JPS63127366A - Analyzer for simultaneous linear equations system - Google Patents

Abstract

PURPOSE:To effectively utilize a parallel computer, by performing forward/ backward substitute calculation for an inverse matrix operation by an approximate matrix with a parallel processing. CONSTITUTION:A coefficient matrix degeneration circuit 1 which inputs the coefficient matrix of a simultaneous linear equation system and a coefficient vector obtained by the discrete approximation of a partial differential equation and degenerates the coefficient matrix to the approximate matrix in which the inverse matrix operation can be performed, a forward/back substitution circuit 2 which performs the inverse matrix operation of the approximate matrix generated at the circuit 1 on a vector data from a recursive calculation circuit 3 and outputs it to a circuit 5, and the circuit 5 are provided. And assuming an equation Au=f as an expression I corresponding to the division (u)=[u1...um] of an unknown vector (u), each of diagonal blocks A11, A22,...A is imperfectly LU-degenerated. It is assumed that the matrix imperfectly LU-degenerated of Aii (i=1-m) is expressed as LiiUii=Mii, and the approximate matrix for the inverse calculation of the approximate matrix M is expressed as an expres sion II. In such way, it is possible to perform the inverse matrix operation of the approximate matrix M at every diagonal matrix Mii.

Description

DETAILED DESCRIPTION OF THE INVENTION [Field of Industrial Application] The present invention relates to a system for analyzing simultaneous linear equations, and particularly to a system for analyzing simultaneous linear equations using a finite element analysis method.

[Conventional technology]

Conventional techniques include performing incomplete Choleski decomposition or incomplete LU decomposition (hereinafter collectively referred to as incomplete LU decomposition) on all nonzero elements of a coefficient matrix, and inverting the resulting matrix. A method of creating an approximate matrix for manipulation is known. That is, according to the configuration of the present invention (FIG. 1),
Incomplete LU of all non-zero elements of coefficient matrix in coefficient matrix decomposition circuit
Decomposition is performed, and a forward/backward substitution circuit performs an inverse matrix operation using the approximate matrix subjected to the incomplete LU decomposition.

The LLI decomposition of matrix A transforms A into a lower triangular matrix and an upper triangular matrix U
The product LU (=A>) is decomposed into the product LU (=A>. For example, the unknown vector U in the equation of
-"f (forward substitution) and u=LJ-'v (backward substitution), but in simultaneous linear equations obtained by discrete approximation of partial differential equations, the coefficient matrix A is sparse (all matrix elements are ), whereas L
The U-decomposed matrix has more nonzero elements than that. Normally, the occurrence of this non-zero element is called I'1ll-in, but in large-scale problems, the increase in memory usage and calculation amount due to this fill-in becomes a bottleneck, making it virtually impossible to calculate. becomes. Therefore, in the incomplete LU decomposition method, LU decomposition is performed while ignoring this fill-in part (approximating it to zero). Therefore, the matrix obtained by performing incomplete LU decomposition on all nonzero elements of the coefficient matrix.

U has the same lateral paths as the upper bead corner part and the upper bead corner part of A, respectively.

This conventional technology is described in detail in ``Supercomputers, Applications to Scientific and Technical Calculations'' (published by Kenro Murata et al.).

[Problem that the invention seeks to solve] For example, multiple processors (PIJI).

..., m)) to form simultaneous linear equations A ll
・To solve 1″, use the unknown vector I・11−[+11.
..., Uo], 11 blocks u1(i・1
．． ..., m), and each ul is calculated by PU.

In forward/backward substitution calculations, non-zero elements located in off-diagonal parts of the matrix mean feedback of the calculation results. For example, if the diagonally adjacent matrix elements are non-zero, the assignment calculation will be a sequential calculation in which the calculation of the next row begins only after the calculation of the previous row is completed. Therefore, in the conventional method of performing incomplete LU decomposition for all non-zero elements of a coefficient matrix, forward and backward substitution calculations using L and U are performed for each processor P II + (i・1..., m). This results in sequential processing, which hinders speeding up by parallel processing.

Therefore, the problem to be solved by the present invention, in other words, the object of the present invention is to provide a system for analyzing simultaneous linear equations that improves the above-mentioned problems by applying a procedure that enables parallel processing by a plurality of processors. There is a particular thing.

Means for Solving 11 Problems] The simultaneous linear equation analysis device of the present invention solves i of a partial differential equation.
A coefficient matrix decomposition circuit takes the coefficient matrix and coefficient vector of the simultaneous linear equations obtained by the four-loss approximation as input data, and decomposes the coefficient matrix into an approximate matrix that can be inverted, and the vector data input from the iterative calculation circuit described later. It has a forward/backward substitution circuit that performs an inverse matrix operation on the approximate matrix generated by the coefficient matrix decomposition circuit and outputs it to the iterative calculation circuit, and an iterative calculation circuit connected thereto, which calculates approximate solutions of the simultaneous linear equations. An apparatus for analyzing simultaneous linear equations as an output, wherein the coefficient matrix decomposition circuit divides the coefficient matrix into block matrices, and calculates an incomplete Cbol for each diagonal block matrix.
eski decomposition or incomplete LU decomposition, and the forward and backward substitution circuit performs an inverse matrix operation of the approximate matrix by performing forward and backward substitution using each incompletely decomposed diagonal block matrix on the vector data input from the iterative calculation circuit. It is configured as follows.

[Effect]

In the present invention, the unknown vector U is divided U−[IJ
l, = . A22.・
... Perform incomplete LU decomposition of each A m111. A
Let the incomplete LU decomposition matrix of ll be L,,U,,=
M, , , and an approximate matrix (this is M) for inverse matrix calculation is defined as follows.

In this way, the inverse matrix operation of the approximate matrix M is performed using each diagonal matrix M
It can be done independently for each. That is, (blank space below) Therefore, forward/backward assignment calculation by M, v=M-'g, is independent forward/backward assignment calculation by each processor PUP (i・1. . . , a+), V+"Mz-' g+ (jl+-
J).

〔Example〕

Hereinafter, the present invention will be explained with reference to the drawings.

FIG. 1 is a basic configuration diagram showing one embodiment of the present invention, and FIG. 2 is a device configuration diagram. In the same figure, the simultaneous linear equation analysis device includes m processors pu1. ...PU.

, and each Ptl can access the local memory LMI and the shared memories SM, -, and SMI. In addition, the control processor CP has all the shared memories SM, (i
・1, ..., m) can be accessed. Coefficient matrix decomposition circuit 1 in FIG. Forward/backward substitution circuit 22 Iteration calculation circuit 3
are configured independently within each PUI.

In the simultaneous linear equations Au=f, A is assumed to be a symmetric fixed matrix, and the iterative calculation circuit uses the conjugate gradient method. Also assume the structure of the equation. Equations obtained by the finite element method, finite difference method, etc. have this structure. In order to solve this equation using the apparatus shown in FIG. 2, the submatrix A 1 = [A 1l-IA 11A
11+1], partial solids 1 to 1f, to local memory LM
, the approximate solution vectors i and u1 are allocated to the shared memory SM+-1°SM', and the calculation of each ui is performed by each PU.

Do it with

PUI uses a coefficient matrix decomposition circuit to take in A11 from LM, performs incomplete LU decomposition into "L-th IL", and converts this into LM,
write to. In addition, in the forward/backward substitution circuit, for a vector input from the iterative calculation circuit, for example, g++, V+''(
Output LzLth ")-' g+.

FIG. 3 shows a flowchart in the iterative calculation circuit of the PU. The basic flow follows the preconditioned conjugate gradient method. 301 u ++ - t + ur + u ++
t represents the (1) initial value. PUI is 301
'C''ul-1, ul+1 can be taken in from SM, -, ,SM, respectively.

Similarly, pI is SMI-t, SM+Mz at 302.312
By writing to , each PUI is 305 and P + - t + p
l+l can be taken directly. At 302.308, the vector is sent to the forward/backward substitution circuit to obtain a result multiplied by M-1. As mentioned above, this calculation can be performed in parallel on each PU. 303.304, inner product r= (p, r>
-4: In the calculation of (pI+r +)=n7 +, each PUI calculates 11, and the CP calculates their sum. The same applies to 307, 308 and 309.310. C
P makes a convergence judgment based on ε=εε■, and if convergence occurs, instructs each PUI to terminate (311).

〔Effect of the invention〕

As is clear from the above description, according to the present invention, forward and backward substitution calculations for inverse matrix operations using approximate matrices can be processed in parallel, and a parallel computer comprising a plurality of processors can be effectively utilized. Since the range in which incomplete LU decomposition is performed is limited to the diagonal block portion, it is expected that the narrower the block division, the lower the convergence compared to the conventional method in which decomposition is performed on the entire coefficient matrix. However, it has been experimentally confirmed that even in the worst case where the number of processors is equal to the number of elements of the equation, that is, one unknown corresponds to one block, the number of iterations increases by about three times. Normally, the number of processors is small and the blocks are therefore large, so within that range, deterioration in convergence is hardly a problem.

[Brief explanation of the drawing]

FIG. 1 is a basic configuration diagram showing an embodiment of the present invention, FIG. 2 is a structural diagram of a parallel computing device, and FIG. 3 is a flow diagram showing a processing procedure of an iterative computing circuit. l... Coefficient matrix decomposition circuit, 2... Forward/backward substitution circuit, 3... Iterative calculation circuit, 11... Input data, 12
... Matrix decomposed of approximate matrix M of A, 13... Vector (temporarily expressed as g), 14... Vector v (-M
-1g or M''-'g), 15...Equation Au
-$ / Figure 11: Person Tepada (Matrix A, Beatle f) 12:
Acl)M)? #pysuMn L U inla γf7
hE;tDsuI3:Be7tonrezuf4:Beftor VC=M-”j Father M7t3) 15:Approximate solution of Fang Yaojin Au=f8ttlecon 2nd CP:Bun)Lo)L processor sH:$waste Upstream PU: Allocessor unit LN: Local memory

Claims (1)

[Claims] A coefficient matrix decomposition circuit that takes as input data the coefficient matrix and coefficient vector of a system of linear equations obtained by discrete approximation of a partial differential equation, and decomposes the coefficient matrix into an approximate matrix that can be inverted, and input from an iterative calculation circuit described later. It has a forward/backward substitution circuit that performs an inverse matrix operation on the approximate matrix generated by the coefficient matrix decomposition circuit on vector data and outputs it to the iterative calculation circuit, and an iterative calculation circuit connected thereto, A system for analyzing simultaneous linear equations that outputs an approximate solution, wherein the coefficient matrix decomposition circuit divides the coefficient matrix into block matrices, and performs incomplete Choleski decomposition or incomplete LU decomposition on each diagonal block matrix. The forward and backward substitution circuit performs an inverse matrix operation of the approximate matrix by performing forward and backward substitution using each diagonal block matrix incompletely decomposed into the vector data input from the iterative calculation circuit. Simultaneous linear equation analysis device.