JPS6158076A - Simultaneous primary equation simulator - Google Patents

Abstract

PURPOSE:To make possible the calculation for a short time by utilizing the band efficiency matrix of simultaneous primary equation and dividing into several blocks and processing by each processor. CONSTITUTION:A simulator consists of an all control processor T, (m) number of a subordinate processor S and an auxiliary storage device M. The auxiliary storage device memorizes each asymptotic value of the unknown number vector phi1, phi2-phim. A subordinate processos Sj defines a vector asymptotic value phij(k), -m(k) k times to input into support storage device Mj according to it. After all vector asymptonic value phi(k), phim(k) are recognized, each subordinate proces sor changes j into j + 1 to calculate [phij(k+1)] from [phij(k)]. The result is inputted into the auxiliary storage device [Mj]. The all control processor T carries out the synchronous control and spreading discrimination every time.

Description

[Detailed Description of the Invention] (g) Technical Field The present invention relates to a Jλ system structure of a simultaneous linear equation simulator.

(Conventional Technology) When simulating physical phenomena described by partial differential equations using a digital computer, it is necessary to numerically solve simultaneous linear equations.

Simultaneous linear equations can always be solved because it is clearly known how to solve them.

However, if the number of elements of the simultaneous linear equations becomes large, a large amount of calculation time is required.

The number of unknowns, that is, the number of independent equations, is called an element. Letting this be n, the number of operations increases on the order of nn.

Let the unknowns be Xl, X2,...X, and let the coefficient be a
If ij, then the n-dimensional simultaneous equations are: ): aijxj= bt (1)j=
It can be written as t. To solve this, the value of the matrix of coefficients Δ = det lat j l (2)
It is necessary to find the value Aj of the matrix obtained by replacing the j-th column vector of [ai j ) with (bl). This is an exact solution. In order to find the value of an nXn matrix, we must calculate 11 products and add them together.There are (n+1) matrices to be calculated, so we can calculate the (n+1)i terms just by calculating the products. must be calculated.

When the number n becomes large, this becomes a huge number, and even a high-performance computer requires a large amount of calculation time.

Of course, in many cases, the coefficient (aij) is often 0, and in reality, such a large number of calculations are not required. For one equation, the number of non-zero numerical coefficients is 2
In many cases, there are only one, three, or four.

However, even in such a case, the number of operations is n−n
It increases in proportion to the degree.

(1) Purpose The present invention focuses on the fact that the coefficient matrix of the obtained simultaneous linear equations is often a band matrix, divides it into several blocks, and processes each block with a separate processor. The aim is to provide a parallel simulator that can perform calculations in a short time.

) When dealing with constituent physical phenomena, the physical quantities are displacement, velocity, acceleration,
It is often a vector quantity such as angular acceleration, angular velocity, vortex, etc. It is a two-dimensional or three-dimensional vector. One vector consists of three unknowns (Xl, X2, X3).

However, equations that describe physical phenomena are often expressed in the form of three-dimensional vectors, such as Maxwell's equations and Navier-Stokes equations.

Then, one coefficient matrix corresponds to one physical quantity vector, and several products of matrices and vectors are collected to form three equations of the simultaneous linear equations.

That is, in equation (1), an equation in which Xj is replaced with vector φJ, aij is a matrix, and bl is a vector often holds true. For example, φj is a three-dimensional vector, aij
is a (3×3) matrix, and bi is a three-dimensional vector.

Then, instead of equation (1), j=l becomes. This equation is n simultaneous linear equations.

Since the Aij matrix and φj are vectors, this operation is of course performed as follows. For simplicity,
Omit the suffixes of i and j, and write the matrix elements of A as (apq
If we write the components of l and φ as [xq l, Aφ =
Σ apq Xq (5) q:l.

This is the case for three-dimensional vectors, but in many cases, a vector consisting of a large number of three-dimensional vectors can be expressed as φj, and can be written in the form of equation (4). After all, there are cases in which the entire equation can be rewritten using m h-element vectors.

mh = n (5). φ1, φ2, . . . , φm are h-element vectors, and matrix A is an (hXh) matrix. Instead of formula (5), the product of A and φ becomes Aφ = Σapq

Any n-dimensional simultaneous linear equation expressed by equation (1) can always be rewritten into equations such as (7) and (8).

When dealing with physical phenomena, it is often the case that only two or three of the coefficient matrix of vector φj in equation (8) are not O. This is the important point.

Vectors φ1, φ2, etc. with components of Sokote, h
..., regarding φm, it is possible to formulate a band matrix equation of the following form.

This is called simultaneous linear equations with banded matrices.

In order to create this formula, we must consider the selection of (Xql as a component of φj, the order of the formula, etc.).

The feature of this formula is that for φj other than j=1 and m, φ
There is always one expression that contains only three vectors: j, φj's before it, φj+1 after it, and j=1, j=m
Regarding φj, there is one expression each including φj and φj+s or φj and φj−+.

Therefore, in equation (9), in the first equation, the coefficient of φj is written as Aj, and the coefficient of φj-1 before that is written as Uj,
The subsequent coefficient of φj+l is written as Lj.

Formula (a) is only a typical example. Regarding φ1 and φm, a term for φm and a term for φ1 may be further included. In other words, all equations may include three adjacent (cyclically) vectors.

Furthermore, the number of vectors included in one equation does not necessarily have to be three; there may be four or five vectors. This does not mean that subsequent processing is more complicated.

In creating equation (9), it is not necessary to simply block the original equation into m h-element equations, but to connect the preceding and succeeding vectors using one equation. Adjacent vectors appear in the equation in sequence.

However, even if this condition is imposed, the expression of equation (9) is not uniquely determined, and a degree of freedom remains. (9)
Each of the equations may be multiplied by a differential coefficient to add or subtract, or may be multiplied by a matrix to add or subtract. Such transformations are free.

One important issue is the issue of convergence. bi
, the formula for bj-H is Ujφj-1+Ajφj+Ljφj+t
(10) Ui+tφj+Aj+tφj+
It has the term t+Lj+tφi+z (11).

In order to obtain convergence, the condition should be imposed. However, A indicates the inverse matrix of A. This calculation is a matrix multiplication of (hxh), and the value of the determinant of the matrix product is equal to the product of the determinants of the respective matrices, but equation (12) is small. This formula states that the value (determinant) of the determinant of the product of the coefficient matrices of the diagonal ring is greater than the product of the determinant of the coefficient matrices of the off-diagonal terms. Since equation (13) spans two equations, the third and j+1st equations,
A similar check must be performed (m-1) times. As a sufficient condition to satisfy equation (13), inside the j-th equation l d6t 1Ajl I >
I dot ILjI l (1,4)l aet
1Ajl l > l dot IJI l
It is sufficient if (15) holds true.

This may not happen immediately. In that case, the product of the adjacent expression multiplied by a constant is added to or subtracted from the j-th expression so that (14) and (15) hold true.

FIG. 1 shows the configuration of a simulator according to the present invention.

The simulator consists of one overall control processor T, m subordinate processors S, and m auxiliary storage devices M.

The auxiliary storage device stores unknown vectors φ1, φ2, and
..., memorize the asymptotic value of φm.

The dependent processor s3 moves the term φj to the left side of equation (9) and the other terms to the right side, and calculates the value of φj by repeating processing.

The number of repetitions will be expressed as φ(k).

The j-th subordinate processor Sj is Ajφj(k) = bj −Ujφj−1(k
-1) -Ljφi+t(ic-t) (16)
Using the asymptotic formula, the (k-1)th result φj-1
(k-1), φj+1(k-1), the k-th result φ
Calculate j(k).

All subordinate processors perform the operation (16) at the same time.

Among the necessary data, Aj, 1)j, Uj, L
Since j is a constant matrix, it is stored in advance. (k-1
)-th data is transferred from the adjacent auxiliary storage device to φj−1,
Read and use φj+1. This is the diagonally upward arrow pointing from the auxiliary storage device to the subordinate processor in FIG.

The dependent processor Sj has the vector asymptotic value φj(k
) is input into the corresponding auxiliary storage device Mj. φj(k-1) disappears from the memory of Mj, and φj(
k) will be newly man-powered. The downward arrow in Figure 1 indicates this.

In this way, the k-th vector asymptotic value φ, (k),...
・Since all of φm(k) is known, each subordinate processor performs the same calculation by changing j in equation (16) to j+t and calculating {φ
{φj(k+1)) is calculated from j(k)).

This result is input to the auxiliary storage device [Mj].

In this way, similar calculations are repeated, but as shown in FIG. 2, synchronization control and convergence determination are performed each time. this is,
This is carried out by the overall control processor T.

Synchronous control is used to match the timing of calculations in each subordinate processor.

Convergence judgment is an operation to find the difference between the (k-1)th asymptotic value and the kth asymptotic value of the j-th vector. If this is smaller than a predetermined smaller vector ε, the j-th vector is Determine that it has converged sufficiently near the strict solution 1 φj(k) −φj(+c−i)l< ε
(17) ``Ro-de Junior High II-
Pass ↓ person 14-! l river η1 i − ] −
, li prayer ← T 4 ba ? Since it is performed every 9 minutes, there is a slight difference in the time.

This is what is meant by the different lengths of the arrows in Figure 2.

If even one of the convergence determination results φj does not satisfy equation (17), the same operation is repeated one more time. When all φj satisfy equation (17), the KS iterative operation is finished. Vector obtained here {φj)
is the solution to the simultaneous linear equations.

The initial value of {φj) {φj(1) l can be given in any way.

For example, all may be O vectors. In this way, any constant may be used as the initial value.

It is desirable to choose an initial value that converges quickly, but in order to do so, for example, divide it into two equations and remove the U and L terms before and after.
b, (18) U2φ1+A2φ2 = b2 ・(19
), you can create a reduced equation and find the solution. It is also possible to start with this solution as an initial value.

The method mentioned earlier starting from 0 vector is k=2 and φj(2) = Aj bj
(20) How to select the initial value.

No matter what initial value is chosen, as long as (14) and (15) hold, it will converge toward an exact solution. However, the speed of convergence depends on how the initial values are selected.

) Simultaneous linear equations whose effect coefficient matrix is a band matrix can be solved in a short time.

This is because simultaneous linear equations with large elements are decomposed into small simultaneous linear equations with m elements, and these are processed and solved in parallel.

[Brief explanation of drawings]

FIG. 1 is a block diagram showing the configuration of a simultaneous linear equations simulator of the present invention. FIG. 2 is an explanatory diagram showing the operation of the simultaneous linear equations simulator of the present invention in which parallel processing is repeated. T ・・・・・・・・・ Overall control Bromo Nsa S ・
...... Dependent processor M ......
・・・ Auxiliary storage device φ, ~φm ・・・・・・ Unknown vector Inventor: Osamu Kaoru Miki
Male 1mi;

Claims (1)

[Claims] Consists of one overall control processor T, m subordinate processors S_1, S_2, ..., S_m, and m auxiliary storage devices M_1, M_2, ..., M_m, and an element When we have a large simultaneous linear equation with a band matrix as a coefficient matrix and an unknown vector as {φj}, the j-th equation is the j-th unknown vector φj, the preceding and following vectors φj−1,
φj+1, and the value of the coefficient determinant of φj det|Aj
| is the value det of the coefficient matrix of φj−1, φj+1 |Uj|
, det|Lj|, and the j-th auxiliary storage device M_j stores the asymptotic value φj(k) of the j-th unknown vector φj. For {φj}, an appropriate initial value {φj(1
)}, and the subordinate processor {Sj} and auxiliary storage {
Mj} is the asymptotic value of the iterative unknown vector {φj(k)}
, and the j-th subordinate processor S_j is m
The j-th expression among the block expressions and the auxiliary storage {M
} The (k-1)th asymptotic value φj-1(
k) and φj+1(k-1), etc., the k-th asymptotic value φ
Calculate j(k) and store φj(k) in the auxiliary storage device M_j.
The overall control processor T performs synchronous control so that the k-th calculation of {φJ(k)} is simultaneously processed in parallel by all subordinate processors {Sj}, and Newly calculated k-th asymptotic value φj
Convergence judgment is performed by comparing the absolute value of the difference between (k) and (k-1)th asymptotic value with a predetermined infinitesimal vector ε, and when all vectors {φj} have converged, this is converted into a system of linear equations. A simulator of simultaneous linear equations characterized by the solution of .