JPH04247570A - Parallel processing arithmetic device for simultaneous linear equations - Google Patents

Abstract

PURPOSE:To efficiently execute an entire processing at high speed by transfer ring data between a secondary storage device and the memory of a processor group while the processor group parallelly executes calculation by using the secondary storage device to store the coefficient matrix of a simultaneous linear equations. CONSTITUTION:The first H rows of the coefficient matrix are transmitted from a secondary storage device 3 through a main processor 1 to a processor group 2. When the number of processors is defined as (n), each processor takes charge of H/n rows. When data are received, the processor group 2 starts calculation and when the first row is calculated, the processor in charge of this row calculates the next charged row and simultaneously transmits the data of the first row completing LU decomposition through a link 5 and the main processor 1 to the secondary storage device 3. The main processor 1 receives the data of the (H+1)th row from the secondary storage device 3 and transmits it to the processor group 2, and the LU decomposition is advanced.

Description

[Detailed description of the invention]

0001

[Industrial Application Field] The present invention is a parallel processing method that uses a parallel processing device to process large-scale simultaneous linear equations that occur when performing the finite element method or the finite difference method at high speed and with a simple configuration. It relates to arithmetic devices.

0002

[Prior Art] When solving simultaneous linear equations using the finite element method or the finite difference method using a parallel processing arithmetic unit, the conjugate gradient method is often used as the solution algorithm (for example, Japanese Patent Laid-Open No. 63-95568 , Japanese Patent Publication No. 63-127
Publication No. 366). This is because the conjugate gradient method requires only data on non-zero elements of the coefficient matrix, so all the necessary data could be stored in the memory of a group of processors that perform parallel processing. However, the conjugate gradient method, which is an iterative method, takes a very long time to find a solution.
Alternatively, the solution may not be found without convergence.

On the other hand, there are also examples of solving simultaneous linear equations using the LU decomposition method on a parallel processing arithmetic unit. Since the LU decomposition method is a direct method, the time required to solve the problem depends only on the dimension and bandwidth of the coefficient matrix (described later), and a solution can always be obtained.

0004

[Problem to be Solved by the Invention] However, there is one problem when using the LU decomposition method, and that is when the number of dimensions of the coefficient matrix is large, it is necessary to store all the data of the coefficient matrix on the memory of the processor group. It becomes difficult. For this reason, the coefficient matrix is often stored on a secondary storage device;
Data I/O between the storage device and the processor group that performs parallel processing takes time, which reduces the performance of parallel processing (for example, "Matrix Numerical Computation Method" by Hayato Togawa (Ohmsha) 1971 Published in 1985, pp. 52-53, "PAX Computer" by Tsutomu Hoshino (Ohmsha), published in 1985, chapters 3-4).

Therefore, the present invention aims to improve parallel processing performance when using the LU decomposition method as a solution algorithm for simultaneous linear equations, even when the coefficient matrix data cannot be stored in the memory of a group of processors and a secondary storage device is used. It is an object of the present invention to provide a parallel processing arithmetic device capable of solving simultaneous linear equations at high speed without deterioration.

[0006]

[Means for Solving the Problems] In order to solve the above problems, the present invention provides a parallel processing arithmetic device that solves simultaneous linear equations using the LU decomposition method, and a secondary storage device that stores the coefficient matrix of the simultaneous linear equations. , a main processor connected to the secondary storage device, and n processors each connected to the main processor via a link, each processor receiving data sent from the secondary storage device through the main processor. A processor group is connected to the processor group and is in charge of calculating every nth row of the coefficient matrix to be generated, and data is transferred between the processor group and the secondary storage device while the processor group performs the calculation. and a memory adapted to perform the transfer.

[0007] Each processor of the processor group receives ln rows (l=1, 2, 3) of the coefficient matrix sent from the secondary storage device through the main processor.
,...) can be configured to take charge of every calculation.

[0008] Furthermore, each processor in the processor group may be configured to be in charge of calculating every nth column of the coefficient matrix sent from the secondary storage device through the main processor.

Furthermore, each processor of the processor group receives ln columns (l=1, 2, 3) of the coefficient matrix sent from the secondary storage device through the main processor.
,...) can also be configured to take charge of every calculation.

0010

[Operation] In the present invention, a secondary storage device that stores coefficient matrices of simultaneous linear equations is used, and while a group of processors performs calculations in parallel, data is stored between this secondary storage device and the memory of the processor group. Since it is configured to transfer data, the overall processing can be performed efficiently and at high speed.

[0011]

DESCRIPTION OF THE PREFERRED EMBODIMENTS The present invention will be described below with reference to the drawings.

First, the simultaneous linear equations to be solved are shown in equation (1).

[0013]Ax=b (1)

A is a coefficient matrix, b is a right-hand side vector,
x is the solution vector. The coefficient matrix A is shown in FIG. In the finite element method and finite difference method, nonzero elements of A often exist only near the diagonal, and a matrix in which nonzero elements exist only within a certain distance from the diagonal is called a band matrix. , the width where non-zero elements exist is called the bandwidth. The matrix in FIG. 2 is a band matrix. Here, M represents the dimension of the matrix, and W represents the bandwidth. H is the distance between the diagonal element and the farthest nonzero element from the diagonal, and is called the half-bandwidth, and there is a relationship of W=2H-1. A solution is obtained by applying the LU decomposition method to this band matrix. The LU decomposition method is a method of decomposing the coefficient matrix A into an upper triangular matrix U and a lower triangular matrix L and transforming equation (1) as follows.

LUx=b (2)

[
Then, by providing an intermediate vector y, the solution vector x can be calculated as follows.

[0017]Ly=b (3)

[0018] Ux=y (4)

Since L is a lower triangular matrix, y can be found as follows.

for i=1 to Myi=bi for j=1 to i-1 yi=yi-bj*Lij next j yi=yi/Lii next i
(5)

Furthermore, since U is an upper triangular matrix, the solution vector x can be obtained as follows.

for i=M to 1xi=yi for j=i+1 to M xi=xi-yj*Uij next j xi=xi/Uii next i
(6)

Forward substitution of equation (5), (6)
The expression is called backward substitution. Note that equations (5) and (6) are based on Bas
ic, but this is used to show the calculation method, and the present invention does not use this program.

[0024] Upper triangular matrix U, lower triangular matrix L
are calculated using the following equations (7). In addition, equation (7) also
Although it is written in Basic, this program is not used in the present invention. The initial values of U and L are the upper triangular part and lower triangular part of A, respectively (however, the diagonal elements are included in U).

for i=1 to MLii=1 for j=1 to H−1 Li+j, i=Li+j, i/Ui, ifor k=1
to j−1 Li+j, i+k=Li+j, i+k−Ui, i+k*
Li+j,i next k for k=j to H-1 Ui+j,i+k=Ui+j,i+k-Ui,i+k*
Li+j,i next k next j next i
(7)

The calculation of this equation (7) is as follows: (5
), it takes much more time than calculating equation (6), and occupies most of the time to solve the simultaneous linear equations.

In this embodiment, the calculation of equation (7) is parallelized.

FIG. 1 shows the hardware system of this embodiment. In the figure, 1 is a main processor, 2 is a processor group consisting of n processors, 3 is a secondary storage device, 4 is a memory, and 5 is a link. The processors in the processor group 2 are configured to be able to perform calculations and exchange data with the outside via the link 5.

In this embodiment, each processor of processor group 2 calculates every nth row of coefficient matrices U and L. FIG. 3 shows an example of the correspondence between each processor (processor numbers 0, 1, 2, and 3) and rows when there are four processors. Each processor calculates equation (7) only for the elements in the row for which it is responsible.

The arithmetic processing method in this embodiment will be explained with reference to the flowchart of FIG.

The data required to calculate one step of the outermost loop (loop counter i) in equation (7) is shown in FIG.
This is the shaded area shown in . The portion to the upper left of the shadow is a portion where LU decomposition has already been completed. Therefore, processor group 2 stores only the rows including the shaded portions in memory, and performs one-step calculation of equation (7).

For example, in the i-th step, the i-th to i+H-1 rows are calculated. i+j row (1≦j<
The elements that the processor in charge of H) must calculate are Li+j, i+k (0≦k<j), Ui+j
, i+k (j≦k<H), and the data necessary for the calculation is Li+j, i+k (0≦k<j) from equation (7).
, Ui+j,i+k(j≦k<H) and Ui,i+k(
0≦k<H). Therefore, the processor in charge of row i+j can perform the calculation of equation (7) if it has the data of row i+j and the data of row i. Therefore, in the i-th step, the processor that has the i-row data (is in charge of the i-row) transfers the i-row data to all processors, and each processor that receives it transfers the data to its respective Perform row calculations (Figure 5 ■). When the i-th step is completed, the i-th row is no longer needed for the calculation of equation (7), so it is sent to the secondary storage device 3 (Fig. 5 ■), and the i+H row is sent from the secondary storage device 3. Then, the i+1th step begins. Naturally, if i+H is larger than M,
There is no data transfer (Figure 5■).

According to the method described above, processor group 2
The amount of data stored in the memory 4 remains on the order of H2, and unless the coefficient matrix is extremely large,
It fits in memory 4. Then, in the i-th step, the i-th row is only referenced and not updated, so while the processor group 2 is performing calculations, the i-th row is stored in the secondary storage 3.
For example, if a disk is used as the secondary storage device 3, disk I/O and calculation will be performed in parallel, and the disk I/O will be processed in parallel. The time is hidden in the calculation time (see Figure 4). That is, in this step, the row where the LU decomposition ends is calculated first, so it can be sent to disk while the following rows are being calculated. In addition, ■, ■, ■ in Figure 5
processing is performed simultaneously.

When simultaneous linear equations are solved using the LU decomposition method in the system shown in FIG. 1, the first H rows of the coefficient matrix are first sent from the secondary storage device 3 to the processor group 2 through the main processor 1. The number of processors is n
Then, each processor is responsible for H/n (rounded up) rows. Upon receiving the data, processor group 2 immediately starts calculating the first step of equation (7). When the calculation of the first row is performed, the processor in charge of this row calculates the next row, and at the same time
The data of the first line that has been decomposed is sent to the secondary storage device 3 via the link 5 and the main processor 1 (actually, the first line data is
The row is not updated by calculation, but is only referenced when calculating rows 2 to H). Furthermore, the main processor 1 receives data from the H+1 row from the secondary storage device 3 and sends it to the processor group 2 via the link 5. Proceeding with the LU decomposition in this way, in step i, the processor group 2 finishes decomposing the i-th row and sends it to the secondary storage device 3, and receives the i+H row from the secondary storage device 3. In the case of i<MH, no data is received, and only the data of the line that has been decomposed is sent to the secondary storage device 3 after calculation.

In the above embodiment, a case was explained in which each processor has a row in charge of calculation every n rows, but as shown in FIG. Or, it can be generalized to every ln rows (l=1
, 2, 3, . . .), the algorithm remains almost the same, only the order of processors that perform data I/O with the secondary storage device 3 and the order of processors that transfer data to other processors change.

Further, in the above embodiment, parallelization was performed by allocating rows to each processor, but there is also a method of performing parallelization by allocating columns. From equation (7), the data required to calculate each element in the i-th step of LU decomposition exists in the column containing that element and in the i-th column. (responsible for column i)
Equation (7) can be parallelized if the processor transfers the data of the i column to all processors, and each processor that receives the data calculates the column it is responsible for. When parallelization is performed on columns, the correspondence system between processors and columns is l columns every ln columns (l=1, 2, 3, . . . ).

[0037]

[Table 1]

[0038]

[Table 2]

Table 1 shows the relationship between the number of processors and calculation time when simultaneous linear equations are actually solved using the present invention. Further, Table 2 shows a similar relationship when the calculation of LU decomposition and the I/O between the secondary storage device and the processor group are performed sequentially. It can be seen that the CPU efficiency (speed ratio/number of CPUs) is higher in Table 1 than in Table 2 when the number of CPUs is increased.

[0040]

According to the present invention, the processor group of the parallel processing arithmetic unit can perform LU decomposition simply by having a memory sufficient to store a part of the coefficient matrix of the simultaneous linear equations. Furthermore, since communication between the secondary storage device that stores the coefficient matrix and the processor group is performed at the same time as the LU decomposition calculation, the I/O time with the secondary storage device is not visible, and therefore the overall processing can be done quickly.

[Brief explanation of the drawing]

FIG. 1 is a block diagram showing the configuration of a parallel processing arithmetic device according to an embodiment of the present invention.

FIG. 2 is an explanatory diagram of dimensions, bandwidth, etc. of a coefficient matrix.

FIG. 3 is an explanatory diagram showing the correspondence between rows of a coefficient matrix and each processor of a processor group.

FIG. 4 is an explanatory diagram showing the exchange of matrix data with a secondary storage device when performing one step of LU decomposition.

FIG. 5 is a flowchart illustrating LU decomposition processing.

FIG. 6 is an explanatory diagram showing another correspondence relationship between rows of a coefficient matrix and each processor of a processor group.

[Explanation of symbols]

1 Main processor 2 Processor group 3 Secondary storage device 4 Memory 5 Link

Claims (4)

[Claims] 1. A parallel processing arithmetic device for solving simultaneous linear equations using an LU decomposition method, comprising: a secondary storage device for storing a coefficient matrix of the simultaneous linear equations; and a main processor connected to the secondary storage device. It consists of n processors connected to the main processor via a link, each processor being responsible for calculating every nth row of the coefficient matrix sent from the secondary storage device through the main processor. and a memory connected to the processor group and configured to transfer data to and from the secondary storage device while the processor group performs the calculation. parallel processing arithmetic unit. 2. A parallel processing arithmetic device for solving simultaneous linear equations using an LU decomposition method, comprising: a secondary storage device that stores a coefficient matrix of the simultaneous linear equations; and a main processor connected to the secondary storage device. Consisting of n processors connected to the main processor via links, each processor receives ln rows (l=1, 2, 3) of the coefficient matrix sent from the secondary storage device through the main processor. ,...) connected to the processor group and configured to transfer data between the processor group and the secondary storage device while the processor group performs the calculation. A parallel processing arithmetic device for simultaneous linear equations, which is equipped with an integrated memory. 3. A parallel processing arithmetic device for solving simultaneous linear equations using an LU decomposition method, comprising: a secondary storage device that stores a coefficient matrix of the simultaneous linear equations; and a main processor connected to the secondary storage device. It consists of n processors connected to the main processor via a link, each processor being responsible for calculating every nth column of the coefficient matrix sent from the secondary storage device through the main processor. and a memory connected to the processor group and configured to transfer data to and from the secondary storage device while the processor group performs the calculation. parallel processing arithmetic unit. 4. A parallel processing arithmetic device for solving simultaneous linear equations using an LU decomposition method, comprising: a secondary storage device for storing a coefficient matrix of the simultaneous linear equations; and a main processor connected to the secondary storage device. It consists of n processors connected to the main processor via a link, and each processor receives ln columns (l=1, 2, 3) of the coefficient matrix sent from the secondary storage device through the main processor. ,...) connected to the processor group and configured to transfer data between the processor group and the secondary storage device while the processor group performs the calculation. A parallel processing arithmetic device for simultaneous linear equations, which is equipped with an integrated memory.