JP2643110B2 - A fast method for solving simultaneous linear equations using an auxiliary storage device - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION With the advent of a supercomputer and an extended storage device which is a semiconductor high-speed storage device, the present invention has been developed to realize simultaneous high-speed simultaneous linear equations having a dense matrix that is too large to fit in a main storage device. The present invention relates to a high-speed calculation method for a large-scale simultaneous linear equation in the field of numerical calculation, which has been able to calculate in a short time.

[0002]

2. Description of the Related Art As a method of calculating simultaneous linear equations using an extended storage device, a calculation method by Tsuda et al. (Tsuda,
T. Okabe, Y .; : Use of Semico
nductor Extended Storage
as Extended Main Storage
for Large-Scale Supercomp
uting, Proceedings of the
Second International Conf
erence on Supercomputing,
May 1987, Santa Clara (Sup
ercomputing '87), Vol. 1, pp.
176-183, International Sup
ercomputing Institute. ). In this method, while exchanging data between an extended storage and a main storage, a coefficient matrix is transformed into L
The U-decomposition is performed to obtain a solution of the simultaneous linear equation. Hereinafter, this calculation method will be described. Considering a system of linear equations 1 as shown in FIG. 6, the coefficients aij (i = 1, 2, 2,
.., N, j = 1, 2,..., N) and bi (i = 1, 2,
, N) and the solution xi (i = 1, 2,..., N) is calculated by the computer shown in FIG. This computer is composed of a central processing unit 2, a main storage device 3 directly accessed by the central processing unit 2, an auxiliary storage device 4 capable of exchanging data with the main storage device 3, and particularly a semiconductor as the auxiliary storage device 4. It has an extended storage device 12 which is a high-speed auxiliary storage device. Starting from a state in which the coefficients of the simultaneous linear equation 1 are stored in the extended storage device 12 in a format as shown in FIG. 8, finally, while exchanging data between the extended storage device 12 and the main storage device 3, Obtaining the converted data as shown in FIG. 9 on the extended storage device 12 is the LU decomposition step which is a main part of the present calculation method. Here, lij and uij in FIG. 9 give the corresponding elements of the n-th square lower triangular matrix L and the n-th square upper triangular matrix U, respectively, and the matrix product of the matrix L and the matrix U is expressed by the simultaneous linear equation 1
Is given as a coefficient matrix 11 having the coefficients of It is assumed that all diagonal elements of the n-th square lower triangular matrix L are 1. As described above, the coefficient matrix 11 of the simultaneous linear equation 1 is 2
If it is decomposed into the product of two triangular matrices L and U, the solution of the simultaneous linear equation 1 can be easily calculated by forward and backward substitution. 9 is obtained by the calculation method shown in FIG. When the initial value of aij is calculated as an element of the coefficient matrix 11 according to the calculation method of FIG.
Obtains the elements of the matrix 15 after LU decomposition shown in FIG. This calculation method is called the inner product form Gauss method (or jki form-GAXPY Gauss method from the calculation order of subscripts). Note that, for simplicity, FIG. 10 shows a calculation procedure to which a speeding-up method by unrolling described later is not applied. The description of the calculation method partially uses the notation used in the computer language PASCAL.

FIG. 11 shows a reference relation of data in the calculation of the k-th column in the inner product form Gaussian method. The feature of the inner product form Gauss method is that when obtaining the final calculation result of the k-th column, only the lower triangular part on the left side of the first, second, ..., k-1 columns and the k-th column itself are required. That is. Now, as shown in FIG. 7, a reference data storage area 5 and an update data storage area 6 are considered as two types of storage areas on the main storage 3. Here, the data read from the auxiliary storage device 4 onto the reference data storage area 5 is only referred to in the processing, is not changed, and is therefore deleted by the next reading. On the other hand, the value of the data read into the update data area 6 is changed by the processing, and the processing result is stored in the auxiliary storage device 4 associated before the next reading is performed.
Shall be returned to the top. In an embodiment to be described later, the reference data storage area 5 and the update data storage area 6
Change the usage of. To efficiently transfer data between the main storage device 3 and the auxiliary storage device 4, the coefficient matrix 11 of the simultaneous linear equation 1 stored in the auxiliary storage device 4 (in this case, the extended storage device 12) is stored. Are page-divided into fixed columns as shown in FIG. 8, and are referred to as a first page 7, a second page 8, a third page 9,..., An N-th page 10, respectively, and data transfer is performed in page units. In FIG. 8, m represents the number of columns included in one page. However, the last page, Nth
The page 10 includes m or less and 1 or more columns determined by the dimension of the coefficient matrix 11. To read the first page 7, the second page 8,..., The N-th page 10 of the coefficient matrix 11 into the update data storage area 6 in FIG. 7, and perform the conversion shown in FIG. The calculation method shown in FIG. 12 may be used depending on the characteristics of the inner product form Gauss method described above.

A calculation method based on the inner product Gauss method as described above is said to be good when the page division method as shown in FIG. 8 is used because the number of times of input / output can be reduced.

[0005]

On the other hand, in a supercomputer, a technique called unrolling is used to speed up the calculation. Unrolling refers to, for example, converting the calculation method (a) in FIG. 4 to the calculation method (b), thereby reducing the number of times of data storage in the storage area represented by ai from 2n to n, and increasing the speed. Method. FIG. 4 shows an example of two-stage unrolling.

When such an unrolling method is applied, the inner product form Gaussian method is not always the optimal method in terms of calculation speed. This is what Robert et al.
t, Y. and Sguazzero, P.M. : The
LU decomposition algorithm
m and it's effective FORTRA
N implementation on IBM 3
090 Vector Multiprocessor,
IBM Tech. Rep. , ICE-0006, Ma
rch 1987). The inventor has also conducted a similar experiment on the supercomputer SX-2A, and has confirmed that the inner product form Gaussian method is significantly inferior to the outer product form Gaussian method in terms of calculation speed when the unrolling method is used. . For example, 10
When solving the system of linear equations of 00 element on the main memory, the inner product form Gaussian method is 500-
600 MFLOPS (1 MFLOPS is 100 per second
Although the calculation speed is of the order of the number of times of performing floating point operations for 10,000 times), the cross product Gaussian method can achieve a calculation speed exceeding 900 MFLOPS. This is because the outer product form Gaussian method can reduce the number of times of data storage by the unrolling compared to the inner product form Gaussian method. Therefore, a problem that the calculation method for calculating the solution of the simultaneous linear equations while exchanging the data between the main storage device and the expansion storage device by extending the inner product form Gauss method is not necessarily the optimum calculation method with respect to the calculation speed. have. On the other hand, although the expansion storage device is high-speed, it is more difficult to increase its capacity than a low-speed disk device, and the available storage capacity is limited. Therefore, when trying to solve a system of linear equations having a coefficient matrix large enough to exceed the available expanded storage capacity, another auxiliary storage device (for example, a high-speed disk device) that is slower than the expanded storage device ) Must be used. In this case, as much of the data transfer between the main storage device and the auxiliary storage device as possible is performed between the extended storage device and the main storage device, and the rest is performed between the other auxiliary storage device and the main storage device. It is necessary to devise something.
However, the conventional calculation method does not take the above problem into consideration. Furthermore, the conventional calculation method has a problem that it lacks versatility because it does not assume a combination with another operation such as a matrix multiplication operation using a large-scale matrix. SUMMARY OF THE INVENTION The present invention has been made to solve the above-described problems, and an object of the present invention is to provide a high-speed calculation of a solution of a system of linear equations using an auxiliary storage device that measures storage efficiency by a Gaussian cross product method using unrolling. Is to provide a scheme.

[0007]

SUMMARY OF THE INVENTION According to the present invention, a high-speed system for solving simultaneous linear equations using an auxiliary storage device includes a central processing unit, a main storage device directly accessed by the central processing unit, a main storage device and a data storage device. In a computer having an auxiliary storage device capable of exchanging data, in particular, a computer having an extended storage device, which is a high-speed auxiliary storage device constituted by a semiconductor as the auxiliary storage device, a coefficient matrix corresponding to a coefficient of a simultaneous linear equation on the auxiliary storage device Is given first, the coefficient matrix is decomposed into a product of two triangular matrices, and a solution of the simultaneous linear equation is calculated. In decomposing into a product of matrices, the coefficient matrix is divided so as to include a fixed number of columns, and the divided units are called pages, and the first page, the second page,
The page is set as a processing unit as. However, the last page is not the same size as the other pages, but includes extra elements according to the size of the coefficient matrix.

[0008] One or more pages of the coefficient matrix are read from the auxiliary storage device to the main storage device, the read data on the main storage device is processed, and the result is converted to a corresponding page of the auxiliary storage device. return.

The storage area on the main storage device is divided into two, a reference data storage area and an update data storage area, and the data read from the auxiliary storage device into the reference data storage area is updated only with its own data. The process is performed, and is referred to for updating data read into the update data area described below through the process. Therefore, the page is returned to the corresponding page in the auxiliary storage device after all the related data has been updated. On the other hand, the data read into the update data storage area is updated with reference to the data read into the reference data storage area, and is returned to the corresponding page of the auxiliary storage device.

When the coefficient matrix is composed of N pages and is called a first page, a second page,..., An Nth page in order, the order of data exchange between the auxiliary storage device and the main storage device is the first. The page is read from the auxiliary storage device into the storage area for reference data and processed. The second page, the third page,..., And the Nth page are read from the auxiliary storage device to the storage area for update data, processed, and returned to the corresponding page of the auxiliary storage device.

The first page is returned from the reference data storage area to the corresponding page of the auxiliary storage device.

The second page is read from the auxiliary storage device into the reference data storage area and processed. The third page, the fourth page,..., And the Nth page are read in order from the auxiliary storage device to the update data storage area, processed, and returned to the corresponding page of the auxiliary storage device.

The second page is returned from the storage area for reference data to the corresponding page in the auxiliary storage device.

Reads the N-1 page from the auxiliary storage device into the reference data storage area and processes it.

The Nth page is read from the auxiliary storage device into the update data area, processed, and returned to the corresponding page in the auxiliary storage device.

The N-1th page is returned from the reference data storage area to the corresponding page of the auxiliary storage device, in order.
This is a calculation method, and when the auxiliary storage device includes an extended storage device, the process proceeds while holding, on the extended storage device, only enough pages to fit in the extended storage device in reverse order from the Nth page of the coefficient matrix. This is a calculation method.
Only the small matrix corresponding to the part to be referred to and updated in the subsequent processing is extracted, and this is regarded as a new matrix, and the above-described processing in (1) and (2) is performed. Is a calculation method for expressing the coefficient matrix as a product of two triangular matrices and calculating the solution of the simultaneous linear equations, treating the data on the auxiliary storage device as a direct organization file, and Is set to such a size that an element included in one column of the coefficient matrix just fits.

[0017]

According to the present invention, a high-speed calculation method for solving a system of linear equations using an auxiliary storage device is divided into a reference data storage region and an update data storage region by a cross product Gaussian method. Since the storage area is extended and used by exchanging data between them, high-speed calculation is possible.

[0018]

EXAMPLE A first example of the present calculation method will be described. This calculation method is described in Robert et al.
and Sguazzero, P.M. : The LU d
ecological algorithm an
d its effective FORTRAN i
implementation on IBM 3090
Vector Multiprocessor, IBM
Tech. Rep. ICE-0006, March
1987) is improved for a large-scale matrix by using the outer product form Gaussian method, which can be calculated by a computer having an auxiliary storage device 4 as shown in FIG. First, the outer product Gaussian method will be described. Considering the simultaneous linear equation 1 as shown in FIG. 6, the coefficient aij (i = 1,
2, ..., n, j = 1,2, ..., n) and bi (i = 1,
2, ..., n) to give a solution xi (i = 1,2, ..., n)
Consider calculating Starting from a state in which the coefficients of the simultaneous linear equation 1 are stored in the extended storage device 12 in a format as shown in FIG. 8, finally, while exchanging data between the extended storage device 12 and the main storage device 3, Obtaining the converted data as shown in FIG. 9 on the extended storage device 12 is the LU decomposition step which is a main part of the present calculation method. However, lij and uij in FIG.
Corresponding elements of an nth-order square lower triangular matrix L and an nth-order upper square triangular matrix U are given, and a matrix product of the matrix L and the matrix U gives a coefficient matrix 11 having coefficients of the simultaneous linear equation 1 as components. It is assumed that all diagonal elements of the n-th square lower triangular matrix L are 1. If the coefficient matrix 11 of the simultaneous linear equation 1 is decomposed into the product of the two triangular matrices L and U as described above, the solution of the simultaneous linear equation 1 can be easily calculated by forward and backward substitution. 9 is obtained by the calculation method shown in FIG. When the initial value of aij is calculated as an element of the coefficient matrix 11 according to the calculation method of FIG.
For ij, the elements of the matrix 15 after LU decomposition shown in FIG. 9 are obtained. This calculation method is called the outer product form Gauss method (or kji form-SXPY Gauss method from the calculation order of the subscript). In addition, for simplicity, a calculation procedure to which the high-speed technique by unrolling is not applied is shown. FIG. 3 shows a reference relation of data in the calculation of the k-th step of the outer product form Gaussian method. The feature of the Gaussian cross product method is that, in the calculation of the k-th step, as shown in FIG.
This means that only the -k + 1 order lower right small square matrix part is needed, and the update is performed only in the nk order lower right square matrix part 16. In general, even when the unrolling method described above is used, the property that the portion to be processed is reduced and can be represented by a lower right small square matrix does not change as the step proceeds. Therefore, the coefficient matrix 11 of the simultaneous linear equation 1 is divided into pages as shown in FIG. 8, and data is exchanged between the extended storage device 12 and the main storage device 3 on the extended storage device 12 as shown in FIG. To perform the conversion shown in FIG. 9, the calculation method shown in FIG. 1 may be executed. Here, the reference data storage area 5 and the update data storage area 6, which are two storage areas on the main storage device, are selectively used as follows. The data read from the auxiliary storage device 4 into the reference data storage area 5 is updated only with its own data, and is referred to for updating the data read into the update data area 6 through the processing.
Therefore, the page is returned to the corresponding page in the auxiliary storage device 4 after all the related data has been updated. On the other hand, for the data read into the update data storage area 6, the value is updated with reference to the data read into the reference data storage area 5, and the data is returned to the corresponding page of the auxiliary storage device 4. The calculation procedure shown in FIG. 1 is realized on a supercomputer SX-2A having an extended storage from among several types of operation pipeline system vector computers, and the calculation of the solution of the 5000-element simultaneous linear equation is performed by 1GFLOPS (1 (The speed of performing one billion floating-point operations per second). The storage capacity on the main storage device used for this calculation is about 25 Mbytes, and this value is about 1 of the main storage capacity required when calculating the solution of this simultaneous linear equation using only the main storage device 3. / 8. The theoretical peak performance of SX-2A is 1.3 GFLOP
S, and a high-speed performance very close to the theoretical performance was realized. The embodiment described above corresponds to a case where only the extended storage device 12 is used as the auxiliary storage device 4.

As a second embodiment, a case where the coefficient matrix 11 is too large to fit in the extended storage device 12 will be described. In this case, a slower disk device must be considered as the auxiliary storage device 4. However, since the difference in data transfer speed between the extended storage 12 and the disk device is remarkable, many data transfers are performed in the extended storage device. What is necessary is just to use the auxiliary storage device 4 properly, as performed on the 12 side. Due to the feature of the Gaussian elimination method described above, the number of data transfers increases as the coefficient matrix 11 of the simultaneous linear equation 1 is divided into pages, from the first page to the last page (the Nth page). Therefore, high-speed processing can be realized by performing processing such that pages that only enter the extended storage device in the reverse order from the Nth page are stored in the extended storage 12 and the remaining portion is retained on the disk. This calculation method is based on the procedure described in the first embodiment.
2 can be easily realized by rewriting the exchange between the disk drive and the main storage device 3 so as to partially change the exchange between the disk device and the main storage device 3. For example, when the first page to the p-1 page of the coefficient matrix 11 of the simultaneous linear equation 1 are provided on the disk device and the p pages to the N page are provided on the extended storage device as the initial state, the initial state In order to obtain the result of the conversion shown in FIG. 9 with the same page arrangement as in FIG. 9, the calculation method in FIG. 1 may be rewritten as shown in FIG. It is to be noted that the above-described calculation method is changed so as to correspond to a case where the coefficient matrix 11 of the simultaneous linear equation 1 is entirely on the disk device as an initial state or a case where all the final results are to be stored on the disk device. Easy.

As a third embodiment, consider the case where the ratio of the data transfer time between the auxiliary storage device 4 and the main storage device 3 to the total calculation time is large. In this case, it is important to reduce the data transfer amount between the auxiliary storage device 4 and the main storage device 3. Earlier, in the outer product form Gaussian elimination method, the part to be referred and updated is the lower right small square matrix part that shrinks as the processing progresses, and the other parts in the page are unnecessary for subsequent processing . Therefore, if only the lower right sub-matrix necessary for the future processing is extracted, this is regarded as one matrix, and the method described in the first or second embodiment is applied, the data transfer amount can be reduced. it can. This method uses, for example, the extended storage device 12 as the auxiliary storage device 4.
This is an effective method to reduce the amount of data transfer in a computer configuration that does not have the above.

Finally, as a fourth embodiment, as a storage format of data in the auxiliary storage device 4, in the above embodiment,
It is assumed that a direct organization file is used. In a direct-organization file, a unit called a record is used as a unit of a storage area, and reading and writing can be performed on an arbitrary record. Conventionally, as the size of this record, the size of one page when the coefficient matrix 11 of the simultaneous linear equation 1 is divided into pages has been adopted. However, other calculations and the solution of the simultaneous linear equation A problem arises when calculating by combining. For example, considering the case of calculating the product of two large-scale matrices and solving a system of linear equations using the calculated product as a coefficient matrix, in general, the optimal page size when calculating the product of the matrices and The optimum page size when calculating the solution of the system of linear equations differs. However, once a record length is determined in a direct organization file, it cannot be changed later. Therefore, it is not preferable to set the record length to the size of the page. Here, this problem was solved by taking the size of one column of the coefficient matrix as the record length, and reading and writing data for one page continuously by reading and writing a plurality of records corresponding to one page. This makes it possible to set an optimum page size for each process even when the process is combined with another process.

Although not described in the present embodiment, a similar calculation method can be adopted when a partial pivoting operation which is usually performed when solving simultaneous linear equations is performed.

[0023]

According to the present calculation method, a large-scale simultaneous linear equation can be calculated at high speed. As a result, not only various engineering fields such as structural analysis, fluid dynamics, electromagnetics, and nuclear power, but also, It can contribute to speeding up numerical calculations in many application fields.

[Brief description of the drawings]

FIG. 1 is an explanatory diagram of a high-speed calculation method for solving simultaneous linear equations using an auxiliary storage device of the present invention.

FIG. 2 is a diagram showing a calculation method of each element in the outer product form Gaussian method of the present invention.

FIG. 3 is a diagram illustrating a k-th data reference relationship in the outer product form Gaussian method of the present invention.

FIG. 4 is an explanatory diagram of an unrolling method.

FIG. 5 is an explanatory diagram of a second embodiment of the present invention.

FIG. 6 is a diagram showing simultaneous linear equations for which a solution is to be obtained.

FIG. 7 is a diagram showing a computer configuration.

FIG. 8 is an explanatory diagram of page division of simultaneous linear equations.

FIG. 9 is a diagram showing a correspondence relationship between each element of a modulus matrix of a system of linear equations and an element of an LU-decomposed matrix;

FIG. 10 is a diagram showing a calculation method of each element in a conventional inner product form Gaussian method.

FIG. 11 is a diagram showing a calculation method of each element in calculation of a k-th column in a conventional inner product form Gaussian method.

FIG. 12 is a diagram illustrating a conventional calculation method.

[Explanation of symbols]

DESCRIPTION OF SYMBOLS 1 ... Simultaneous linear equation, 2 ... Central processing unit, 3 ... Main storage device, 4 ... Auxiliary storage device, 5 ... Storage area for reference data, 6
... storage area for update data, 7 ... first page of coefficient matrix divided into pages, 8 ... second page, 9 ... third page, 10 ... Nth page (last page), 11 ... simultaneous 1
Coefficient matrix of the following equation, 12... Expanded storage device, 13... Element of n-th square upper triangular matrix U, 14.
, 15... Arrangement of each element after LU decomposition, 16.
The area updated at the step number 17; the area referenced at the k-th step.

Claims (4)

(57) [Claims] 1. A central processing unit, a main storage device directly accessed by the central processing unit, and an auxiliary storage device capable of exchanging data with the main storage device, in particular, a high-speed auxiliary storage device composed of a semiconductor as the auxiliary storage device. In a computer having a certain extended storage device, numerical data of a coefficient matrix corresponding to a coefficient of a system of linear equations is given to an auxiliary storage device,
First, there is provided a calculation method for decomposing the coefficient matrix into a product of two triangular matrices and calculating a solution of a system of linear equations. In decomposing the coefficient matrix of the system of linear equations into a product of two triangular matrices, , The coefficient matrix is divided so as to include a fixed number of columns, the divided units are called pages, and the first page, the second page,...
However, the last page is not the same size as the other pages, but includes extra elements according to the size of the coefficient matrix. One or more pages of the coefficient matrix are read from the auxiliary storage device to the main storage device, the read data on the main storage device is processed, and the result is returned to the corresponding page of the auxiliary storage device. The storage area on the main storage device is divided into two, a reference data storage area and an update data storage area, and the data read from the auxiliary storage apparatus into the reference data storage area is subjected to update processing only with its own data. And is referred to for updating the data read into the update data area described below through the processing.
After all related data has been updated, the page is returned to the corresponding page in the auxiliary storage device. On the other hand, for the data read into the update data storage area, the data read into the reference data storage area is referred to. To update the value,
Return to the corresponding page in the auxiliary storage device. When the coefficient matrix is composed of N pages and is called a first page, a second page,...
The first page is read from the auxiliary storage device into the storage area for reference data, and the order of data exchange with the main storage device is processed. The second page, the third page,..., The Nth page are read in order from the auxiliary storage device to the update data storage area,
Process and return to the corresponding page in the auxiliary storage device. The first page is returned from the reference data storage area to the corresponding page of the auxiliary storage device. The second page is read from the auxiliary storage device to the storage area for reference data and processed. The third page, the fourth page,..., The Nth page are read in order from the auxiliary storage device to the storage area for update data, processed, and returned to the corresponding page of the auxiliary storage device. The second page is returned from the reference data storage area to the corresponding page of the auxiliary storage device. ... Read the N-1th page from the auxiliary storage device into the reference data storage area and process it. The Nth page is read from the auxiliary storage device into the update data area, processed, and returned to the corresponding page in the auxiliary storage device. The (N-1) th page is returned from the reference data storage area to the corresponding page of the auxiliary storage device. A high-speed calculation method for solving simultaneous linear equations using an auxiliary storage device. 2. When the auxiliary storage device includes an extended storage device, the process proceeds while holding, on the extended storage device, only enough pages to fit in the extended storage device in reverse order from the Nth page of the coefficient matrix. A high-speed calculation method for solving simultaneous linear equations using the auxiliary storage device according to claim 1. 3. During processing, only a small matrix corresponding to a part to be referred to / updated in the subsequent processing is extracted, and this is regarded as a new matrix, and the processing is performed. 3. The auxiliary storage device according to claim 1, wherein the coefficient matrix is represented as a product of two triangular matrices, and a solution of the simultaneous linear equation is calculated. High-speed calculation method for solving simultaneous linear equations. 4. The data processing apparatus according to claim 1, wherein the data on the auxiliary storage device is treated as a direct organization file, and a record length thereof is set to a size just enough to include an element included in one column of the coefficient matrix. A high-speed calculation method for solving a system of linear equations using the auxiliary storage device according to the first, second, and third terms.