FAST CALCULATION OF INVERSE MATRICES OCCURRING IN SQUARED-RECTANGLE CALCULATION

Transcription

1 R911 Philips Res. Repts 30, 329*-336*, 1975 Issue in honour of C. J. Bouwkamp FAST CALCULATION OF INVERSE MATRICES OCCURRING IN SQUARED-RECTANGLE CALCULATION 1. Introduction by A. J. W. DUIJVESTIJN Technological University Twente Enschede, The Netherlands (Received January 22, 1975) In my thesis 1) a method is described how to generate so-called c-nets automatically. I used a code ofthe network from which a planar representation can immediately be found; this code defines the net uniquely. From this code one can easily obtain the mesh-mesh incidence matrix INe automatically. By removing one row and its corresponding column one obtains a matrix C that plays an important role in the determination of the currents in the network when an accumulator is placed in one of the branches assuming resistors of 1 n in all of the branches. For the determination of the currents one needs the inverse of C. In my thesis this inverse was obtained by means of Gaussian elimination using integer arithmetic. Each element in the calculation was denoted by a pair of integers (although one common denominator per row). In order to avoid extensive growth of the integers it was necessary to determine highest common divisors that could be removed from the occurring integers. Obviously the determination of the inverse of C is rather time-consuming. In this paper a new method is presented that lis much faster than the abovedescribed method. To achieve this I use the fact that a network NI with a corresponding matrix CN! can be obtained from a network No having one branch less than NI' If CNo -1 happens to be known one can obtain CNl-1 without completely inverting the matrix. 2. Outline of the method In order to explain the method I use a reference network R and its dual 11 as an example. It is shown in fig. 1. The reference net R has the following characteristics: number of nodes number of meshes number of branches complexity K =9, M=8, B -:- 15, C = 1600.

2 330* A. J. W. DUIJVESTIJN Original A code of the original network R is A code of its dual ft is Fig. 1. Reference network R and its dual R. Dual For definitions of the code and the complexity I refer to ref. 1. From the code one can obtain the mesh-mesh incidence matrix INCR The matrix CR has been obtained from INCR by removing row 8 and column 8. The complexity of R is denoted by CR' The matrices INCR, CR and CR CR- 1 are given in fig. 2. We now assume that we want to construct a network N of order 16 with K = 10 and M = 8 by adding branch l3 to the dual network and dualizing back. Note that INC N differs from INC R in four corresponding elements namely: First of all we change the reference system of independent meshes. In the reference net R we have used the meshes 1, 2, 3, 4, 5, 6 and 7. We remove either 1 or 3. Let us remove mesh 3 and introduce mesh 8. The matrix with independent meshes 1,2,8,4, 5, 6, 7 is denoted by CR"' It can be shown that CR" = rcr I", where r is a square transformation matrix consisting of elements that only take the value 1,0 and -1 2). TI is the transpose of r. Hence it follows (CR")-1 = Cr-1)! CR- 1 F:», The inverse (CR")-1 can therefore be obtained from CR-1 by th~ following formulae: qlj = PIJ +P«- Pit - PtJ> i =1= t,j -=I t qlt = Pu - PIt> i =1= t

3 INVERSE MATRICES IN SQUARED-RECTANGLE CALCULATION 331* INC R CR er- 1 Fig. 2. INC R, CR and CR CR- 1 for the reference net R. CR and where t is the mesh that has been removed from CR and PIJ = element of CR- 1 and qlj = element of (er*)-1. In our example t = 3. Using these relations we obtain the following result: CR (CR*)-1 = The matrices CR* and en * only differ in one element, namely' the one with indices 1, 1.

4 332* A. J. W. DUIJVESTIJN I Fig.3. When we finally interchange meshes 1 and 7, we arrive at the results shown in fig. 3. The matrices are denoted by CR, C/o and CR CCR )-1. The matrix C/;'R )-1 is partitioned in the following way: CR CCR )-1 = t::t \~:j, where ;3 = ;2 = and ;2 t is the transpose of ;2' The complexity C N of the new network equals the determinant of CN "', It can therefore be obtained from C N = CR + ;3 = = The new inverse (/;,/.)-1 can be obtained from the following formula as can be verified easily:

5 INVERSE MATRICES IN SQUARED-RECTANGLE CALCULATION 333* Apparently el C N - e2 e2 t must be divisible by CR' The result is CN (CN )-l = Finally we interchange the meshes 1 and 7 and as a last step we remove mesh 8 and introduce mesh 3 again. We then have completed our task. The result is CN CN- 1 = Sofar we have only added branches in the dual network. We also want to add branches in the original network. One could think of dualizing the network but I could not easily obtain the corresponding inverse matrix of the dual network from the original network. Therefore I tried to derive the inverse of the new network from the inverse of the original network only. 3. Adding branches in the original network When we add a branch in the original network it means that we split a mesh. When for example a branch 25 is added, mesh 8 is split into meshes 8 and 9. Note that mesh 8 is not one of the independent meshes in CR' In case we split an independent mesh we have to change the reference system by the method described in sec. 2 such that this mesh is not one of the independent meshes.

6 334* A. J. W. DUIJVESTIJN Let us denote the new network by Nz. We then obtain the following meshmesh incidence matrix INC Nz : INC N2 = ~ 1-1 ~-1 ~ \ \-1 "9" =ï =i =ï... =ï ~ 4 For the determination of currents we remove mesh 9 and obtain CN2: The matrix CN2 is partitioned as follows: -1 aa Al = a, a Az = 4 and CRhas the same meaning as in sec

7 INVERSE MATRICES IN SQUARED-RECTANGLE CALCULATION 335* If we put gr = CR CR- 1 and gr Al = BI we can easily show that In our example we have: Finally we find CR Al = while AI' gr Al = 3232 and C N2 = 4 x = ~I C N2 CN2-1 = Significance of the method In 1962 we calculated all c-nets of orders up to and including 19 on the PASCAL and STEVIN computers of Philips' computing centre. The c-nets were made available on punched cards and punched paper tape. Furthermore all nets of order 20 were generated but kept in the computer. We investigated whether these nets produced any perfect squared-square of order 19.We found only a large number of imperfect squared-squares of order 19.In 1968 and 1969 IBM Netherlands gave me the opportunity to use the IBM 7094 at Rijswijk to tackle this problem. D. Severein wrote 'programs in 7094 assembly code according to the method described in my thesis. As a result the nets of orders up to and including 20 are available now on magnetic tape.

8 336* A. J. W. DUIJVESTIJN We found that the number of c-nets of order 20 is less than or equal to The following approach is now feasible. First we calculate for each net N of order 20 the matrix CN-1 by means of Gaussian elimination. Next we generate the nets of order 21, 22, 23 and 24 and search for existence of possible squared-. square solutions. To achieve this we investigate CN- 1 directly for the orders 21,22 and 23. For the nets of order 24 we only calculate the complexity and remember that squared-square solutions can only be obtained from a net with a complexity of the form 2kA 2, where k and A are integers. The nets of order 21 will not be generated in one run. We rather generate the complete tree of nets and search this tree. By doing this we only need to store four nets with inverse matrices. Obviously we will not store the nets on magnetic tape. Acknowledgement I am grateful to IBM Netherlands, which gave me the opportunity to use the IBM Furthemore I am indebted to Ir. J. G. Rietveld of IBM for his encouragement. I want to thank D. Severein of the Computing Centre of the Technological University Twente for carrying out the enormous task of programming and testing the large programs in the IBM 7094 assembly code. REFERENCES 1) A. J. W. Duijvestijn, Thesis, Eindhoven, 1962 (Philips Res. Repts 17, ,1962). 2) G. Kron, Tensor analysis of networks, John Wiley & Sons, Inc., New York, 1939.