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Decay Rates of Inverses of Banded M-Matrices that are Near to Toeplitz Matrices
Victor Eijkhout* and Ben Pohnanf
Department of Mathematics University of Nijmegen The Netherlatw!~
Submitted by Richard A. Bmaldi
ABSTRACT
The decaying behavior of inverses of positive definite band matrices is analysed for M-matrices that are in some sense close to Toeplitz matrices. Estimates based on the factorization are derived that are better than existing ones, in particular for nonsymmetric matrices. Some examples are given.
1. INTRODUCTION
It is a well-known fact that inverses of positive definite (from here on abbreviated as “pd”) band matrices exhibit an exponential decay of their elements away from the main diagonal. More precisely, if A is a pd band matrix (i.e., Aij = 0 for j > i + p, and j<i-p_ where p_,p+>,O),there exist constants c, X z 0 such that for all i, j
IA;‘1 < chli-jl.
These constants are determined by the bandwidth and the spectrum of A (in particular, if the spectral condition number of A is large, the decay factor X
*Sponsored by Control Data Corporation through its PACER fellowship program. ‘Sponsored by the Dutch Organisation for the Advancement of Pure Research (ZWO).
LINEAR ALGEBRA AND ITS APPLZCATZONS 109247-277 (1988) 247
(0 ELsevier Sdence Publishing Co., Inc., 1988 655 Avenue. of the Americas, New York, NY 10010 00%4-3795/88/$3.50
248 VICTOR EIJKHOUT AND BEN POLMAN
may be so near to one as to make the decay indiscernible), but are indepen- dent of its order. This formula, however, fails to account for two facts:
(1) in many cases the sequences j suggesting an .expression like
c, A;’ have an oscillatory character,
k=l
for some integer I; (2) if the matrix is nonsymmetric (in particular if p _ f p + ), decay rates
are not equal in the lower and upper triangular part of the inverse. In this paper we consider the decay of inverses of banded M-matrices
satisfying some additional constraint (such matrices appear as the pivot blocks during incomplete block factorizations [2]). It will be proved that under this constraint (which is satisfied by diagonally dominant Toeplitz M-matrices) to a given matrix X there exists a Toeplitz matrix 2 = ( zi _ j) such that:
(1) ZZX-‘20 (by >, >, etc. we mean entrywise comparison throughout this paper); in some sense this Z is the best approximating Toeplitz matrix (see Section 3.1). In the case that X is a Toeplitz matrix we give a theorem proving the accuracy of this approximation. We thus expect the theory developed here to be of value for matrices close enough to being Toeplitz matrices.
(2) There exist cl+) ,..., ciz), cl-) ,..., cir), &+) ,..., ti;+), A’;),. . ., x’[_) (generally complex) such that
P+
j > i: zi_i = c ci+)j$yi, k=l
P-
j<i: zi_i = C ci-)x(i)i-j, I
k-l
In Section 2 we show how this approximation is generated, and we give estimates for the coefficients involved; in Section 3 we will discuss how well 2 approximates X- ’ b y giving some limit theorems and showing a couple of numerical illustrations.
INVERSES OF BANDED M-MATRICES 249
2. APPROXIMATION OF INVERSES
In this section we start of by giving a precise definition of the intended class of matrices in terms of a set of coefficients deriving from Gaussian factorization (Section 2.1); these constants are estimated for Toeplitz matrices in Section 2.2. In Section 2.3 we generate from them the approximation to the inverse; its decay is considered in Sections 2.4, 2.5, and 2.6.
2.1. Factorization We define the class of factored Toeplitz&ounded M-matrices (or FI’BM-
matrices) as those M-matrices the factors of which are bounded by diagonally dominant Toephtz forms.
DEFINITION 2.1. A banded M-matrix X (xii=0 for j > i + P+ and j < i - p_, where p_ , p, > 0) is called an FTBiU-mutrix if, when factored
X=(D-G)D-‘(D-H)
with D diagonal, G strictly lower, and H strictly upper triangular nonnega- tive matrices, there exist constants d,, g, ,..., gP_, h, ,..., h,+ such that
O<d,,<Dii Vi, (14
j=l ,...,p_: gj Z Gi+j,i W (lb)
j=l ,***, p+: hi>, Hi,i+j Vi (lc)
satisfying
d,- f&z,-, d,- E h,>O. i-1 i-l
(14
If we consider the family {Xc”)} of n X n banded Toeplitz matrices with a given (fixed) set of coefficients, their factorizations are increasingly “Toeplit&lce” in an obvious sense. This suggests taking for d,, g,, . . . , h,+
the limit values along the bands of the factors. We wilI prove that if the Xc”) are diagonaIly dominan t, this choice (which is the sharpest one independent
250 VICTOR EIJKHOUT AND BEN POLMAN
of n) satisfies (la,b,c,d), and we show a relation between the diagonal dominance of Xc”) and the set d,, g,, . . . , h,+.
LEMMA 2.2. The class of FTBM-matrices contains the diagonally domi- nant Toeplitz matrices that are M-matrices; if
x = toeplitz( - x+ ,...) - x-1, x0, - x1,. ..) - XP+ )
=(D-G)LT1(D-H),
then there is a choice of d,, g, ,..., g,_, h, ,..., h,+ satisfying (la,b,c,d) such that
xo - iFoxi = (do - Cgi)d,‘(do - Chi).
Proof. Gaussian elimination gives the following recurrences for the elements of D, G, H:
min(p-, P+ ) Dii = xii - C Gi,i-kDi_\,i-kHi-k,i; (24
k=l
for k=p_ ,..., max(p_,p+) we have Gi+k,i=Xi+k,i; for k=l,..., p_ -1
p--k
Gi+k,i = ‘i+k,i + C Gi+k,i-lDi~~,i-lHi-l,i; 1=1
(2b)
for k= p+,..., ma.x(p_,p+) we have Hi,i+k=Xi,i+k; for k=l,...9p+-l
P+ -k
Hi,i+k = Xi,i+k + C Gi,i-ID~:,,-IHi-I,i+k. (2c)
l-1
As X is an M-matrix, the pivots in its factorization are positive, i.e.,
Dii>O Vi.
INVERSES OF BANDED M-MATRICES 251
Induction then yields
Gi+k,i 2 Gi-l+k,i-lZ ‘-k,
Boundedness of the sequences i I+ Gi+k,i and i I-+ Hi, i +k for k = 1 , . . . , min( p _ , p + ) uniformly in the dimension n of X follows from the fact that
mW-- . P+)
xnn = Dnn + c Gn,n-kDii,n-kHn-k,n k-l
mwp-. P+ ) mink-. P+ ) =j c Gn,n-tHn-k,nG c G,,n-kD?k,n-kDn-p,n-pHin-k.n
k-l k-l
=D n-p.n-p(x”” - %J
<x2 =x2 9X” 0,
where p = min( p_ , p, ). Boundedness of the Dii away from zero then follows from (2a):
mwp- 9 P+ ) D,,>O - x0> c Gi,i-kDL\,i-kHi-k,k
k-l
X1X-l * Di_l,,_l’ -a
*0
We get Dj, 1 do, Gr+j,, t g,, Hj,j+i t hi, he= do, g,, 4 satisfy the ha
252
equations
VICTOR EIJKHOUT AND BEN POLMAN
d,=x,-
p- -i &+khk gi=x_i+ c ~
k=l do ’
-i gkhi+k hi=ri+Pk d.
k=l 0
A little algebra’ yields
P+
i=l
(34
(3b)
(3c)
i)
lbt p=max(p_,p+),mdlet g,=Oforp_ <i6p and hi=OforP+ii~p.Then
d, x0 - 5 x__i - 5 xi i-l i-l
min(p_ . P+ ) =d;+ c gkhk-dokgi+ 5 Pm~-‘gi+thi-do&++ 5 P&A+r
k-l i-l i-l k-l i-l i-l k-l
cd;+ k g,h,-d&g+ 5 k&khi-do&a,+ 2 2 ahi k-l i-l k-l i-l i-l k-l i-k+1
P P P P i P \/ P
INVERSES OF BANDED M-MATRICES 253
If x0 is increased by a small amount, the 1.h.s. (which is positive) increases. From the recursion formulae it follows that d, will also increase, whereas all gi and hi will decrease. Hence both factors of the r.h.s. will increase; as their product increases and they are of equal sign, they must both be positive. m
2.2. Some Elementary Estimates for Toeplitz Matrices In the case X = (xi_ j) the coefficients introduced in (la,b,c) are readily
estimated. From (3a) we estimate d,:
gkhk do=Xo-Cd
0
> x - (~gkmhk) 0 0
do
ax,-do
3 do>:.
On the other hand, the same equality gives
d,<x,-=. X0
In the symmetric case (Xi = X-i for all
gi >, gi+j (au j a O)V
p-i gkgi+k gi=x_i+ c -
k-l d0
i, so hi = gi for all i) and provided
[from (4)l
do
i 1
l/2
e g,< - &
X-i,
254 VICTOR EIJKI-IOUT AND BEN POLMAN
where S = x,, - 2Xx,. In the general (nonsymmetric) case (but assume p + = P _ ), if gi 2 gi+ jt
p-i gi+khk
gi=X_i+ C - k=l do
= x_i + x0 -do<ci+2.
Likewise, if hi >, hi+j,
2.3. Approximate Inverse Given the factorization of a matrix, an approximation to its inverse can
be generated using the limit form of an algorithm to compute the inverse exactly [S].
LEMMA 2.3. Let X be an FTBM-mutrix, and let do, g, ,... ,gp_, h I,. . . , h,+ be as in Definition 2.1. Then:
(i) The system
P+ hk
zi= c zi-k-, k-l d0
ha a unique solution.
i=l ,..., p- >
i=l ,..*> P+*
(5b)
(54
INVERSES OF BANDED M-MATRICES w
(ii) Zf (5b) and (SC) are extended to generate z_ i for i > p _ and zi for i > p,, we get
z,>q>o Vi # 0, (6)
(iii) and the matrix Z = toeplitz(. . . , z__~, z_ I, zo,zl, z2.. . .) satisfies
z>x-‘>o. (7)
Proof. (i): Consider the system
do -g, *.* -gp_
do -g, ..* -g,_
-h,+ ... -h, do
-h,+ .a. -h, -h, do
2-p
z-1 ZO
2 P+
=
0
iI 1 *
0
The 1.h.s. matrix is a strictly diagonally dominant M-matrix, whence its inverseispositive,~oz~~Ofori= -p_,...,p+.
(ii): Now suppose z,, = maxizi for some i, > 0. Then from (SC) it follows
that
P+ Z&-k do= c -
k-l ‘i,
h,< 2 h,, k=l
which contradicts the assumptions on do and hi. Therefore
20 > zi Vial.
In a similar manner Equation (5b) leads to
20 ’ zi Vi< -1.
256 VICTOR EIJKHOUT AND BEN POLMAN
(iii): The inverse of X = (D - G)D- ‘(D - H) can be computed using the recurrences
x,1= l 1 min(p_ , n - i)
K+iT c XLr!+Gi+k,ir 1, tt k=l ’
(94
1 min(p_,n-i+j) x:? ,=p c
“‘-’ Di-j,i-j k_l X7’ r,i-j+G-j+k,i-j, W
X7! 1 min(p+,n - i)
r,:+j = 6 c iii i+k'IT;lk,i+j' k=l ’
PC)
From the first line it follows that
X,; = l/D,,, < l/d, < za.
Induction on i and j (where the order is determined by the fact that X;’ can be computed if all X,r have been computed for r < t < n and s Q u < n with the exception of X;’ itself) then leads to
x;;+j < zj ViVj > 0. n
2.4. Decay Rates The same set of coefficients that was used to generate the approximation
to the inverse can be employed in two polynomials the roots of which describe the decaying behavior of the approximation.
Let X be an FTBM-matrix, let da, g, ,..., g,_, h, ,..., h,+ be as in the definition, let . . . ,z_~, zo, zl,. . . be defined by (5a,b,c), and let the polynomi-
INVERSES OF BANDED M-MATRICES 257
als f_ , f+ (of degree p_ , p, respectively) be defined by
P_
f_(z) = d,rP- - c gixp-, (104 i-l
P+
f+(x) = doxP+ - c hpP+ -i. (lob) i-l
THEOREM 2.4. Suppose f_, f+ have no zeros with multiplicity greater than one. Then:
(i) There exists a set of coefficients cl-‘,..., ciz’, c{+) ,..., c::’ such that
iq= 5 cp#p, k-l
P,
zj= 2 c~+)llJ:)j, jzl, k-l
where PI;), yc) are zeros off- , f+.
(ii) AU Irf;)l< 1 and IclJ;“I < 1
(iii) L..et II’;‘, c(r) be the roots off_ , f + with largest modulus. Then
(a) p\-),,~y) and their associuted coejjL%nts cl-‘, cl+) are positive
real; (b) &)>Ipj;)l &k=2,...,p_, and~I+)>I~~)l fork=%...,p+;
(c) pp < 1 - f_(l)/fL(l) and p’l” < I- f+(l)/fi(l).
Proof. As all assertions regarding f_ , f+ are independent and com- pletely analogous, we will only prove one of each pair.
(i): The sequence i e zi was generated from the linear recurrence with constant coefficients
d,z, = E Zi_khk. k-l
258 VICTOR EIJKHOUT AND BEN POLMAN
If f+ has no zeros with multiplicity greater than one, zi can be written as
P+
zi = c ppy, k-l
where the ci+) are chosen to accommodate za, . . . , zp+ _1. (ii): From d, > CR+lhk it follows that
r>l - f+W>O
X6-1 * f+W{ >O if peven,
-=O if podd.
Hence all roots have modulus less than one. (iii): As zj > 0 (all j 2 0), the root with maximal modulus p(;t) and the
associated coefficient cl+) must be positive real, and no other roots with the same modulus exist. All derivatives of f, are positive in the interval cli+)<x<l,so f+ isconvexand
f+(l) +i+‘r l- f;(l) *
REMARK. The condition that the zeros of f_ , f+ have multiplicity one imposes no severe restrictions on the theory developed here. In the case of a zero with higher multiplicity we can replace some gi (which was an estimate after all) by gi + E for some small enough value of E.
2.5. Estimation of Decay Rates In the symmetric case it is possible, provided the matrix is near enough to
being a Toeplitz matrix, to give a bound on pi that can be derived from the coefficients of the matrix.
Let
X0 = minXii t
2 ’ i
= max - X,+j i, j=l ,...,p,
INVERSES OF BANDED M-MATRICES 259
andsupposeO<d=x, -27,x,.I_.et Jo,g’l(j=l,...,p)bederived[using Equation (3)] by factorizing the matrix
a=toeplitz( -xp ,*.., -Xl,Xo, -x1 ,... 9 -xp),
and let fil be the zero with largest modulus of f(x) = Joxp - XP_lg”5~p-? Then PI > ~1~. Now [see Equation (4)]
d=-&o-Egj)2, 0
so with a = (d/$y2 and using the rather crude estimate Xgj 2 I( j/p)gj, we obtain
CEj =
[
l- ;
l/2
i II d”, = (1 - +I0 0
UZj d 1-a d”o-cgj a p.
Hence
Al) c&J - xgj 8’41-fo=1-ppdh_~(p_ j)g,
Jo - cgj 1 cl-
p(Jo-~g’i)+~j~~ gl-
p : lvap cl-“.
P a
260 VICTOR EIJKHOUT AND BEN POLMAN
If the dimension of 2 is sufficiently large, a can be estimated in terms of the eigenvalues of r7:
d=X,,(Q, 2x,-d=h,,(X)
3 (y= ( ;)1’2> ( t)1’2- (,+&J”.
A slightly sharper estimate is obtained by taking a?0 < rO - x,“/x, or even d”, < xc - cx;/x().
Estimates involving the condition of the matrix in a similar manner have been derived before (see [4] and references cited therein), but these decay rates are (although admittedly sharper than the one developed in this section) considerably less accurate than those derived from the factorization. It remains an open question whether the latter decay rates have a similar dependence on the condition.
In the asymmetric case of (say) a,, - X:dj > JO - Ckj the estimate
= dad
gives, again,
so an estimate of the fastest-decaying half of the inverse is obtained. It is a matter of regret that our approach does not lead to an estimate of the slower-decaying half that can be derived directly from the matrix.
2.6. Estimation of Coefficients Apart from asymptotic decay rates one would like to know (asymptotic)
absolute magnitudes of elements of the inverse. Thus if z j = Z[_ rckpL and ~1 i is the root with largest modulus, one would like an estimate of ci [which is real: see Theorem 2.4(ii.i)]. As z. = cp,ick, we would have ci < z. if all ck could be proved to have a positive real part. The authors conjecture that this is indeed the case (and their opinion is strengthened by an extensive computer search that failed to deliver counterexamples) if in addition gi > g,+r for-all i=l,...,p-1. Sofar we have not been able to prove this.
In Section 2.6.1 we will restrict ourselves to giving exact expressions for all ck; in Section 2.6.2 we estimate one of the terms involved if gi > gi + i is satisfied.
INVERSES OF BANDED M-MATRICES 261
2.6.1. General Fmmulae. If the formula zi = C[,lcky;C is extended to generate 2 j for j < 0, then with
T=
B=
1-p $P . . . pp
the equation T = BCD holds, so for each 1~ k < p we have
c,=(BCe,),=(TD-‘eJ,= 5 Tli(D-‘)ik i-l
I , D=
C=diag(c,,...,c,),
where
COfaCk,( D) =
I-lP
. . .
. . .
p-1 Pl
pP-’ P
1 i-2 . . . p1 *.*
Denoting Vandermonde determinants by
n-1 x1 . . . Xl 1
A(q,...,x,) = i . . . * . .
r,“-’ *** X” 1
262
we have [S]
VICTOR EIJKI-IOUT AND BEN POLMAN
IQ=( -1)‘p’%(~1,...r/4p)
=(-l)‘p’2’Ld~~s<p(cL,-k) .
and
where
sk,p-i(pl ,.**,f-lk-lr~k+l~*..~ PA = c llj, . . . Pip-,
1~j,<...~j*<k<j,+,<‘..<j,-,~~
is the coefficient of xi - ’ in
( _ ly<-J& *) = ( -“+&~x-y,‘““‘~ jfk
Furthermore
_ c-1>“-’ n (pk-pj) ’ j*k
INVERSES OF BANDED M-MATRICES
and the coefficients a j of
satisfy
’ gp-j xl’- c -%J
j-0 4
263
g p -1 a,=--CLk ,
d0
h,_, = 1
and
aj=Pk -‘(“j_l+y). j=l,..., p-l,
which implies 0 < a, < *. - < (I~_~ = 1 if pk is real. Thus
ck= i zi_l( - I)~+“( _ l)lp/2l( _ l)l(p-WJ( _ l)k-’
i-l
x(_l)i-l(_l)p-l ‘i-1 njtk(pk-pj)
1
= nj+k(pk-pj)
2.6.2. Coefficients to Real Roots. In cases where pk is real, the sum C~:&Z~ can be made slightI y more explicit under the additional assumption g,>,g,>, *** zgp.
Let tj=zjaj for j=O,...,p-1. Then
t,=x’pil($+yq, j=o )..., p-l,
where t _ 1/z_ 1 denotes zero.
264 VICTOR EIJKHOUT AND BEN POLMAN
Define ‘T[ = doCTLJ-‘(Zj+l/Zj)tj. Then
p-1 p-1 c ziai= iTotj=$ i-0
The Tl satisfy
p-l-2
tj-l+ C ‘j+lgp-j
j=O
=p;‘(T,+,+S,), Z-0 ,..., p-l,
where S, = X$‘:~-‘zj+lg,_j satisfy
p-1-(l-l)
s,= c Zj+(Z-l)gp-j+l j - 1
p-1-(l-l)
G c zj+(l-l)gp-j= I-l-z~-lgpe S j-l
By induction it now follows that
Tp_l< S,_l i /L;5-gp f: “p-jji2 Pir+m. j-l j-2 m-0
Using So = CTT,‘z,g,_, = zpdO, we find
INVERSES OF BANDED M-MATRICES 265
Thus we arrive at an estimate for ck:
3. THE APPROPIUATENESS OF THE APPROXlMATION
For each matrix X there exists in a trivial sense a “best” approximation to its inverse by a Toeplitz form, namely Z=(Z,_~), ~~__~=rnax,X,r_‘i+~,~. There exist applications, however (such as the numerical solutiou of partial differential equations), where X is a member of an infinite family of matrices stemming from the same problem, all members being related but of different dimension. Thus it makes sense to ask for a set . . . , z_ lr zo, z 1,, . . such that 2 = ( z~__~) is both an upper bound for all members, and optimal in the limit of infinite dimension.
We will prove optimality for two classes {Xc”)} of FTBM-matrices (which allow dimension independent formulation) in the sense that the difference between the inverses and the generated approximation is per element a vanishing function of the matrix dimension, i.e., lim n _ ,Xi,:?_i = zi. Convergence of decay rates of Xc”) - ’ to those of 2 (again, pointwise) then follows as a corollary. The latter of these two classes contains the FTBM- matrices that are Toeplitz-matrices.
Some numerical examples will show that even for fairly small matrix dimensions accurate estimates of decay rates can be obtained.
3.1. Limit ThQurem8 We first consider those FTBM-matrices the factors of which are Toeplitz
forms.
LEMMA 3.1. Let D, G, H be Toeplitz w&rices with dim.enskm indepen- dfmt coejj%Aents
G i+k.l= gk, H f.f+k = h k’ Dii = do
206 VICTOR EIJKHOUT AND BEN POLMAN
sutisfying (Id), and let X (“)=(D-G)D-‘(D-H), which makes Xc”) an FTBM-matrix. Let, firthennme, Z = toeplitz(. . . , z_ 1, zo, zl,. . .) be gener- ated by Equation (5). Then
(i) the sequence n * Xi:)+ j -’ (ii) for given i we huve’lim
is monotonically increasing for all i, j; ,_+,Xi:$,-‘= zj unifmly in j.
Proof. (i): Monotonicity and boundedness are proved by induction using Equation (9).
(ii): As the limit values lim n _ ,X:,:5 j -’ satisfy the system (5) (which has a unique solution), they must be equal to the zj. Uniformity follows from the fact that the quantities
s,!:;j=xiy’, i+l+j-l-x;“>+j-l, j#O,
satisfy the same recurrences as Xi,“,< j - ’ if j # 0, from which one can derive
spy > spy+ j Vj + 0
in a manner analogous to the proof that z0 > zj for j # 0. n
Pointwise convergence can also be proved (though in a more restricted sense) for a broader class of FTBM-matrices; this class contains the FTBM- matrices that are Toeplitz-matrices.
LEMMA 3.2. Let n-D”,,, nHGn+k,n (for k=l,..., p_) and nc, H ,,,n+k (for k=L..., P,) be seq-a such that 4, i 4, Gn+k,,, t & H n,n+k t h,, and d,, gk, h, satisfy (Id); furthermore bt DC”), G(“), H(“) be n x n matrices such that
afid h XC”) = (DC”) -
wing d,, gk, h, then
G(“))D(“-tD(“) - H(“)). Zf the zj are generated
unifmly in j.
INVERSES OF BANDED M-MATRICES 267
Proof. Let n1 be such that 2; generated by (5) using d,i s D,,,.nl,
gi+k,.i s Gn,+k.n,r ad hi,i+k E Hn,,n,+k satisfy
E z;zzj--,
2
and let no = n, + n2, where ns is such that Y$y!j generated by (9) using
dii = Dn,, no’ gi+k,i E Gn,+k,nop and hi,i+k G Hno,+,+k Satisfy
From monotonicity of n I+ D”,,, n I+ Gn+k,n, and n I+ Hn,n+k it follows that
andtaking n>,n,=n,+n2gives
The existence of nr for a specific value of j follows from the continuous dependence of all zI on the matrix coefficients in (5); the choice can even be uniformly in j, as only a finite number of j’s (namely those for which z j >, e/2) need be considered. A similar argument gives the existence of n, uniformly in j. n
REMAmc. The monotonicity conditions of the previous lemma can easily be relaxed to
limDii=d,, i
Dii>d, (di),
etc.
For decay rates we get uniformly in j.
LEMMA 3.3. Dejh
the same convergence results, only this time not
nlfm for au i, j the 611;’ converge to 0 under the conditions of Lamus (3.1) and (3.2).
268 VICTOR EIJKHOUT AND BEN POLMAN
3.2. Numerical Illustratiom To illustrate the theory developed above we have computed the inverses
and decay matrices [X, = (aij) with Si,i+k = X$+,/X,;/+,_, and Si+k,i =
xtr+lk,i/xtT;‘k-l,i for k > 0] of some 15 X 15 band matrices. The matrices used were
(1) x = toephtz( - 1,5, - l), (2) X = toeplitz( - 0.9,5, - 1. l), (3) x = toephtz( - 0.5, - 1,5, - 1,0.5), (4) X = toeplitz( - 0.5, - 1,5, - 1. - 0.5) + 0.1 x [random pentadiagonal
O(l)l*
The results are shown in Table 1. Of these matrices decay rates were predicted:
(a) By inspection of the factorization: pf is the largest root of f(x) = d,xP - E~_bIhi~P-i, where da, h,,. .., h, satisfy the conditions of Definition 2.1.
(II) Using the a priori estimate (see Section 2.5) CL,,, = 1 - a/p, where (Y = [2/(1+ X,,/X,,>]“2 and eigenvalues are estimated by the Gershgorin circle theorem.
(c) Using the main theorem of [4]: let [a, b] contain the spectrum of A*A; let 9 = (@ - l)/(@ f 1) and po = 9’/m~(p~~p+); then IA,<‘1 d
cpD . li-jf The bounds on the spectrum were derived by Gershgoti estimates.
OBSERVATIONS.
Matrix (1): For symmetric tridiagonal Toeplitz matrices we know that
both Pr and PD are asymptotically exact; this example shows that they are accurate even for fairly small matrix dimensions. This is the only case for which pD can be proved to be exact. As was to be expected from the discussion in Section 2.5, p,,, is slightly off.
Matrix (2): For both halves of the inverse of this unsymmetric matrix pr is extremely accurate (note that for tridiagonal Toeplitz matrices X the sequences k c-, XT_‘~,~ and k c-) Xi,:_k are geometric progressions); the a priori estimate p, is not too far off the actual (fastest; see the end of Section 2.5) decay rate of the lower triangle; the estimate pD is rather crude, due to the inadequacy of Gershgorin estimates of the spectrum of A*A.
Matrix (3): For matrices with p > 1 the oscillatory behavior of the inverse and decay matrices becomes apparent. Thus we can only remark that pf is in overall accordance with the actual results; pD is considerably off, and pm is even more so.
Tab
le 1
con
tinu
ed
0.46
69
0.4652
0.4531
0.4207
4.9373
-1.0371
- 0.5433
0.0000
0.0009
0.0000
0.0060
0.0000
0.0600
0.0000
0.0000
0.0000
O.CNOO
0.0090
0.0000
0.2182
0.0546
0.0383
0.0148
0.0073
0.0032
0.46
76
0.46
49
0.45
32
0.42
06
-1.0371
4.9790
-0.9821
-0.5314
0.0000
O.OOW
0.0060
0.0000
O.oooO
0.0006
0.0000
0.0000
0.0000
O.OMO
0.0000
0.0546
0.2283
0.064-M
0.0393
0.0153
0.0076
0.4662
0.4655
0.4529
0.4208
- 0.5433
-0.9821
4.9484
-0.9612
-0.4562
0.0000
0.0900
OSKXKI
0.0000
0.6000
0.0000
0.0000
0.0009
O.OODO
0.0000
0.0383
0.0604
0.2333
0.0595
0.0371
0.0147
0.46
90
0.46
42
0.45
36
0.4205
0.46
34
0.46
69
0.4522
0.4211
0.4750
0.4613
0.4550
0.4198
0.4516 0.5014
0.4729 0.4495
0.4494 0.4609
0.4225 0.4170
Matrix(4)
0.4062 0.6374 0.2674 -
0.2657 0.6211 0.3610
0.4993 0.4042 0.6352 0.2657 -
0.2570 0.5892
0.4378 0.4869 0.3932 0.6211 0.2570 -
0.2264
0.4284 0.4054 0.4544 0.3610 0.5892 0.2284 -
0.0000 0.0000 0.0000
-0.5314
0.0006 0.0000
-0.9612 - 0.4562 0.0000
5.0451 -0.9918 -0.5010
-0.9918
4.9536 - 0.9569
-0.5010 -0.9569
5.0920
0.0000 -0.5188 - 1.0334
0.0009 0.0000 -0.4841
0.0000 0.0000 0.0090
0.0000 0.0009 O.OOC!O
0.0600 0.0000 0.0000
0.0006 0.0000 0.0060
0.0000 0.0000 0.0006
0.0900 0.0009 0.0009
0.0000 0.0000 0.0000
o.OOOo 0.0000
0.0009 o.OOoO
o.oOoO 0.0600
0.0090 0.0000
- 0.5188 o.ooOa
-1.0334 -0.4841
5.0041 - 1.0288
- 1.0268 4.9785
-0.4832 - 1.0234
0.0000 - 0.5282
0.0000 o.OoOo
0.0000 0.0900
0.0600 0.0000
o.ooOO o.oOOO
0.0000
o.OOoo
Inversematrix
0.00
00
0.0000 0.0000
o.oo
oo o.oo
oo o.oo
oo o.oo
oo
0.0000 0.0000
o.oo
oo o
.ooo
o o.o
ooo o
.ooo
o 0.0000 0.0000
o.oo
oo o.
oooo
o.oo
oo o.
oooo
0.0000 0.0000
o.oo
oo o.
oooo
o.o
ooo o
.ooo
o 0.0000 0.6000
o.oo
oo o
.ooo
o o.o
ooo o
.oo4
lo
0.0000 0.0000
o.oo
oo o
.ooo
o o.o
ooo o
.ooo
o -0.4832
0.0090
o.oo
oo o.
oooo
o.o
ooo o
.ooo
o - 1.0234 - 0.5282
o.oo
oo o
.ooo
o o.o
ooo o
.ooo
o 5.0616 -1.0245 - 0.4740 0.0000 o.OOoO 0.0090
-1.0245
5.0490 -1.0498-0.4930 o.OoOo o.OoOO
- 0.4740 -1.0498
5.0645- l.OlB- 0.4715 o.OOOo
0.0009 - 0.4930 - 1.0128 4.996s 0.9626-
0.4508
0.0000 0.0000 - 0.4715- 0.9626 4.9330-
0.9578-
0.0000 0.0000 0.0900- 0.4508- 0.9578 4.9094-
0.0000 O.OCOO o.oOoO o.OoOO- 0.5362- 1.0122 0.
0000
0.0000
0.00
00
0.0000
!mJo
o 0.
0000
0.0000
0.00
00
0.0000
0.0000
0.5362
1.0122
5.0492
0.0146 0.0073 0.0032 0.0016
0.0007 0.0003 0.0002 0.0001 o.OOoO o.OooO o.oooO o.ooOO
0.0393 0.0153 0.0076 0.0036
0.0017 0.0008 0.0004 0.0002 0.0901 o.OoOO 0.0009 0.0006
0.0595 0.0371 0.0147 0.0077
0.0034 0.0016 0.0008 0.0004 0.0002 0.0001 0.0000 0.0000
0.2293 0.0604 0.0379 0.0161
0.0079 0.0036 0.0018 0.0008 0.0004 0.0092 0.0001 0.0009
0.0604 0.2331 0.0595 0.0406
0.0162 0.0081 0.0038 0.0017 o.ooo8 0.0004 0.0002 0.0001
0.0379 0.0595 0.2278 0.0628
0.0393 0.0159 0.0083 0.0036 0.0017 0.0008 0.0003 0.0001
Inverse
matrix
0.0015 0.0007
0.0035 0.0017
0.0081 0.0037
0.0162 0.0061
0.0398 0.0162
0.0624 0.0398
0.2329 0.0624
0.0624 0.2329
0.0398 0.0624
0.0162 0.0398
0.0081 0.0162
0.0037 0.0081
0.0017 0.0037
0.0008 0.0017
0.0003 0.0007
Decaymatrix
0.4774 0.4632
0.4517 0.4759
0.5023 0.4521
0.4067 0.5021
0.6382 0.4068
0.2679 0.6382
-
0.2679
0.2679
-
0.6382 0.2679
0.4068 0.6381
0.5019 0.4067
0.2137 0.0512 0.0351 0.0139 0.0071 0.0032
0.0512 0.2260 0.0595 0.0364 0.0156 0.0078
0.0351 0.0595 0.2317 0.0618 0.0396 0.0161
0.0139 0.0384 0.0618 0.2326 0.0623 0.0398
0.0071 0.0156 0.0396 0.0623 0.2329 0.0624
0.0032 0.0078 0.0161 0.0398 0.0624 0.2329
0.0015 0.0035 0.0081 0.0162 0.0398 0.0624
0.0007 0.0017 0.0037 0.0081 0.0162 0.0396
0.0003 0.0008 0.0017 0.0037 0.0081 0.0162
0.0002 0.0004 0.0008 0.0917 0.0037 0.0081
0.0001 0.0002 0.0004 0.0008 0.0017 0.0037
0.0090 0.0091 0.0002 0.0004 0.0008 0.0017
O.OOCQ 0.0000 O.cQol 0.0902 0.0004 0.0008
O.OOM
O.CCMO 0.0000 0.0001 0.0002 O.OfIO4
0.0000 0.0000 0.0000 O.OOOG 0.0001 0.0002
- 0.23
93
0.68
59
0.3951
0.5098
0.4490
0.4774
0.4632
0.4700
0.4666
0.4682
0.23
93
- 0.2635
0.64
49
0.40
50
0.5032
0.4517
0.4759
0.4639
0.4697
0.4667
0.68
59
0.26
35
- 0.2668
0.6399
0.4063
0.50!?3
0.4521
0.4757
0.4639
0.4695
0.39
51
0.64
49
0.26
68
- 0.2877
0.6385
0.4067
0.5021
0.4522
0.4756
0.4639
0.5098
0.4050
0.6399
0.2677
-
0.2679
0.6382
0.4068
0.5020
0.4522
0.4755
0.4490
0.5032
0.4063
0.6385
0.2679
-
0.2679
0.6382
0.4068
0.5020
0.4521
0.0003 0.0002 0.0001 0.0000 o.
oooo
o.oo
oo o.
oooo
0.0008 0.0004 0.0002 0.0001 o.
oooo
o.oo
oo o.
oooo
0.0017 0.0008 0.0004 0.0002 0.0001 0.0000 0.0000
0.0037 0.0017 0.0008 0.0004 0.0002 0.0001 o.OOoa
0.0081 0.0037 0.0017 0.0008 0.0004 0.0002 0.0001
0.0162 0.0081 0.0037 0.0017 0.0008 0.0004 0.0002
0.0398 0.0162 0.0081 0.0037 0.0017 0.0008 0.0003
0.0624 0.0398 0.0162 0.0081 0.0037 0.0017 0.0007
0.2329 0.0624 0.0398 0.0162 0.0081 0.0035 0.0015
0.0624 0.2329 0.0624 0.0398 0.0161 0.0078 0.0032
0.0398 0.0624 0.2329 0.0623 0.0396 0.0156 0.0071
0.0162 0.0398 0.0623 0.2326 0.0618 0.0384 0.0139
0.0081 0.0161 0.0396 0.0618 0.2317 0.0595 0.0351
0.0035 0.0078 0.0156 0.0384 0.0595 0.2260 0.0512
0.0015 0.0032 0.0071 0.0139 0.0351 0.0512 0.2137
0.4700
0.4639
0.4757
0.4522
0.5020
0.4068
0.6382
0.2679
-
0.2679
0.6380
0.4666
0.4697
0.46
39
0.4756
0.4522
0.5020
0.4068
0.6381
0.2679
-
0.2678
0.4682
0.4667
0.4695
0.4639
0.4755
0.4521
0.5019
0.4067
0.6380
0.2678
-
0.46
69 0.4652 0.4531 0.4207
0.4676 0.4649 0.4532 0.4206
0.4662 0.4655 0.4529 0.4208
0.4690 0.4642 0.4536 0.4205
0.4634 0.4669 0.4522 0.4211
0.4750 0.4613 0.4550 0.4198
0.4516 0.4729 0.4494 0.4225
0.5014 0.4495 0.4609 0.4170
0.4062 0.4993 0.4378 0.4284
0.6374 0.4042 0.4869 0.4054
0.2674 0.6352 0.3932 0.4#%4
Tab
le
1
0.18
78
0.18
78
0.18
78
0.18
78
0.18
78
0.18
78
0.18
78
0.18
78
0.18
78
C 1
878
0.18
77
0.18
74
0.18
00
5.00
90
- 1.
0900
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0.
0000
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1.00
90
5.00
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0.50
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0.00
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1.09
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0.50
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0.00
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o.oo
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0.18
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0.22
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Mat
rix
(3)
0.22
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276 VICTOR EIJKHOUT AND BEN POLMAN
Matrix (4): Similar remarks to those for matrix (3) apply here, but how pLf is also somewhat inaccurate. This is due to the fact that the estimates d o,. . . determining the approximation to the inverse can be based on (i.e., are attained at) elements in radically different places in the triangular factors of the matrix. These extreme elements, however, have (only in an informal sense) a ‘localized’ effect on the actual inverse.
The results for the decay rates are as follows:
Decay rates
Matrix (a) (b) (c)
(1) (2)
(3) (4)
.2687 .2094 .2687
.1877 (I) .2096 .5143
.2295 (r)
.4678 .6772 .5773
.4956 .6967 .5888
Some conclusions are:
(1) In the symmetric case the Demko estimates are systematically better than those based on Section 2.5; they can be exact for p = 1, and for larger p they are pessimistic
(2) In the asymmetric case the Den&o estimates are very pessimistic and potentially difficult to derive, as A*A need not be diagonally dominant, which inhibits application of Gershgorin circle theorems
(3) The estimates based on the factorizations are accurate, even for p > 1 and asymmetric matrices.
3.3. Other Approaches to Exponential Decay Our results concerning the decaying behavior of the inverse of a banded
matrix are closely related to a theorem of Barrett [3], who proved that if T = (ti_ .), then T-’ is upper pbanded (i.e., TL~+~ = 0 for j > p) if the t, can be d escribed as certain sums of powers of roots of a polynomial; the sum of the multiplicities of the roots equals p. This same behavior could have been derived (for general pd banded matrices) in at least two other ways, which we sketch here (rather roughly):
(1) Any banded matrix can be partitioned to block tridiagonal form; if the matrix is pbanded, the blocks are square and have dimension p. Factorizing this block matrix and computing its inverse by a block form of (9) gives a decay proportional to the pth root of a constant related to the condition of the matrix. Compare this with the estimate p < 1 - a/p z (I -
INVERSES OF BANDED M-MATRICES 277
a) i/P in Section 2.5; the (1 - a)i/P was derived in a different manner by Den&o et al. 141.
(2) An early result of Asphmd [l] (used by Barrett [3]) states that a matrix is upper pbanded if the upper p-minors (i.e., the determinants of those matrices having a main diagonal up or above the main diagonal of the matrix) of its inverse are zero. If the inverse was Toeplitz, this can be used to give pterm characteristic recurrences.
4. CONCLUSION
We have developed a way to approximate the inverses of certain pd band matrices by Toeplitz matrices the decaying behavior of which closely resem- bles that of the actual inverses. If the factorization of the band matrices is known, the exponential decay can be estimated far more accurately than with the currently best available results [4]. This is particularly interesting when applied to incomplete blockwise factorizations, where it is desirable to have an accurate estimate of the error made in approximating the inverse of a banded matrix by a banded part of that inverse [2]. As one algorithm to compute banded parts of inverses [(S)] uses the factorization, one can dynamically obtain good estimates of the error.
The authors wish to thank Prvfasm 0. Ax&son for helpful suggestions regarding the content and presentatbn of this munusmipt.
REFERENCES
E. Asplund, Inverses of matrices { aij } which satisfy aij = 0 for j > i + p, Math. Scmd. 7:57-60 (1959). 0. Axehon and B. Polman, On approximate factorization methods for block matrices suitable for vector and parallel processors, Linem Algebra Appl. 77:3-26
(1Qw. W. W. Barrett, Toephz matrices with banded inverses, Linear Algebra Appl. 57:131-145 (1984). S. Den&o, W. F. Moss, and P. W. Smith, Decay rates for inverse of band matrices, Math. Camp. 43:491-499 (1984). F. Neiss, Lktetmhunten utd Mahizen, Springer, Berlin, 1967. K. Takahishi, J. Fagan, and M. S. Chen, Formation of a sparse bus impedance matrix and its application to short circuit study, in 8th PICA Ccmjknce proceed- ings, Minneapolis, 1973, pp. 63-69.
&c&xxi 29 September 1986; jhal manumipt accepted 13 January 1988
