JP2953929B2 - Modified Cholesky decomposition calculator - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a numerical calculation apparatus for performing a numerical calculation using an electronic computer, and more particularly to a modified Cholesky decomposition calculation apparatus for calculating a modified Cholesky decomposition.

[0002]

2. Description of the Related Art A method of reducing the calculation of a modified Cholesky factorization of a given symmetric matrix into the calculation of a small matrix constituting the matrix, that is, a block matrix, is called a blocking process. Is reduced to the product and difference of the blocking matrices.

[0003] The modified Cholesky method is known as a method of modifying a given matrix by modified Cholesky decomposition, and the following method is used to calculate the modified Cholesky method at high speed by an electronic computer. For details, see Jay Dongeera, "A Set of Level 3 Basic Linear Algibra Subprograms" (Reference 1: J.DO)
NGARRA `` A Set of Level 3 Basic Linear Algebra
Subprograms '' ACMTransaction on Mathematical Softw
are, Vol. 16, No. 1, March, 1990).

First, the matrices A, L, D and U are

[0005]

[0006] As shown in FIG. A11,
L11 and U11 are square matrices of the same order M, and M is hereinafter referred to as a block size. Then, A = LDU is

[0007]

Can be rewritten into the following three equations. L11,
L21, U11 and U12 can be calculated by the modified Cholesky method. Since the use of the results of this is the left-hand side of equation (3) can be calculated, which was divided again regarded as new matrix A block performing the above calculation. The calculation of the modified Cholesky decomposition by repeating this process is the modified Cholesky decomposition by block processing. In this method, the calculation on the left side of Equation (3) occupies most of the calculation amount. Therefore, the problem of calculating the modified Cholesky decomposition at high speed results in the problem of calculating the blocking matrix A22-L21D11U12 (4) at high speed. Here the A K × K matrix and the K × M matrix and U 12 of L21 and M × K matrix, K is varies in the course of calculation, the block size M does not change in the course of calculations.
The calculation speed of the modified Cholesky decomposition depends on the configuration of the central processing unit registers, the configuration of the hardware of the computer such as the capacity of the cache memory, and the given matrix A.
And the block size M. Of these, only the block size can be set artificially, but according to Literature 1, the calculation is conventionally fixed to the block size showing the highest calculation speed on average for various matrix orders. Therefore, the maximum calculation speed that can be realized by the computer used is not always obtained for any order of the matrix.

[0009]

SUMMARY OF THE INVENTION An object of the present invention is to provide a modified Cholesky decomposition calculation in which an optimum block size is automatically selected for a given matrix so that a modified Cholesky decomposition can be calculated. It is to provide a device.

[0010]

SUMMARY OF THE INVENTION The modified Cholesky of the present invention.
The decomposition calculator calculates the symmetric matrix as a lower triangular matrix L, a diagonal matrix
Using a transpose matrix U of D and L, a matrix in LDU format
For calculating the modified Cholesky factorization represented by
Wherein the computing device comprises an input device, an output device and data
The input device comprises a processing unit and a storage device.
Input device, and the output device outputs the
Output a matrix in LDU format, wherein the storage device
Record the relationship data that records the relationship between the size and the degree N of the matrix.
The block size determination data storage unit and the symmetric
From the matrix storage unit that stores the matrix and the matrix in the LDU format
The data processing unit is configured to store the pair in a matrix storage unit.
The order of the symmetric matrix with reference to the
The block size in the relational data that matches N
Means for determining a block size to be selected;
To the block size determined by the size determination means.
The symmetric matrix is divided, a modified Cholesky operation is performed, and the LD
Calculate a matrix in U format and store it in the matrix storage section in the storage section.
And a modified Cholesky decomposition calculations means for storing.

[0011]

[0012]

Next, the present invention will be described with reference to the drawings.

FIG . 1 is a block diagram showing an embodiment of the present invention.
It is. This embodiment uses an input device 1 such as a keyboard.
, A data processing device 2 and a storage device 3 for storing data
And an output device 4 such as a display.

The data processing unit 2 includes a block size determining means 21 and a modified Cholesky decomposition calculating means 22.
The storage device 3 includes a block size determination data storage unit 31 and a matrix storage unit 32.

Block size determination data storage unit 31
Stores in advance data on what block size is to be used for the fastest modified Cholesky decomposition for several orders N of several matrices. Matrix storage unit 32
Immediately after the start of the operation, the matrix A to be decomposed is stored, but its contents are updated with the operation of the data processing device 2.

Next, the operation of this embodiment will be described with reference to FIGS.

The matrix A given from the input device 1 is stored in the matrix storage unit 32, and the order N of the matrix A is given to the block size determining means 21 (step A1). The block size determining means 21 compares the lower limits L and N of the order of the matrix for performing the predetermined blocking process (step A2). If N <L, the modified Cholesky decomposition calculating means 22 calculates the modified Cholesky decomposition without using the blocking process (step A6). If N <L, the block size determination means 21 stores N in the matrix storage unit 32
(Step A3). Then, when a suitable optimal block size is found, the value is given to the modified Cholesky decomposition calculation means 22. If not found, a default block size is given to the modified Cholesky decomposition calculating means 22 (step A5). Then, the modified Cholesky decomposition calculating means 22 performs the modified Cholesky decomposition on the matrix A with the block size given by the blocking process shown in Document 1 as described in the section of the related art, and stores the result in a matrix. It is stored in the section 32 (step A4). To save storage space, the result of the modified Cholesky decomposition is overwritten on the input matrix A data. Regarding the storage method, the lower triangular part of the matrix L is stored in the lower triangular part of the matrix A, and the upper triangular part of the matrix U is stored in the upper triangular part of the matrix A excluding the diagonal elements. Finally, the output device 4 outputs the calculation result of the modified Cholesky decomposition stored in the matrix storage unit 32 (step A).
7).

[0018]

As described above, while the fixed block size is used in the calculation of the modified Cholesky decomposition by the conventional block processing, the present invention uses the block size determining means and the block size determining data. By providing the storage unit, the optimum block size is automatically selected, and the calculation speed of the modified Cholesky decomposition is improved.
Further, the present invention can be applied to an apparatus for calculation other than the modified Cholesky decomposition using the blocking process.

[Brief description of the drawings]

FIG. 1 is a block diagram of one embodiment of the present invention.

FIG. 2 is a flowchart showing the operation of the embodiment of FIG.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 Input device 2 Data processing device 3 Storage device 4 Output device 21 Block size determination means 22 Modified Cholesky decomposition calculation means 31 Block size determination data storage unit 32 Matrix storage unit

────────────────────────────────────────────────── ─── Continuation of the front page (56) References JP-A-3-229363 (JP, A) JP-A-6-75988 (JP, A) (58) Fields investigated (Int.Cl. ⁶ , DB name) G06F 17/12 G06F 17/16 JICST file (JOIS)

Claims (1)

(57) [Claims] 1. A symmetric matrix is defined as a lower triangular matrix L, a diagonal matrix D,
And L are transposed into a matrix in LDU format
Calculation device for calculating the modified Cholesky decomposition
Therefore , the computing device is referred to as an input device, an output device, and a data processing unit.
The input device inputs the symmetric matrix.
The output device is a matrix of the LDU format in the storage device
And the storage device stores the relationship between the block size and the matrix order N.
For determining the block size for storing the relation data that records
A data storage unit, the symmetric matrix and the LDU matrix
The data processing unit is configured to store the symmetric matrix in the matrix storage unit and
Refer to the relational data and agree with the degree N of the symmetric matrix.
Block for selecting the block size in the relational data
Lock size determining means, and block size determining means
The symmetric matrix to the block size determined by
Divide and perform modified Cholesky operation to obtain the line in LDU format
Fixing a column and storing it in the matrix storage unit in the storage unit
Positive Cholesky decomposition calculation means
Modified Cholesky decomposition calculator.