Abstract:
One or more groups, into which a circuit simulator may partition an overall circuit, that are determined to belong to a feedback loop, also known as cycles, may be merged into a single group. The length of the loop, which is the number of groups in the loop, determines whether or not the groups of a loop will be merged into a single group. More particularly, loops of a length less than or equal to a number are merged. The number may be specified by the user, or otherwise determined. Once a merged group is formed, its inputs and outputs are determined, and it is treated like any other previously existing group. Preferably, not all the groups are merged into a single group.

Description:
TECHNICAL FIELD 
     This invention relates to circuit simulators, and more particularly, to circuit simulators which partition the overall circuit into disjoint groups of components which are separately solved during the simulation process. 
     BACKGROUND OF THE INVENTION 
     It is well known in the art that it is desirable to simulate the operation of a circuit, e.g., to determine how the circuit will perform prior to physically building it. One prior art simulation technique develops a mathematical model to represent the circuit, which is then solved to indicate the operation of the circuit. This model is developed by entering into the circuit simulator a representation of the circuit, such as may be produced as the output of a schematic capture system, e.g. in the form of a netlist. The elements of a netlist generally correspond one to one with the schematic of the circuit to be simulated as it was input into the schematic capture system. 
     Next, the overall circuit is partitioned into disjoint groups of components as a function of circuit connectivity as described in the netlist. When a circuit is partitioned for eventual simulation, the various resulting groups of components typically have many inputs and outputs, each of which connect to others of the groups, or are inputs or outputs of the overall circuit. For purposes herein, the term “group” includes either the components which result from a partition of the circuit, or a behavioral model of a circuit which is merely a software description of the operation of a circuit without being embodied in a particular circuit implementation of the behavioral model. It is possible that one or more feedback loops, also known as cycles, will be formed among the interconnected groups, a loop being a sequence of groups such that each group in the sequence has an output connected to an input of the next group in the sequence and the last group in the sequence has an output connected to an input of the first group in the sequence. 
     The simulator has access to mathematical models for—i.e., equations and, if necessary, parameters representing—the devices included within the netlist, and, after partitioning, these models are employed to represent the individual devices. The resulting equations of groups are then separately solved during the simulation process. By solving each of the groups separately the computation burden of solving the circuit is reduced. Furthermore, this solution takes into account feedback within the group. Feedback among the groups is taken into account by circulating events, but such feedback cannot be solved in all cases. 
     Note that the circuit simulator may be implemented by a computer with appropriate software. One such commercially available simulator is ATTSIM available from Lucent Technologies, Inc., the documentation of which is incorporated herein by reference. 
     SUMMARY OF THE INVENTION 
     We have recognized that, disadvantageously, the presence of feedback loops among the groups can result in either multiple re-evaluations of many groups, or worse, oscillation, preventing further simulation. Therefore, in accordance with the principles of the invention, groups which are determined to belong to a loop may be merged into a single group. The length of the loop, which is the number of groups in the loop, determines whether or not the groups of a loop will be merged into a single group. More particularly, loops of a length less than or equal to a number are merged. The number may be specified by the user, or otherwise determined. Once a merged group is formed, its inputs and outputs are determined, and it is treated like any other previously existing group. In other words, the equations for the merged group are derived from the components in the merged group just as for any originally existing group. Preferably, not all the groups are merged into a single group, which would eliminate the advantages of partitioning the circuit to simulating it. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     In the drawing: 
     FIG. 1 shows an exemplary network of groups that results when an exemplary circuit (not shown) is initially partitioned for simulation; 
     FIG. 2 shows a group graph for the network of groups shown in FIG. 1; 
     FIG. 3 shows the group graph of the exemplary network of FIG. 1 after detecting and eliminating loops of length two has been performed; 
     FIG. 4 shows the group graph of the exemplary network of FIG. 1 after detecting and eliminating loops of both length two and length three has been performed; 
     FIG. 5 shows the group graph of the exemplary network of FIG. 1 after all loops have been removed; and 
     FIG. 6 shows a generalized flow chart of a process for implementing the invention. 
    
    
     DETAILED DESCRIPTION 
     FIG. 1 shows an exemplary network of groups that results when an exemplary circuit (not shown) is initially partitioned for simulation. In FIG. 1, groups G1 through G10 are shown and they are connected by signals S1 through S6, S75, S78 and S8 through S10. 
     When a circuit is partitioned into groups for simulation, there are two types of feedback that may result. The first is feedback within a group, and the second is feedback among the groups. Typically, e.g., in ATTSIM, feedback within a group is handled during the iterative solution of the differential equations which represent the circuits within the group. This feedback is taken into account essentially by simultaneous solution of the appropriate equations. Feedback among the groups is taken into account by the technique of circulating events but such feedback cannot be solved in all cases. 
     Feedback among the groups can arise because there is explicit feedback in the design, of which the designer is aware, or because of implicit feedback in the partitioned design, of which the designer is unaware. For example, a set-reset (S-R) latch built from two NAND gates has explicit feedback. If the NAND gates are built from MOSFETS, by default two groups will be created, with the output of each connected to an input of the other. Although a circuit designer may know about the internal feedback within an S-R latch, the designer may not be aware that the latch was partitioned into two cross-coupled groups. Similarly, if a large circuit has complicated behavior due to an effective internal state machine, partitioning may result in the creation of many groups, with there being feedback among the groups of which the circuit designer was not explicitly aware. 
     In the present invention, loops, starting with the shortest, i.e., of length two, if any, are identified, and then increasingly longer loops are identified, up to some limit, which may be determined experimentally. Since long loops can arise when there is naturally occurring feedback in the design, it is often unnecessary to merge such long loops, since in the limit, the entire design would be merged into a single group, entirely defeating the purpose of partitioning. Also, note that, for purposes of the invention, self feedback, which is a loop of length one, which occurs when a group has an output connected to one of its own inputs, is not important, and is, therefore, ignored and treated as if it didn&#39;t exist. 
     Identifying feedback among the groups is equivalent to finding loops of groups. A loop, also known as a cycle, is defined as a sequence of groups such that each group in the sequence has an output connected to an input of the next group in the sequence, and the last group in the sequence has an output connected to an input of the first group in the sequence. The number of groups in the sequence is the order or length of the loops. FIG. 1 shows four loops: two of length 2, the first consisting of groups G2 and G3 and the second consisting of G9 and G10; one of length 3, consisting of G5, G6, and G7; and one of length 8, consisting of G1, G2, G3, G4, G5, G6, G7, and G8. 
     In accordance with the principles of the invention, loops of increasing length, from two-group loops up to an n-group loop, where n is a parameter, are found. There be any number, including zero, of loops of any given length n. The groups forming a loop are then merged into a new group. 
     The interconnection of groups by signals can be described as a graph, the so-called group graph. In a group graph the groups are the nodes of the graph, and there is an edge of the graph from group i to group j if there is at least one signal which connects an output pin of group i to an input pin of group j. Thus, the group graph is a directed graph. Edges which correspond to self feedback are omitted. As a result, the graph has no edges from any node to itself. 
     FIG. 2 shows the corresponding group graph for the network of groups shown in FIG.  1 . It is possible that one signal in the network gives rise to more than one edge in the graph. This occurs when a signal fans out to more than one group. For example signal S3 fans out to groups G4 and G2, thus giving rise to edges E34 and E32. For purposes herein, fanout from a group to itself group does not result in an edge in the graph from a node to itself. 
     In accordance with an aspect of the invention, the group graph, in turn, is represented by a square adjacency matrix, A, which has one row and column for each group into which the circuit is currently partitioned. It has a non-null entry in row i, column j, if the corresponding graph has an edge from group i to group j. Thus, A ij  is non-null if there is at least one signal from group i to group j. Because, as noted above, self feedback is ignored, A has no non-null diagonal elements. A is often a so-called “sparse matrix”, since each group typically connects to only a few of the other groups. Table 1 shows the adjacency matrix for the sample group graph of FIG.  2 . 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Adjacency matrix A for exemplary graph 
               
             
          
           
               
                   
                 G1 
                 G2 
                 G3 
                 G4 
                 G5 
                 G6 
                 G7 
                 G8 
                 G9 
                 G10 
               
               
                   
                   
               
             
          
           
               
                 G1 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 G2 
                 0 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 G3 
                 0 
                 1 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 G4 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
                 0 
                 1 
                 0 
               
               
                 G5 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
               
               
                 G6 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
                 0 
               
               
                 G7 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 G8 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 G9 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1 
               
               
                  G10 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
               
               
                   
               
             
          
         
       
     
     The adjacency matrix, A, can be derived from an incidence matrix, M, defined as follows: M has one row, i, for each group and one column, j, for each signal in the circuit that is an input or an output of any group. M ij  is+1 if signal j is an output of group i, −1 if signal j is an input to group i but not also an output, and 0 otherwise. Since as noted, each group typically connects to only a few of the other groups, matrix M, like the adjacency matrix A, is often a sparse matrix. Incidence matrix M can be derived directly from the input-output lists for each group, which are developed in a well known manner as part of the partitioning process. 
     The adjacency matrix A may be developed by taking the product of the incidence matrix by its transpose, i.e., A=M·M T , alternatively written as, A ij =ΣM ik ·M jk , where the non-commuting inner product operator is (+1)·(−1)=1, other products being 0, and where 0+0=0, other sums being 1. To state the same in words, group i is adjacent to group j if there is at least one signal S k  which is an output of group i, i.e., M ik =+1 and an input to group j, i.e., M jk =−1. That is to say, there is a path from group i to group j if there is at least one signal which is an output of group i and an input of group j. Table 2 is the incidence matrix M for the exemplary network of groups shown in FIG.  1 . It can be shown that A=(M&gt;0)·(M T &lt;0). 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Incidence Matrix M for exemplary graph 
               
             
          
           
               
                   
                 S1 
                 S2 
                 S3 
                 S4 
                 S5 
                 S6 
                 S75 
                 S78 
                 S8 
                 S9 
                 S10 
               
               
                   
                   
               
             
          
           
               
                 G1 
                   1 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                 −1 
                   0 
                   0 
               
               
                 G2 
                 −1 
                   1 
                 −1 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
               
               
                 G3 
                   0 
                 −1 
                   1 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
               
               
                 G4 
                   0 
                   0 
                 −1 
                   1 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
               
               
                 G5 
                   0 
                   0 
                   0 
                 −1 
                   1 
                   0 
                 −1 
                   0 
                   0 
                   0 
                   0 
               
               
                 G6 
                   0 
                   0 
                   0 
                   0 
                 −1 
                   1 
                   0 
                   0 
                   0 
                   0 
                   0 
               
               
                 G7 
                   0 
                   0 
                   0 
                   0 
                   0 
                 −1 
                   1 
                   1 
                   0 
                   0 
                   0 
               
               
                 G8 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                 −1 
                   1 
                   0 
                   0 
               
               
                 G9 
                   0 
                   0 
                   0 
                 −1 
                   0 
                   0 
                   0 
                   0 
                   0 
                   1 
                 −1 
               
               
                  G10 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                   0 
                 −1 
                   1 
               
               
                   
               
             
          
         
       
     
     In accordance with an aspect of the invention, the diagonal elements of powers of the adjacency matrix A are used to identify loops. More specifically, the non-null diagonal elements of the matrix that are produced when matrix A is raised to a power indicate the existence of a path from group i to itself through one or more other groups, i.e., a loop, with a length equal to the power to which A is raised. For example, where the power is 2, the resulting matrix is A 2 . Since A ij  is non-null when there is a path from group i to group j, the matrix A 2  has non-zero elements if there is a path from group i to group j through some intermediate group k. In other words, there is a group k such that some signal is an output of group i and an input to group k, and another signal which is an output of group k and an input to group j. If A 2   ij  is non-null, there is a path from group i to itself through some group k, in other words, group i is in a loop of length two. Thus, the non-null diagonal elements of A 2  identify loops of length two. 
     In any circuit there may be more than one loop of length two. Subsets of the non-null diagonal elements identify disjoint loops of length two. To find the subsets of non-null diagonal elements it is necessary to save the contributions to the terms of the inner product when doing the multiplication of A by itself to find A 2 . It is necessary to save the set of intermediate groups k on the path from i to j, and to identify paths having the same set of groups. This generalizes to higher powers of A. In a similar fashion, the non-null diagonal elements of higher powers of A correspond to longer loops. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Square of Adjacency matrix for exemplary graph (A 2 ) 
               
             
          
           
               
                   
                 G1 
                 G2 
                 G3 
                 G4 
                 G5 
                 G6 
                 G7 
                 G8 
                 G9 
                 G10 
               
               
                   
                   
               
             
          
           
               
                 G1 
                 0 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 G2 
                 0 
                 
                   1 
                 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 G3 
                 0 
                 0 
                 
                   1 
                 
                 0 
                 1 
                 0 
                 0 
                 0 
                 1 
                 0 
               
               
                 G4 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
                 0 
                 1 
               
               
                 G5 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
                 0 
               
               
                 G6 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 G7 
                 1 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
               
               
                 G8 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 G9 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 
                   1 
                 
                 0 
               
               
                  G10 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 
                   1 
                 
               
               
                   
               
             
          
         
       
     
     Table 3 shows the matrix A 2  for the exemplary network of groups shown in FIG.  1 . The non-null diagonal elements of Table 3 are shown in bold. The terms of the inner product which gave rise to both A 2   2,2  and A 2   3,3  are G2, G3, while the terms which gave rise to both A 2   9,9  and A 2   10,10  are G9, G10. Hence G2 and G3 are in a loop of length two, and G9 and G10 are likewise in a separate loop of length two. 
     Further higher powers of A are obtained by multiplying the previously obtained power of A by A again. Non-null diagonal elements of the higher powers indicate longer loops. For example, Table 4, which shows A 3  for the exemplary network of groups shown in FIG. 1, indicates the loop formed by G5, G6, and G7. The inner product terms giving rise to A 3   5,5  are A 2   5,7  and A 7,5 . Those for A 3   6,6  are A 2   6,5  and A 2   5,6  comes from A 6,7 ·A 7,5  yielding the same set of three groups). Similarly, the groups involved in the inner product yielding A 3   7,7  also reduce to the same loop. Hence there is only one loop of length three consisting of groups G5, G6, and G7. The formation of higher powers of A, including saving the terms of the inner products for the diagonal elements, is repeated until loops of the desired length are found. In no case can a loop be longer than the total number of groups. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Square of Adjacency matrix for exemplary graph (A 3 ) 
               
             
          
           
               
                   
                 G1 
                 G2 
                 G3 
                 G4 
                 G5 
                 G6 
                 G7 
                 G8 
                 G9 
                 G10 
               
               
                   
                   
               
             
          
           
               
                 G1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 G2 
                 0 
                 0 
                 1 
                 0 
                 1 
                 0 
                 0 
                 0 
                 1 
                 0 
               
               
                 G3 
                 0 
                 1 
                 0 
                 1 
                 0 
                 1 
                 0 
                 0 
                 0 
                 1 
               
               
                 G4 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
                 1 
                 0 
               
               
                 G5 
                 0 
                 0 
                 0 
                 0 
                 
                   1 
                 
                 0 
                 0 
                 1 
                 0 
                 0 
               
               
                 G6 
                 1 
                 0 
                 0 
                 0 
                 0 
                 
                   1 
                 
                 0 
                 0 
                 0 
                 0 
               
               
                 G7 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
                 
                   1 
                 
                 0 
                 0 
                 0 
               
               
                 G8 
                 0 
                 0 
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 G9 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1 
               
               
                  G10 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 1 
                 0 
               
               
                   
               
             
          
         
       
     
     In the event that a group appears in more than one subset of groups corresponding to the terms of the inner product from which a non-null diagonal element of a power of A was formed, this indicates that the group is part of more than one loop of the same length. In such a case, the various separate subsets of groups for each of the loops in which the common group appears are replaced by a single group having as its members the union of the groups corresponding to the subsets of groups in which the common group appears. For example, if there exists two loops of length three, one including groups A, B, and C, the other including groups A, D, and E, then when groups of length three are determined a union will be formed of all five groups A, B, C, D, and E. Subsequently, all the groups in the several loops of the same length that have a common group will be merged, as described hereinbelow. 
     In accordance with the principles of the invention, after loops of the desired length are identified, the groups in each loop are merged into a single group. This operation has two parts: a) the elements in all of the groups in the loops have to be combined; and b) the boundary signals of the merged group are established, with their directions appropriately determined. 
     Combining the elements of all groups on a loop is performed by taking the union of the elements of each of the groups to be merged. 
     To determine the boundary signals, and the directions thereof, for the merged group first an initial set S, which is the union of boundary signals of all of the groups to be merged is developed, i.e., S={S k }. Next, each signal S k  of set S is processed in turn. 
     If signal S k  connects to only groups in the set to be merged, i.e., groups on the loop, and is an input to at least one and an output of at least one, then signal S k  has become an internal signal of the merged group, i.e., it is not connected to any point outside the merged group. Therefore, signal S k  is not a boundary signal of the merged group and it is removed from set S. 
     If signal S k  is not an output of any of the groups in the set to be merged, then signal S k  is designated as an input of the merged group. Otherwise, if signal S k  connects to a group not on the loop, or to a behavioral model, then signal S k  is designated as an output of the merged group. Note that this is because such a signal S k  cannot be an input of the merged group since it was an output of one of the groups being merged, and therefore could not have also been an output of any other group or model. 
     After detecting and eliminating loops of length two, the group graph of the exemplary network of FIG. 1 is shown in FIG.  3 . Note that the signals corresponding to edges E23, E32, E109, and E910, which were edges in the group graph of FIG. 2 for the fully partitioned network of FIG. 1 have, in FIG. 3, become internal to their respective merged groups. Signal S1 corresponding to edge E12 has been designated an input to merged group G2/G3 and signal S4 corresponding to edge E49 has been designated an input to merged group G9/G10. Similarly, signal S3 corresponding to edge E34 has been designated an output of merged group G2/G3. 
     Similarly, FIG. 4 shows the group graph of the exemplary network of FIG. 1 after detection and elimination of both loops of length two and loops of length three. Likewise, FIG. 5 shows the group graph of the exemplary network of FIG. 1 after all loops have been removed. 
     In one embodiment of the invention, the matrix operations described above are not performed using any standard or sparse matrix package. Instead, the adjacency matrix A, and its powers, A n , are represented by a list of their rows, which consist of lists of group identifiers, e.g., represented by pointers, which correspond to their non-empty columns. In other words, each row of the nth power of the adjacency matrix is the list of groups reachable from each group in n-steps. The inner products are performed by employing nested for loops, with implicit indexing controlled by comparison of group pointers. A union operation is used to identify loops corresponding to common subsets of groups of each given length. Only the original adjacency matrix, A, and the current power are retained. Each subsequent power is obtained from the previous power by adding the additional groups reachable by one additional edge of the original adjacency matrix; this corresponds to multiplication by adjacency matrix A. The subsets of common groups on a loop are accumulated only when computing the diagonal elements of each new power. 
     Some possible policies for applying feedback elimination are: 1) remove all loops up to a fixed length; 2) remove only all instances of the shortest loops; or 3) remove all loops, which, as noted above, may negate the benefits of partitioning. 
     FIG. 6 shows a generalized flow chart of a process for implementing the foregoing in an exemplary manner. The process begins in step  601  when the simulator is activated to simulate a circuit. Next, in step  603 , the circuit, e.g., its netlist, is loaded into the circuit simulator. Thereafter, the partitioning of the circuit in the conventional manner is performed in step  605 . 
     The process then undertakes to identify the loops that exist among the groups. To this end, in step  607 , the incidence matrix M is derived from the group input/output lists which were developed as part of the partitioning process, and, in step  609 , the adjacency matrix A is derived from the incidence matrix M by computing A=M·M T . 
     Next, the loops of groups which resulted from the partitioning process and having a length of less than or equal to N, a predetermined number, e.g., selected by the user, are merged. To this end, the value of a variable n, which determines the loop length that is currently being processed, is initialized to two, in step  611 . Then, in step  613 , the nth power of A is determined, e.g., by computing A n =A n−1 ·A. The non-null diagonal elements of A n , which indicate groups within loops of length n, and which particular groups belong to which particular loops, if any, are identified in step  615 . In other words, the subsets of groups that give rise to non-zero diagonal elements of A n  are identified. If there are two or more loops that have a group that is common to those loops, i.e., a contribution is made by such a group to a non-zero diagonal element that belongs to the subsets identifying such loops, then all the groups of those loops will be identified as part of a single subset of groups. 
     Thereafter, conditional branch point  617  tests to determine if n&lt;N. If the test result in step  617  is YES, control passes to step  619 , in which n is incremented. Control then passes to conditional branch point  621 , which tests to determine if n&lt;G, where G is the number of groups produced as a result of the original partitioning of the circuit. If the test result in step  621  is YES, indicating that greater length loops may still be found, control passes back to step  613 , and the process continues as described above. 
     If the test result in step  617  is NO, or the test result in step  621  is NO, which indicates that all the loops of groups from the original partitioning which have a length of less than or equal to the lesser of G or N have been identified, control passes to step  623  to begin the process of merging the groups of each loop into a single group. More specifically, in step  623 , unique subsets of groups which are part of a particular loop and correspond to the non-null diagonal elements of A n  accumulated in step  615  are identified. Next, the elements of the identified groups are merged, which is achieved by: a) combining the elements of the identified groups on the loop in step  625 ; and then b) determining the group boundary signals, and their appropriate direction; in step  627 . Conditional branch point  629  tests to determine if there remain any unique subset of groups that correspond to a loop that have not yet been processed. If the test result in step  629  is YES, control passes back to step  623  to process the groups on the next as-yet-unprocessed loop. If the test result in step  629  is NO, control passes to step  631 , in which the circuit is simulated as it is currently partitioned into groups, which may include original groups as well as merged groups. The process then exits in step  633 . 
     The foregoing merely illustrates the principles of the inventions. It will thus be appreciated that those skilled in the art will be able to devise various arrangements which, although not explicitly described or shown herein, embody the principles of the invention and are included within its spirit and scope.