JP2008129861A - Numerical decomposition method in matrix - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a numerical decomposition method in a matrix of QL decomposition algorithm which reduces computational complexity and power consumption. <P>SOLUTION: In relation to QL decomposition which decomposes a matrix H input into a matrix Q and a lower triangular matrix L, the numerical decomposition method has; a process for inputting the matrix H and a square matrix G which is obtained by multiplying a complex conjugate transposed matrix H<SP>H</SP>of the matrix and the matrix H, giving each element of the matrix H to an empty matrix Q to form a matrix Q, and giving each element of a lower triangular area of the square matrix G to an empty triangular matrix L to form a lower triangular matrix L; a process to calculate each row vector of the matrix L by reading the lower triangular matrix L, calculating a submatrix comprising only non-zero elements in each row vector in units of row vector; a process to find each column vector of the matrix Q by reading the matrix Q from a memory unit and calculating each column vector; and a process to repeat the process for obtaining the matrixes L and Q in the same loop the number of times corresponding to the number of columns of the matrix H. <P>COPYRIGHT: (C)2008,JPO&INPIT

Description

  The present invention belongs to a matrix QL decomposition calculation method in the field of numerical calculation, and in any digital signal processing field (for example, digital communication, image processing, sound processing, sonar system, laser system, etc.) The present invention relates to a numerical decomposition method in a matrix applied to the decomposition into two, that is, a lower triangular matrix L.

In the field of numerical calculation, there is a calculation algorithm for decomposing a matrix H into two, a matrix Q and a lower triangular matrix L, and the calculation image is shown in FIG.
In FIG. 7, a conventional QR decomposition algorithm decomposes a matrix H into two matrices, a matrix Q and a lower triangular matrix L, based on the following equation (1). Here, the matrix H represents an R row and T column matrix, the matrix Q represents an R row and T column matrix, and the lower triangular matrix L represents a T row and T column matrix.

As shown in the above equation (1), algorithms for decomposing the matrix Q and the lower triangular matrix L include the Classical Gram-Schmidt QR decomposition method, Modified Gram-Schmidt QR decomposition method, Householder QR decomposition method, and Given QR decomposition method. Yes (for example, see Non-Patent Document 1).
Gene H. Golub, Charles F. Van Loan, Matrix Computations ThirdEdition. Hohns Hopkins published

However, the QR decomposition method shown in Non-Patent Document 1 has a problem that a large amount of processing is required because it is performed by repeating QR decomposition of a matrix in the GR algorithm.
Further, in the conventional example, when the algorithm is realized by an actual physical circuit, the circuit scale is proportional to the required processing calculation amount, so that the required circuit scale becomes large and the apparatus can be downsized. Have difficulty.
Further, in the conventional example, as described above, the power consumption is increased in proportion to the increase in the circuit scale. When used in a portable device that operates on a battery, power consumption is quick. Can not be used for a long time.

  The present invention has been made in view of such circumstances. When the input matrices G and H are known, the required processing calculation amount is reduced and the circuit scale is reduced as compared with the conventional QR decomposition algorithm. It is an object of the present invention to provide a numerical decomposition method for a matrix of QL decomposition algorithms that can reduce the size and reduce power consumption.

The numerical decomposition method for a matrix according to the present invention is a numerical decomposition method for performing QL decomposition that decomposes an input matrix H into a matrix Q and a lower triangular matrix L, and initialization means is stored in the storage unit. A matrix H to be decomposed and a square matrix G obtained by multiplying the complex conjugate transpose matrix H ^H of the matrix and the matrix H are input, and each element of the matrix H is assigned to the empty matrix Q. An initializing process of setting the provisional matrix Q and the elements of the lower triangular portion of the square matrix G to the empty triangular matrix L to form the provisional lower triangular matrix L, and a matrix L operation unit from the storage unit The temporary lower triangular matrix L is read out, and the lower lower triangular matrix L is subordinate to the row vector unit, from the lower part to the upper part, with respect to the submatrix having only non-zero elements in each row vector, the elements below the corresponding row vector Are sequentially subtracted at a predetermined ratio (for example, the first The matrix L operation process in which each row vector of the calculated matrix L is stored in the storage unit by performing the processing (3) to the processing (5) in the embodiment (3) to (5) in the embodiment, and the matrix Q operation unit stores the memory The temporary matrix Q is read from the unit, and the temporary matrix Q is subtracted from each element of the column vector in units of column vectors, and an operation of sequentially subtracting elements of a predetermined column vector is performed. The matrix Q operation process for storing the vector in the storage unit, and the operation processing unit performs the matrix L operation process and the matrix Q operation process in the same processing loop, and this processing loop is performed a number of times corresponding to the number of columns of the matrix H. Iterating (T operation loops from the process (2) to the process (11) in the first embodiment), an operation process for reading out and outputting the matrix Q and the lower triangular matrix L obtained by decomposing the matrix H from the storage unit; Characterized by having That.

The numerical decomposition method for a matrix according to the present invention is a numerical decomposition method for performing QL decomposition that decomposes an input matrix H into a matrix Q and a lower triangular matrix L, and initialization means is stored in the storage unit. A matrix H to be decomposed and a square matrix G obtained by multiplying the complex conjugate transpose matrix H ^H of the matrix and the matrix H are input, and each element of the matrix H is assigned to the empty matrix Q. An initializing process of setting the provisional matrix Q and the elements of the lower triangular portion of the square matrix G to the empty triangular matrix L to form the provisional lower triangular matrix L, and a matrix L operation unit from the storage unit The temporary lower triangular matrix L is read out, and the lower lower triangular matrix L is subordinate to the row vector unit, from the lower part to the upper part, with respect to the submatrix having only non-zero elements in each row vector, the elements below the corresponding row vector Are sequentially subtracted at a predetermined ratio (for example, the second A matrix R calculation process (processing in the second embodiment (processing in the second embodiment)) is performed by performing processing (3) to processing (5) in the embodiment (calculation of equation (3)) and storing each row vector of the calculated matrix L in the storage unit. 2) to T (operation loop of process (7)), and the matrix Q operation unit reads the provisional matrix Q from the storage unit, and the provisional matrix Q is obtained from each element of the column vector in units of column vectors. The matrix Q calculation process in which each column vector of the calculated matrix Q is stored in the storage unit by performing an operation of sequentially subtracting elements of a predetermined column vector (from processing (8) to processing (12) in the second embodiment) And T computation loops), and the computation processing unit performs the matrix L computation process and the matrix Q computation process in independent processing loops, and each processing loop is repeated a number of times corresponding to the number of columns of the matrix H. , Matrix Q which decomposes matrix H, and An arithmetic processing step of reading out and outputting the triangular matrix L from the storage unit, and the arithmetic processing unit executes the iterative loop of the matrix Q arithmetic step after the iterative loop of the matrix L arithmetic step ends. And

The numerical decomposition method for a matrix according to the present invention is a numerical decomposition method for performing QL decomposition that decomposes an input matrix H into a matrix Q and a lower triangular matrix L, and initialization means is stored in the storage unit. A matrix H to be decomposed and a square matrix G obtained by multiplying the complex conjugate transpose matrix H ^H of the matrix and the matrix H are input, and each element of the matrix H is assigned to the empty matrix Q. An initializing process of assigning each element of the upper triangular portion of the square matrix G to the empty triangular matrix U as the temporary matrix Q and setting it as the upper triangular matrix U, and a matrix L calculation unit from the storage unit to the temporary matrix An upper triangular matrix U is read out, and the upper triangular matrix U is calculated in a column vector unit, from the right end to the left end direction, with respect to a partial matrix of only non-zero elements in each column vector, and each column of the calculated matrix U is calculated. Matrix U operation process for storing vector in storage (third embodiment) And the matrix L calculation unit reads the matrix U after the predetermined number of iterations of the processing loop and reads out the matrix U and performs complex conjugate transposition processing on the matrix U. And the complex conjugate transposition process stored in the storage unit as the lower triangular matrix L, and the matrix Q calculation unit reads the temporary matrix Q from the storage unit, and the temporary matrix Q is a column vector. A matrix Q operation process in which, in each unit, an operation of sequentially subtracting an element of a predetermined column vector from each element of the column vector and storing each column vector of the calculated matrix Q in a storage unit; The matrix U computation process and the matrix Q computation process are performed in the same processing loop (the processing loop from processing (2) to processing (11) in the third embodiment), and this processing loop corresponds to the number of columns of the matrix H. Repeat as many times as Decomposed matrix Q and the lower triangular matrix L columns H, and having an arithmetic processing step of outputting from the storage unit.

The numerical decomposition method for a matrix according to the present invention is a numerical decomposition method for performing QL decomposition that decomposes an input matrix H into a matrix Q and a lower triangular matrix L, and initialization means is stored in the storage unit. A matrix H to be decomposed and a square matrix G obtained by multiplying the complex conjugate transpose matrix H ^H of the matrix and the matrix H are input, and each element of the matrix H is assigned to the empty matrix Q. An initializing process of assigning each element of the upper triangular portion of the square matrix G to the empty triangular matrix U as the temporary matrix Q and setting it as the upper triangular matrix U, and a matrix L calculation unit from the storage unit to the temporary matrix An upper triangular matrix U is read out, and the upper triangular matrix U is calculated in a column vector unit, from the right end to the left end direction, with respect to a partial matrix of only non-zero elements in each column vector, and each column of the calculated matrix U is calculated. Matrix U operation process for storing vector in storage (fourth embodiment) And the matrix L calculation unit reads the matrix U after the predetermined number of repetitions of the processing loop, and performs complex conjugate transposition processing on the matrix R. And the complex conjugate transposition process stored in the storage unit as the lower triangular matrix L, and the matrix Q calculation unit reads the temporary matrix Q from the storage unit, and the temporary matrix Q is a column vector. A matrix Q operation process in which, in each unit, an operation of sequentially subtracting an element of a predetermined column vector from each element of the column vector and storing each column vector of the calculated matrix Q in a storage unit; The matrix L operation process and the matrix Q operation process (operation loops from the processing (9) to the processing (14) in the fourth embodiment) are performed in different processing loops, and each processing loop is performed in the column of the matrix H. Number of times corresponding to the number Ri returns, decomposed matrix Q and the lower triangular matrix L and matrix H, and having an arithmetic processing step of outputting from the storage unit.

As described above, according to the algorithm of the present invention, since the entire matrix is calculated for each submatrix, the required processing amount is the conventional algorithm (Classical Gram-Schmidt QR decomposition method, Modified Gram-Schmidt QR decomposition method). , Householder QR decomposition method, Given QR decomposition method, etc.).
Further, according to the algorithm of the present invention, a circuit for the QL decomposition method can be realized on a small scale. That is, since the circuit scale is proportional to the required amount of calculation, the amount of calculation is smaller than that of the conventional example, so the required circuit scale of the matrix QL decomposition is also smaller than that of the conventional device, and is smaller than that of the conventional device. And lower manufacturing costs can be realized.

Further, according to the algorithm of the present invention, as described above, since the required calculation amount in the QL decomposition is small, the current consumption in the configured circuit is reduced as compared with the conventional circuit. The effect that the lifetime of the battery of a portable apparatus can be extended and portability can be improved is acquired.
In addition, according to the algorithm of the present invention, as described above, the circuit scale is reduced, low power consumption and low manufacturing cost are possible, and this is suitable for mass production.

Numerical decomposition method in matrix of present invention is determined by a numerical calculation of the matrix H ^H complex conjugate transpose for the matrix H of the decomposed, by multiplying the a matrix H matrix H ^H, calculates the square matrix G, this This is an algorithm that decomposes the matrix H into the matrix Q and the lower triangular matrix L using the matrix H and the square matrix G. As will be described later, a matrix G created from the matrix H (G = H ^H H + γ: γ is a parameter such as noise standard variance / signal standard variance) is input, and T times (T is the number of column vectors of the matrix H) is cyclic. (Loop calculation) generates matrices Q and L (lower triangular matrix) where H = QL.

The present invention decomposes the matrix H into two matrices Q and a lower triangular matrix L on the premise that G = H ^H H + γ is known for the matrix H in the complex domain (or real number domain).
That is, the present invention uses only the matrix H and decomposes the matrix H into two matrices, the matrix Q and the matrix L, which requires a large amount of computation, a large circuit size, a large power consumption, etc. It solves the drawbacks.
That is, the image of the calculation is performed according to the flow shown in FIG. 1, and the matrix A and the matrix G are input. As shown in the following equation (2), the matrix L is changed to the bottom row for each row. To the top row, and the matrix Q is calculated sequentially from the right side to the left side for each column. In this equation (2), q _t (1 ≦ t ≦ T) represents the t th column vector of the matrix Q, and l _t (1 ≦ t ≦ T) represents the t th row vector of the matrix L. ing.

<First Embodiment>
Hereinafter, a numerical decomposition system according to a first embodiment of the present invention will be described with reference to the drawings. FIG. 2 is a block diagram showing a configuration example of the numerical decomposition system according to the embodiment.
The numerical decomposition system according to the first embodiment includes an input unit 1, an initialization unit 4, a matrix L operation unit 5, a matrix Q operation unit 6, a storage unit 7, and an operation control unit 8.

The input unit 1 stores a known value in the matrix H of the R row and T column to be decomposed and the square matrix G of T row and T column in the storage unit 7. That is, the following description will be given on the assumption that the matrix H and the matrix G (= H ^H H) are known and these matrices are input from an external device. Further, the matrix H and the matrix G to which the input unit 1 is input may be directly output to the initialization unit 4.
The initialization unit 4 sets a temporary matrix used in the matrix L calculation unit 5 and the matrix Q calculation unit 6. That is, the initialization unit 4 assigns the matrix H to be decomposed to the empty matrix Q (the elements of the matrix H are assigned to the positions of the corresponding elements of the empty matrix Q (specified by rows and columns). Initialize the matrix L by overwriting the element data) to give the matrix Q and assigning the lower triangular matrix of the square matrix to the empty matrix L (overwriting the element data at the corresponding position) .

The matrix L calculation unit 5 reads out the row vector L (i, 1: i) of the corresponding i-th row stored in the storage unit 7 according to the set numerical value i, that is, the data of the row vector l _i , The scalar l _{k, i} ^* is calculated from the elements l _{k, i in} the k-th row and the i-th column of the matrix L, a new row vector is calculated by the following equation (3), and the position read from the storage unit 7 (Ie, address) is overwritten, and the row vector data is changed.
Here, the row vector L (i, 1: i) represents a partial matrix from the first column component to the i-th column component in the row vector of the i-th row, and the scalar l _{k, i} ^* represents the matrix L The complex conjugate of the element in the k-th row and the i-th column is shown. That is, for each column vector unit, only a partial matrix composed of non-zero element parts is calculated.
L (i, 1: i): = L (i, 1: i) −l _{k, i} ^* · L (k, 1: i) (3)

That is, the matrix L calculation unit 5 multiplies the row vector L (k, 1: i) by the scalar l _{k, i} ^* , subtracts the multiplication result from the row vector, and obtains the subtraction result as a new value. As the row vector L (i, 1: i), the storage unit 7 is overwritten at the read position.
At this time, the matrix R operation unit 4 decrements k and repeats the above expression (3) between T and i + 1, that is, in the range of i + 1 ≦ k ≦ T.

Further, the matrix L calculation unit 5 reads the finally obtained row vector L (i, 1: i) from the storage unit 7 as shown in the following equation (4) _{, and} the square root of the scalar l _{i, i} . Is divided by (l _{i, i} ) ^1/2 , and this division result is stored in the storage unit 7 as a final row vector L (i, 1: i) as an operation result. This scalar l _{i, i} indicates the i-th element and i-th element of the matrix L.
L (i, 1: i): = L (i, 1: i) / (l _{i, i} ) ^1/2 (4)

The matrix Q calculation unit 6 reads the i-th column vector Q (:, i) of the matrix Q from the storage unit 7 according to the set numerical value i, and as shown in the following equation (5), the scalar l _i, divided by _i, the division result a new column vector Q (:, i) as written in the read position in the memory unit 7. This scalar is the element in the i-th row and the i-th column in the matrix L.
Q (:, i): = Q (:, i) / l _{i, i} (5)
The matrix Q calculation unit 6 reads the column vector Q (:, j) from the storage unit 7, calculates a new column vector Q (:, j) by the following formula (6), Store in the read position. Here, the scalar l _{i, j} is an element in the i-th row and j-th column in the matrix L.
Q (:, j): = Q (:, j) -l _{i, j} · Q (:, i) (6)

That is, the matrix Q calculation unit 6 multiplies the elements of the column vector Q (:, i) by a scalar, subtracts the multiplication result from the column vector Q (:, j), and adds the subtraction result to a new value. As the column vector Q (:, j), the data is updated by writing the data in the read position of the storage unit 7.
At this time, the matrix Q calculation unit 6 sets j as an initial value i + 1, and repeats the above equation (6) between i + 1 ≦ j ≦ T while incrementing each time the process is performed.
The arithmetic control unit 8 controls the order of arithmetic processing in the matrix R arithmetic unit 5 and the matrix Q arithmetic unit 6, and controls the number of arithmetic processes in each arithmetic unit, that is, controls the arithmetic loop.

Next, the operation of the numerical decomposition system according to the first embodiment will be described with reference to FIGS. FIG. 3 is a table showing the processing of each part in the algorithm of the numerical decomposition system of the first embodiment, and the processing of (1) to (11) is sequentially performed. Here, it is assumed that the operations of the input unit 1, the complex conjugate transposed matrix generation unit 2 and the square matrix generation unit 3 are omitted, and the matrix H and the square matrix G are respectively obtained and stored in the storage unit 7.
In the process (1), the initialization unit 4 reads the matrix H to be divided from the storage unit 7, and for each empty matrix Q, each element of the matrix H is designated by the corresponding row number and column number. , I.e., the value of the matrix H is assigned to the empty matrix Q, and the empty matrix Q is set to an R row and T column matrix.
The initialization unit 4 reads the matrix G from the storage unit 7, extracts the lower triangular part from the matrix G, and writes it in the position specified by the corresponding row number and column number of the empty matrix L, that is, the empty matrix An element of the lower triangular part of the matrix G is given to L, and the matrix L is a matrix L that is a lower triangular matrix whose lower triangular part is a non-zero element.

Next, in the process (2), the operation control unit 8 extracts the number T of columns and rows of the matrix L that is a square matrix, and the number of repeated operations in the matrix L operation unit 5 and the matrix Q operation circuit 6. Is set to T, the numerical value i is initialized to “1”, and the arithmetic processing in the matrix L arithmetic unit 5 and the matrix Q arithmetic circuit 6 from processing (2) to processing (11) is performed. Repeat for ≦ T.
Here, for example, after the processing of one calculation loop is completed, the numerical value i is decremented (subtract 1) in the processing (11), and the numerical value i is in the range of 1 ≦ i ≦ T in the processing (2). If it is detected that the value is within the range, the process proceeds to the next process (3). On the other hand, the numerical value i has fallen below the numerical value 1, that is, has become 0 ( If it is out of range, the processing loop of processing (2) to processing (11) (matrix division calculation loop) is terminated, and the matrix L and matrix Q stored in the storage unit 7 at that time are stored. , Respectively, are output as a lower triangular matrix L and a matrix Q obtained by decomposing the matrix H.

Next, in the process (3), the matrix L operation unit 5 receives the numerical value i from the operation control unit 8, initializes the numerical value k to “T”, and performs the operations from the process (3) to the process (5). The loop is decremented every time the computation loop is completed, and the number k is repeated from T to i + 1 (that is, i + 1 ≦ k ≦ T). For each computation loop, (2) ), The row vectors (submatrices) of the leftmost row vector l ₁ to row vector l _T in the order of the row vector from the bottom row vector to the top row vector of the matrix, that is, from row vector l _T to row vector l _1. The row vector is calculated in order across from. Here, the row vector l ₁ is the same as L (1, 1: T).

Here, within the range of i + 1 ≦ k ≦ T, even when i = 1, k = 1 does not hold, so the row vector l ₁ in the first row remains the row vector in the first row of the matrix G. And
Therefore, after the numerical value k = 1, when the matrix L calculation unit 5 detects that the numerical value k is in the range of i + 1 ≦ k ≦ T, the process proceeds to the process (4).

Next, in the process (4), the matrix L calculation unit 5 reads the partial matrix L (i, 1: i) in the row vector l _i indicated by the numerical value i from the matrix L stored in the storage unit 7.
Then, the matrix L calculation unit 5 calculates the expression (3) for the partial matrix L (i, 1: i), writes the read position in the storage unit 7, and advances the process to the process (5). .
Next, in the process (5), the matrix L calculation unit 5 decrements the numerical value k for each calculation loop and advances the process to the process (3).

  Next, the matrix L calculation unit 5 detects whether or not the numerical value k is in the range of i + 1 ≦ k ≦ T in the process (3). 4), when it is detected that the numerical value k has fallen below the numerical value i + 1 (out of the range), the processing loop (processing loop for generating the matrix R) of processing (3) to processing (5) is performed. The process is terminated, and the process proceeds to process (6).

  Next, in the process (6), the matrix R calculation unit 5 reads the partial matrix L (i, 1: i) in the row vector corresponding to the numerical value i at this time from the matrix L stored in the storage unit 7. , (4) calculation processing is performed, the calculation result is overwritten on the read position of the storage unit 7, the data is changed, and the processing proceeds to processing (7).

Next, in the process (7), the matrix Q calculation unit 6 receives the numerical value i from the calculation control unit 8, and from the matrix Q stored in the storage unit 7, the column vector q _i corresponding to the numerical value _i (that is, Q (:, i)) is read, the processing shown in equation (5) is performed, that is _{, the} column vector q _i is divided by the scalar l _{i, i} , and the result of division is written in the read position of the storage unit 7. The data is changed, and the process proceeds to process (8).

Next, in the process (8), the matrix Q calculation unit 6 initializes the numerical value j to “1”, and advances the process to the process (8).
Hereinafter, the matrix Q computation unit 6 repeatedly performs the computation loop from the processing (8) to the processing (10) while the numerical value j is 1 ≦ j ≦ i−1, and for each computation loop, Each column vector (submatrix) of the column vector q ₁ to the row vector q _{i−1 on} the leftmost side of the equation (2) is changed in order from the rightmost column vector to the leftmost column vector, that is, the column vector q _{i. The} column vector is calculated in order from ₋₁ to the column vector q ₁ . Here, the column vector qj is the same as Q (:, j).

Next, in the process (9), the matrix Q calculation unit 6 does not execute the process of the expression (6) for Q (:, i) from the range of 1 ≦ j ≦ i−1. Only the process of (7) is performed for the i-th column of Q. That is, only the process (7) is performed on the column vector q _i of the i-th column, and the processes from the process (8) to the process (10) are not performed.
Then, the matrix Q calculation unit 6 reads the column vector q _j indicated by the numerical value j, that is, the partial matrix Q (,: ij) from the matrix Q stored in the storage unit 7.
Then, the matrix Q calculation unit 6 performs the calculation of the expression (6) on the partial matrix Q (,: i), writes it in the read position in the storage unit 7, and advances the processing to the process (10).
Next, in the process (10), the matrix Q calculation unit 6 increments the numerical value j for each calculation loop and advances the process to the process (8).

  Next, the matrix Q calculation unit 6 detects whether or not the numerical value j is within the range of 1 ≦ j ≦ i−1 in the process (8). When the processing is shifted to the processing (9), on the other hand, when it is detected that the numerical value j exceeds the numerical value i-1 (out of the range), the operation loop (generation of the matrix Q) of the processing (8) to processing (10) is performed. ) Is terminated, and the process proceeds to process (11).

<Second Embodiment>
The numerical decomposition system according to the second embodiment has the same configuration as that of the first embodiment. The difference between the second embodiment and the first embodiment is the calculation order of the matrix L calculation unit 5 and the matrix Q calculation unit 6 of the calculation control unit 8. That is, in the first embodiment, the algorithm is such that an operation loop for generating the matrix L and an operation loop for generating the matrix Q are executed in the same operation loop.

On the other hand, in the second embodiment, the calculation control unit 8 controls the calculation loop for generating the matrix L and the calculation loop for generating the matrix Q independently in order, that is, generation of the matrix L. After the operation loop is completed and the divided matrix L is obtained, the operation loop for generating the matrix Q is started to generate the matrix Q, and each of the matrix L and the matrix Q obtained by decomposing the matrix H is generated. It is an algorithm that performs separately.
Therefore, the operations of the other components (input unit 1, matrix L calculation unit 5, matrix Q calculation unit 6 and storage unit 7) other than the calculation control unit 8 and the initialization unit 4 are the same as in the second embodiment. The configuration is the same as that of the first embodiment.

Hereinafter, the operation of the numerical decomposition system according to the second embodiment will be described with reference to FIGS. 2 and 4. FIG. 4 is a table showing the processing of each part in the algorithm of the numerical decomposition system of the second embodiment, and the processing of (1) to (13) is sequentially performed. Here, it is assumed that the operations of the input unit 1, the complex conjugate transposed matrix generation unit 2 and the square matrix generation unit 3 are omitted, and the matrix H and the square matrix G are respectively obtained and stored in the storage unit 7.
In the process (1), the initialization unit 4 reads the matrix H to be divided from the storage unit 7, and for each empty matrix Q, each element of the matrix H is designated by the corresponding row number and column number. , I.e., the value of the matrix H is assigned to the empty matrix Q, and the empty matrix Q is set to an R row and T column matrix.
The initialization unit 4 reads the matrix G from the storage unit 7 and writes each element of the matrix G to the empty matrix L at the position specified by the corresponding row number and column number, that is, the empty matrix L. Is assigned a matrix G value, and the empty matrix L is a square matrix of T rows and T columns.

  Next, in the process (2), the operation control unit 8 extracts the number T of the number of columns and the number of rows of the matrix L that is a square matrix, and the number of repetition operations in the matrix L operation circuit 5 and the matrix Q operation circuit 6. Is set to T, the numerical value i is initialized to “T”, and the arithmetic processing in the matrix L arithmetic circuit 5 from processing (2) to processing (7) is performed between the numerical value i and T, that is, 1 ≦ i. Repeat for ≦ T. Here, for example, after one process is completed, the numerical value i is decremented (subtract 1) in the process (7), and the numerical value i is in the range of 1 ≦ i ≦ T in the process (2). If it is detected that the numerical value i is within the range, the process proceeds to the next process (3), while the numerical value i is less than 1 ( If it is out of range, the processing loop of processing (2) to processing (7) (the processing loop for generating the matrix L) is terminated, and the matrix L stored in the storage unit 7 at that time is replaced with the matrix L H is output as a decomposed matrix Q, and the process proceeds to process (8).

Next, the processing from the processing (3) to the processing (6) is the same as the processing (3) to the processing (6) in the first embodiment, and the description of the processing is omitted.
In the process (7), the arithmetic control unit 8 decrements the numerical value i (subtracts 1) and advances the process to the process (2).
Next, in the process (2), the arithmetic control unit 8 detects whether or not the numerical value i is within the range of 1 ≦ i ≦ T. On the other hand, when it is detected that the numerical value i is below the numerical value 1 (out of the range), it is stored in the storage unit 7 as the matrix L at this time, and the processing is advanced to processing (8). .

  Next, in process (8), the operation control unit 8 extracts the number T of the matrix Q, which is a matrix of R rows and T columns, and sets the number of repetition operations in the matrix Q operation circuit 6 to T, The numerical value i is initialized to “T”, and the calculation processing in the matrix L calculation unit 5 from processing (8) to processing (13) is performed in the range where the numerical value i is from T to 1, that is, 1 ≦ i ≦ T. Repeat. Here, for example, after the processing of one calculation loop is completed, the numerical value i is decremented (1 is subtracted) in the processing (13), and the numerical value i is in the range of 1 ≦ i ≦ T in the processing (8). If it is within the range, i.e. whether it is within the range or not, the process moves to the next process (9), while the numerical value i is below the numerical value T. When (out of range) is detected, the processing loop of processing (9) to processing (12) (the processing loop for generating the matrix Q) is terminated, and the matrix Q stored in the storage unit 7 at that time is The matrix H is output as a decomposed matrix Q, and the matrix H decomposition process ends. The processes (9), (10), (11), and (12) are the same as the processes (7), (8), (9), and (10) in the first embodiment, respectively. Is omitted.

That is, in the processing (2) to processing (7), the matrix L is converted into the row vector of the bottom row of the expression (2) in the operation loop T times (number of times corresponding to the number of rows and columns of the square matrix G). The calculation is performed in order from l _T to the row vector l ₁ in the top row.
Further, in the processing (8) to processing (13), the matrix Q is replaced by the column in the rightmost column of the expression (2) in the T times (number of times corresponding to the number of rows and columns of the square matrix G). from the vector q _T, until column vector q ₁ of the leftmost column is calculated sequentially.

<Third Embodiment>
Hereinafter, a numerical decomposition system according to a third embodiment of the present invention will be described with reference to the drawings. FIG. 5 is a block diagram showing a configuration example of the numerical decomposition system according to the embodiment. The configuration is the same as in the first embodiment, but the function of each configuration is different.
Therefore, as in the first embodiment, the numerical decomposition system according to the third embodiment includes an input unit 1, an initialization unit 4, a matrix L operation unit 5, a matrix Q operation unit 6, a storage unit 7, and an operation control unit. 8.

The input unit 1 stores a known value in the matrix H of the R row and T column to be decomposed and the square matrix G of T row and T column in the storage unit 7. That is, the following description will be given on the assumption that the matrix H and the matrix G (= H ^H H) are known and these matrices are input from an external device. Further, the matrix H and the matrix G to which the input unit 1 is input may be directly output to the initial lower part 4.
The initialization unit 4 sets a temporary matrix used in the matrix L calculation unit 5 and the matrix Q calculation unit 6. That is, the initialization unit 4 assigns a matrix H to be decomposed to the empty matrix Q (overwriting element data at the corresponding position) to form a matrix Q, and a square matrix for the empty matrix U Is initialized (overwriting element data at the corresponding position) and initialized as a matrix U.

The matrix L calculation unit 5 reads out the data of the column vector U (1; i, i) of the corresponding i-th column stored in the storage unit 7 according to the set numerical value i, and scalar u _{i, k} ^*. Then, a new column vector is calculated by the following equation (7), the position read from the storage unit 7 is overwritten, and the column vector data is changed.
Here, the column vector U (1; i, i) represents a partial matrix from the first row component to the i-th row component in the column vector of the i-th column, and the scalar u _{i, k} ^* represents the matrix U. The complex conjugate of the element in the i-th row and the k-th column is shown. Here, since L = U ^H , this corresponds to sequentially calculating the uppermost row of the row vector of the matrix L in order from the lowermost portion.
U (1: i, i): = U (1: i, i) −u _{i, k} ^* · U (1: i, k) (7)

That is, the matrix L operation unit 5 multiplies the row vector U (1: i, k) by the scalar u _{i, k} ^* and subtracts the multiplication result from the row vector U (1: i, i). Then, the subtraction result is overwritten at the read position in the storage unit 7 as a new column vector U (1: i, i).
At this time, the matrix L calculation unit 4 decrements k and repeats the above equation (7) within a range from T to i + 1, that is, between i + 1 ≦ k ≦ T.

Further, the matrix L calculation unit 5 reads the finally obtained column vector U (1: i, i) from the storage unit 7 as shown in the following equation (8), and calculates the square root of the scalars u _{i and i} . (U _{i, i} ) ^1/2 and the obtained division result is stored in the storage unit 7 as a final column vector U (1: i, i). The scalars u _{i and i} indicate i-th element and i-th element of the matrix U.
U (1: i, i): = U (1: i, i) / (u _{i, i} ) ^1/2 (8)

The matrix Q calculation unit 6 reads the i-th column vector Q (:, i) of the matrix Q from the storage unit 7 according to the set numerical value i, and represents a scalar u _i, divided by _i, the division result a new column vector Q (:, i) as written in the read position in the memory unit 7. The scalars u _{i and i} are elements in the i-th row and the i-th column in the matrix U.
Q (:, i): = Q (:, i) / u _{i, i} (9)
The matrix Q calculation unit 6 reads the column vector Q (:, j) from the storage unit 7, calculates a new column vector Q (:, j) by the following equation (10), Store in the read position. Here, the scalar u _{i, j} ^* is a complex conjugate of the element in the i-th row and the j-th column in the matrix U.
Q (:, j): = Q (:, j) −u _{i, j} ^* · Q (:, i) (10)

That is, the matrix Q calculation unit 6 multiplies the elements of the column vector Q (:, i) by a scalar u _{i, j} ^* , and subtracts the multiplication result from the column vector Q (:, j). The subtraction result is written as a new column vector Q (:, j) at the read position in the storage unit 7 and the data is updated.
At this time, the matrix Q calculation unit 6 sets j as an initial value 1 and repeats the above equation (6) between 1 ≦ j ≦ i−1 while incrementing each time the processing is performed.
The arithmetic control unit 8 controls the order of arithmetic processing in the matrix L arithmetic unit 5 and the matrix Q arithmetic unit 6, and controls the number of arithmetic processes in each arithmetic unit, that is, controls the arithmetic loop.

Next, the operation of the numerical decomposition system according to the third embodiment will be described with reference to FIGS. FIG. 5 is a table showing the processing of each part in the algorithm of the numerical decomposition system of the third embodiment, and the processes (1) to (13) are sequentially performed. Here, it is assumed that the operations of the input unit 1, the complex conjugate transposed matrix generation unit 2 and the square matrix generation unit 3 are omitted, and the matrix H and the square matrix G are respectively obtained and stored in the storage unit 7.
In the process (1), the initialization unit 4 reads the matrix H to be divided from the storage unit 7, and for each empty matrix Q, each element of the matrix H is designated by the corresponding row number and column number. , I.e., the value of the matrix H is given to the empty matrix Q, and the empty matrix Q is set to a matrix of R rows and T columns.
The initialization unit 4 reads the matrix G from the storage unit 7, extracts the upper triangular portion from the matrix G, and writes it in the position specified by the corresponding row number and column number of the empty matrix U, that is, the empty matrix The element of the upper triangular part of the matrix G is given to U, and the matrix U is an upper triangular matrix whose upper triangular part is a non-zero element.

Next, in the process (2), the operation control unit 8 extracts the number T of the number of columns and the number of rows of the matrix U which is a square matrix, and the number of repeated operations in the matrix R operation circuit 5 and the matrix Q operation circuit 6. Is set to T, the numerical value i is initialized to “T”, and the arithmetic processing in the matrix L arithmetic circuit 5 and the matrix Q arithmetic circuit 6 from processing (2) to processing (11) is performed. Up to the above range, ie, 1 ≦ i ≦ T, is repeated. Here, for example, after the processing of one calculation loop is completed, the numerical value i is decremented (subtract 1) in the processing (11), and the numerical value i is in the range of 1 ≦ i ≦ T in the processing (2). Is detected, i is 0 or not, and if it is within the range, the process proceeds to the next process (3), while the numerical value i is lower than the numerical value 1. If it is detected (out of range), the processing loop of processing (2) to processing (11) (matrix division calculation loop) is terminated, and the processing proceeds to processing (12).
Further, the matrix L calculation unit 5 calculates L = U ^H in the process (12), that is, performs complex conjugate transpose processing on the obtained upper triangular matrix U to obtain the lower triangular matrix L Is calculated and stored in the storage unit 7.
Next, the arithmetic control unit 8 outputs the matrix L and the matrix Q stored in the storage unit 7 at that time as an upper triangular matrix L and a matrix Q obtained by decomposing the matrix H, respectively.

Next, in the process (3), the matrix R calculation unit 5 receives the numerical value i from the calculation control unit 8, initializes the numerical value k to “1”, and performs the calculation from the process (3) to the process (5). The loop is repeated while the numerical value k is in the range of T to i + 1, i.e., i + 1 ≦ k ≦ T, and each column vector of the column vector u ₁ to column vector u _T of the matrix U is expressed by equation (7) for each operation loop. to determine the (submatrix), the column vector rightmost portion of the matrix in the order of the column vectors of the leftmost portion, toward the column vector u ₁ column vectors u _T, sequentially calculating a column vector. Here, a partial matrix consisting of all rows component column vector u _T is, U: is the same as (1 T, T).

Here, since the case of k = i does not hold within the range of i + 1 ≦ k ≦ T, k = 1 does not hold even when i = 1, and the column vector u _{1 in the} first column is ₁ of the matrix G. Keep the row vector of the column.
When the matrix L calculation unit 5 detects that the numerical value k is within the range of i + 1 ≦ k ≦ T, the matrix L calculation unit 5 advances the processing to the processing (4).

Next, in the process (4), the matrix L calculation unit 5 reads the partial matrix U (1: i, i) in the column vector u _i indicated by the numerical value i from the matrix U stored in the storage unit 7.
Then, the matrix L calculation unit 5 calculates the expression (7) for the partial matrix U (1: i, i), writes the read position in the storage unit 7, and advances the processing to the process (5). .
Next, in the process (5), the matrix R calculation unit 5 decrements the numerical value k for each calculation loop and advances the process to the process (3).

  Next, the matrix L calculation unit 5 detects whether or not the numerical value k is in the range of i + 1 ≦ k ≦ T in the process (3). 4) The processing is shifted to 4). On the other hand, when it is detected that the numerical value k is less than 0 (out of the range), the processing loop (processing loop for generating the matrix U) of processing (3) to processing (5) is terminated Then, the process proceeds to process (6).

  Next, in the process (6), the matrix R calculation unit 5 reads out the partial matrix U (1: i, i) in the row vector corresponding to the numerical value i at this time from the matrix U stored in the storage unit 7. , (8) calculation processing is performed, the calculation result is overwritten on the read position of the storage unit 7, the data is changed, and the processing is moved to processing (7).

Next, in the process (7), the matrix Q calculation unit 6 receives the numerical value i from the calculation control unit 8 and reads the column vector q _i corresponding to the numerical value i from the matrix Q stored in the storage unit 7. The processing shown in the equation (9) is performed, that is _{, the} column vector q _i is divided by the scalar u _{i, i} , the division result is written in the read position of the storage unit 7, the data is changed, and the processing is performed ( Go to 8). Here, the column vector q _i is equal to the submatrix Q (:, i).

Next, in the process (8), the matrix Q calculation unit 6 initializes the numerical value j to “1”, and advances the process to the process (8).
Hereinafter, the matrix Q computation unit 6 repeatedly performs the computation loop from the processing (8) to the processing (10) while the numerical value j is 1 ≦ j ≦ i−1, and for each computation loop, Each column vector (submatrix) from the column vector q _T on the rightmost side to the column vector q _{1 on} the leftmost side of the equation (2) is expressed in the order from the column vector on the rightmost side to the column vector on the leftmost side of the matrix. toward the column vector q ₁ from the _T, in order to calculate a column vector. As described above, the column vector qj is the same as Q (:, j).

Next, in the process (9), since the process of the formula (10) is performed in the range of 1 ≦ j ≦ i−1, the matrix Q calculation unit 6 does not perform the process in the case of i = 1 and performs the calculation. In the loop, only the process (7) is performed on the first column of the matrix Q. That is, the processing from the processing (8) to the processing (10) is not performed on the first column vector q1.
Then, the matrix Q calculation unit 6 reads the column vector u _j indicated by the numerical value j, that is, the partial matrix Q (,: j) from the matrix Q stored in the storage unit 7.
Then, the matrix Q calculation unit 6 performs the calculation of the expression (10) on the partial matrix Q (,: j), writes it in the read position in the storage unit 7, and advances the processing to the process (10).
Next, in the process (10), the matrix Q calculation unit 6 increments the numerical value j for each calculation loop and advances the process to the process (8).

Next, the matrix Q calculation unit 6 detects whether or not the numerical value j is within the range of 1 ≦ j ≦ i−1 in the process (8). When the processing is shifted to the processing (8), on the other hand, when it is detected that the numerical value j exceeds the numerical value i-1 (out of the range), the operation loop (generation of the matrix Q) of the processing (8) to processing (10) is performed. ) Is terminated, and the process proceeds to process (11).
Next, in the process (11), the arithmetic control unit 8 decrements the numerical value i (subtracts 1) and advances the process to the process (2).
Then, in the process (2), the arithmetic control unit 8 detects whether or not the numerical value i is within the range of 1 ≦ i ≦ T, and if it is detected that the numerical value i is within the range, the next process (3) On the other hand, when it is detected that the numerical value i is less than 1 (becomes 0) (out of the range), the processing loop of processing (2) to processing (11) (matrix division calculation loop) ) And the process proceeds to process (12).

Next, the matrix R operation unit 5 performs an operation of L = U ^H in the processing (12), that is, performs complex conjugate transposition processing on the obtained upper triangular matrix U, and obtains a matrix of the lower triangular matrix. L is calculated and stored in the storage unit 7.
After the complex conjugate transposition processing described above, the arithmetic control unit 8 reads the matrix Q and the matrix L from the storage unit 7 and outputs them as the decomposition result of the matrix H.

<Fourth Embodiment>
The numerical decomposition system according to the fourth embodiment has the same configuration as that of the third embodiment. The difference between the fourth embodiment and the third embodiment is the calculation order of the matrix L calculation unit 5 and the matrix Q calculation unit 6 of the calculation control unit 8. That is, in the third embodiment, an algorithm is executed in which an operation loop for generating the matrix L and an operation loop for generating the matrix Q are executed in the same operation loop.

On the other hand, in the fourth embodiment, the calculation control unit 8 controls the calculation loop for generating the matrix L and the calculation loop for generating the matrix Q independently in order, that is, generation of the matrix L. After the operation loop is completed and the divided matrix L is obtained, the operation loop for generating the matrix Q is started to generate the matrix Q, and each of the matrix L and the matrix Q obtained by decomposing the matrix H is generated. It is an algorithm that performs separately.
Therefore, the operations of the other constituent elements (input unit 1, initialization unit 4, and storage unit 7) other than the matrix L calculation unit 5, the matrix Q calculation unit 6, and the calculation control unit 8 are the same as in the fourth embodiment. The configuration is the same as that of the third embodiment.

Hereinafter, in the fourth embodiment of the present invention, only the configuration of functions different from the third embodiment will be described. The configuration itself of the fourth embodiment is the same as that of the third embodiment, but the function of each component is different.
The matrix L calculation unit 5 reads the data of the column vector U (1: i, i) of the corresponding i-th column stored in the storage unit 7 with the set numerical value i, and at the same time, scalar u _{i, k} ^* Then, a new column vector is calculated by the following equation (7), the position read from the storage unit 7 is overwritten, and the column vector data is changed.
Here, the column vector U (1: i, i) represents a partial matrix from the first row component to the i-th row component in the column vector of the i-th column, and the scalar u _{i, k} ^* represents the matrix U. The complex conjugate of the element in the i-th row and the k-th column is shown. Here, since L = U ^H , this corresponds to sequentially calculating the top row of the row vector of the matrix U in order from the bottom.

That is, the matrix R operation unit 5 multiplies the column vector U (1: i, i) by a scalar u _{i, k} ^* , subtracts the multiplication result from the column vector, and then adds the subtraction result to a new value. As the column vector LU (1: i, i), the storage unit 7 is overwritten at the read position.
At this time, the matrix R calculation unit 4 decrements k and repeats the above expression (7) within a range of k from T to i + 1, i.e., i + 1 ≦ k ≦ T.

Further, the matrix L calculation unit 5 reads the finally obtained column vector U (1: i, i) from the storage unit 7 as shown in the above equation (8), and uses the square root of the scalars u _{i and i} . Divide by a certain (u _{i, i} ) ^1/2 and store the obtained division result in the storage unit 7 as a final column vector U (1: i, i). The scalars u _{i and i} indicate the i-th element and the i-th element of the matrix L, respectively.

The matrix Q calculation unit 6 reads the i-th column vector Q (:, i) of the matrix Q from the storage unit 7 with the set numerical value i, and, as shown in the above equation (5), the scalar l _{i, i} The result of division is written as a new column vector Q (:, i) at the position read in the storage unit 7. The scalar l _{i, i} is an element in the i-th row and the i-th column in the matrix L.
Further, the matrix Q calculation unit 6 reads the column vector Q (:, j) from the storage unit 7, calculates a new column vector Q (:, j) by the above equation (6), and reads the storage unit 7. Remember the location Here, the scalar l _{i, j} is a complex conjugate of the element in the i-th row and the j-th column in the matrix L.
That is, the matrix Q operation unit 6 multiplies the element of the column vector Q (:, i) by the scalar l _{i, j} , and subtracts the multiplication result from the column vector Q (:, j). The result is written as a new column vector Q (:, j) at the read position in the storage unit 7 and the data is updated.
At this time, the matrix Q calculation unit 6 sets j as an initial value 1 and repeats the above equation (12) between 1 ≦ j ≦ i−1 while incrementing each time the process is performed.
The arithmetic control unit 8 controls the order of arithmetic processing in the matrix L arithmetic unit 5 and the matrix Q arithmetic unit 6, and controls the number of arithmetic processes in each arithmetic unit, that is, controls the arithmetic loop.

Next, the operation of the numerical decomposition system according to the fourth embodiment will be described with reference to FIGS. FIG. 6 is a table showing the processing of each part in the algorithm of the numerical decomposition system of the fourth embodiment, and the processing of (1) to (14) is sequentially performed. Here, it is assumed that the operations of the input unit 1, the complex conjugate transposed matrix generation unit 2 and the square matrix generation unit 3 are omitted, and the matrix H and the square matrix G are respectively obtained and stored in the storage unit 7.
Since the process (1) performed by the initialization unit 4 in the process (1) is the same as that in the third embodiment, the description thereof is omitted.

Next, in the process (2), the operation control unit 8 extracts the number T of the number of columns and the number of rows of the matrix U which is a square matrix, and the number of repeated operations in the matrix L operation circuit 5 and the matrix Q operation circuit 6. Is set to T, the numerical value i is initialized to “T”, and the arithmetic processing in the matrix L arithmetic circuit 5 and the matrix Q arithmetic circuit 6 from processing (2) to processing (7) is performed. Up to the above range, ie, 1 ≦ i ≦ T, is repeated.
Here, for example, after the processing of one calculation loop is completed, the numerical value i is decremented (subtracted from 1) in the processing (7), and the numerical value i is 1 ≦ i ≦ T in the processing (2). Whether it is within the range or 0 is detected. If it is within the range, the process proceeds to the next process (3), while the numerical value i is less than 1 ( If it is out of range), the processing loop of processing (2) to processing (7) (matrix partitioning calculation loop) is terminated, and the processing proceeds to processing (8).
Further, the matrix L operation unit 5 performs an operation of L = U ^H in the process (8), that is, performs complex conjugate transposition processing on the obtained upper triangular matrix U, and obtains a lower triangular matrix L Is calculated and stored in the storage unit 7.

Since the next processing (3) to processing (5) are the same as the processing (3) to processing (5) in the third embodiment, description thereof will be omitted.
Next, in the process (7), the arithmetic control unit 8 decrements the numerical value i (subtracts 1) and advances the process to the process (2).
Next, in the process (2), the arithmetic control unit 8 detects whether or not the numerical value i is within the range of 1 ≦ i ≦ T. On the other hand, when it is detected that the numerical value i is 0 (out of range), the matrix U is stored in the storage unit 7 as an upper triangular matrix, and the process proceeds to process (8). .

Next, in the process (8), the matrix R operation unit 5 performs an operation of L = U ^H , that is, performs a complex conjugate transposition process on the obtained upper triangular matrix U, thereby obtaining a matrix of a lower triangular matrix L is calculated and stored in the storage unit 7, and the process proceeds to process (9).

Next, in process (9), the operation control unit 8 extracts the number T of the matrix Q, which is a matrix of R rows and T columns, and sets the number of repetition operations in the matrix Q operation circuit 6 as T. The numerical value i is initialized to “T” and the arithmetic processing in the matrix Q arithmetic circuit 6 from the processing (9) to the processing (14) is performed in the range where the numerical value i is from T to 1, that is, 1 ≦ i ≦ T. Repeat for a while. Here, for example, after the processing of one calculation loop is completed, the numerical value i is decremented (subtracted from 1) in the processing (14), and the numerical value i is in the range of 1 ≦ i ≦ T in the processing (9). Is detected, that is, whether it is within the range. If it is detected that the value is within the range, the process proceeds to the next process (10), while the numerical value i is less than 1 (range Is detected), the processing loop of the processing (9) to processing (14) (the processing loop for generating the matrix Q) is terminated, and the matrix Q stored in the storage unit 7 at that time is replaced with the matrix H. Is output as a decomposed matrix Q, and the decomposition process of the matrix H is terminated. The processes (10), (11), (12), and (13) are the same as the processes (7), (8), (9), and (10) in the third embodiment, respectively. Is omitted. However, the formula used in the process (10) is the formula (5), and the formula used in the process (12) is the formula (6).

According to the first to fourth embodiments described above, there are the following effects.
-The algorithm in each embodiment of the present invention has a required processing complexity compared to the conventional algorithms (Classical Gram-Schmidt QR decomposition method, Modified Gram-Schmidt QR decomposition method, Householder QR decomposition method, Given QR decomposition method, etc.) Less.
The algorithm in each embodiment of the present invention can realize a circuit for the QL decomposition method on a small scale. That is, since the circuit scale is proportional to the required amount of calculation, the amount of calculation is smaller than that of the conventional example, so the required circuit scale of the matrix QL decomposition is also smaller than that of the conventional device, and is smaller than that of the conventional device. Can be realized.

-Since the algorithm in each embodiment of this invention can reduce a circuit scale as mentioned above, it can reduce power consumption compared with the past. In addition, since the amount of calculation is reduced, the circuit operation time can be reduced, the power consumption can be reduced as compared with the prior art, and a portable device having a QL decomposition calculation function that operates on a battery or the like. Use time can be extended.
-By each effect mentioned above, compared with a prior art example, the implementation to the apparatus of a QL decomposition | disassembly calculation function can be performed easily.
-Due to the above-described effects, miniaturization, low power consumption, low manufacturing cost, and the like are improved as compared with the conventional example, so that economical mass production can be performed.

Next, definitions of terms used in equations and matrices in the algorithm of each embodiment described above are shown below.
Since the matrix and the scalar are defined every time, they are omitted, and hereinafter, the matrix A and the element (scalar) of the matrix A will be described as a.
ai or A (i,:) ... i-th row vector of matrix A aj or A (:, j) ... j-th column vector of matrix A aij ... element of i-th row and j-th column of matrix A A ( n: m, i: j): Submatrix in the range of the n-th to m-th rows and the i-th to j-th columns of the matrix A AH: the complex conjugate transposed matrix of the matrix A AT: the transposed matrix of the matrix A ‖b ‖ ... secondary norm of vector b a * ... complex conjugate of scalar a | a | ... absolute value of scalar a a: = a ... represents update processing of a

  The program for realizing the function of the numerical decomposition system in FIG. 1 is recorded on a computer-readable recording medium, and the program recorded on the recording medium is read into the computer system and executed, thereby executing the QL of the matrix. A decomposition process may be performed. Here, the “computer system” includes an OS and hardware such as peripheral devices. The “computer system” includes a WWW system having a homepage providing environment (or display environment). The “computer-readable recording medium” refers to a storage device such as a flexible medium, a magneto-optical disk, a portable medium such as a ROM and a CD-ROM, and a hard disk incorporated in a computer system. Further, the “computer-readable recording medium” refers to a volatile memory (RAM) in a computer system that becomes a server or a client when a program is transmitted via a network such as the Internet or a communication line such as a telephone line. In addition, those holding programs for a certain period of time are also included.

  The program may be transmitted from a computer system storing the program in a storage device or the like to another computer system via a transmission medium or by a transmission wave in the transmission medium. Here, the “transmission medium” for transmitting the program refers to a medium having a function of transmitting information, such as a network (communication network) such as the Internet or a communication line (communication line) such as a telephone line. The program may be for realizing a part of the functions described above. Furthermore, what can implement | achieve the function mentioned above in combination with the program already recorded on the computer system, and what is called a difference file (difference program) may be sufficient.

It is a conceptual diagram explaining the algorithm of QL decomposition | disassembly used in the 1st-4th embodiment of this invention. It is a block diagram which shows the structural example of the numerical decomposition system by the 1st-4th embodiment of this invention. It is a table which shows the order of a process explaining the algorithm of QL decomposition | disassembly in the 1st Embodiment of this invention. It is a table which shows the order of a process explaining the algorithm of QL decomposition | disassembly in the 2nd Embodiment of this invention. It is a table which shows the order of a process explaining the algorithm of QL decomposition | disassembly in the 3rd Embodiment of this invention. It is a table which shows the order of a process explaining the algorithm of QL decomposition | disassembly in the 4th Embodiment of this invention. It is a conceptual diagram explaining the algorithm of QL decomposition | disassembly used in a prior art example.

Explanation of symbols

  DESCRIPTION OF SYMBOLS 1 ... Input part 2 ... Complex conjugate transposed matrix production | generation part 3 ... Square matrix production | generation part 4 ... Initialization part 5 ... Matrix L production | generation part 6 ... Matrix Q production | generation part 7 ... Memory | storage part 8 ... Operation control part

Claims (4)

A numerical decomposition method for performing QL decomposition that decomposes an input matrix H into a matrix Q and a lower triangular matrix L,
The initialization means inputs the matrix H to be decomposed stored in the storage unit and the square matrix G obtained by multiplying the matrix H by the complex conjugate transpose matrix H ^H of the matrix, Initialization process of assigning elements to the empty matrix Q to give a temporary matrix Q and assigning each element of the lower triangular part of the square matrix G to the empty triangular matrix L to give a temporary lower triangular matrix L When,
A matrix L calculation unit reads the temporary lower triangular matrix L from the storage unit, and the temporary lower triangular matrix L is obtained by subtracting only the non-zero elements in each row vector from the bottom to the top in units of row vectors. A matrix L operation process of sequentially subtracting elements below the corresponding row vector at a predetermined ratio, and storing each row vector of the calculated matrix L in a storage unit;
A matrix Q operation unit reads the temporary matrix Q from the storage unit, performs an operation of sequentially subtracting the elements of a predetermined column vector from each element of the column vector in units of column vectors. A matrix Q calculation process for storing each column vector of the generated matrix Q in a storage unit;
An arithmetic processing unit performs the matrix L calculation process and the matrix Q calculation process in the same processing loop, and repeats this processing loop a number of times corresponding to the number of columns of the matrix H, and the matrix Q and the lower triangular matrix obtained by decomposing the matrix H A numerical decomposition method for a matrix, comprising: an arithmetic processing step of reading L from a storage unit and outputting the same. A numerical decomposition method for performing QL decomposition that decomposes an input matrix H into a matrix Q and a lower triangular matrix L,
The initialization means inputs the matrix H to be decomposed stored in the storage unit and the square matrix G obtained by multiplying the matrix H by the complex conjugate transpose matrix H ^H of the matrix, Initialization process of assigning elements to the empty matrix Q to give a temporary matrix Q and assigning each element of the lower triangular part of the square matrix G to the empty triangular matrix L to give a temporary lower triangular matrix L When,
A matrix L calculation unit reads the temporary lower triangular matrix L from the storage unit, and the temporary lower triangular matrix L is obtained by subtracting only the non-zero elements in each row vector from the bottom to the top in units of row vectors. A matrix L operation process of sequentially subtracting elements above the corresponding row vector at a predetermined ratio, and storing each row vector of the calculated matrix L in a storage unit;
A matrix Q operation unit reads the temporary matrix Q from the storage unit, performs an operation of sequentially subtracting the elements of a predetermined column vector from each element of the column vector in units of column vectors. A matrix Q calculation process for storing each column vector of the generated matrix Q in a storage unit;
An arithmetic processing unit performs the matrix L calculation process and the matrix Q calculation process in independent processing loops, repeats each processing loop a number of times corresponding to the number of columns of the matrix H, and decomposes the matrix H into the matrix Q and A lower triangular matrix L that is read from the storage unit and output, and
A numerical decomposition method for a matrix, wherein the arithmetic processing unit executes a repetition loop of a matrix Q calculation process after a repetition loop of a matrix L calculation process is completed. A numerical decomposition method for performing QL decomposition that decomposes an input matrix H into a matrix Q and a lower triangular matrix L,
The initialization means inputs the matrix H to be decomposed stored in the storage unit and the square matrix G obtained by multiplying the matrix H by the complex conjugate transpose matrix H ^H of the matrix, An initialization process in which elements are assigned to the empty matrix Q to form a temporary matrix Q, and each element of the upper triangular portion of the square matrix G is assigned to the empty triangular matrix U to form an upper triangular matrix U;
A matrix L calculation unit reads the provisional upper triangular matrix U from the storage unit, and the upper triangular matrix U with respect to a partial matrix of only non-zero elements in each column vector from the right end to the left end in units of column vectors. A matrix U calculation process for performing calculation and storing each column vector of the calculated matrix U in a storage unit;
After the matrix L calculation unit finishes repeating the processing loop a predetermined number of times, the matrix U is read, complex conjugate transposition processing is performed on the matrix U, and the calculation result is stored as a lower triangular matrix L in the storage unit. Memorizing complex conjugate transposition process;
A matrix Q operation unit reads the temporary matrix Q from the storage unit, performs an operation of sequentially subtracting the elements of a predetermined column vector from each element of the column vector in units of column vectors. A matrix Q calculation process for storing each column vector of the generated matrix Q in a storage unit;
An arithmetic processing unit performs the matrix U calculation process and the matrix Q calculation process in the same processing loop, and repeats this processing loop a number of times corresponding to the number of columns of the matrix H, and the matrix Q and the lower triangular matrix obtained by decomposing the matrix H A numerical decomposition method for a matrix, comprising: an arithmetic processing step of reading L from a storage unit and outputting the same. A numerical decomposition method for performing QL decomposition that decomposes an input matrix H into a matrix Q and a lower triangular matrix L,
The initialization means inputs the matrix H to be decomposed stored in the storage unit and the square matrix G obtained by multiplying the matrix H by the complex conjugate transpose matrix H ^H of the matrix, An initialization process in which elements are assigned to the empty matrix Q to form a temporary matrix Q, and each element of the upper triangular portion of the square matrix G is assigned to the empty triangular matrix U to form an upper triangular matrix U;
A matrix L calculation unit reads the provisional upper triangular matrix U from the storage unit, and the upper triangular matrix U with respect to a partial matrix of only non-zero elements in each column vector from the right end to the left end in units of column vectors. A matrix U calculation process for performing calculation and storing each column vector of the calculated matrix U in a storage unit;
After the matrix L calculation unit finishes repeating the processing loop a predetermined number of times, the matrix U is read out, complex conjugate transposition processing is performed on the matrix R, and the calculation result is stored as a lower triangular matrix L in the storage unit. Memorizing complex conjugate transposition process;
A matrix Q operation unit reads the temporary matrix Q from the storage unit, performs an operation of sequentially subtracting the elements of a predetermined column vector from each element of the column vector in units of column vectors. A matrix Q calculation process for storing each column vector of the generated matrix Q in a storage unit;
An arithmetic processing unit performs the matrix L calculation process and the matrix Q calculation process in different processing loops, and repeats each processing loop a number of times corresponding to the number of columns of the matrix H, thereby decomposing the matrix H and the matrix Q and A numerical decomposition method for a matrix, comprising: an arithmetic processing step of reading out and outputting a lower triangular matrix L from a storage unit.

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