The Kreiss Matrix Theorem asserts the uniform equivalence over all N x N matrices of power boundedness and a certain resolvent estimate. It is shown that the ratio of the constants in these two conditions grows linearly with N, and the optimal proportionality factor is obtained up to a factor of 2. Analogous results are also given for the related problem involving matrix exponentials. The proofs make use of a lemma that may be of independent interest, which bounds the arch length of the image of a circle in the complex plane under a rational function

Leveque, R. J.

Trefethen, L. N.

English

NASA Technical Reports Server

N AS.A Contractor Report 172177 
ICASE 
NASA-CR-172177 
19830026388 
ON THE RESOLVENT CONDITION IN THE KREISS MATRIX THEOREM 
Randall J. LeVeque 
and 
Lloyd N. Trefethen 
Contract No. NASl-17070 
July 1983 
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING 
NASA Langley Research Center~ Hampton, Virginia 23665 
Operated by the Universities Space Research Association 
N/\SI\ 
National Aeronautics and 
Space Administration 
Langley IResearch Center 
Hampton, Virginia 23665 
111111111111111111111111111111111111111111111 
NF02530 
.U-\NGLEY RESEAf,(>' ., fIJi 
U3RAI'Y. I\I/\SA 
~jJI.M:)TON, V!fKlINIA 
https://ntrs.nasa.gov/search.jsp?R=19830026388 2020-03-21T00:59:14+00:00Z
ON THE RESOLVENT CONDITION IN THE KREISS MATRIX THEOREM 
Randall J. LeVeque * 
Department of Mathematics 
University of California at Los Angeles 
and 
. * Lloyd N. Trefethen 
Courant Institute of Mathematical Sciences 
New York University 
ABSTRACT 
The Kreiss Matrix Theorem asserts the uniform equivalence over all 
N x N matrices of power boundedness and a certain resolvent estimate. We 
show that the ratio of the constants in these two conditions grows linearly 
with N, and we obtain the optimal proportionality factor up to a factor of 
2. Analogous. results are also given for the related problem involving matrix 
exponentials eAt. The proofs make use of a lemma that may be of independent 
interest, which bounds the arc length of the image of a circle in the complex 
plane under a rational function. 
AMS Subject Classification: primary 39All; secondary 15A45,30AIO. 
* Research supported by NSF Mathematical Sciences Postdoctoral 
Fellowships, by the Courant Institute of Mathematical Sciences, and by the 
National Aeronautics and Space Admlnistration under Contract No. NASl-17070 
while the authors were in residence at the Institute for Computer Applications 
in Science and Engineering, NASA Langley Research Center, Hampton, VA 23665. 
i 
1. Introduction 
Let A be an N x N matrix that satisfies the power boundedness 
condition 
p(A) ::: n sup IIA II < 00, (1) 
n ~ 0 
where II-II = 11-11 2 • By a power se.ries expansion it is readily verified that 
A then also satisfies the resolvent condition 
r(A) = sup (izl-1)II(zI-A)-1 11 <00, 
Izl>l 
(2) 
and moreover r(A)';; p(A). One of the assertions of the Kreiss Matrix Theorem 
[3,4,:7] is that the converse is also valid: if r(A) < 00, then p(A) < 00 
also, and p(A) can be bounded in terms of Nand r(A) but otherwise 
indeplandently of A. This resul t is useful in proofs of stability theorems 
for finite difference approximations to partial differential equations. 
In this note we resolve an old question that has been contributed to most 
recently by Tadmor [8]: given Nand r(A), how large can p(A) be? 
According to Tadmor, Kreiss's original proof in [4] unwinds to give a far from 
sharp bound 
N 
p(A) ~ [r(A)]N (VA) 
which subsequent improvements by Morton, Strang, and Miller lowered to 
2 
e
9N 
r(A) (VA). 
2 
A few years ago Strang (private communication) observed that a paper of Laptev 
[5] implicitly derives a much more reasonable estimate [3] 
p(A) ~ 32eN2r (A) (VA). 1f 
Finally Tadmor's proof, which makes use of an elegant Cauchy integral argument 
adapted}Jrom Laptev, yields a bound that is linear in N, 
p(A) ~ 32eNr(A) (VA) 1f (3) 
Tadmor conjectures that a linear dependence as in (3) is the best possible. 
However, up to now the strongest growth of p(A) with r(A) attained by an 
example has been logarithmic, i.e., p(A) '" r(A)logN [6] • 
First we will show that Tadmor's conjecture is correct, by exhibiting a 
family of matrices {AN} for which p(~) ~ eNr(~) as ~oo. 'By refining 
the Cauchy integral argument, we will then show that for arbitrary matrices 
(3) can be sharpened to p(A) ~ 2eNr(A). (Our proof is essentially Tadmor'sf 
but gains the factor 16/1f over his by dealing with complex functions 
directly rather than taking real and imaginary parts.) Together these results 
establish that eN is the optimal constant of proportionality relating 
p(A) to r(A) except for a possible factor of 2. The final sectiort will 
prove analogous results for the continuous problem involving matrix 
exponentials eAt. 
3 
2. J!:xs.ple with p(~) '" eNr(AN)· 
Consider the N x N Jordan matrix 
A D ~ = [ y ~] 0 y 0 
with N ) 3, y ) N (these constraints could be relaxed considerably). For 
this matrix one has nAnn = yn for n ~ N-1 and nAnn = 0 otherwise, so 
A is power bounded with 
peA) = yN-1 (4) 
On the other hand the resolvent matrix is 
1 y/z (y/z)2 (y /z)N-1 
1 y/z (y/z)2 
• (zI-:A)-1 1 = -z· 
1 
From the fact that nBIl <; IIBlil for any upper-triangular ToepUtz matrix B, 
we obtain with a little·calculation the estimates 
By (2), one therefore has 
I N-1( -1 -N N } rCA) <: max sup (p-1)y 1-p/y) p ,sup (p-1)2 /p • 1<:p<;y /2 p)y /2 
This maximum is attained at a point p = 1+N-1+0( N-2) , where the estimate 
becom4~s 
4 
(5) 
since y ~ N. Comparing (4) and (5) shows that for this example one has 
p(~) ~ (eN-const)r(~), (6) 
as required. 
3. Proof of p(A) .. 2eNr(A) for all A. 
Theorem 1. Let A be an N x N matrix with r(A) < 00.. Then 
p(A) .;; 2eNr(A). (7) 
Remark. The factor of 2 is probably unnecessary; seethe remark after 
the Lemma in the Appendix. 
Proof. Suppose r(A) < 00. The matrix An can be written in terms of 
the resolvent by means of a Cauchy integral .(see[2] ,pp. 555--577) 
n 1 J n -1 A :: - Z (zI-A) dz 2~i ' (8) 
where the contour of integration is any curve enclosing the eigenvalues of A, 
which must all'.l:ie in I z I .. 1 since r(A) < 00. Let u and vbe arbitrary 
unit N-vectors, i.e., lIuli = IIvll = 1. Then 
* n 1 n 
v A u = 2~iJz q(z)dz 
5 
where. q(z) =: v*(zI-A)-1u • Integrating by parts gives 
* n --1 J n+1 , 
v A u = 2~i(n+1) z . q (z)dz. 
Let the contour of integration be taken as 
has I zn+1 1 ,; e, and there follows the bound 
1 Izl = 1 + n+1. On this path one 
Iv * A nu I ( e J 1 I qP (z) II dz I • 2~(n+1) Iz 1=1+ _ 
n+1 
Now ai~ verifled on p. 155 in [8], q is a rational function of degree N. By 
the l,emma in the Appendix, the integral above is accordingly bounded by 4~N 
times the supremum of Iq(z)1 on Izl = 1 + 1/(n+1), and by (2) this supremum 
is at most (n+l)r(A). Hence we obtain 
* n Iv A \Jl1 ~ 2eNr(A). 
Since II Ann is the supremum of I v*Anu I over all unit vectors u and v, 
this proves the theorem. I 
4. k!alogouB Results for eAt 
l~or prolblems that are continuous in time rather than discrete, stability 
depends on the boundedness of a family of matrix exponentials eAt (t)O) 
rathelr than of powers Correspondingly, the resolvent of A is of 
inten~st for z in the right half plane rather than outside the unit 
circlc~. FoHowing (1) and (2), def:f.ne 
6 
and 
peA) At = suplle II 
R(A) = 
t)O 
-1 
sup Rezll(zI-A) II. 
Rez)O 
(9.) 
(10) 
As before, one has R(A)" peA) readily, this time as a consequence of the 
-1 00 -zt At Laplace transform formula (d-A) = J e edt. The continuous form of the 
o 
Kreiss matrix theorem asserts that conversely, peA) can be bounded in terms 
of R(A) and N, independently of A. We make this sharp by essentially the 
same argument used before: 
Theorem 2. Let A be an N x N matrix with R(A) < 00. Then 
peA) " 2eNR(A). (11) 
Remark. Again the factor of 2 is probably unnecessary. The 
!,.aptev/Tadmor estimate has a constant 32e/Tf, as in (3). 
Proof. In analogy to (8), one has now 
At 1 J zt -1 
e = 2Tfi e (zI-A) dz, 
where the contour of integration can be taken as any line Rez = ~>o. 
Integration by parts gives 
with q(z) = v*(zI-A)-lu • Taking the contour ~ =.! . now leads to the desired t 
bound (11), again by the use of the lemma in the Appendix. I 
Constructing an example to prove that (11) is sharp, on the other hand, 
is trickier than it was in the power-boundedness case. The following example 
achieves growth proportional to IN, not N. Omitting details, define 
Then one has 
A = ~ = 
At 
e 
-t 
e 
1 
-1 Y 
-1 
yt 
1 
y 
yt 
-1 Y 
-1 
N-l (yt) 
(N-l)! 
1 
• 
• 
For large y, this matrix achieves; maximum norm near t = N, where it is 
dominated by the upper-right entry, with magnitude approximately 
e-NNN-1yN-l N-l p( A __ ) '" '" L_ · (12) -~ (N-l) ! 12'1rN 
For the second estimate we have used Stirling's formula. On the other hand 
the rlesolvent: matrix is 
1 -1 (zI-A) =---
z + 1 
1 
y 
z + 1 
1 
y 
z + 1 
• 
1 
7 
8 
For large y, Rez times the norm of this is maximized near z= lIN ,where 
again the upper-right entry dominates and one has 
N-I 
'" -y--. 
eN 
Comparing (12) and (13) shows that in this example one has 
P(AN)/R(AN) '" ~ e. 
Appendix - LeJlllla on arc length of a rational function ,on aclrcle. 
Let S be any circle or line in the complex plane, and define the 
L 
"" 
norms over S by Hill = J If(z)lldzl, 1If1l"" =suplf(z)l. 
S s 
(13) 
The 
following lemma provided the key argument in proving Theorems 1 and 2. For 
the case of a polynomial the result is a corollary of Bernstein's inequality 
[1], IIq'lI .. Nllqll 
"" "" 
for S {z:lzl=l}, but the extension to rational 
functions appears to be new. Since II q'1I 1 represents the arc length of the 
image of S under q, the lemma has a simple geometric meaning. The 
example q(z) = b(z-s), where b(z) is any finite Blaschke .productofdegree 
N (such as z -N) and s is the center of S, shows that it is sharp except 
for a factor of 2. 
Lemma. Let q be a rational function of degree Nwith no poles on 
S. Then 
Remark. We believe that the bound is valid with a factor 21T instead of 
41T, but have been unable to prove this. 
F'roof. Since the composition of q with a Mobius transformation is 
again a rational function of type N, we can assume without loss of generality 
that S is the unit circle. Define g(z) to be the angle of the tangent 
to q(S) at q(z), i.e. 
g(z) = arg[zq'(z)J. 
Let TV[gJ be the total variation of g over S. The lemma is a consequence 
of the! following two facts: 
(a) IIq'1I 1 <; TV[gJllqlloo' 
(b) TV[gJ ~ 41TN. 
The proof of (a) is a matter of integration by parts: 
~ IIqll P Ig'(z) Ildzl = IIqll TV[gJ. 
00 00 
1'0 prove (b), note that q' iel of rational type (2N-l,2N), so zq'(z) 
is of rational type (2N,2N) and can be written as a product 
This implies 
9 
10 
and therefore 
2N ~z+bk 
TV[g] " L TV[ arg( +d)]" 41fN. I 
k=1 ckz k 
Acknowledgments and remark. We are grateful to Paul Garabedian and Eitan 
Tadmor for valuable discussions. Tadmor has pointed out that if A is an 
arbitrary but fixed bounded operator on t2 (infinite matrix) with 
r(A) < co, then the· arguments of [6] can be adapted to show that nAnn may 
grow as n + co in proportion to log n,but it is not known if it can grow 
faster. Our example of §2 apparently sheds no light on this question, for 
if one seeks a family {~} with r(~) uniformly bounded in N, the numbers 
YN have to satisfy YN <: 1 + O(l/N), which implies that p( AN) is also 
uniformly bounded. 
11 
Re.ferences 
[1] N. de Bruijn, "Inequalities concerning polynomials in the complex 
domaLin," Indag. Math. 9 (1947), 59l-598. 
[2] N. Dunford and J. Schwartz, Linear Operators I, Wiley-Interscience, 
195j' • 
[3] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral 
Methods: Theory and Applications, SIAM, 1977. 
Kreiss, "Ueber die Stabilitaetsdefinition fuer 
Differenzengleichungen die partielle Differentialgleichungen 
approximieren," BIT 2 (1962), 153-181. 
[5] G. 1. Laptev, "Conditions for the uniform well-posedness of the 
Cauchy problem for systems of equations," Soviet Math. Dokl. 16 
(1975), 65-69. 
[6] C. A. McCarthy and J. Schwartz, "On the norm of a finite Boolean 
algebra of projections and applications to theorems of Kreiss and 
Morton," Comm. Pure Appl. Math. 18 (1967), 389-396. 
[7] R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-
Value Problems, Wiley-Interscience, 1967. 
12 
[8] E. Tadmor, "The equivalence of L2-stability, the resolvent cooc:U,tion,. 
and strict H-stability," Lin. Alg. and its Applies. 41 (1981), 151-
159. 
1. Report No. 
NASA CR-172177 I 2. Government,_A_c_c_ess_io_n_N_o_. ____ . __ +-_3_. _R_ec_iP_ie_n_t'_s _Ca_ta_log_N_O_. _____ -..j' 
5. J~Pf; ~'tj~3 4. Title and Subtitle 
On the Resolvent Condition in the Kreiss Matrix Theorem 
7. Author(s) 
Randall J. LeVeque and Lloyd N. Trefethen 
6. Performing Organization Code 
8. Performing Organization Report No. 
83-35 
1-____________________________________ -; 10. Work Unit No. 
9. Performing Organization Name and Address 
Institute for Computer Applications in Science 
and Engineering 
Mail Stop l32C, NASA Langely Research Center 
Hampton, VA 23665 
12. Sponsoring A!lency Name and Address 
National Seronautics and Space Administration 
Washington, D.C. 20545 
11. Contract or Grant No. 
NASl-l7070 
13. Type of Report and Period Covered 
Contractor report 
14. Sponsoring Agency Code 
15. Supplementary Notes Additional support: NSF Mathematical Sciences Postdoctoral Fellowships, 
Courant Institute of Mathematical Sciences. 
Langley Technical Monitor: Robert H. Tolson, Final report 
16. Abstract 
The Kreiss Matrix Theorem asserts the uniform equivalence over all N x N matrices 
of power boundedness and a certain resolvent estimate. We show that the ratio of the 
constants in these two conditions grows linearly with N, and we obtain the optimal 
proportionality factor up to a factor of 2. Analogous results are also given for the 
related problem involving matrix exponentials e t. The proofs make use of a lennna that 
may be of independent interest, which bounds the arch length of the image of a circle ill 
the complex plane under a rational function. 
17. Key Words (S.uggested by Author(s)) 
Kreiss matrix theorem 
resolvent condition 
stability 
19. Security Classif. (of this report) 
Unclassified 
1 B. Distribution Statement 
64 Numerical Analysis 
Unclassified-Unlimited 
20. Security Classif. (of this page) 
Unclassified 
21. No. of Pages 
14 
22. Price 
A02 
N-305 For sale by the National Technical Information Service, Springfield, Virginia 22161 
End of Docurnent 


On the resolvent condition in the Kreiss matrix theorem

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server