JPH03229363A - Parallel solving system for sparse matrix - Google Patents

Abstract

PURPOSE:To decrease the communication quantity between PEs and to execute the processing of the whole at a high speed by forming a block diagonal matrix with an edge by rearranging the line and the row of a sparse matrix, and thereafter, transmitting a value obtained by an LU resolution processing without a communication by the PE at every partial matrix to the PE concerned and executing the LU resolution processing of the whole. CONSTITUTION:With regard to a given sparse matrix 1, it is formed to a block diagonal matrix 2 with an edge by rearranging a line and a line, and a row and a row, and thereafter, partial matrixes of a block (A1) and edge blocks (L1, U1) on the diagonal in the block diagonal matrix 2 with an edge are allocated to a PE (processor), respectively. Subsequently, the PE determines the value obtained by an LU resolution processing of the partial matrix in the block with out a communication to the other PE, and transmits it to the PE concerned in a block (M) which is influenced by its partial matrix so that the LU resolution processing of the whole can be executed. In such a way, by decreasing the communication quantity between the PEs, the processing of the whole can be executed at a high speed.

Description

[Detailed Description of the Invention] [Summary] Regarding a parallel solution method for solving sparse matrices in parallel, after rearranging the rows and columns of a sparse matrix to form a block diagonal matrix with edges, each submatrix is solved by PE without communication. L
Send the U-decomposed value Ml to the corresponding PE and calculate the entire LU
The purpose of decomposition processing is to reduce the amount of communication between PEs and speed up the overall processing. The submatrices of the diagonal pro, j (A□), and edge blocks (L+, u+) on the diagonal of the edged block diagonal matrix were assigned to PH, and the PE processed the LU decomposition of the submatrices within the block. The configuration is such that the value M1 is obtained without communication with other PEs, and this is sent to the corresponding PH in the block CM that is affected by the partial matrix to perform the entire LU decomposition process.

[Industrial application field]

The present invention relates to a parallel solution method for solving sparse matrices in parallel, and there is a demand for speeding up by a parallel computer in single-circuit simulation and the like. These simulations require solving large sparse matrices thousands to tens of thousands of times.

Since this sparse matrix is a simultaneous linear equation, it is difficult to efficiently solve it on a parallel computer, especially a parallel computer using distributed memory, and it is desired to solve it at high speed.

[Prior art and problems to be solved by the invention] Conventionally, as a method for quickly executing LU decomposition, which is a typical method for solving a sparse matrix, as shown in FIG. Check the number of updates and references for each element in advance and create tables (data for elements of lower triangular matrices, data for elements of upper triangular matrices, data for pivots, etc.), and send and receive data according to these tables when solving a solution. - A method has been proposed that performs calculations and solves matrices at high speed. The method shown in FIG. 5 is efficient because there is no need to synchronize the PEs during matrix solving and calculations can be executed one after another.

However, in the method shown in Figure 5, each element is basically a separate PE (
The basic processing of matrix solution (A=AB*C, or A=AlB type calculation')
There was a problem in that communication occurred between PEs for every The purpose is to reduce the amount of communication between PEs and speed up the overall processing by transmitting the value M1 obtained by LU decomposition processing by the PE to the corresponding PH without communication for each matrix and performing the overall LtJ decomposition processing. There is.

[Means to solve problems]

FIG. 1 shows the principle configuration and explanatory diagram of the present invention.

In FIG. 1, a sparse matrix 1 schematically represents a matrix on which LU decomposition processing is to be performed. here,
The black dots are nonzero elements.

The green block diagonal matrix 2 is a schematic representation of a matrix in which the sparse matrix 1 is rearranged row by row and column by column, and non-zero elements are gathered at points.

The block (A,) on the diagonal is an arbitrary block (A+) (1=1 to n) on the diagonal of the edged block diagonal matrix 2, and is a submatrix block assigned to the PE. .

Block (M) is a block affected by the submatrix of block (A+) on the diagonal.

[Effect]

As shown in FIG. 1, the present invention rearranges rows and rows and columns of a given sparse matrix 1 to form an edged block diagonal matrix 2, and then
The diagonal block (A, ) and the edge block (
A submatrix of Ll 101 ) is assigned to each PE, and the PE calculates the value Ml obtained by LtJ decomposition of the submatrix within the block with other PEs i! This is then sent to @the PE in the block CM that is affected by the submatrix, and the entire L tJ decomposition process is performed.

Therefore, after rearranging the rows and columns of the sparse matrix to form a bordered block diagonal matrix, the PE performs LU decomposition processing for each submatrix without communication, and sends the value M to the corresponding PE to perform the entire LU decomposition processing. By doing so, it is possible to reduce the amount of communication between PEs (and speed up the overall processing).

〔Example〕

Next, the configuration and operation of Embodiment I of the present invention will be sequentially explained in detail using FIGS. 1 to 4.

FIG. 1(a) shows the original sparse matrix l.

This is a schematic representation of a matrix (sparse matrix) representing simultaneous linear equations for which LU decomposition processing is to be performed, and black dots are non-zero elements.

Figure 1 <c> shows the edged pro-diagonal matrix 2 after rearrangement. This is a matrix with elements gathered in the shaded area. Corresponding to this rearrangement of rows and columns, the row direction pointer and column direction pointer in FIG. 4(a) are changed.

FIG. 1(c) shows a matrix processed by IPE.

In this case, the submatrices necessary for performing LU decomposition processing on an arbitrary block (AI) on the diagonal of the block diagonal matrix 20 (1=1 or n) shown in FIG. The LtJ decomposition processing is performed on the sub-matrix divided into small parts within one PE without communication between PEs, thereby reducing the overall amount of communication between PEs.

Incidentally, the values U and Ll of the submatrix of arbitrary AI (AI) are obtained by the following equations (1) and (2).

IJr =U+ -L-r XU+-+ -ol-!
XU+-z- (hereinafter omitted) =U+ (because the second term and subsequent terms are zero) ・・・・a) LL
=L+ L-+ XL+-+ LL-z XL
+-z (hereinafter omitted) =L+ (because the second term and subsequent terms are zero)... (2) Therefore, the submatrix of block (AI) in Figure 1 (B) becomes as shown in Figure 1 (C). Ml can be found using the following formula (m) using the submatrix of .

k=1 Here, n represents the number of blocks on the diagonal part, m represents the number of elements in the block on the edge part, ■ represents 1 to n, and n
, represents the matrix dimension of block ■.

FIG. 1(d) shows the creation of a coupling matrix M from Ml calculated for submatrices within each PE. This calculates the overall coupling matrix M using the following equation (4) from M calculated and transmitted within each PE.

M=ΣM, ・ ・ ・ ・ ・ ・ ・ −・ ・
-. . . . (4) 1=1 Next, the operation of the configuration shown in FIG. 1 will be explained using FIG. 2.

In FIG. 2(a), ■ rearranges the matrix to form a bordered block diagonal matrix 2. This is done by rearranging the rows and columns of the original Snowy matrix l in Figure 1 (a).
Figure (b) Convert to bordered block diagonal matrix 2.

At @, the matrix data to be handled is passed to each PE.

This transfers the data of the submatrix of an arbitrary block (AI) to the PE corresponding to ffl in FIG. 1(b).

■ is the LU
Check the number of updates and destination during disassembly.

[Phase] creates an update count table and a destination table when solving the coupling matrix (set the update count and destination in the destination table in advance in FIG. 4(b)).

Through the above preprocessing, any block (
A1) Set the matrix data, number of updates, destination, etc. to the PH that receives the LU decomposition processing of each submatrix, and (d) set the number of updates, destination, etc. of the coupling matrix.

In FIG. 2(b), (2) executes U decomposition within the block. At this time, after completing the predetermined processing for the connection matrix, the value is transmitted to the destination PH. This is shown in Figure 1 (
For the matrix (submatrix) processed by IPR in c), calculate the value M1 of the submatrix using the formula (j), and apply this to the block CM of the transmission destination (preset destination that is affected).
) to the corresponding PE.

(2) performs update processing upon receiving the coupling matrix element. After receiving updates for a predetermined number of times, they are sent to the destination PE. In this case, when @ receives a value M obtained by LtJ decomposition of a joint matrix element (submatrix), the old value is updated to a new value using equation (4). After updating for the specified number of updates,
Send to destination PE (affected preset destination PE).

Next, a specific example of the configuration of one embodiment of the present invention will be explained in detail using FIG.

In the diagram, ■ is the LU of the submatrix of block A1
The decomposed value M is obtained. In this case, the value M1 obtained by performing the LU decomposition process of the submatrix (submatrix shown in the lower left of Fig. 3) of any block A of the diagonal matrix 2 in the upper left edged matrix is calculated using the illustrated formula ( (See Yuu).

■ is the location (processor,
PE) (to the PEs affected by the value MI in block M).

In ■, the PE in block M that received the transmission in ■ calculates the value M obtained by LU decomposition processing of the entire block using the illustrated formula (see formula (4) already mentioned). At this time, all PEs perform cooperative processing. , as shown in the upper right explanatory diagram, that is, in ■, send from b in the diagram (the value of the partial matrix in the lower left) and from C in the diagram (the value of U in the upper right submatrix) The corresponding PE that received the transmission bids the one received first in the reception buffer, multiplies it by the element received later, and subtracts it from the old value <(a-bxC
(Perform ■)). Through the above processing, LU decomposition processing is performed for each submatrix, and this result is sent to other affected PEs to perform LU decomposition processing on the whole.

FIG. 4 shows an example data structure of the present invention.

The fourth part (a) shows a submatrix table in the PE.

This is a table that stores information on each element of the matrix processed by ipE in FIG.

Here, when converting from the original sparse matrix in FIG. 1(a) to the edged block diagonal matrix 2 in FIG. 1(b), the row direction pointer and column direction pointer are changed. In addition, the number of updates, destination, etc. are set in advance at 0 in FIG. 2 (a).

FIG. 4(b) shows a table for connection matrices.

This is a table that stores information on each element of the coupling matrix M from M calculated in each PE in FIG. 1(d). The number of updates, destination, etc. are set in advance using @ in FIG. 2 (A).

〔Effect of the invention〕

As explained above, according to the present invention, the sparse matrix 1
After rearranging the rows and columns of to create a green block diagonal matrix 2, each submatrix is assigned to a PE and communicated with other PEs. Send it to the corresponding PE in 1., IJ of the whole
Since a configuration that performs disassembly processing is adopted, the amount of communication between PEs can be reduced and the overall processing speed can be increased.

[Brief explanation of drawings]

Fig. 1 is a diagram explaining the principle configuration of the present invention, Fig. 2 is a 70-chart explaining the operation of the present invention, Fig. 3 is a diagram showing the configuration and explanatory diagram of one embodiment of the present invention, and Fig. 4 is a data structure of the present invention. For example, the fifth leap represents a conventional matrix solving method. In the figure, 1 represents a sparse matrix, and 2 represents a green block diagonal matrix.

Claims (1)

[Claims] In a parallel solution method for solving sparse matrices in parallel, after rearranging the rows and columns of a given sparse matrix (1) to create a diagonal edged block matrix (2), the edged block pair The diagonal block of the angular matrix (2) (
A_1) and the submatrix of the edge block (L_1, U_1) are assigned to each PE, and the PE calculates the value M_1 obtained by LU decomposition of the submatrix within the block without communicating with other PEs, and calculates the influence of the submatrix in question. 1. A parallel solution method for sparse matrices, characterized in that the sparse matrix is transmitted to a corresponding PE in a receiving block (M) to perform overall LU decomposition processing.