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A special matrix equation and its application in Microelectronics
A Special Matrix Equation and Its Application in Microelectronics Ferenc Szidarovszky Department
of Systems and Industrial Engineering
University of Arizona Tucson, Arizona 8.5721
and
Olgierd A. Palusinski Department of Electrical and Computer Engineering University of Arizona Tucson, Arizond 8572i
Transmitted
by L. Ducksteifi
ABSTRACT
A special matrix equation is examined, which arises in designing electrical interconnections between microelectronic circuits and systems. The equation is first rewritten as a fixed point problem. A globally convergent iteration method is then proposed and monotone convergence is proved. A sufficient and necessary condition for the existence of the solution is presented. The solution can be used to minimize the reflection coefficients of the active signals.
1.
INTRODUCTION
Signal transmission in interconnections of high speed electronic circuits and systems involves such wave propagation phenomena as delay, reflection at the boundaries, and electromagnetic coupling between the lines (cross-talk). Reflections generate undesirable signals, which may lead to faults in operations; therefore the transmitting and receiving circuits are usually designed to minimize such reflections. Parallel interconnections are modeled as coupled transmission lines, and in the case of perfect materials they are characterized by wave equations, which involve coefficients typically represented in the form of inductance (LJ, and capacitance (c) matrices. These matrices are used to calculate the characteristic APPLIED 0 Elsevier
MATHEMATICS
AND COMPUTATION
64: 115-l 19 (1994)
115
Science Inc., 1994
655 Avenue of the Americas,
New York, NY 10010
0096-3003/94/$7.00
116
F. SZIDAROVSZKY
AND 0. PALUSINSKI
admittance matrix JJ = L-1 (Lc) ‘j2, which is useful in estimating various signal propagation properties of the interconnections. Very important phenomena occurring in the systems are the signal reflections on line boundaries. Reflections can be described in terms of reflection coefficients defined as the ratios of amplitudes of incident and reflected waves. In multiconductor transmission systems, which involve several waves, the reflections can therefore be determined using the matrix & of reflection coefficients. This matrix for the case of resistive terminations can be computed as: R=
(M+X)-‘(M-II),
(1)
where & is the diagonal admittance matrix representing the load of the resistive terminating network. Physics indicate that matrix @ has non-positive off-diagonal elements and a positive diagonal and a nonnegative inverse with positive diagonal. Hence & is an M-matrix [I]. In (1) we wish to select an appropriate matrix X which eliminates all elements of &. However, there are n2 equations and only n unknown diagonal elements of X because in practical situations & is diagonal. The diagonal elements of X represent the conductances of the terminating elements. In practice it is usually required that the reflection coefficients of the active signals are equal to zero. That is, the diagonal of matrix & must contain only zero elements. For more physical details see, for example, [2]. In summary, this problem can be mathematically formulated as follows. Given an M-matrix A4, find a diagonal matrix X with positive diagonal such that matrix (M+X)-’ (M -&) has zero diagonal. We can rewrite this problem as a system of nonlinear algebraic equations and use some iteration algorithm, such as the multivariable Newton method [3]. Let n be the size of the matrices under consideration, and let mij and rij denote the (i, j) elements of M and & respectively. If Xi is the i th diagonal element of X, then (1) implies that for i = 1, 2, . . . 3nandjfi,
?Tlii
-Xi
=
2
mierei
e=i t+1
and n
Wlij
=
C
mitrtj
+
(mii + Xi)rij.
r=1
r#i,j Therefore we have n2 equations for the n2 unknowns xi and rij (j # i). HOWever, the Newton method has only local convergence, and therefore a sufficiently good initial approximation is needed, which is hard to get. In this paper we develop a globally and monotonically convergent iteration algorithm to compute an appropriate matrix X. Thus we can eliminate the difficult problem of finding sufficiently good initial approximations.
117
A Matrix Equation in Microelectronics 2.
THE ALGORITHM Notice first that from (l), &=
(M+XJ’(M+X-2X)
=@2(&+x)-‘x;
therefore all diagonal elements of (M + XJ-‘X must be equal to l/2. This observation implies that the diagonal elements of X are nonzero and those of (M +X)-l are 1/2x1, 1/2x2,. . . , 1/2~~. For the sake of simplicity, introduce the point to point mapping E: RnXn + RnX” such that for any n x n matrix 4 = @ii), I;@) = diag(alt, . . . , a,,). That is, ‘(A) is defined as the diagonal of 4. The above observation can be rewritten as F((@+x)-‘)
=diag
,
&,...,& n>
which is equivalent
to the following n-dimensional
x = g-q@ This problem is obviously iteration procedure
equivalent
p+‘)
=
fixed point problem:
+s)-1).
to our original problem.
g-1 ((AJ + p-l)
starting from the initial approximation
(2) Consider now the
(3)
& (0) = Q.
THEOREM. For all k 2 0, Xck+‘) > &ck). The iteration sequence is bounded if and only ifthe$xedpointproblem (2) has at least one nonnegative solution. Zn this case, the iteration sequence converges to the smallest nonnegative solution of
(2).
PROOF. The assumptions on matrix M imply that X(‘) 2 0, that is, X(l) > x(O). It is well known [l] that (M + X)-t decreases in 3; therefore mapping mcreases in X. Assume next that for some C(X) = (1/2@((M +a-‘) k 1_Xtk) > Xtk-l) Then -.
p+‘)
= G(p))
2 G(p-1))
= X(k).
118
F. SUDAROVSZKY
AND 0. PALUSINSKI
Let X* be a nonnegative fixed point of problem (2). We prove next that for all k, XCk) < X*. Assume that for some k, X@-‘) ( X*. Then since X (0) = _0 < _ x+, -X(k) = @k-l))
5 &“)
= x*.
Assume first that sequence {XC”)} 1s ’ b ounded. The monotonicity of this sequence implies that it converges. Let X** denote the limit. The continuity of mapping G implies that X** is a fixed point of G and from the above observation we conclude that x** I X*. Hence ,** is the smallest nonnegative fixed point. Assume next that mapping G has a nonnegative fixed point. Then it is a bound for sequence (@)}, which completes the proof. n
EXAMPLE.
We illustrate method (3) in the case of matrix
Select -X(O) = 0; then a simple computer program gives the results:
x*=
(
9.8703 0 0
0 9.8703 0
0 0 19.8912
Eleven initial iterates are presented in Table 1.
Assume that a nonnegative diagonal matrix x(O) is found such REMARK. that G(x(‘)) 5 X (‘1 . Then similarly to the proof of the theorem one can show that sequence {XC”)} 1s ’ d ecreasing, and converges to the largest fixed point X* such that X* 5 x(O). Since any fixed point satisfies the above condition with equality, we have proved that nonnegative fixed point exists if and only if there is a nonnegative diagonal matrix X such that G(X) 5 X.
119
A Matrix Equation in Microelectronics
TABLE 1. NUMERICALRESULTS
1 2 3 4 5 6 7 8 9 10
3.
4.8668 7.3458 8.5983 9.2397 9.5478 9.7079 9.7886 9.8292 9.8496 9.8599
4.8668 7.3458 8.5983 9.2397 9.5478 9.7079 9.7886 9.8292 9.8496 9.8599
9.8826 14.8666 17.3702 18.6267 19.2569 19.5731 19.7327 19.8112 19.8511 19.8711
CONCLUSIONS
In practical applications the combination of the method of the previous section with the multivariable Newton method is the procedure to be used. It is well known that the Newton method converges if the initial approximation is good enough, and the convergence is very fast. Therefore one should apply first method (3), starting from the initial solution x(O) = 0 until a sufficiently good approximation Xck) is obtained. And then starting from X ck) as initial solution, the multivariable Newton method has to be applied until the difference between two consecutive approximations becomes smaller than a given error tolerance. Finally we mention that our iteration method is a point-to-point version of the general scheme discussed in [4]. REFERENCES J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Non-linear Equations in Several Variables, Academic Press, New York, 1970. C. S. Chang, Electrical design of signal lines for multilayer printed circuit boards, IBMJ. Rex Dev. 32547-657
(1988).
F. Szidarovszky and S. Yakowitz, Principles and Procedures ofNumericalAnalysis, Plenum Press, New York/London, 1978. I. K. Argyros and F. Szidarovszky, On the monotone convergence of algorithmic models, Appl. Math. Comput. 48: 167-176 (1992).
of Systems and Industrial Engineering
University of Arizona Tucson, Arizona 8.5721
and
Olgierd A. Palusinski Department of Electrical and Computer Engineering University of Arizona Tucson, Arizond 8572i
Transmitted
by L. Ducksteifi
ABSTRACT
A special matrix equation is examined, which arises in designing electrical interconnections between microelectronic circuits and systems. The equation is first rewritten as a fixed point problem. A globally convergent iteration method is then proposed and monotone convergence is proved. A sufficient and necessary condition for the existence of the solution is presented. The solution can be used to minimize the reflection coefficients of the active signals.
1.
INTRODUCTION
Signal transmission in interconnections of high speed electronic circuits and systems involves such wave propagation phenomena as delay, reflection at the boundaries, and electromagnetic coupling between the lines (cross-talk). Reflections generate undesirable signals, which may lead to faults in operations; therefore the transmitting and receiving circuits are usually designed to minimize such reflections. Parallel interconnections are modeled as coupled transmission lines, and in the case of perfect materials they are characterized by wave equations, which involve coefficients typically represented in the form of inductance (LJ, and capacitance (c) matrices. These matrices are used to calculate the characteristic APPLIED 0 Elsevier
MATHEMATICS
AND COMPUTATION
64: 115-l 19 (1994)
115
Science Inc., 1994
655 Avenue of the Americas,
New York, NY 10010
0096-3003/94/$7.00
116
F. SZIDAROVSZKY
AND 0. PALUSINSKI
admittance matrix JJ = L-1 (Lc) ‘j2, which is useful in estimating various signal propagation properties of the interconnections. Very important phenomena occurring in the systems are the signal reflections on line boundaries. Reflections can be described in terms of reflection coefficients defined as the ratios of amplitudes of incident and reflected waves. In multiconductor transmission systems, which involve several waves, the reflections can therefore be determined using the matrix & of reflection coefficients. This matrix for the case of resistive terminations can be computed as: R=
(M+X)-‘(M-II),
(1)
where & is the diagonal admittance matrix representing the load of the resistive terminating network. Physics indicate that matrix @ has non-positive off-diagonal elements and a positive diagonal and a nonnegative inverse with positive diagonal. Hence & is an M-matrix [I]. In (1) we wish to select an appropriate matrix X which eliminates all elements of &. However, there are n2 equations and only n unknown diagonal elements of X because in practical situations & is diagonal. The diagonal elements of X represent the conductances of the terminating elements. In practice it is usually required that the reflection coefficients of the active signals are equal to zero. That is, the diagonal of matrix & must contain only zero elements. For more physical details see, for example, [2]. In summary, this problem can be mathematically formulated as follows. Given an M-matrix A4, find a diagonal matrix X with positive diagonal such that matrix (M+X)-’ (M -&) has zero diagonal. We can rewrite this problem as a system of nonlinear algebraic equations and use some iteration algorithm, such as the multivariable Newton method [3]. Let n be the size of the matrices under consideration, and let mij and rij denote the (i, j) elements of M and & respectively. If Xi is the i th diagonal element of X, then (1) implies that for i = 1, 2, . . . 3nandjfi,
?Tlii
-Xi
=
2
mierei
e=i t+1
and n
Wlij
=
C
mitrtj
+
(mii + Xi)rij.
r=1
r#i,j Therefore we have n2 equations for the n2 unknowns xi and rij (j # i). HOWever, the Newton method has only local convergence, and therefore a sufficiently good initial approximation is needed, which is hard to get. In this paper we develop a globally and monotonically convergent iteration algorithm to compute an appropriate matrix X. Thus we can eliminate the difficult problem of finding sufficiently good initial approximations.
117
A Matrix Equation in Microelectronics 2.
THE ALGORITHM Notice first that from (l), &=
(M+XJ’(M+X-2X)
=@2(&+x)-‘x;
therefore all diagonal elements of (M + XJ-‘X must be equal to l/2. This observation implies that the diagonal elements of X are nonzero and those of (M +X)-l are 1/2x1, 1/2x2,. . . , 1/2~~. For the sake of simplicity, introduce the point to point mapping E: RnXn + RnX” such that for any n x n matrix 4 = @ii), I;@) = diag(alt, . . . , a,,). That is, ‘(A) is defined as the diagonal of 4. The above observation can be rewritten as F((@+x)-‘)
=diag
,
&,...,& n>
which is equivalent
to the following n-dimensional
x = g-q@ This problem is obviously iteration procedure
equivalent
p+‘)
=
fixed point problem:
+s)-1).
to our original problem.
g-1 ((AJ + p-l)
starting from the initial approximation
(2) Consider now the
(3)
& (0) = Q.
THEOREM. For all k 2 0, Xck+‘) > &ck). The iteration sequence is bounded if and only ifthe$xedpointproblem (2) has at least one nonnegative solution. Zn this case, the iteration sequence converges to the smallest nonnegative solution of
(2).
PROOF. The assumptions on matrix M imply that X(‘) 2 0, that is, X(l) > x(O). It is well known [l] that (M + X)-t decreases in 3; therefore mapping mcreases in X. Assume next that for some C(X) = (1/2@((M +a-‘) k 1_Xtk) > Xtk-l) Then -.
p+‘)
= G(p))
2 G(p-1))
= X(k).
118
F. SUDAROVSZKY
AND 0. PALUSINSKI
Let X* be a nonnegative fixed point of problem (2). We prove next that for all k, XCk) < X*. Assume that for some k, X@-‘) ( X*. Then since X (0) = _0 < _ x+, -X(k) = @k-l))
5 &“)
= x*.
Assume first that sequence {XC”)} 1s ’ b ounded. The monotonicity of this sequence implies that it converges. Let X** denote the limit. The continuity of mapping G implies that X** is a fixed point of G and from the above observation we conclude that x** I X*. Hence ,** is the smallest nonnegative fixed point. Assume next that mapping G has a nonnegative fixed point. Then it is a bound for sequence (@)}, which completes the proof. n
EXAMPLE.
We illustrate method (3) in the case of matrix
Select -X(O) = 0; then a simple computer program gives the results:
x*=
(
9.8703 0 0
0 9.8703 0
0 0 19.8912
Eleven initial iterates are presented in Table 1.
Assume that a nonnegative diagonal matrix x(O) is found such REMARK. that G(x(‘)) 5 X (‘1 . Then similarly to the proof of the theorem one can show that sequence {XC”)} 1s ’ d ecreasing, and converges to the largest fixed point X* such that X* 5 x(O). Since any fixed point satisfies the above condition with equality, we have proved that nonnegative fixed point exists if and only if there is a nonnegative diagonal matrix X such that G(X) 5 X.
119
A Matrix Equation in Microelectronics
TABLE 1. NUMERICALRESULTS
1 2 3 4 5 6 7 8 9 10
3.
4.8668 7.3458 8.5983 9.2397 9.5478 9.7079 9.7886 9.8292 9.8496 9.8599
4.8668 7.3458 8.5983 9.2397 9.5478 9.7079 9.7886 9.8292 9.8496 9.8599
9.8826 14.8666 17.3702 18.6267 19.2569 19.5731 19.7327 19.8112 19.8511 19.8711
CONCLUSIONS
In practical applications the combination of the method of the previous section with the multivariable Newton method is the procedure to be used. It is well known that the Newton method converges if the initial approximation is good enough, and the convergence is very fast. Therefore one should apply first method (3), starting from the initial solution x(O) = 0 until a sufficiently good approximation Xck) is obtained. And then starting from X ck) as initial solution, the multivariable Newton method has to be applied until the difference between two consecutive approximations becomes smaller than a given error tolerance. Finally we mention that our iteration method is a point-to-point version of the general scheme discussed in [4]. REFERENCES J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Non-linear Equations in Several Variables, Academic Press, New York, 1970. C. S. Chang, Electrical design of signal lines for multilayer printed circuit boards, IBMJ. Rex Dev. 32547-657
(1988).
F. Szidarovszky and S. Yakowitz, Principles and Procedures ofNumericalAnalysis, Plenum Press, New York/London, 1978. I. K. Argyros and F. Szidarovszky, On the monotone convergence of algorithmic models, Appl. Math. Comput. 48: 167-176 (1992).