PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 11, November 1996
EXISTENCE FOR A MATRIX EQUATION ARISING IN MICROELECTRONICS
JONQ JUANG
(Communicated by David H. Sharp)
Abstract. In this paper we rigorously show the existence of solutions of a matrix equation which arises in the design of micro electronical circuits. This equation was studied by Szidarovszky and Palusinsk [Appl. Math. Comput.
64, 115-119(1994)], who also presented an iterative algorithm for its solution.
We show, via an example, that this algorithm could converge extremely slow in certian cases. The solution can then be used to minimize the reflection coefficients of the active signals.
1. Introduction Consider the following matrix equation of the form
R = (M + X)−1(M− X). (1)
Here M is an M -matrix [3], which is given. A matrix A is called an M -matrix if A is invertible, A−1 is nonnegative (in the componentwise sense), and aij ≤ 0 for all i, j = 1,· · · , n, i 6= j. The unknown matrices R and X have the following constraints:
(C1) The matrix X = diag (x1, x2,· · · , xn), xi> 0 for all i = 1, 2,· · · , n. (C2) The diagonal of matrix R contains only zero elements.
Equation (1) arises in microelectronics. The matrix M is the characteristic admittance matrix, which represents various signal propagation properties of the interconnections of high speed electronic circuits and systems. The diagonal admit-tance matrix X gives the load of the resistive terminating network. The reflection matrix R describes the ratios of the amplitudes of the incident and reflected waves. Physically, matrix M has non-positive off-diagonal elements, a positive diagonal and a nonnegative inverse with positive diagonal. In practice, we wish to select the load of the resistive terminating elements so that the reflection coefficients of the active signals are equal to zero. That is, given an M -matrix M , find a diago-nal matrix X with positive diagodiago-nal such that R = (M + X)−1(M − X) has zero diagonal.
Received by the editors March 10, 1995 and, in revised form, May 22, 1995. 1991 Mathematics Subject Classification. Primary 78A25, 15A24.
Key words and phrases. Microelectronics, M -matrix, a priori bounds, degree theory.
The work is partially supported by the National Science Council of Taiwan, R. O. C. c
1996 American Mathematical Society
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3478 JONQ JUANG
Some numerical procedures for solving equation (1) were proposed in [2]. Note that the boundedness of the sequence in [2] has not been established. For more physical details see, for example, [1, 2].
The purpose of this paper is two-fold. First, a priori upper and lower bounds for X are obtained. The existence of solutions to (1) will be established via degree theory. Second, an example is given to illustrate that the algorithm in [2] could converge extremely slow in certain cases.
2. Main results
Define the mapping F : Rn×n → Rn×n such that for any n× n matrix A = (aij), F(A) = diag (a11, a22,· · · , ann). We note, as in [2], that, from (1),
R = (M + X)−1(M + X− 2X) = I − 2(M + X)−1X.
Therefore, all diagonal elements of (M + X)−1X must be equal to 12. Equation (1) can thus be decoupled as
F((M + X)−1) = diag ( 1 2x1 , 1 2x2 .· · · , 1 2xn ), which is equivalent to the following fixed point problem:
X = 1 2F
−1((M + X)−1) := G(X).
(2)
Notation. 1. Let A be the set of all n×n matrices with only nonnegative elements. 2. Let A, B,∈ Rn×n; we write A≥ B if A − B ∈ A.
To be complete, we recall the following well-known results, see e.g., 2.4.10 of [3] and Theorem 13.2.11 of [4] respectively.
Theorem 1. Let two n× n matrices Ai, i = 1, 2, be, respectively, decomposed as
Ai = Di− Bi, where Di, i = 1, 2, are diagonal parts of Ai, i = 1, 2. Suppose A1 is
an M -matrix, D1≤ D2 and B1≥ B2. Then A2 is an M -matrix and A−12 ≤ A−11 .
Theorem 2. Let D ⊂ Rn be an open bounded set. Suppose that φ : ¯D → Rn is
continuous. Assume that no solution of φ(x) = p lies on ∂D. Then the following hold:
(i) Homotopy Invariance. Let Ht be a homotopy, and suppose that Ht(x) 6= p
for any x∈ ∂D and t ∈ [0, 1]. Then d(Ht, p, D) is independent of t. (ii) d(I, p, D) = 1 if p∈ D, d(I, p, D) = 0 if p /∈ D.
(iii) If d(φ, p, D)6= 0, the equation φ(x) = p has at least one solution in D. Consider the following one-parameter family of equations:
X = 12F−1((M + tX)−1) := Gt(X), 0≤ t ≤ 1. (3)
We next establish a priori bounds for X and all t.
Lemma 1. Let X ∈ A be a solution of (3). Then 12F−1(M−1)≤ X ≤ F (M) for all 0≤ t ≤ 1.
Proof. Let X be as assumed. Using Theorem 1, we see that
X = 1 2F −1((M + tX)−1)≤ 1 2F −1((M + X)−1) ≤ 12F−1((F (M ) + X)−1) = 1 2(F (M ) + X).
A MATRIX EQUATION IN MICROELECTRONICS 3479
Consequently, the a priori upper bound as asserted follows. The a priori lower bound can be obtained by dropping the term tX.
Theorem 3. There exist R and X satisfying the constraints (C1) and (C2) and equation (1). Moreover, for any solution X∈ A of equation (2), the corresponding solution R of (1) has the property that−R ∈ A.
Proof. Let D = {X ∈ Rn×n : 1
4F−1(M−1) < X < 2F (M )}. D is evidently a
non-empty bounded open subset of Rn×n, and G
t: ¯D → Rn×n is continuous. It is also clear, via Lemma 1, that if X− GtX = 0 for X ∈ ¯D, then X ∈ D. The preparations for the use of degree theory are now complete. Consider the homotopy Ht= I− Gt. Hence by homotopy invariance
d(I− G0, 0, D) = d(I− G1, 0, D). But d(I− G0, 0, D) = d(I, 1 2F −1(M−1), D) = 1
by Theorem 2-ii, as12F−1(M−1)∈ D. The first assertion of the theorem now follows from Theorem 2-iii. The last assertion of the theorem follows from the constraints (C2) and the fact that (M + X)−1≥ 0.
Our second result deals with the asymptotic convergence rate of the algorithm in [2]. Consider now the iteration procedure:
X(k+1)= G(X(k)), (4a)
X(0)= 0. (4b)
Applying the theorem in [2], we obtain that {X(k)} is a bounded, increasing se-quence. Hence, it converges upward to a limit, say X∗. Note that in this case X∗ is the smallest positive solution of (2). In the following, we shall illustrate, via an example, that such an algorithm can be extremely slow to converge. Let
M = 1 −(1 − ε) −(1 − ε) 1 ,
where ε is a small positive parameter. Clearly, M is an M -matrix. Writing equation (2) in component form, we obtain that
x1= (1 + x1)(1 + x2)− (1 − ε)2 2(1 + x2) := f1(x1, x2; ε), (5a) x2= (1 + x1)(1 + x2)− (1 − ε)2 2(1 + x2) := f2(x1, x2; ε). (5b)
A simple calculation gives that the unique positive solution x∗ = (x∗1, x∗2) to (5) is x∗1= x∗2=p1− (1 − ε)2. Define : f : R2→ R2by
f (x; ε) = (f1(x; ε), f2(x; ε)),
(5c)
where x = (x1, x2).
Then the iteration procedure (4) is equivalent to x(k+1)= f (x(k); ε), (6a)
x(0) = 0. (6b)
3480 JONQ JUANG
Note that the asymptotic convergence rate of the algorithm is determined by the spectral radius σ(f0(x∗; ε)) of the Jacobian matrix f0(x∗; ε). A direct calculation yields that
σ(f0(x∗; ε)) = 1 2[1 +
(1− ε)2
(1 +p1− (1 − ε)2)] = 1− h(ε),
where h(ε) > 0 and h(ε) = o(ε12).
We also note that if M is a diagonal matrix, the convergence rate of (4) is 12. It is observed that as M becomes less diagonally dominate, the convergence rate of the algorithm deteriorates.
In conclusion, we note that if the initial sequence X(0) is chosen to be F(M ),
then {X(k)} is decreasing and bounded below. These observations are a direct
consequence of Lemma 1. The sequence then converges to the largest positive solution X∗∗ of (2). Applying Lemma 1, we obtain the following corollary. Corollary. Let R∗ = (rij∗) and r∗∗ = (r∗∗ij), respectively, be the corresponding solutions of equation (1) with respect to the solutions X∗ and X∗∗ of equation (2). We further assume that R = (rij) is the corresponding solution of (1) associated
with any positive solution X of (2). Then rij∗∗≤ rij ≤ r∗ij≤ 0 for all i, j. Acknowledgment
The suggestions by the referee on the improvement of this paper are greatly appreciated.
References
1. C. S. Chang, Electrical design of signal lines for multilayer printed circuit boards, IBM J. Res. Dev. 32(1988) 647-657.
2. F. Szidarovszky and O. A. Palusinski, A special matrix equation and its application in
mi-croelectronics, Appl. Math. Comput. 64(1994) 115-119. MR 95e:94065
3. J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several
variables, Academic Press, New York, 1970. MR 42:8686
4. V. Hutson and J. S. Pym, Applications of functional analysis and operator theory, Academic Press, New York, 1980. MR 81i:46001
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China
E-mail address: [email protected]
