Modelling and numerical solution of multibranched DC rail traction power systems

Y . Cai M.R. Irving S.H. Case

Indexing terms ' Gauss elimination, Traction power network, Zollenkopfbifactorisaiion

Abstract: Based on the method of nodal voltage circuit analysis, a complex DC-fed traction power network with multibranched lines has been mod- elled. The model also incorporates detailed return circuits so that rail potentials can be calculated, a requirement which arises because of safety and stray current considerations. The introduction of branch joints and divided rail cells greatly increases the quantity of circuit nodes and gives rise to a large sparse conductance matrix. After establishing a set of simultaneous linear equations describing the behaviour of the network, consider- ation is given to the most suitable method of solu- tion to prevent excessive computation times and storage requirements. Sample calculation results for train, substation busbar and branch joint volt- ages and currents are given, together with rail potential profiles.

1 Introduction

The development of computer-based simulations to study the performance characteristics of DC rail transit systems has a history spanning more than a decade [l-31. In this paper, which concentrates on the power network solution rather than the detailed train model, emphasis is given to the mathematical solution of a large-scale DC traction network described by a sparse conductance matrix. A multiladder DC traction network model is constructed using lumped circuit components. The electrical analysis approach for solving DC traction circuits in an existing simulation package is reviewed and enhamements to the method for the efficient solution of large-scale circuits introduced. The application of Zollenkopf's bifactorisa- tion method to the analysis of complex DC-fed traction power networks is described and illustrated. The associ- ated optimally ordered pivotal sequence, packed com- puter storage scheme and special programming techniques are also described. Finally, sample calculation results are presented and comparisons are made among three programs using alternative methods.

(0 IEE, 1995 Paper 2118B (P2). first received 16th January and in revised form 7th June 1995 Y . Cai and M.R Irving are with the Department of Electrical Engineer- ing and Electronics, Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom S.H. Case is with Balfour Kilpatrick Limited (a BICC company), Power Construction Division, PO Box 12, Acornfield Road, Kirkby, Liverpool L33 7TY, United Kingdom

I E E Proc.-Electr. Power Appl., Vol. 142, N o . 5, September 1995

A DC-fed traction power network can be modelled by defining nodal voltage equations relating the node con- ductances and node voltages to the node currents [4]. This allows matrix algebra to be used to find numerical solutions. In general, methods for solving a set of simul- taneous linear equations can be divided into two cate- gories: direct and iterative [SI. The direct methods are based on straightforward manipulation of the equations. They are very accurate particularly if the pivotal elements of the coefficient matrix are dominant. However, they require large computer storage space and computation time if sparsity techniques are not used. The simplest direct method is Gauss elimination, where the coefficient matrix is transformed into a triangular matrix. The unknown variables are then evaluated by back- substitution. Iterative methods solve the equations by successive approximations until the results are within an acceptable accuracy limit. These methods are often not preferred owing to the possibility of slow convergence or even divergence occurring.

In 1988, a PC-based simulation program Railpower was developed by Balfour Beatty for studying relatively simple unbranched traction power systems. Subsequently, it was decided to develop new simulation software capable of handling multibranched networks, and incorp- orating a detailed model of the return circuit, thus enabling rail potential profiles to be calculated at every time step. (This last requirement was driven partly by the recent trend in Britain towards a renaissance of tram systems, and the attendant concern regarding possible stray current corrosion). By introducing these refinements the quantity of circuit nodes is greatly increased and a large sparse conductance matrix is produced. It is clear that the matrix solution stage is critical, because this is the most time consuming part of the traction power network analysis. Therefore it is essential to find an effi- cient means of solution which avoids the wasteful storage and processing of the large number of matrix elements which are known to remain zero.

In the existing Railpower package, a modified Gauss elimination technique, termed target Gauss elimination, has been applied to manipulate the traction circuit con- ductance matrix exploiting its sparsity. This method and the related computing techniques were originally devel- oped by Dr. J.G. Yu while at Balfour Beatty [SI. The method is highly efficient but is restricted to a symmetri- cal conductance matrix with two off-diagonal terms in each row or column. A large proportion of the conduct- ance matrices arising in traction power systems can be arranged to satisfy these constraints by careful number- ing of the circuit nodes.

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In a ladder-like double-track traction power network, approximately 85% of the nodes are connected to only two others, and these are assembled to form a submatrix

positlve busbar

down track

Fig. 1 rail system

Sectional multiladder network representotion of double-trock

in which target Gauss elimination can be implemented to minimise the computation time and storage. The sub- matrices formed by the remaining nodes with more than two nonzero elements in a row or column may not use the target Gauss elimination since new nonzero matrix coefficients (or ‘fill-ins’) are created which cannot be stored in the target elimination data structure. To com- plete the solution, standard Gauss elimination with back- ward substitution and a full matrix storage scheme can therefore be applied to solve the remaining simultaneous equations. However, it is obviously not economical to deal with many zero operations even though the remain- ing nodes are only 15% of the total. This overhead can be very significant when a large traction power network is being studied.

There is a significant benefit to be obtained if an efi- cient method of dealing with the submatrix, which includes many zero terms can be devised. Many previous results are available on the subject of the solution of sparse systems of equations [7, 81. The bifactorisation algorithm introduced by Zollenkopf has been successfully used for power system analysis and has been shown to achieve an advantage over triangular decomposition [9 ] . The bifactorisation method requires less storage, less indexing and simpler arithmetic operations to solve the equations from a knowledge of the factors, and therefore its application in traction power networks would appear to be attractive. Three methods for the numerical solu- tion of multibranched DC traction power systems have been investigated and programmed.

Method 1. target Gauss elimination + standard Gauss elimination: This uses the target Gauss elimination for the majority of the matrix and the standard Gauss elim- ination for the remaining submatrices (for which the target elimination scheme is not applicable).

Method 2. target Gauss elimination + bijiactorisution ulgo- rithm: This uses the target Gauss elimination as before, but uses the Zollenkopf bifactorisation algorithm for the remaining sparse submatrices solution.

Method 3. hijactorisation algorithm: This uses the Zoll- enkopf bifactorisation method for the whole matrix solu- tion.

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2 Multiladder, two-track system model

The components of the DC-powered traction system are modelled in terms of resistances and current sources, and the interconnection of these components constitutes a nodal electrical circuit. The traction return circuit is modelled by dividing the track into a number of cells which are generally equal in length. Each cell forms a pi-circuit with a series resistance R and shunt leakage conductance G to represent the distributed parameters. (Each track is assumed to consist of two similar rails in parallel, giving half the longitudinal resistance of a single rail). The values of R and G may vary from section to section along the rails. For most practical purposes, a cell length of 100 m is considered sufficient for accurate DC traction network solutions. Smaller values can be used if required, but this will increase substantially the number of equations. Most rail nodes, designated ‘normal’ rail nodes only connect to two other nodes, while rail nodes at substations, crossbonds, branches or trains are termed ‘special’ rail nodes and connect with three others. It is assumed that no train exactly coincides with a sub- station, a branch joint or a crossbond. If this occurs in practice, the situation can easily be resolved by artificially moving the train a very small distance which does not have a significant effect on the results or cause any solu- tion stability problems.

A substation is represented by a Thevenin equivalent voltage source (constant voltage in series with a resistance) which is transformed within the model into a Norton equivalent current source. There are two circuit nodes incorporated in the substation. The positive busbar node is connected with five others: four feeder nodes and the negative busbar node. The negative busbar node is connected with three others: two rail nodes and the positive busbar node. The model representation of the trains may vary depending on the iterative solution method adopted. However, the numerical solution tech- niques presented here are equally applicable to the various train models and the various iterative techniques available for determining the interaction between trains and the supply system. In the Railpower package, for example, the trains can be modelled by a voltage- dependent resistance, a voltage-dependent ideal current source, a voltage-dependent Norton equivalent current source or a Thevenin equivalent voltage source.

It can be seen that the system model for each track is in the form of a double ladder. The two double ladders are interconnected at locations where the substations are situated or the tracks are crossbonded. Some systems incorporate paralleling stations where the overhead con- ductors (or third rails) associated with the two tracks are connected together to improve power distribution. These paralleling stations would also constitute points of inter- connection between the two double ladders.

Branch joints and paralleling stations are modelled respectively by three or four switches which can, if required, be opened to facilitate the simulation of various outage conditions. A branch joint node is linked with three other nodes: two on the trunk line and one on the branch line. The model allows for several double or single track branch lines to be connected to the trunk line. A paralleling station node will be connected with four other nodes (two in each double ladder).

Based on the circuit model described, and the principle of nodal analysis, three computer programs have been developed to obtain the voltages at the pantographs (or collector shoes) of the trains and substation busbars, the

I E E Proc.-Electr. Power Appl., Vol. 142, Nu. 5, September 1995

rail potential profiles, and the currents at the trains and substations of a DC-fed traction power system. The same procedure for constructing a traction power circuit is

L J

t consider alternative matrix A

REDUCE (numerical factorisation 1

consider alternative

vector b

i

-

-

ASSEMBLE I (formulate problem equations)

ORDER I (establish elimination sequence) I

applied in all three programs. Fig. 1 shows a sectional diagram of a ladder-like DC traction circuit model with one substation and five trains.

3 Solution by combined target and conventional Gauss elimination

The node voltages are derived by setting up the conduct- ance matrix and solving the associated linear equations. Target Gauss elimination and conventional Gauss elim- ination are combined in the program to solve the circuit nodal equations. A circuit model is established by the data input routine which gives the system dimensions and locations of all the traction elements. The next task is to number the nodes in a logical way to facilitate sub- sequent identification and manipulation. The conduct- ance matrix is symmetrical and is also sparse because each node is directly connected to only a few others. An algorithm which stores and manipulates only nonzero elements can be produced by considering node number- ing and exploiting regularity. The conductance matrix is split into six submatrices each containing different types of nodes according to the following node numbering sequence :

(1) normal rail nodes (2) special rail nodes (3) substation negative busbar nodes (4) train nodes ( 5 ) branch joint nodes (6) paralleling station and substation positive busbar

nodes

The first submatrix stores the normal rail nodes with two off-diagonal elements in each row. Four submatrices rep- resenting the special rail nodes, substation negative busbar nodes, train nodes and branch joint nodes are then combined together to form a submatrix with three off-diagonal elements in each row. The paralleling station and substation positive busbar nodes are stored in a sub- matrix with five off-diagonal elements in each row. The fifth element for the paralleling station node in this sub- matrix is set to zero since only four other circuit nodes are connected to it. Additional index arrays enable identi- fication of the column location of each nonzero off-

I E E Proc.-Elertr. Power Appl., Vol. 142, N o . 5, September 1995

diagonal element in the corresponding conductance matrix.

The target row is found by an element column number in the index array corresponding to the matrix with two off-diagonal elements. The element in the target row is then eliminated by searching the address contained in the index array. New elements created during the elimination process are assigned to the storage locations vacated by the eliminated elements. Since only one location is avail- able in the target row after the elimination process, this target Gauss elimination method is limited to the sub- matrix with two nonzero elements in each row or column. Standard Gauss elimination with backward sub- stitution is applied at this stage to solve the remaining simultaneous equations with a full matrix storage scheme.

It is evident that a lot of memory space and computa- tion time must be wasted in storing zero elements and performing zero operations. It is therefore important to consider more efficient matrix solution methods for the solution of the remaining submatrices. Advantage can be taken of sparsity to reduce the amount of computer storage and the duration of the computing time. To enable the efficient use of sparsity in order to minimise the number of nonzero elements introduced during the elimination process, it is necessary to employ special techniques and programming.

4 Implementation of Zollenkopf's algorithm

Zollenkopf introduced a new form of the Gauss elim- ination process. Instead of systematically eliminating rows in the matrix, he forms a sequence of elementary matrices from the original matrix, and shows that by multiplying these together, the inverse would be formed. Given a vector for which the solution is desired, the solu- tion could then be obtained merely by successive elemen- tary matrix multiplications. These multiplications are simple because each elementary matrix has a small number of nonzero elements in one row or column of an otherwise unit matrix. This approach is called bifactorisa- tion. This method is particularly suitable for sparse coef- ficient matrices that have dominant diagonal elements and that are either symmetrical, or, if not symmetrical, have a symmetrical sparsity structure. The interested reader is referred to the original paper by Zollenkopf for further details of the bifactorisation algorithm [SI.

The problems of sparsity-directed programming are associated not only with the basic aspects of factorisation and numerical solution but also with the problem of storing and identifying the nonzero elements. Zoll- enkopf s paper contains flowcharts which accurately describe the bifactorisation method. The flowcharts incorporate an algorithm for near-optimal sparsity order- ing of the elimination steps, which is based on selecting the next column from amongst those with the minimum number of nonzero elements. For overall efficiency, an ordering process is initially undertaken, which establishes the elimination sequence and produces data structures required to hold the elimination factor matrices. All numerical calculations are performed subsequently in the reduction and solution processes. A major advantage of this approach is that the elimination sequence and data structures d o not need to be altered unless the sparsity structure of the original matrix is changed, consequently the ordering process does not have to be repeated for each numerical elimination. Each new set of results can be generated by loading the matrix and dependent vector

325

with values, and performing the reduction and solution routines.

A Fortran program using Zollenkopf's bifactorisation method to solve a large sparse set of linear equations has been implemented [lo]. The program is composed of four subroutines which perform the four distinct aspects of formulating the problem equations, establishing the sequence of elimination, factorising the coefficient matrix and solving the problem numerically. A set of n linear equations can be expressed in matrix notation as A x = b, where x is a column vector of the n unknowns and b is a known vector with at least one nonzero element. For the same network configuration the bifactorisation program can be used in an iterative mode to provide solutions for different coefficient values A and right-hand-side vectors b as shown in Fig. 2.

5 Solution using bifactorisation algorithm

To achieve the most efficient DC traction power circuit solution, the bifactorisation method has been studied and its suitability in conjunction with traction power net- works has been investigated. Two Fortran programs (referred to as method 2 and method 3) have been devel- oped which incorporate the Zollenkopf subprograms.

Method 2 involves the circuit solution methods of target Gauss elimination and bifactorisation. After per- forming the target Gauss elimination process, two further submatrices remain with three or five nonzero elements in each row. The value and column location of the nonzero elements in these submatrices may vary during the elimination process. Using the bifactorisation method instead of standard full matrix Gauss elimination is the first aspect to be considered. To fulfil the requirements for the subroutine which formulates the Zollenkopf equa- tions, the remaining submatrices are converted into node and branch data. The node numbers at the sending and receiving end of each branch are identified from the column addresses of the nonzero diagonal and off- diagonal elements in each row. The branch conductance can easily be calculated. The next step is to call the four subroutines of the bifactorisation program.

A further Fortran program (method 3) has also been written using the bifactorisation method throughout. The input requirements are the same as for the original program. The circuit model is established by a ladder-like nodal network with different types of node connected together, such as rail nodes, train nodes, substation nodes and so on. The conductance between adjacent nodes is evaluated in the program and a branch with a switch is placed at the terminals, junctions and paralleling stations to represent the connection status. The circuit node num- bering is started from the first to second ladder (rails to catenaries or third rails) according to the following sequence:

(1) rail nodes (2) negative busbar nodes (3) train and branch joint nodes (4) substation positive busbar and paralleling station

nodes

The following branch conductances and sending and receiving end node numbers of each branch are required in the program. The branch terms are dealt with in the following sequence:

(I) branches along the rails (2) rail terminal branches (3) rail crossbond branches

326

(4) branches between substation negative busbar and

(5) branches along the catenaries or third rails (6) catenary or third rail terminal branches (7) train branches connected between catenary (or

third rail) and running rail (8) branches between substation positive and negative

busbar

Since an optimal ordering process is performed in the bifactorisation matrix solution method, the numbering sequence of the circuit nodes and branches does not affect the efficiency of the resultant sparse matrix oper- ation. Having assembled the node and branch data, the bifactorisation method can be applied by calling four subroutines. All the node voltages are available at the end of this procedure. The first and second ladder volt- ages are then extracted following the nodal sequence.

rail

6 Calculation results

To illustrate the computational effort required by the three methods, a DC-fed double-track traction network with a branch line has been modelled. Three substations with 18 operating trains are located on both tracks of a trunk line and branch line. The trains are assumed to be drawing a power of 750 kW which is a typical average value for a 750 V light rail vehicle. This is a reasonably representative DC traction system which is used as a benchmark. A schematic diagram of this DC traction supply system is shown in Fig. 3.

1 station5 A I A

trunk h e

up track I';t down track

branch line trains are shaded

.-"Tubyation*

Fig. 3 Schematic diagram of DC traction supply system

The three programs, using the network solution methods of target Gauss elimination, standard Gauss elimination, bifactorisation and their combinations have been applied to this model system. All three programs achieve the same calculated results. Table 1 shows the

Table 1 : Substation voltages and currents

Number Positive Negative Current

v v A 1 643.0 -22.3 4231.5 2 628.5 -19.2 5112.8 3 646.7 -20.5 4140.1

calculated substation positive and negative busbar volt- ages and feeding currents.

Train voltages and currents are obtained and listed in Table 2. The two lowest train voltages 513.1 and 513.8 V correspond to trains 11 and 4 which are at the centre of the trunk line and relatively far from the substations. The

I E E Proc-Electr. Power Appl . , Vol. 142, No. 5 , September 1995

sum of the substation feeding currents is equal to the total train current allowing for rounding errors.

The rail potential distribution along each track can also be calculated at the same time. It is seen from Figs. 4

Table 2: Calculated train voltages and currents

Number Voltage Current

1 625.4 833.9 2 570.8 761.1 3 531.7 709.0 4 513.8 685.1 5 550.9 734.5 6 563.6 751.5 7 556.5 742.0 8 606.4 808.5 9 624.2 832.3

10 548.1 730.7 11 513.1 684 1 12 530.5 7073 13 562.5 750.0 14 571.5 7620 15 547.6 730.1 16 525.8 701.1 17 566.3 755.1 18 604.4 805.9

V A

substation substation

0

Trunk line up-track rail potential profile

distance

Fig. 4

and 5 that the rail voltages at the substation positions are depressed owing to the return currents flowing to the substations through these nodes.

substation 3

- 5 0 g -5

\

7 -

-15 -20L

0

Branch line up-track rail potential profile dlstance

Fig. 5

Table 3: General comparison of three methods

Program Method CPU time

s 1 TGE+SGE 1.52 2 TGE+ZBF 0.26 3 ZBF 0.76

7

A comparison has been made to illustrate the relative efficiency of the three programs with the emphasis mainly on computational speed since computer memory is now relatively inexpensive. Three matrix solution methods appear in Tables 3-5 with TGE, S C E and ZBF standing for the target Gauss elimination, standard Gauss elim- ination and Zollenkopf's bifactorisation.

The three programs have been run on a Digital VAX8600 machine and the run times are shown in Table 3. It is evident that by introducing Zollenkopf's method

Comparison of the three methods

Table 5: Comparison of Zollenkopf subproblem character- istics in Droarams 2 and 3

Program Nodes Branches Ratio Nonzeros Time

2 113 171 1.51 455 0.06 3 753 813 1.08 2380 0.61

n b bln s

the efficiency of program 2 has been significantly improved compared with program 1. Program 2 is six times faster than program 1 and program 3 is two times faster than program 1. Another aspect which is apparent is that the target elimination technique is the most effi- cient method for solving a sparse matrix with the special feature of only two nonzero elements in each row or column and having a symmetrical structure. The sequence of circuit nodes are very well organised in the TGE before performing the elimination so that less com- puting time is spent on ordering during the process. More details of the CPU time consumed in each sub- process are given in Table 4.

All three programs took the same amount of time (0.15 s) to assemble input data and establish the circuit model. It is also informative to consider the character- istics of the subproblems considered by the bifactorisa- tion procedure in the case of programs 2 and 3. Table 5 shows the Zollenkopf subproblems considered by the bifactorisation procedure in the case of programs 2 and 3. Tables 5 shows the Zollenkopf subproblem sizes and cor- responding time taken (the total time for ASSEMBLE, ORDER, REDUCE and SOLVE).

Zollenkopf's method involves four subprocesses: data formulation, ordering, factorisation and solution. Separating these various aspects into distinct subroutines offers a very significant advantage. For example, when solutions are required for several right-hand vectors, the same factorised matrix can be used. Similarly, when several solutions are required for different coefficient matrices having the same sparsity structure, the original pivoting order can be used for all the required solutions. Table 4 shows that most of the computing time in bifac- torisation is spent in the ordering process (0.48 s in program 3). For large systems this component is by far the greatest. Tables 4 and 5 show that when the system size is increased, the ordering time is very significant. Consequently, it is clear that if similar problems are to be

Table 4: Computation time for each task

Program Construct TGE SGE ZBF model

ASSEMBLE ORDER REDUCE SOLVE

S s s s s s s 1 0.1 5 0.05 1.32 2 0.1 5 0.05 0.01 0.02 0.02 0.01 3 0.1 5 0.05 0.48 0.07 0.01

327 I E E Proc.-Elertr. Power Appl., Vol. 142, No. 5 , September 1995

solved then, once the order has been determined, the time taken to calculate the solution for new coefficient matrices or new known vectors becomes minimal, even for very large systems.

The DC traction power flow problem is formulated in its basic analytical form with the network represented by linear, bilateral and lumped parameters. However, the train power and voltage constraints make the problem nonlinear and the numerical solution must therefore be iterative in nature. The conductance matrix iterative method which represents the trains as resistances, changes the conductance matrix but not the structure in each iterative step and only requires repeating the REDUCE and SOLVE subprocesses (0.08 s) in program 3. However, program 2 requires repeating the target Gauss elimination and the whole bifactorisation process in each iteration (a total time of 0.11 s). The current vector iterative approach using an ideal current source train model takes full advantage of the bifdCtOriSatiOn circuit solution method by only executing SOLVE (0.01 s), the last of the four subroutines, in each iterative step, solving the same equations but with different right- hand-side vectors. If this iterative process is used and requires more than six iterations for the traction power system simulation, program 3 becomes attractive com- pared with program 2. Furthermore, the introduction of train regenerative braking creates additional nonlinear constraints when there is a surplus of power produced by the regenerative braking trains and in this case program 3 is computationally very attractive.

8 Conclusions

A complex multiladder, multibranched DC traction power system has been modelled and solved using differ- ent circuit solution methods. Where branch lines and multiple rail ‘cells’ are introduced in the model, the con- ductance matrix size is dramatically increased and conse- quently an efficient sparse matrix solution technique is required. Three programs have been developed using various solution techniques. It is evident that the imple- mentation of Zollenkopf‘s bifactorisation method has sig- nificantly improved program efficiency. The advantage of

separation of the various aspects into four subroutines in the bifactorisation program would result an important time saving when the iterative process is incorporated into a complete DC traction system simulation. There- fore, program 3 using the bifactorisation matrix solution method is recommended as an electrical analysis routine for the DC traction simulation software.

9 Acknowledgment

The authors thank the Directors of Balfour Beatty Power Construction Limited for permission to publish this work.

10 References

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2 RAMBUKWELLA, N.B., MELLITT, B.. GOODMAN, C.J., and MOUNEIMNE, Z.S.: ‘Traction equipment modelling and the power network solution for DC supplied transit power system studies’. IEE international conference on Electric railway .systems/br a new century. London, September 1987, pp. 218-224

3 YU, J.G., and GOODMAN, C.J.: ‘Modelling of rail potential rise and leakage current in DC rail transit systems’. IEE colloquium on Stray currenf efecfc uJDC railways and tramways, London, October 1990, pp. 221-226

4 TALUKDAR, S.N., and KOO, R.L.: ’The analysis of electrified ground transportation networks’, IEEE Trans., 1977, PAS-%, ( I ) . pp. 240-247

5 BRAMELLER, A., ALLAN, R.N., and HAMAM. Y.M.: ’Sparsity ~ its practical application to systems analysis’ (Pitman, 1976)

6 YU, J.G.: ‘Electrical analysis routine for DC traction systems‘. Balfour Beatty, internal report, July 1988

7 TINNEY, W.F., and WALKER, J.W.: ‘Direct solutions of sparse network equations by optimally ordered triangular factorization‘. IEEE Proc., 1967, 55, ( I 1). pp. 1801-1809

8 BERRY, R.D.: ‘An optimal ordering of electronic circuit equations for a sparse matrix solution’, IEEE Trans., 1971, CT-18. ( l ) , pp. 40-50

9 ZOLLENKOPF, K.: ‘Bi-factorization ~ Basic computational algo- rithm and programming techniques’, in REID, J.K.: ‘Large sparse sets of linear equations’ (Academic, 1971), pp. 75-96

I0 CHEUNG, C.H.: ‘Subroutines for solving large sparse equations with symmetric real coeflicients’. Brunel University. internal report. revision 2.0.1, July 1985

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