Backward Errors
In this section we discuss the sensitivity of the matrix sign function sub
ject to p erturbations Based on Fr echet derivatives Kenney and Laub
presented a rst order p erturbation theory for the matrix sign function via
the solution of a Sylvester equation Mathias derives an expression for
the Fr echet derivative using the Schur decomp osition Kato s encyclop edic
monograph includes an extensive study of series representations and
of p erturbation b ounds for eigenpro jections In this section we derive an
expression for the Fr echet derivative using integral formulas
For a p erturbation matrix E we give estimates for sign A E in terms
of p owers of kE k Partition E conformally with as
k n k
k E E
H
Q EQ
n k E E
Consider rst the relatively simple case in which A is blo ck diagonal
Lemma Suppose A is block diagonal
A
A
A
n n
where A C and A C Partition the perturbation E R
conformally with A as
E E
E
E E
If kE k is su ciently smal l then
F
sign A E sign A O kE k
F
where F and F satisfy the Sylvester equations
A F F A E
F A A F E
Proof Note that the eigenvalues of A E have negative real part
and the eigenvalues of A E have p ositive real part In the de nition
choose the contour to enclose A and A E but neither
A nor A E So
Z
dz I z I A E sign A E
i
Z
z I A z I A E z I A dz I
i
O kE k
sign A F O kE k
where
Z
z I A E z I A dz F
i
Partitioning F conformally with E and A then we have
Z
F z I A E z I A dz
i
Z
z I A E z I A dz F
i
Z
F z I A E z I A dz
i
Z
F z I A E z I A dz
i
As in F and F are the solutions to the Sylvester equations
and The contour encloses no eigenvalues of A so z I
A E z I A is analytic inside and F
We rst prove that F in the case that A is diagonalizable say
A X X where diag Then
n k
Z
F X z I X E X z I dz X
i
R
Each comp onent of the ab ove integral is of the form c z z
j
dz for some constant c If j k then this is the integral of a residue
k
free holomorphic function and hence it vanishes If j k then
Z Z
c c
dz dz
z z z z
i j i j i j
The general case follows by taking limits of the diagonalizable case and using
the dominated convergence theorem
Theorem Let the Schur form of A be given as in and let E be as in
If kE k is su ciently smal l then
sign A E sign A E sign A E sign A O kE k
t p
where
Y E Y
21
E
H
E Q Q
t
E
H
E Q Q
p
E
E satis es the Sylvester equation
A E E A E
Y satis es and E satis es
E Y YE Y YE
E A A E E
Y
I
then Proof If S
I
A A A
S S
A A
and
YE YE E Y YE Y
21 22 11 21
E E E E
S S
E Y
21
E E
E E
It follows from Lemma that
I E
H
O kE k sign SQ A E QS
I
E
Since sign S AS S sign A S multiplying QS on the left side and
H
SQ on the right side of the ab ove equation we have
Y E Y
21
Y E E
H
sign A E sign A Q Q O kE k
E E Y
It is easy to verify that
Y E Y
21
Y E E
E E Y
Y E Y
21
I Y I Y
E
I I
E
E
The theorem follows from
I Y
H H
Q sign A Q sign Q AQ
I
Of course Theorem also gives rst order p erturbations for the pro jec
tions P P A and P P A
Corollary Let the Schur form of A be given as in and let E be as in
If kE k is su ciently smal l then
P A E
P A E sign A E sign A O kE k
t p
E P A P A E P A P A P A
t p
O kE k
P A E P A I E P A I O kE k
t p
where E and E are as in the statement of Theorem
t p
Taking norms in Theorem gives rst order p erturbation b ounds
Corollary Let the Schur form of A be given as in E as in and
let sep A A then the rst order perturbation of the matrix
sign function stated in Theorem is bounded by
kA k
kE sign A E sign A k kE k
t p
The corollary follows from the sum of the ab ove b ounds
On rst examination Corollary is discouraging It shows that calculat
ing the matrix sign function may b e more ill condi tioned than nding bases
of the stable and unstable invariant subspace If the matrix A whose Schur
decomp osition app ears in is p erturb ed to A E then the stable invari
ant subspace Im Q is p erturb ed to Im Q Q E where kE k kE k
q q
Corollary and the following example show that k sign A E k
may di er from sign A by a factor of which may b e much larger than
kE k
Example Let
A
E
The matrix A is already in Schur form so sep A A If
then we have
sign A
p
sign A E
The di erence is
sign A E sign A O
Perturbing A to A E do es indeed p erturb the matrix sign function by a
factor of
Of course there is no rounding error in Example so the stable invariant
subspace of A E is also the stable invariant subspace of sign A E
and in particular evaluating the matrix sign function exactly has done no
more damage than p erturbing A The stable invariant subspace of A is
V A Im the stable invariant subspace of A E and sign A E
is
O V A E Im Im
p
2
For a general small p erturbation matrix E the angle b etween V A
and V A E is of order no larger than O The matrix
sign function and the projections P and P may be signi cantly more
il l conditioned than the stable and unstable invariant subspaces Never the
less we argue in the next section that despite the p ossible p o or conditioning
of the matrix sign function the invariant subspaces are usually preserved
ab out as accurately as their native conditioning p ermits
Perturbation Theory for Invariant Subspaces
of the Matrix Sign Function
In this section we discuss the accuracy of the computation of the stable
invariant subspace of A via the matrix sign function
An easy rst observation is that if a backward stable metho d was used
to compute the matrix sign function then the computed value of sign A is
the exact value of sign A E for some p erturbation matrix E prop ortional
to the precision of the arithmetic The exact stable invariant subspace of
sign A E is also an invariant subspace of A E
However in general we can not guarantee that the computed value of
sign A is exactly the value of sign A E for a small p erturbation E
We probably can not even represent such sign A E within the limits of
nite precision arithmetic The b est that can b e hop ed for is to compute
sign A E F for some small p erturbation matrices E and F Consider
now the e ect of the hypothetical forward error F
Let A have Schur form and let E b e a p erturbation matrix partitioned
conformally as in Let Q b e the rst k columns of Q and Q b e the
remaining n k columns If
kE k kA k kE k
sep A A kE k kE k
then A has stable invariant subspace V A Im Q and A E has an
invariant subspace Im Q Q W where W satis es
kE k
kW k
sep A A kE k kE k
The singular values of W are the tangents of the canonical angles
b etween V Im Q and Im Q Q W In particular the canonical
angles are at most of order sep A A
For simplicity of notation ignore for the moment the backward error
matrix E and consider only the forward error Let B sign A F where
F represents the forward error in evaluating the matrix sign function and A
has Schur form Let sign A and F have the forms
k n k
k I Y
H
Q sign A Q
n k I
and
k n k
k F F
H
Q FQ
n k F F
where Q is the unitary factor from the Schur decomp osition of A Now
consider the stable invariant subspace V A V sign A Im Q
If kF k kY k kF k sep I I kF k kF k then p erturbing
sign A to sign A F p erturbs the invariant subspace Im Q to Im Q
Q W where kW k kF k kF k kF k If kF k
s s
k sign A k then by and
kF k k sign A k
s
kY k
A
kY k
kA k
sep A A
Since sep A A kAk W ob eys the b ounds
F s
kA k
12
sep A A
11 22
kW k
s
kF k kF k
kAk
F
O
sep A A
Comparing with we see that p erturbing the computed value of
sign A by a relative error to a nearby sign matrix disturbs the stable
invariant subspace no more than twice as much as p erturbing the original
data A by a relative error of size might
In order to illustrate the results we give a comparison of our p erturbation
b ounds and the b ounds given by Bai and Demmel for b oth the matrix
sign function and the invariant subspaces in the case of Example The
distance to the ill p osed problem
d min A iI
A min
where A iI is the smallest singular value of A iI in which
min
p
is real and i leads to overestimating b ounds in Since d
A
the b ounds given in are resp ectively O for the matrix sign function
and O for the invariant subspaces
A Posteriori Backward and Forward Error
Bounds
A priori backward and forward error b ounds for evaluation of the matrix
sign function remain elusive even for the simplest algorithms However it
isn t di cult to derive a p osteriori error b ounds for b oth backward and
forward error
We will need the following lemma to estimate the distance b etween a
matrix S and sign S
n n
Lemma If S R has no eigenvalue with zero real part and
k sign S S I k then k sign S S k kS S k
Proof Let F sign S S The matrices F S and sign S commute
so
I sign S S F S SF F
This implies that
S S S F
F
Taking norms and using kFS k k sign S S I k we get
kS S k kF k kF k
and the lemma follows
It is clear from the pro of of the Lemma that sign S S S S
is asymptotically correct as k sign S S k tends to zero The b ound in the
lemma tends to over estimate smaller values of k sign S S k by a factor of
two
Supp ose that a numerical pro cedure for evaluating sign A applied to a
n n n n
matrix A R pro duces an approximation S R Consider nding
n n n n
small norm solutions E R and F R to
sign A E S F
Of course E and F are not uniquely determined by Common algo
rithms for evaluating sign A like Newton s metho d for the square ro ot of I
guarantee that S is very nearly a square ro ot of I i e S is a close ap
proximation of sign S In the following theorem we have arbitrarily taken
F sign S S
Theorem If k sign S S I k then admits a solution with
kF k kS S k and
kSA AS k kE k
kS S k
kAk kAk
The right hand side of is easily computed or estimated from the known
values of A and S but it is sub ject to subtractive cancellation of signi cant
digits
Proof The matrices S F and A E commute so an underdetermined
consistent system of equations for E in terms of S A and F sign S S
is
E S F S F E sign S A A sign S SA AS FA AF
Let
I Y
H
U sign S U
I
b e a Schur decomp osition of sign S whose unitary factor is U and whose
H
triangular factor is on the left hand side of Partition U EU and
H
U AU conformally with the right hand side of as
E E
H
U EU
E E
and
A A
H
U AU
A A
H
Multiplying on the left by U and on the right by U and partitioning
gives
YE E Y YE E YA A Y YA A
E E Y A A Y
A hop efully small norm solution for E is
A Y YA A E E
H
U EU
A E E
For this choice of E we have
kE k k sign S A A sign S k
kSA AS k kFA AF k
kSA AS k kS S k kAk
from which the lemma follows
Lemma and Theorem agree well with intuition In order to assure
small forward error S must b e a go o d approximate square ro ot of I and in
addition to assure small backward error S must nearly commute with the
original data matrix A Newton s metho d for a ro ot of X I tends to do
a go o d job of b oth Note that in general Newton s metho d makes a
p o or algorithm to nd a square ro ot of a matrix The square ro ot of I is a
sp ecial case See for details
The Newton Iteration for the Computation of
the Matrix Sign Function
There are several numerical metho ds for computing the matrix sign func
tion Among the simplest and most commonly used is the Newton
Raphson metho d for a ro ot of X I starting with initial guess X A
It is easily implemented using matrix inversion subroutines from
widely available high quality linear algebra packages like LAPACK
It has b een extensively studied and many variations have b een suggested
Algorithm Newton Iteration without scaling
X A
FOR k
X X X
k k
k
If A has no eigenvalues on the imaginary axis then Algorithm converges
globally and lo cally quadratically in a neighborho o d of sign A Al
though the iteration ultimately converges rapidly initially convergence may
b e slow However the initial convergence rate may b e improved by scaling
Theorem shows that the rst order p erturbation of sign A may b e as
large as k sign A k where is the relative uncertainty in A If there is no
other uncertainty then is at least as large as the unit round of the nite
precision arithmetic Thus it is reasonable to stop the Newton iteration
when
kX X k c kX k
k k k
The ad hoc constant c is chosen in order to avoid extreme situations e g
c n Exp erience shows furthermore that it is often advantageous to
take an extra step of the iteration after the stopping criterion is satis ed
In exact arithmetic the stable and unstable invariant subspaces of the
iterates X are the same as those of A However in nite precision arith
k
metic rounding errors p erturb these subspaces The numerical stability of
the Newton iteration for computing the stable invariant subspace has b een
analyzed in we give an improved error b ound here
Let X and X b e resp ectively the computed k th and k st iterate
of the Newton iteration starting from
A A
H
X A Q Q
A
Supp ose that X and X have the form
X X X X
H H
X Q Q X Q Q
E X E X
A successful rounding error analysis must establish the relationship b etween
and E In order to do so we assume that some stable algorithm is E
applied to compute the inverse X in the Newton iteration More precisely
we assume that X satis es
X E X E
X X
E X
Z
where
kE k c kX k
X
kE k c kX k kX k
Z
for some constant c Note that this is a nontrivial assumption Ordinarily
if Gaussian elimination with partial pivoting is used to compute the inverse
the ab ove error b ound can b e shown to hold only for each column separately
Write E and E as
X Z
E E
H
E Q Q
X
E E
E E
H
E Q Q
Z
E E
The following theorem b ounds kE k and indirectly the p erturbation in the
stable invariant subspace
Theorem Let X X E and E be as in and If
X Z
k k and c kX kkX c kX kkX
k kX k c kX k kkX kE k c kX k kX
where c is as in then
k kkX kX
k kE kE k c kX k c kX k kX k
Proof We start with In fact the relationship b etween E and E
follows from applying the explicit formula for the inverse of X E in
X
H
Q X E Q
X
X X X X X E E X X X
c c
X X E E X
c c
Here
X X E
X X E
X X E
X X X E E X
c
Then
X X E E X E X X X X
c
X X X E X X
c
E E E E X E E X
c
X X X E
c
Using the Neumann lemma that if kB k then k I B k kB k
we have
k kX k kX
kX k kX k
kX kkE k c kX kkX k
The following inequalities are established similarly
kX k kX k
kX k kX k c kX k
k kX k kX
kX k
c
kkX k k kE k kE k kX kX
Inserting these inequalities in we obtain
k kkX kX
k kE kE k c kX k kE k
The b ound in Theorem is stronger than the b ound of Byers in A
step of Newton iteration is backward stable if and only if
kE k kE k
sep X X
sep X X
is dominated by X The term sep X
X X X X
sep
In order to guarantee numerical stability the factors in the b ound of Theo
k and kX k kX k should b e not so large as to violate kkX rem kX
the inequality
1 1
X X X X
22 11
22 11
sep
kE k k kE
sep X X
Roughly sp eaking to have numerical stability throughout the algorithm
neither kX kkX k nor kX k kX k should b e much larger than
sep A A
The following example from shows violation of inequality which
explains the numerical instability
Example Let
A
A A
T
b e a real matrix and let A A Form R
E A
T
and A QRQ where the orthogonal matrix Q is chosen to b e the unitary
factor of the QR factorization of a matrix with entries chosen randomly
uniformly distributed in the interval The parameter is taken as
so that there are two eigenvalues of A close to the imaginary
axis from the left and right side The entries of A are chosen randomly
uniformly distributed in the interval to o The entries of E are chosen
randomly uniformly distributed in the interval eps where eps
is the machine precision
In this example sep A A and A
min
The following table shows the evolution of kE k sep X X
during the Newton iteration starting with X A and X R resp ectively
where E is as in The norm is taken to b e the norm
k kE k sep X X sep X X
A R
e e e
e e e
e e e
e e e
e e e
e e
e e
e e
e e
e e
e e
e e
e e
e e
e e
e e
e e
e e
e e
e e
e e
e e
k kA k inequality k kA Because kA
is violated in the rst step of the Newton iteration for starting matrix
A which is shown in the rst column of the table Newton s metho d never
recovers from this
It is remarkable however that Newton s metho d applied to R directly
seems to recover from the loss in accuracy in the rst step The second
column shows that although kE k sep X X at the
rst step it is reduced by the factor every step until it reaches
which is approximately kE k sep A A Observing that in this
case the p erturbation E in E as in is zero and kE k is dominated by
Z
k It is surprising to see that from the second step E X kE k kX
on kX E X k is as small as eps since A and A do not explicitly
after the rst step E X app ear in the term X
By our analysis the Newton iteration may b e unstable when X is ill
k
conditioned To overcome this di culty the Newton iteration may b e carried
out with a shift along the imaginary line In this case we have to use complex
arithmetic
Algorithm Newton Iteration With Shift
X A iI
FOR k
X X X
k k
k
END
The real parameter is chosen such that A iI is not small
min
For Example when is taken to b e we have kE k sep X X
for k Then by our analysis the computed invariant
subspace is guaranteed to b e accurate
We can combine Algorithm and Algorithm in the following way
Algorithm Computing the Stable Invariant Subspace
Cal l Algorithm with the stopping criterion and get X
k
Perform a QR factorization of X I and partition Q Q Q
k
H
Stability test If k Q AQ k kAk n kX k where is the
k
machine precision then use Q as the orthonormal basis of the computed
stable invariant subspace Otherwise call Algorithm and start the step
again
Numerical Exp eriments
In this section we demonstrate the theoretical results of this pap er with
some numerical exp eriments The numerical algorithms were implemented
in MATLAB on a HP workstation with eps
The stopping criterion for the Newton iteration is as in with c
and an extra step of the Newton iteration is p erformed after the stopping
criterion is satis ed
Example This example is devoted to demonstrate the validity of our
stopping criterion We constructed a matrix
H
A QRQ
where Q is a random unitary matrix and R an upp er triangular matrix
with diagonal elements i i
a parameter in the k k p osition and zero everywhere else We
chose such that the norm k sign A k varies from small to large The
typical numerical b ehavior of log kX X k is that it go es down
k k
and then b ecomes staionary This b ehavior is shown in the following graph
for the cases and in which k sign A k is
and resp ectively The Newton iteration with our
stopping criterion stops at the th step for and at the th step for
and 10
5
0 alpha=50
-5 alpha=20
log10(||X_{k+1}-X_k||_1) -10
alpha=0 -15
-20 0 5 10 15 20 25 30 35 40 45 50
the number of iterations
Example In this test random matrices of size were consid
H
ered In all cases the condition kQ A Q k kAk n eps kX k is
k
satis ed which indicates by the p erturbation analysis in Section that the
computed stable invariant subspace is acceptable
H
In Example however we have kQ A Q k kAk n
eps kX k and hence the computed stable invariant
k
subspace is not acceptable However when the Newton iteration is started
with X A iI the stability condition is satis ed
Conclusions
We have given a rst order p erturbation theory for the matrix sign function
and an error analysis for Netwon s metho d to compute it This analysis
suggests that computing the stable or unstable invariant subspace of a
matrix with the Newton iteration in most circumstances yields results as
go o d as those obtained from the Schur form
Acknowledgments
The authors would like to express their thanks to N Higham for valuable
comments on an earlier draft of the pap er and Z Bai and P Benner for
helpful discussions
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