Abstract: One embodiment of the present invention provides a system that uses a block-partitioned technique to efficiently solve a system of linear equations. The system first receives a matrix that specifies the system of linear equations to be used in performing a time-based simulation. This matrix includes a static portion containing entries that remain fixed over multiple time steps in the time-based simulation, as well as a dynamic portion containing entries that change between time steps in the time-based simulation. Next, the system performs the time-based simulation, wherein performing the time-based simulation involves solving the system of linear equations for each time step in the time-based simulation. In solving the system of linear equations, the system factorizes the static portion of the matrix only once, and reuses the factorization of the static portion in performing an overall factorization of the matrix for each time step of the time-based simulation.

Abstract: One embodiment of the present invention provides a system that uses a block-partitioned technique to efficiently solve a system of linear equations. The system first receives a matrix that specifies the system of linear equations to be used in performing a time-based simulation. This matrix includes a static portion containing entries that remain fixed over multiple time steps in the time-based simulation, as well as a dynamic portion containing entries that change between time steps in the time-based simulation. Next, the system performs the time-based stimulation, wherein performing the time-based simulation involves solving the system of linear equations for each time step in the time-based simulation. In solving the system of linear equations, the system factorizes the static portion of the matrix only once, and reuses the factorization of the static portion in performing an overall factorization of the matrix for each time step of the time-based simulation.

Abstract: A system that efficiently performs a CMOD operation in solving a system of equations involving a sparse coefficient matrix by identifying supernodes in the sparse matrix. Each supernode comprises a set of contiguous columns having a substantially similar pattern of non-zero elements. The system performs a CMOD operation on each supernode, by determining a structure for the supernode, and computing a function of the structure. The system uses a one-dimensional trapezoidal representation for the supernode during the CMOD operation, if the result of the function is lower than a threshold value, and otherwise uses a two-dimensional rectangular representation for the supernode. The function of the structure of the supernode is a function of a number of computational operations involved in computing a lower-triangular sub-block portion of the supernode and a number of computational operations involved in computing a rectangular sub-block portion of the supernode.

Abstract: One embodiment of the present invention provides a system for efficiently perform a modification (cmod) operation in solving a system of linear algebraic equations involving a sparse coefficient matrix. The system operates by identifying supernodes in the sparse matrix, wherein each supernode comprises a set of contiguous columns having a substantially similar pattern of non-zero elements. In solving the equation, the system performs a cmod operation between a source supernode and a destination supernode. As part of this cmod operation, the system determines a subset of the source supernode that will be used in creating an update for the destination supernode. The system partitions the subset into a plurality of tiles, each tile being a rectangular shape of fixed dimensions chosen so as to substantially optimize a computational performance of the cmod operation on a particular computer architecture.

Abstract: A hybrid system for efficiently performing a cmod operation in solving a system of linear algebraic equations involving a sparse coefficient matrix. The system operates by identifying supernodes in the sparse matrix, wherein each supernode comprises a set of contiguous columns having a substantially similar pattern of non-zero elements. In solving the equation, the system performs a column modification (CMOD) operation between a source supernode and a destination supernode. As part of this CMOD operation, the system determines dimensions of the source supernode and the destination supernode. If a result of a function on the dimensions is lower than a threshold value, the system performs the CMOD operation between the source supernode and the destination supernode using a kernel that is written in an architecture-independent high-level language.