OPTIMIZATION PROBLEM SOLVING CALCULATION APPARATUS

Abstract

A calculation apparatus according to an embodiment includes matrix multiplication circuitry, time evolution circuitry, management circuitry, and output circuitry. The matrix multiplication circuitry calculates N second intermediate variables at a first time point by matrix multiplication between N (N>=2) first intermediate variables at the first time point and a preset coefficient matrix in N rows and N columns. The time evolution circuitry calculates N first variables at a second time point and N first intermediate variables at the second time point, the second time point being a time point following one sampling period after the first time point. The management circuitry increments time point from a start time point for each sampling period and controls the above circuitry to perform a process for each time point. The output circuitry outputs N first variables at a preset end time point.

Description

FIELD

Embodiments described herein relate generally to a calculation apparatus.

BACKGROUND

Various algorithms for solving optimization problems using the Ising model have been proposed. Hardware for solving optimization problems using the Ising model also has been proposed.

It is preferable that such hardware for solving optimization problems can solve problems at high speed with a simple configuration. It is also preferable that such hardware for solving optimization problems can handle more variables. It is also preferable that such hardware for solving optimization problems can flexibly deal with change in number of variables to be treated without requiring a major design change.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a configuration diagram of a calculation apparatus according to a first embodiment;

FIG. 2 is a flowchart illustrating a process flow in a computing unit;

FIG. 3 is a configuration diagram of the computing unit according to a second embodiment;

FIG. 4 is a relation diagram of variables and a coefficient matrix in the second embodiment;

FIG. 5 is a diagram illustrating a first example of the configuration of a function computing unit;

FIG. 6 is a diagram illustrating a second example of the configuration of the function computing unit;

FIG. 7 is a diagram illustrating a third example of the configuration of the function computing unit;

FIG. 8 is a diagram illustrating a fourth example of the configuration of the function computing unit;

FIG. 9 is a relation diagram of variables and a coefficient matrix in a third embodiment;

FIG. 10 is a configuration diagram of a matrix multiplication unit according to the third embodiment;

FIG. 11 is a diagram illustrating an implementation example according to the third embodiment;

FIG. 12 is a relation diagram of variables and a coefficient matrix in a fourth embodiment;

FIG. 13 is a configuration diagram of a block matrix multiplication unit according to the fourth embodiment;

FIG. 14 is a relation diagram of variables and a coefficient matrix in a fifth embodiment;

FIG. 15 is a configuration diagram of a submatrix multiplication unit according to the fifth embodiment;

FIG. 16 is a diagram illustrating a block matrix stored in a block matrix memory;

FIG. 17 is a diagram illustrating a submatrix transmitted to a multiply-accumulator;

FIG. 18 is a configuration diagram of the multiply-accumulator;

FIG. 19 is a diagram illustrating parameters and timing in the fifth embodiment;

FIG. 20 is a configuration diagram of a time evolution unit according to a sixth embodiment; and

FIG. 21 is a timing chart of a first intermediate variable and a second intermediate variable.

DETAILED DESCRIPTION

According to an embodiment, a calculation apparatus includes matrix multiplication circuitry, time evolution circuitry, management circuitry, and output circuitry. The matrix multiplication circuitry is configured to calculate N second intermediate variables at a first time point by performing matrix multiplication between N (N is an integer equal to or greater than two) first intermediate variables at the first time point and a coefficient matrix including preset coefficients in N rows and N columns. The time evolution circuitry is configured to calculate, based on the N second intermediate variables at the first time point, N first variables at a second time point and N first intermediate variables at the second time point, the second time point being a time point following one sampling period after the first time point. The management circuitry is configured to increment time point from a start time point for each sampling period and control the matrix multiplication circuitry and the time evolution circuitry to perform a process for each time point. The output circuitry is configured to output N first variables at a preset end time point.

A calculation apparatus 10 according to embodiments will be described in detail below with reference to the drawings. The calculation apparatus 10 according to embodiments aims to solve optimization problems using the Ising model.

Preconditions

First of all, preconditions for a process performed in the calculation apparatus 10 will be described.

Energy E_Ising of the Ising model is given by Eq. (1) below.

$\begin{array}{cc}{E}_{\mathrm{Ising}}=-\frac{1}{2}\sum _{i=0}^{N-1}\sum _{j=0}^{N-1}{J}_{i,j}{s}_i{s}_j+\sum _{i=0}^{N-1}{h}_i{s}_i& \left(1\right)\end{array}$

In Eq. (1), N is the number of spins; s_i is a state of the i^th spin, for example, s_i=±1; s_j is a state of the j^th spin, for example, s_j=±1; i and j are integers equal to or greater than zero and equal to or smaller than (N−1).

In Eq. (1), J_i,j is a coefficient in the i^th row and j^th column included in an N-by-N coefficient matrix J_i,j denotes the interaction between the i^th spin and the j^th spin. The coefficient matrix J is, for example, a real symmetric matrix. The real symmetric matrix is a matrix in which all diagonal entries (diagonal elements) are zero. In Eq. (1), h_i is the i^th coefficient included in N coefficient arrays and denotes the action individually acting on the i^th spin. The problem of searching for a spin state (ground state) with the minimum energy E_Ising in the Ising model is called the Ising problem. A machine solving the Ising problem may be called Ising machine. The coefficient matrices J and h are input to the Ising machine, which calculates and outputs a near optical solution of the ground state or the lower energy.

A classical model of quantum bifurcation machines (hereinafter called classical bifurcation machine) has been proposed. The classical bifurcation machine calculates the optimal solution for Eq. (1), using the equations of motion given by simultaneous ordinary differential equations of Eq.

(2), Eq. (3), and Eq. (4).

$\begin{array}{cc}\frac{{\mathrm{dx}}_j}{\mathrm{dt}}=\frac{\partial H}{\partial y_i}=y_i\left\{D+p\left(t\right)+K\left(x_i^2+y_i^2\right)\right\}-c\sum _{j=0}^{N-1}{J}_{i,j}{y}_j& \left(2\right)\end{array}$ $\begin{array}{cc}\frac{{\mathrm{dy}}_i}{\mathrm{dt}}=-\frac{\partial H}{\partial x_i}=x_i\left\{-D+p\left(t\right)-K\left(x_i^2+y_i^2\right)\right\}-{\mathrm{ch}}_{i}a\left(t\right)+c\sum _{j=0}^{N-1}{J}_{i,j}{x}_j& \left(3\right)\end{array}$ $\begin{array}{cc}H=\sum _{j=0}^{N-1}\left[\frac{D}{2}\left(x_i^2+y_i^2\right)-\frac{p\left(t\right)}{2}\left(x_i^2-y_i^2\right)+\frac{K}{4}{\left(x_i^2+y_i^2\right)}^2+{\mathrm{ch}}_i{x}_{i}a\left(t\right)-\frac{c}{2}\sum _{j=1}^N{J}_{i,j}\left(x_i{x}_j+y_i{y}_j\right)\right]& \left(4\right)\end{array}$

In Eq. (2), Eq. (3), and Eq. (4), N is the number of mass points, an integer equal to or greater than two, and N corresponds to the number of spins; x₁ is a real number denoting the position of the i^th mass point; y_i is a real number denoting the kinetic momentum of the i^th mass point; and i and j are integers from zero to (N−1).

In Eq. (2), Eq. (3), and Eq. (4), J_i,j is a coefficient in the i^th row and j^th column included in a coefficient matrix J including predetermined coefficients in N rows and N columns. The coefficient matrix J is, for example, a real symmetric matrix. In the equations, h_i is the i^th coefficient included in predetermined N coefficient arrays. In Eq. (2), Eq. (3), and Eq. (4), the term h_i may be eliminated.

In Eq. (2), Eq. (3), and Eq. (4), D is, for example, a constant corresponding to detune; c is a constant; and K is, for example, a constant corresponding to the Kerr coefficient. For example, D, c, and K are predetermined.

In Eq. (2), Eq. (3), and Eq. (4), t is a time point; p(t) is, for example, a function of t, a pump rate; a(t) is a function of t, and a(t) is given, for example, by Eq. (5) below.

α(t)=√{square root over (p(t)/K)}  (5)

The classical bifurcation machine updates x₁ and y₁ using Eq. (2) and Eq. (3), every t, by incrementing t by a short time from zero to a sufficiently large value. The classical bifurcation machine then outputs a sign (±1) of the final value of x₁ when t is sufficiently large, as the optimal solution for s_i. In this way, the classical bifurcation machine considers the Ising model as Hamiltonian mechanics, where H in Eq. (2), Eq. (3), and Eq. (4) is Hamiltonian.

Simulated annealing is known as a method of calculating the optimal solution to the Ising model. In this method, a sequential update algorithm is employed. The sequential update algorithm selects a plurality of spins one by one and successively updates them. This sequential update algorithm is not suitable for parallel calculations and hardly achieves higher speed.

In this respect, another possible algorithm is to solve the equation of motion in the classical bifurcation machine using a discrete solution method with a digital calculator. Unlike simulated annealing, this algorithm can simultaneously update a plurality of variables.

However, the classical bifurcation machine has to perform matrix multiplication using the coefficient matrix J with greatest computational complexity in order to calculate x_i and y_i. Moreover, the classical bifurcation machine has to solve the equations of motion given by Eq. (2), Eq. (3), and Eq. (4) by performing the discrete solution method (for example, the fourth-order Runge-Kutta method) with high computational cost. Thus, the classical bifurcation machine suffers from enormous computational complexity.

By contrast, the calculation apparatus 10 according to embodiments calculates the optimal solution of Eq. (1), using the equations of motion given by new simultaneous ordinary differential equations in Eq. (6), Eq. (7), and Eq. (8) below.

$\begin{array}{cc}\frac{{\mathrm{dx}}_i}{\mathrm{dt}}=\frac{\partial H^{\prime }}{\partial y_i}={\mathrm{Dy}}_i& \left(6\right)\end{array}$ $\begin{array}{cc}\frac{{\mathrm{dy}}_i}{\mathrm{dt}}=-\frac{\partial H^{\prime }}{\partial x_i}=\left\{-D+p\left(t\right)-{\mathrm{Kx}}_i^2\right\}{x}_i-{\mathrm{ch}}_{i}a\left(t\right)+c\sum _{j=0}^{N-1}{J}_{i,j}{x}_j& \left(7\right)\end{array}$ $\begin{array}{cc}{H}^{\prime }=\sum _{i=0}^{N-1}\left[\frac{D}{2}\left(x_i^2+y_i^2\right)-\frac{p\left(t\right)}{2}{x}_i^2+\frac{K}{4}{x}_i^4+{\mathrm{ch}}_i{x}_{i}a\left(t\right)-\frac{c}{2}\sum _{j=0}^{N-1}{J}_{i,j}{x}_i{x}_j\right]& \left(8\right)\end{array}$

In Eq. (6), Eq. (7), and Eq. (8), N, x_i, y_i, i, j, J_i, j, h_i, D, c, K, t, p(t), and a(t) are the same as in Eq. (2) to Eq. (4).

The calculation apparatus 10 according to embodiments updates x₁ and y₁ using Eq. (6) and Eq. (7), every t, by incrementing t by a short time from zero to a sufficiently large value. The calculation apparatus 10 according to embodiments then outputs a sign (±1) of the final value of x_i when t is sufficiently large, as the optimal solution for s_i. In this way, the calculation apparatus 10 according to embodiments solves the Ising problem by simulating time evolution of Hamiltonian mechanics, where H in Eq. (8) is Hamiltonian.

In Eq. (6), Eq. (7), and Eq. (8), the matrix multiplication for the coefficient matrix J with greatest computational complexity is included in Eq. (7) but not included in Eq. (6). Therefore, the calculation apparatus according to embodiments has to perform matrix multiplication for the coefficient matrix J with greatest computational complexity only for updating y_i and does not have to perform for updating x_i. In Eq. (6) for calculating the time derivative value of x_i (dx_i/dt), p(t) is erased. Therefore, the calculation apparatus 10 according to embodiments can calculate the optimal solution to the Ising model with small computational complexity.

Eq. (6) for calculating the time derivative value of x_i (dx_i/dt) includes y_i but does not include x_i. Eq. (7) for calculating the time derivative value of y_i (dy_i/dt) includes x₁ but does not include y_i.

In other words, when Eq. (6) and Eq. (7) are used, x_i and y_i are separate from each other in Hamiltonian. Therefore, the calculation apparatus 10 according to embodiments can update x_i and y_i by employing a discrete solution method that is stable with small computational complexity. For example, the calculation apparatus 10 according to embodiments updates x_i and y_i, for example, using the symplectic Euler method. Therefore, the calculation apparatus 10 according to embodiments can calculate the optimal solution to an optimization problem using the Ising model, with simple computation and a simple configuration.

The calculation apparatus 10 according to embodiments may calculate the optimal solution for Eq. (1), using the equations of motion given by new simultaneous ordinary differential equations in Eq. (9) and Eq. (10).

$\begin{array}{cc}\frac{{\mathrm{dx}}_i}{\mathrm{dt}}={\mathrm{Dy}}_i& \left(9\right)\end{array}$ $\begin{array}{cc}\frac{{\mathrm{dy}}_i}{\mathrm{dt}}=\left\{\left[-D+p\left(t\right)\right]\left(1+x_i^n\right)-{\mathrm{Kx}}_i^{n+2}\right\}{x}_i-{\mathrm{ch}}_{i}a\left(t\right)+c\sum _{j=0}^{N-1}{J}_{i,j}{x}_j& \left(10\right)\end{array}$

In Eq. (9) and Eq. (10), N, x_i, y_i, i, j, h_i, D, c, K, t, p(t), and a(t) are the same as in Eq. (2) to Eq. (4). In Eq. (9) and Eq. (10), n is an even number equal to or greater than two.

In this case, the calculation apparatus 10 according to embodiments updates x_i and y_i using Eq. (9) and Eq. (10), every t, by incrementing t by a short time from zero to a sufficiently large value. The calculation apparatus 10 according to embodiments then outputs a sign (±1) of the final value of x_i when t is sufficiently large, as the optimal solution for s_i.

Here, the matrix multiplication for the coefficient matrix J with greatest computational complexity is included in Eq. (9) but not included in Eq. (10). Therefore, also when Eq. (9) and Eq. (10) are used, the calculation apparatus 10 according to embodiments performs matrix multiplication for the coefficient matrix J with greatest computational complexity only for updating y_i and does not have to perform for updating h_i In Eq. (9) for calculating the time derivative value of x_i (dx_i/dt), p(t) is erased. Therefore, also when Eq. (9) and Eq. (10) are used, the calculation apparatus 10 according to embodiments can calculate the optimal solution to an optimization problem using the Ising model with small computational complexity.

Eq. (9) for calculating the time derivative value of x_i (dx_i/dt) includes y_i but does not include x_i. Eq. (10) for calculating the time derivative value of y_i (dy_i/dt) includes x_i but does not include y_i.

That is, also when Eq. (9) and Eq. (10) are used, x_i and y_i are separate from each other in Hamiltonian. Therefore, also when Eq. (9) and Eq. (10) are used, the calculation apparatus 10 according to embodiments can calculate the optimal solution to an optimization problem using the Ising model with simple computation and a simple configuration, in the same manner as when Eq. (6) and Eq. (7) are used.

First Embodiment

The calculation apparatus 10 according to a first embodiment will be described. In the description of embodiments, the device, block, or circuitry having substantially the same function and configuration as the device, block, and circuitry that have been described before are denoted by the same reference signs and will not be further elaborated except for differences.

FIG. 1 is a diagram illustrating a configuration of the calculation apparatus 10 according to the first embodiment. The calculation apparatus 10 is a device that solves an optimization problem using Ising model, using the simultaneous ordinary differential equations in Eq. (7) and Eq. (8) or using the simultaneous ordinary differential equations in Eq. (9) and Eq. (10).

The calculation apparatus 10 includes a computing unit 20, an input unit 22, an output unit 24, and a setting unit 26.

The computing unit 20 is, for example, an arithmetic processing unit including one or more processors such as central processing units (CPU), each including circuitry, and a memory. The computing unit 20 may include circuitry disclosed in the second and following embodiments.

The computing unit 20 increments t, which is a parameter representing a sampling time point, by a short time (dt) from the start time point (for example, 0). The computing unit 20 updates N first variables x_i and N second variables y_i, for each sampling time point, using the simultaneous ordinary differential equations in Eq. (6) and Eq. (7) or the simultaneous ordinary differential equations in Eq. (9) and Eq. (10). The computing unit 20 then outputs N first variables x_i at the end time point T which is a predetermined sampling time point.

The input unit 22 acquires N first variables x_i and N second variables y_i at the start time point and provides the acquired variables to the computing unit 20, prior to the computation process by the computing unit 20. N first variables x_i and N second variables y_i at the start time point may be, for example, values generated by random numbers or may be preset values (for example, all zero or all predetermined values).

The output unit 24 acquires N first variables x_i at the end time point T after the end of the computation process by the computing unit 20. The output unit 24 then outputs a value representing the sign (for example, +1, −1) of N first variables x_i at the end time point T, as the optimal solution of a combination of N spin states in the Ising model.

The setting unit 26 sets the parameters used in the simultaneous ordinary differential equations in Eq. (6) and Eq. (7) and the simultaneous ordinary differential equations in Eq. (9) and Eq. (10) for the computing unit 20, prior to the computation process by the computing unit 20. More specifically, the setting unit 26 sets N, J, h, D, c, K, p(t), and a(t), where

N is an integer equal to or greater than two, representing the number of the first variables and the second variables;

J is an N-by-N coefficient matrix, J_i,j is a coefficient in the i^th row and j^th column included in the coefficient matrix J;

• • h is a coefficient array including N coefficients, h_i is the i^th coefficient in the coefficient array h;
  • D, c, and K are constants;
  • p(t) is a function of the sampling time point; and
  • a(t) is a function of the sampling time point.

The setting unit 26 may further set dt, T, and M, where

• • dt is a constant representing a sampling period (short time);
  • T is a constant representing a sampling time point corresponding to the end time point; and
  • M is an integer equal to or greater than one, representing the number of iterations of computation in Eq. (6) and Eq. (7) or the number of iterations of computation in Eq. (9) and Eq. (10).

The setting unit 26 may optionally change some of these parameters in accordance with the user operation. Alternatively, the setting unit 26 may set these parameters as fixed values rather than changing them for each computation.

FIG. 2 is a flowchart illustrating a process flow in the computing unit 20.

At S11, the computing unit 20 initializes t, p, and a. For example, the computing unit 20 sets all of t, p, and a to zero, where t is a parameter representing a sampling time point, p is a parameter representing the value of p(t) at the time point t, and a is a parameter representing the value of a(t) at the time point t.

Subsequently, the computing unit 20 repeats the process from S13 to S20 until t reaches an end time point T which is preset (the loop process between S12 and S21). When a(t) is an increasing function, the computing unit 20 may repeat the process from S13 to S20 until a reaches a predetermined value or greater.

At S13, the computing unit 20 calculates N first intermediate variables x′_i by multiplying each of N first variables x_i by the sampling period (short time) dt and a preset coefficient c. That is, at S13, the computing unit performs computation of Eq. (21) below.

x′ _i =dt×c×× _i  (21)

Subsequently, at S14, the computing unit 20 calculates N second intermediate variables b₁ by performing matrix multiplication between N first intermediate variables x′_i and the coefficient matrix J including preset coefficients in N rows and N columns. That is, at S14, the computing unit 20 performs computation of Eq. (22) below.

$\begin{array}{cc}{b}_i=\sum _{j=0}^{N-1}{J}_{\mathrm{ij}}\left(\mathrm{dt}\times c\times x_j\right)=\sum _{j=0}^{N-1}{J}_{\mathrm{ij}}{x}_j^{\prime }& \left(22\right)\end{array}$

The computing unit 20 may perform the process at S13 after performing the process at S14. In this case, at S14, the computing unit 20 calculates N values by performing matrix multiplication between N first variables x_i and the coefficient matrix J. Subsequently, at S13, the computing unit 20 calculates N second intermediate variables b₁ by multiplying each of N values calculated at S14 by (dt×c).

Subsequently, at S15, the computing unit 20 updates N second variables y_i by adding the corresponding second intermediate variable b₁ to each of N second variables y_i. That is, at S15, the computing unit 20 performs computation of Eq. (23) below.

y _i +=b _i  (23)

Subsequently, the computing unit 20 performs the process from S17 to S18 iteratively M times (the loop process between S16 to S19). M is an integer equal to or greater than one.

At S17, the computing unit 20 updates the second variable y_i by performing computation according to Eq. (7) or Eq. (10). For example, at S17, the computing unit 20 performs computation of Eq. (24) below when performing computation according to Eq. (7).

y _i +=dt′×[(−D+p−Kx_i²)x_i−c×h_i×a]  (24)

For example, at S17, the computing unit 20 performs computation of Eq. (25) below when performing computation according to Eq. (10). In Eq. (25), n is an even number equal to or greater than two.

y _i +=dt′×{[(−D+p)(1+x_iⁿ)−Kx_iⁿ⁺²]×_i−c×h_i×a}  (25)

Subsequently, at S18, the computing unit 20 updates the first variable x_i by performing computation according to Eq. (6) or Eq. (9). Eq. (6) and Eq. (9) are the same equations. For example, at S18, the computing unit 20 performs computation of Eq. (26) below when performing computation according to Eq. (6) or Eq. (9).

x _i +=dt′×D×y _i  (26)

The loop process from S16 to S19 corresponds to iterative computation in the symplectic Euler method. The computing unit 20 may perform the process at S17 and the process at S18 in reverse. That is, the computing unit 20 may update the second variable y_i after updating the first variable h_i.

After performing the loop process from S16 to S19, the computing unit 20 proceeds to S20. At S20, the computing unit 20 updates t by adding dt to t. The computing unit 20 further updates p and a. For example, the computing unit updates p by adding a preset dp to p. The computing unit 20 also updates a by computing the square root of the updated p.

After performing the process at S20, the computing unit 20 determines whether t is equal to or greater than T. If t is smaller than T, the computing unit 20 returns to S13 and repeats the process from S13. If t is equal to or greater than T, the computing unit 20 terminates this flow.

By performing the process as described above, the calculation apparatus 10 according to the first embodiment can solve an optimization problem, using the simultaneous ordinary differential equations in Eq. (6) and Eq. (7) or the simultaneous ordinary differential equations in Eq. (9) and Eq. (10). The calculation apparatus 10 according to the first embodiment can update the first variable x_i and the second variable y_i at high speed with simple computation or a simple configuration. The calculation apparatus 10 according to the first embodiment therefore can calculate the optimal solution to an optimization problem at a small cost and at high speed.

Second Embodiment

The calculation apparatus 10 according to a second embodiment will be described.

FIG. 3 is a diagram illustrating a configuration of the computing unit 20 according to the second embodiment. In the second embodiment, the calculation apparatus 10 includes circuitry implemented by one or more semiconductor devices. The calculation apparatus 10 may be, for example, a field programmable gate array (FPGA), a gate array, or an application specific integrated circuit (ASIC). The calculation apparatus 10 may partially include a processor.

The computing unit 20 according to the second embodiment includes a matrix multiplication unit 28, a time evolution unit 30, and a management unit 32.

The matrix multiplication unit 28 acquires N first intermediate variables x′_i(t₁) at a first time point t₁ representing any sampling time point from the time evolution unit 30. The matrix multiplication unit 28 calculates N second intermediate variables b_i(t₁) at the first time point t₁ by performing matrix multiplication between N first intermediate variables x′_i(t₁) at the first time point t₁ and the coefficient matrix J.

For example, the matrix multiplication unit 28 includes a coefficient matrix memory 36 and a matrix multiplication executor 38. The coefficient matrix memory 36 stores the coefficient matrix J. The matrix multiplication executor 38 performs matrix multiplication between N first intermediate variables x_i(t₁) at the first time point t₁ and the coefficient matrix J.

The time evolution unit 30 acquires N second intermediate variables b_i(t₁) at the first time point t₁ from the matrix multiplication unit 28. The time evolution unit 30 calculates N first variables x_i(t₂) at a second time point t₂ representing a sampling time point following one sampling period after the first time point t₁, N second variables y_i(t₂) at the second time point t₂, and N first intermediate variables x′_i(t₂) at the second time point t₂, based on N second intermediate variables b_i(t₁) at the first time point t₁.

The management unit 32 increments the sampling time point for each sampling period after the start time point. The management unit 32 then controls the matrix multiplication unit 28 and the time evolution unit 30 to perform the process for each sampling time point.

More specifically, the management unit 32, for example, controls the matrix multiplication unit 28 to calculate N second intermediate variables b_i(t₁) at the first time point t₁. Subsequently, the management unit 32 controls the time evolution unit 30 to calculate N first variables x_i(t₂), N second variables y_i(t₂), and N first intermediate variables x′_i(t₂) at the second time point t₂. Subsequently, the management unit 32 controls the matrix multiplication unit 28 to calculate N second intermediate variables b_i(t₂) at the second time point t₂. Subsequently, the management unit 32 controls the time evolution unit 30 to calculate N first variables x_i(t₃), N second variables y_i(t₃), and N first intermediate variables xvi(t₃) at a third time point t₃ representing a sampling time point following one sampling period after the second time point t₂. In this way, the management unit 32 alternately controls the matrix multiplication unit 28 and the time evolution unit 30 to perform the process while incrementing the sampling time point.

N first variables x_i (t₀) and N second variables y_i(t₀) at the start time point (for example, t₀) are provided in advance by the input unit 22, for example, prior to the computation process.

For example, the time evolution unit 30 includes a first variable memory 40, a second variable memory 42, a first addition unit 44, a function computing unit 46, a first multiplication unit 48, and a first intermediate variable memory 50.

The first variable memory 40 stores N first variables x_i(t₁) at the first time point t₁. The second variable memory 42 stores N second variables y_i(t₁) at the first time point t₁.

The first addition unit 44 acquires N second intermediate variables b_i(t₁) at the first time point t₁ calculated by the matrix multiplication unit 28. The first addition unit 44 updates N second variables y_i(t₂) at the first time point t₁ by adding N second intermediate variables b_i(t₁) at the first time point t₁ to N second variables y_i(t₁) at the first time point t₁ stored in the second variable memory 42. For example, the first addition unit 44 updates N second variables y_i(t₁) in the order of index from the second variable y₀(t₁) with an initial index (i=0) to the second variable y_N-1(T₁) with the last index (i=N−1).

In embodiments, adding N first values and N second values refer to adding values with the same index to generate N third values.

The function computing unit 46 calculates N first variables x_i(t₂) at the second time point t₂ and N second variables y_i(t₂) at the second time point t₂, based on N first variables x_i(t₁) at the first time point t₁ stored in the first variable memory 40 and the updated N second variables y_i(t₁) at the first time point t₁ calculated by the first addition unit 44. For example, the function computing unit 46 calculates N first variables x_i(t₂) and N second variables y_i(t₂) in the order of index from the first variable x₀(t₂) and the second variable y₀(t₂) with the initial index (i=0) to the first variable x_N-1(t₂) and the second variable y_N-1(t₂) with the last index (i=N−1).

The function computing unit 46 writes N first variables x_i(t₂) at the second time point t₂ into the first variable memory 40. For example, the function computing unit 46 writes each of N first variables x_i(t₂) at the second time point t₂ into the first variable memory 40 in order from the first variable x₀(t₂) with the initial index (i=0). The first variable memory 40 is, for example, a dual port memory and can write data into an address while reading data at another address. When the first variable memory 40 is a dual port memory, the function computing unit 46 can write N first variables x_i(t₂) at the second time point t₂ onto the address at which N first variables x_i (t₁) at the first time point t₁ are stored.

The function computing unit 46 writes N second variables y_i(t₂) at the second time point t₂ into the second variable memory 42. For example, the function computing unit 46 writes each of N second variables y_i(t₂) at the second time point t₂ into the second variable memory 42, in order from the second variable y₀(t₂) with the initial index. The second variable memory 42 is, for example, a dual port memory. When the second variable memory 42 is a dual port memory, the function computing unit 46 can write N second variables y_i(t₂) at the second time point t₂ onto the address at which N second variables y_i(t₁) at the first time point t₁ are stored.

The first multiplication unit 48 calculates N first intermediate variables x′_i(t₂) at the second time point t₂ by multiplying each of N first variables x_i(t₂) at the second time point t₂ by a preset value (in the present embodiment, (dt×c)). The first intermediate variable memory 50 stores N first intermediate variables x′_i(t₂) at the second time point t₂ calculated by the first multiplication unit 48. N first intermediate variables x′_i(t₂) at the second time point t₂ are temporarily stored in the first intermediate variable memory 50 and then transmitted to the matrix multiplication unit 28. The first intermediate variable memory 50 holds each of N first intermediate variables x′_i(t₂) at the second time point t₂ for a period from when it is generated by the first multiplication unit 48 to when it is transmitted to the matrix multiplication unit 28. The first intermediate variable memory 50 may include, for example, a first-in first-out (FIFO) memory.

In such a configuration, the matrix multiplication unit 28 performs the process at S14 described in the first embodiment. The first addition unit 44 performs the process at S15. The function computing unit 46 performs the process from S16 to S19. The first multiplication unit 48 performs the process at S13. The management unit 32 performs management in S11, S20, and the loop process between S12 to S21. The computing unit 20 according to the second embodiment therefore can calculate the optimal solution to an optimization problem at a small cost at high speed, as in the first embodiment.

The first multiplication unit 48 may be provided at a stage before the first addition unit 44, instead of the stage before the first intermediate variable memory 50. This change in arrangement corresponds to the process in which S13 and S14 are reversed in the first embodiment. However, when the value (in the present embodiment (dt x c)) by which each of N first variables x_i(t₂) at the second time point t₂ is multiplied is smaller than one, it is preferable that the first multiplication unit 48 is provided at a stage before the first intermediate variable memory 50. The first multiplication unit 48 thus can reduce the number of digits of N first intermediate variables x′_i(t₂) at the second time point t₂, thereby reducing the probability of overflow in the matrix multiplication unit 28.

When the first multiplication unit 48 is provided at a stage before the first addition unit 44, the first intermediate variable memory 50 stores N first variables x_i(t₂) at the second time point t₂ as N first intermediate variables x′_i(t₂) at the second time point t₂. In this case, the first multiplication unit 48 multiplies each of N second intermediate variables b_i(t₁) at the first time point t₁ output from the matrix multiplication unit 28 by a preset value (dt×c). Then, in this case, the first addition unit 44 updates N second variables y_i(t₁) at the first time point t₁ by adding N second intermediate variables b_i(t₁) at the first time point t₁ multiplied by a preset value (dt×c) by the first multiplication unit 48 to N second variables y₁(t₁) at the first time point t₁.

The time evolution unit 30 may output N first intermediate variables x′_i(t₁) at the first time point t₁ as a first intermediate stream X′ including a first number (the first number is an integer equal to or greater than one) of first intermediate variables x′_i(t₁) in one clock cycle to the matrix multiplication unit 28. The matrix multiplication unit 28 may output N second intermediate variables b_i(t₁) at the first time point t₁ as a second intermediate stream B including a second number (the second number is an integer equal to or greater than one) of second intermediate variables b_i(t₁) in one clock cycle to the time evolution unit 30.

Here, a stream refers to time-series data. More specifically, a stream refers to a data row including P data sets in each clock cycle, where N pieces of data are divided into, for example, P data sets (P is an integer equal to or greater than one).

When acquiring such a stream, the matrix multiplication unit 28 and the time evolution unit 30 perform the process in order from data acquired earlier. The matrix multiplication unit 28 and the time evolution unit 30 thus can start the process before acquisition of all the data included in a stream is completed. When calculation of an initial data set of N pieces of data is completed, the matrix multiplication unit 28 and the time evolution unit 30 start successively transmitting the calculated data set. The matrix multiplication unit 28 and the time evolution unit 30 thus can control the next unit to start a process before calculation of all the data is completed.

FIG. 4 is a diagram illustrating the relation between N first intermediate variables x′_i, the coefficient matrix J, and N second intermediate variables b₁ in the second embodiment. The matrix multiplication unit 28 acquires N first intermediate variables (x′₀, x′₁, x′₂, . . . , x′_i, x′_N-1) for each sampling time point. The matrix multiplication unit 28 stores the coefficient matrix (J_0,0, J_0,1, J_0,2, . . . , J_i,j, . . . , J_N-1,N-1) including coefficients in N rows and N columns.

The matrix multiplication unit 28 then calculates N second intermediate variables (b₀, b₁, b₂, . . . , b_i, . . . , b_{N 1}) by performing matrix multiplication between N first intermediate variables (x′₀, x′₁, x′₂, . . . , x′_i, . . . , x′_N-1) and the coefficient matrix (J_0,0, J_0,1, J_0,2, . . . , J_i,j, . . . , J_N-1,N-1).

In the second embodiment, the matrix multiplication unit 28 may perform matrix multiplication by any process. In the second embodiment, the matrix multiplication unit 28 may include, for example, a processor in the inside to perform matrix multiplication under instructions of a program.

FIG. 5 is a diagram illustrating a first example of the configuration of the function computing unit 46. The function computing unit 46 according to the first example includes a first FX computing unit 51-1, a first FX addition unit 52-1, a first FY computing unit 53-1, and a first FY addition unit 54-1.

The first FX computing unit 51-1 calculates N second derivative values by performing a first function operation (FX(x_i)) for each of N first variables at a first time point. The first FX addition unit 52-1 calculates N second update values by adding N second derivative values calculated by the first FX computing unit 51-1 and N updated second variables calculated by the first addition unit 44.

The first FY computing unit 53-1 calculates N first derivative values by performing a second function operation (FY(y_i)) for each of N second update values calculated by the first FX addition unit 52-1. The first FY addition unit 54-1 calculates N first update values by adding N first derivative values calculated by the first FY computing unit 53-1 and N first variables at the first time point.

The function computing unit 46 then provides N first update values calculated by the first FY addition unit 54-1 to the first variable memory 40. The first variable memory stores N first update values calculated by the first FY addition unit 54-1 as N first variables at a second time point.

The function computing unit 46 also provides N second update values calculated by the first FX addition unit 52-1 to the second variable memory 42. The second variable memory 42 stores N second update values calculated by the first FX addition unit 52-1 as N second variables at the second time point.

Here, the first function operation is computation of Eq. (31) below.

FX(x_i)=dt′×[(−D+p−Kx_i²)x_i−c×h_i×a]  (31)

The second function is computation of Eq. (32) below.

FY(y_i)=dt′×D×y_i  (32)

The first function operation may be computation of Eq. (33) below.

FX(x_i)=dt′×55[(−D+p)(1+x_iⁿ)−Kx_iⁿ⁺²]x_i−c×h_i×a}  (33)

In Eqs. (31), (32), and (33), x_i is the i^th first variable of N first variables at the first time point or the i^th first update values of N first update values; and y_i is the i^th second variable of the updated N second variables calculated by the first addition unit 44 or the i^th second update value of N second update values.

dt′ is a preset short time. D, c, and K are preset constants. h_i is a coefficient set for each i. p and a are values increasing at each sampling time point according to a predetermined operation expression.

The function computing unit 46 having such a configuration in the first example can execute the loop process from S16 to S19 in the first embodiment once in the order of S17 to S18. The function computing unit 46 thus can time-evolve N first variables and N second variables at the first time point using the method called symplectic Euler method to calculate N first variables and N second variables at the second time point.

Further, the function computing unit 46 according to the first example may implement the computation by pipeline processing. When pipeline processing is performed, for example, the function computing unit 46 receives one variable pair including the first variable X_IN and the second variable Y_IN in the order of index for each clock cycle. The function computing unit 46 then performs computation for the received variable pair to calculate one variable pair including the first variable X_OUT and the second variable Your computed.

Specifically, when pipeline processing is performed, the function computing unit 46 further includes two stages of X transfer registers 55-1 to 55-2, two stages of Y transfer registers 56-1 to 56-2, one X output register 57, and one Y output register 58.

For each clock cycle, the X transfer register 55-1 on the first stage acquires one first variable X_TN from among N first variables at the first time point from the first variable memory 40 and holds the acquired one first variable X_IN for one clock cycle period. For each clock cycle, the Y transfer register 56-1 on the first stage acquires one second variable Y_IN from among the updated N second variables calculated by the first addition unit 44 and holds the acquired one second variable Y_IN for one clock cycle period. The X transfer register 55-1 on the first stage and the Y transfer register 56-1 on the first stage acquire one first variable and one second variable with the same index i in the same clock cycle.

For each clock cycle, the first FX computing unit 51-1 calculates a second derivative value by performing the first function operation for the first variable X_IN stored in the X transfer register 55-1 on the first stage. For each clock cycle, the first FX addition unit 52-1 calculates the second update value Y₁ by adding the second derivative value calculated by the first FX computing unit 51-1 in the clock cycle and the second variable Y_IN stored in the Y transfer register 56-1 on the first stage.

For each clock cycle, the X transfer register 55-2 on the second stage acquires the first variable X_IN held in the X transfer register 55-1 on the first stage in the immediately preceding clock cycle and holds the acquired first variable X_IN for one clock cycle period. For each clock cycle, the Y transfer register 56-2 on the second stage acquires the second update value Y₁ calculated by the first FX addition unit 52-1 in the immediately preceding clock cycle and holds the acquired second update value Y₁ for one clock cycle period.

For each clock cycle, the first FY computing unit 53-1 calculates a first derivative value by performing the second function operation for the second update value Y₁ stored in the Y transfer register 56-2 on the second stage. For each clock cycle, the first FY addition unit 54-1 calculates the first update value X₁ by adding the first derivative value calculated by the first FY computing unit 53-1 in the clock cycle and the first variable X_IN stored in the X transfer register 55-2 on the second stage. For each clock cycle, the X output register 57 acquires the first update value X₁ calculated by the first FY addition unit 54-1 in the immediately preceding clock cycle and stores the acquired first update value X₁ as one first variable X_OUT at the second time point into the first variable memory 40.

For each clock cycle, the Y output register 58 acquires the second update value Y₁ stored in the Y transfer register 56-2 on the second stage and stores the acquired second update value Y₁ as one second variable Your at the second time point into the second variable memory 42.

By performing such pipeline processing, the function computing unit 46 according to the first example can perform the computation as described below, for each clock cycle.

Y ₁ =FX(X_IN)+Y_IN

X ₁ =FY(Y₁)+X_IN

Y _OUT =Y ₁

X _OUT =X ₁

Such a function computing unit 46 according to the first example can perform a series of computations of first function operation, FX addition, second function operation, and FY addition once for a variable pair including one first variable KIN and one second variable Y_IN with the same index i. Further, the function computing unit 46 according to the first example performs pipeline processing and thus can perform computation for a plurality of variable pairs in parallel. The function computing unit 46 according to the first example thus can complete computation for N variable pairs in a short time.

FIG. 6 is a diagram illustrating a second example of the configuration of the function computing unit 46. The function computing unit 46 may have the configuration of the second example as illustrated in FIG. 6. The function computing unit 46 according to the second example has the same components as in the first example but differs in arrangement of the components. In a description of the second example, the components performing the same operation as in the first example are denoted by the same reference signs and will not be further elaborated except for differences.

The first FY computing unit 53-1 calculates N first derivative values by performing the second function operation (FY(y_i)) for each of the updated N second variables calculated by the first addition unit 44. The first FY addition unit 54-1 calculates N first update values by adding N first derivative values calculated by the first FY computing unit 53-1 and N first variables at the first time point.

The first FX computing unit 51-1 calculates N second derivative values by performing the first function operation (FX(x_i)) for each of N first update values calculated by the first FY addition unit 54-1. The first FX addition unit 52-1 calculates N second update values by adding N second derivative values calculated by the first FX computing unit 51-1 and the updated N second variables calculated by the first addition unit 44.

The function computing unit 46 having such a configuration in the second example can perform the loop process from S16 to S19 in the first embodiment once in the order of S18 to S17. The function computing unit 46 thus can time-evolve N first variables and N second variables at the first time point using a method called symplectic Euler method to calculate N first variables and N second variables at the second time point.

Further, the function computing unit 46 according to the second example may implement computation by pipeline processing. Also in this case, the function computing unit 46 according to the second example has the same components as in the first example but differs in arrangement of the components.

For each clock cycle, the first FY computing unit 53-1 calculates a first derivative value by performing the second function operation for the updated second variable Y_IN stored in the Y transfer register 56-1 on the first stage. For each clock cycle, the first FY addition unit 54-1 calculates the first update value X₁ by adding the first derivative value calculated by the first FY computing unit 53-1 in the clock cycle and the first variable KIN stored in the X transfer register 55-1 on the first stage.

For each clock cycle, the X transfer register 55-2 on the second stage acquires the first update value X₁ calculated by the first FY addition unit 54-1 in the immediately preceding clock cycle and holds the acquired first update value X₁ for one clock cycle period. For each clock cycle, the Y transfer register 56-2 on the second stage acquires the second variable Y_IN held by the Y transfer register 56-1 on the first stage in the immediately preceding clock cycle and holds the acquired second variable Y_IN for one clock cycle period.

For each clock cycle, the first FX computing unit 51-1 calculates a second derivative value by performing the first function operation for the first update value X₁ stored in the X transfer register 55-2 on the second stage. For each clock cycle, the first FX addition unit 52-1 calculates the second update value Y₁ by adding the first derivative value calculated by the first FX computing unit 51-1 in the clock cycle and the second variable Y_IN stored in the Y transfer register 56-2 on the second stage.

By performing such pipeline processing, the function computing unit 46 according to the second example can perform computation as described below, for each clock cycle.

X ₁ =FY(Y_IN)+X_IN

y ₁ =FX(X₁)+Y_IN

X _OUT =X ₁

Y _OUT =Y ₁

Such a function computing unit 46 according to the second example can perform a series of computations of second function operation, FY addition, first function operation, and FX addition once for a variable pair including one first variable X_IN and one second variable Y_IN with the same index i. Further, the function computing unit 46 according to the second example performs pipeline processing and therefore can perform computation for a plurality of variable pairs in parallel. The function computing unit 46 according to the second example thus can complete computation for N variable pairs in a short time.

FIG. 7 is a diagram illustrating a third example of the configuration of the function computing unit 46. The function computing unit 46 may have the configuration of the third example as illustrated in FIG. 7. The function computing unit 46 according to the third example has substantially the same configuration as the first example.

In a description of the third example, the components performing the same operation as in the first example are denoted by the same reference signs and will not be further elaborated except for differences.

The function computing unit 46 according to the third example further includes, in addition to the configuration of the first example, second to M^th (M is an integer equal to or greater than two) (M−1) FX computing units 51-2 to 51-M, second to M^th (M−1) FX addition units 52-2 to 52-M, second to M^th (M−1) FY computing units 53-2 to 53-M, and second to M^th (M−1) FY addition units 54-2 to 54-M.

The m^th FX computing unit 51-m (m is any integer from two to M) calculates N second derivative values by performing the first function operation for each of N first update values calculated by the (m−1)th FY addition unit 54-(m−1). The M^th FX addition unit 52-m calculates new N second update values by adding N second derivative values calculated by the m^th FX computing unit 51-m and N second update values calculated by the (m−1)th FX addition unit 52-(m−1).

The M^th FY computing unit 53-m calculates N first derivative values by performing the second function operation for each of N second update values calculated by the m^th FX addition unit 52-m. The m^th FY addition unit 54-m calculates new N first update values by adding N first derivative values calculated by the m^th FY computing unit 53-m and N first update values calculated by the (m−1)^th FY addition unit 54-(m−1).

The function computing unit 46 then provides N first update values calculated by the M^th FY addition unit 54-M to the first variable memory 40. The first variable memory stores N first update values calculated by the M^th FY addition unit 54-M, as N first variables at the second time point.

The function computing unit 46 also provides N second update values calculated by the M^th FX addition unit 52-M to the second variable memory 42. The second variable memory 42 stores N second update values calculated by the M^th FX addition unit 52-M, as N second variables at the second time point.

The function computing unit 46 having such a configuration in the third example can perform the loop process from S16 to S19 in the first embodiment, M times, in the order of S17 to S18. The function computing unit 46 thus can time-evolve N first variables and N second variables at the first time point using a method called symplectic Euler method to calculate N first variables and N second variables at the second time point.

Further, the function computing unit 46 according to the third example may implement computation by pipeline processing. In this case, the function computing unit 46 further includes 2M stages of X transfer registers 55-1 to 55-2M, 2M stages of Y transfer registers 56-1 to 56-2M, one X output register 57, and one Y output register 58. In this configuration, the configuration illustrated in the first example performs the same operation as in the first example, except for the X output register 57 and the Y output register 58.

For each clock cycle, the X transfer register 55-(2m−1) on the (2m−1)^th stage acquires the first update value X_m-1 calculated by the (m−1) tri FY addition unit 54-(m−1) in the immediately preceding clock cycle and holds the acquired first update value X_m-1 for one clock cycle period. For each clock cycle, the Y transfer register 56-(2m−1) on the (2m−1)^th stage acquires the second update value Y_m-1 held by the Y transfer register 56-(2m-2) on the (2m-2)^th stage in the immediately preceding clock cycle and holds the acquired one second update value Y_m-1 for one clock cycle period.

For each clock cycle, the m^th FX computing unit 51-m calculates a second derivative value by performing the first function operation for the first update value stored in the X transfer register 55-(2m−1) on the (2m−1)^th stage. For each clock cycle, the m^th FX addition unit 52-m calculates a new second update value Y_m by adding the second derivative value calculated by the m^th FX computing unit 51-m in the clock cycle and the second update value Y_m-1 stored in the Y transfer register 56-(2m−1) on the (2m−1)^th stage.

For each clock cycle, the X transfer register 55-2m on the 2m^th stage acquires the first update value X_m-1 held by the X transfer register 55-(2m−1) on the (2m−1)^th stage in the immediately preceding clock cycle and holds the acquired one first update value X_m-1 for one clock cycle period. For each clock cycle, the Y transfer register 56-2m on the 2m^th stage acquires the second update value Y_m calculated by the m^th FX addition unit 52-m in the immediately preceding clock cycle and holds the acquired second update value Y, for one clock cycle period.

For each clock cycle, the m^th FY computing unit 53-m calculates a first derivative value by performing the second function operation for the second update value Y_m stored in the Y transfer register 56-2m on the 2m^th stage. For each clock cycle, the m^th FY addition unit 54-m calculates a new first update value X_m by adding the first derivative value calculated by the m^th FY computing unit 53-m in the clock cycle and the first update value X_m-1 stored in the X transfer register 55-2m on the 2m^th stage.

For each clock cycle, the X output register 57 acquires the first update value X_M calculated by the M^th FY addition unit 54-M in the immediately preceding clock cycle and stores the acquired first update value X_M as one first variable X_OUT at the second time point into the first variable memory 40.

For each clock cycle, the Y output register 58 acquires the second update value Y_M stored in the Y transfer register 56-2M on the 2M^th stage and stores the acquired second update value Y_M as one second variable Y_OUT at the second time point into the second variable memory 42.

By performing such pipeline processing, the function computing unit 46 according to the third example can perform computation as described below, for each clock cycle.

$Y_1=\mathrm{FX}\left(X_{\mathrm{IN}}\right)+Y_{\mathrm{IN}}{X}_1=\mathrm{FY}\left(Y_1\right)+X_{\mathrm{IN}}\dots Y_m=\mathrm{FX}\left(X_{m1}\right)+Y_{m1}{X}_m=\mathrm{FY}\left(Y_m\right)+X_{m-1}\dots Y_M=\mathrm{FX}\left(X_{M-1}\right)+Y_{M-1}{X}_M=\mathrm{FY}\left(Y_M\right)+X_{M-1}{Y}_{\mathrm{OUT}}=Y_M{X}_{\mathrm{OUT}}=X_M$

Such a function computing unit 46 according to the third example can perform a series of computations of first function operation, FX addition, second function operation, and FY addition, M times, for a variable pair including one first variable X_IN and one second variable Y_IN with the same index i. Further, the function computing unit 46 according to the third example performs pipeline processing and therefore can perform computation for a plurality of variable pairs in parallel. The function computing unit 46 according to the third example thus can complete computation for N variable pairs in a short time.

FIG. 8 is a diagram illustrating a fourth example of the configuration of the function computing unit 46. The function computing unit 46 may have the configuration of the fourth example as illustrated in FIG. 8. The function computing unit 46 according to the fourth example has substantially the same configuration as the second example. In a description of the fourth example, the components performing the same operation as in the second example are denoted by the same reference signs and will not be further elaborated except for differences.

The m^th FY computing unit 53-m (m is any integer from two to M) calculates N first derivative values by performing the second function operation for each of N second update values calculated by the (m−1)^th FX addition unit 52-(m−1). The m^th FY addition unit 54-m calculates N new first update values by adding N first derivative values calculated by the m^th FY computing unit 53-m and N first update values calculated by the (m−1)^th FY addition unit 54-m.

The m^th FX computing unit 51-m calculates N second derivative values by performing the first function operation for each of N first update values calculated by the m^th FY addition unit 54-m. The m^th FX addition unit 52-m calculates N new second update values by adding N second derivative values calculated by the m^th FX computing unit 51-m and N second update values calculated by the (m-1)^th FX addition unit 52-(m−1).

The function computing unit 46 having such a configuration in the fourth example can perform the loop process from S16 to S19 in the first embodiment, M times, in the order of S18 to S17. The function computing unit 46 thus can time-evolve N first variables and N second variables at the first time point using a method called symplectic Euler method to calculate N first variables and N second variables at the second time point.

Further, the function computing unit 46 according to the fourth example may implement computation by pipeline processing. Also in this case, the function computing unit 46 further includes 2M stages of X transfer registers 55-1 to 55-2M, 2M stages of Y transfer registers 56-1 to 56-2M, one X output register 57, and one Y output register 58. In this configuration, the configuration illustrated in the second example performs the same operation as in the second example, except for the X output register 57 and the Y output register 58.

For each clock cycle, the X transfer register 55-(2m−1) on the (2m−1)^th stage acquires the first update value X_m-1 held by the X transfer register 55-(2m-2) on the (2m-2)^th stage in the immediately preceding clock cycle and holds the acquired one first update value X_m-1 for one clock cycle period. For each clock cycle, the Y transfer register 56-(2m−1) on the (2m−1)^th stage acquires the second update value Y_m-1 calculated by the (m−1)^th FX addition unit 52-(m−1) in the immediately preceding clock cycle and holds the acquired second update value Y_m-1 for one clock cycle period.

For each clock cycle, the m^th FY computing unit 53-m calculates a first derivative value by performing the second function operation for the second update value Y_m-1 stored in the Y transfer register 56-(2m−1) on the (2m−1)^th stage. For each clock cycle, the m^th FY addition unit 54-m calculates a new first update value X_m by adding the first derivative value calculated by the m^th FY computing unit 53-m in the clock cycle and the first update value X_{m 1} stored in the X transfer register 55-(2m−1) on the (2m−1)^th stage.

For each clock cycle, the X transfer register 55-2m on the 2m^th stage acquires the first update value X_m calculated by the m^th FY addition unit 54-m in the immediately preceding clock cycle and holds the acquired first update value X_m for one clock cycle period. For each clock cycle, the Y transfer register 56-2m on the 2m^th stage acquires the second update value Y_m-1 held by the Y transfer register 56-(2m−1) on the (2m−1)^th stage in the immediately preceding clock cycle and holds the acquired one second update value Y_m-1 for one clock cycle period.

For each clock cycle, the m′ FX computing unit 51-m calculates a second derivative value by performing the first function operation for the first update value X_m stored in the X transfer register 55-2m on the 2m^th stage. For each clock cycle, the m^th FX addition unit 52-m calculates a new second update value Y_m by adding the second derivative value calculated by the m^th FX computing unit 51-m in the clock cycle and the second update value Y_m-1 stored in the Y transfer register 56-2m on the 2m^th stage.

For each clock cycle, the X output register 57 acquires the first update value X_M stored in the X transfer register 55-2M on the 2M^th stage and stores the acquired first update value X_M as one first variable X_OUT at the second time point into the first variable memory 40.

For each clock cycle, the Y output register 58 acquires the second update value Y_M calculated by the M^th FX addition unit 52-M in the immediately preceding clock cycle and stores the acquired second update value Y_M as one second variable Your at the second time point into the second variable memory 42.

By performing such pipeline processing, the function computing unit 46 according to the fourth example can perform computation as described below, for each clock cycle.

$X_1=\mathrm{FY}\left(Y_{\mathrm{IN}}\right)+X_{\mathrm{IN}}{Y}_1=\mathrm{FX}\left(X_1\right)+Y_{\mathrm{IN}}\dots X_m=\mathrm{FY}\left(Y_{m-1}\right)+X_{m-1}{Y}_m=\mathrm{FX}\left(X_m\right)+Y_{m-1}\dots X_M=\mathrm{FY}\left(Y_{M-1}\right)+X_{M-1}{Y}_M=\mathrm{FX}\left(X_M\right)+Y_{M-1}{X}_{\mathrm{OUT}}=X_M{Y}_{\mathrm{OUT}}=Y_M$

Such a function computing unit 46 according to the fourth example can perform a series of computations of second function operation, FY addition, first function operation, and FX addition, M times, for a variable pair including one first variable X_IN and one second variable Y_IN with the same index i. Further, the function computing unit 46 according to the fourth example performs pipeline processing and therefore can perform computation for a plurality of variable pairs in parallel. The function computing unit 46 according to the fourth example thus can complete computation for N variable pairs in a short time.

Third Embodiment

The calculation apparatus 10 according to a third embodiment will be described.

FIG. 9 is a diagram illustrating the relation between N first intermediate variables the coefficient matrix J, and N second intermediate variables b_i in the third embodiment.

The coefficient matrix J is divided into P_r3 block matrices (JG0, JG1, . . . , JGk, . . . , JGPr3−1). Each of P_r3 block matrices includes coefficients J_i,j in (N/P_r3) rows and N columns. P_r3 is a divisor of N, and k is any integer from zero to P_r3-1.

N second intermediate variables b₁ are divided into P_r3 blocks (BG0, BG1, . . . , BGk, . . . , BGPr3−1). Each of P_r3 blocks further includes (N/P_r3) second intermediate variables b₁. P_r3 blocks are associated with P_r3 block matrices in one-to-one correspondence. For example, the kth block BGk corresponds to the kth block matrix JGk.

In the third embodiment, the matrix multiplication unit 28 calculates P_r3 blocks by separately performing matrix multiplication between N first intermediate variables x′_i and each of P_r3 block matrices.

FIG. 10 is a diagram illustrating a configuration of the matrix multiplication unit 28 according to the third embodiment with the time evolution unit 30.

The matrix multiplication unit 28 includes P_r3 block matrix multiplication units 60 associated with P_r3 block matrices in one-to-one correspondence. Each of P_r3 block matrix multiplication units 60 calculates (N/P_r3) second intermediate variables b₁ included in the corresponding block by performing matrix multiplication between N first intermediate variables x′_i and the corresponding block matrix.

P_r3 block matrix multiplication units 60 have the same configuration. P_r3 block matrix multiplication units 60 may be, for example, circuits implemented in semiconductor devices different from each other.

Each of P_r3 block matrix multiplication units 60 receives a first intermediate stream X′ which is a stream of N first intermediate variables x′_i and a second intermediate stream B which is a stream of N second intermediate variables b₁. Each of P_r3 block matrix multiplication units 60 transmits the first intermediate stream X′ and the second intermediate stream B.

P_r3 block matrix multiplication units 60 are connected in series. An initial one of the P_r3 block matrix multiplication unit 60 receives the first intermediate stream X′ from the time evolution unit 30. The initial block matrix multiplication unit 60 also receives dummy data as the second intermediate stream B. The dummy data may be transmitted from any block. The block matrix multiplication unit 60 except for the initial one receives the first intermediate stream X′ and the second intermediate stream B transmitted from the block matrix multiplication unit 60 on the immediately preceding stage.

The block matrix multiplication unit 60 except for the last transmits the first intermediate stream X′ and the second intermediate stream B to the block matrix multiplication unit 60 on the immediately following stage. The last block matrix multiplication unit 60 transmits the second intermediate stream B to the time evolution unit 30. The first intermediate stream X′ transmitted from the last block matrix multiplication unit 60 is discarded. The first intermediate stream X′ transmitted from the last block matrix multiplication unit 60 may be discarded in the time evolution unit 30 after being received by the time evolution unit 30.

Each of P_r3 block matrix multiplication units 60 includes a buffer 62, a block matrix memory 64, an executor 66, and a selector 68.

The buffer 62 of the initial block matrix multiplication unit 60 acquires the first intermediate stream X′ output from the time evolution unit 30, stores the acquired first intermediate stream X′ for a given time, and then outputs the same. The buffer 62 of the block matrix multiplication unit 60 except for the initial one acquires the first intermediate stream X′ output from the block matrix multiplication unit 60 on the immediately preceding stage, stores the acquired first intermediate stream X′ for a given time, and then outputs the same. The block matrix memory 64 stores the coefficients Jif in (N/P_r3) rows and N columns included in the corresponding block matrix. The executor 66 calculates (N/P_r3) second intermediate variables h_i included in the corresponding block, based on the first intermediate stream X′ stored in the buffer 62 and the block matrix stored in the block matrix memory 64.

Here, the executor 66 outputs (N/P_r3) second intermediate variables b₁ included in the corresponding block as a second intermediate stream B including P_r1 second intermediate variables b₁ in one clock cycle. P_r1 is a divisor of N.

Further, the executor 66 outputs the second intermediate stream B in a clock cycle different from the executor 66 included in another block matrix multiplication unit 60. Therefore, the second intermediate stream B is not output from a plurality of block matrix multiplication units 60 in the same clock cycle.

The selector 68 of the initial block matrix multiplication unit 60 selects and outputs the second intermediate stream B output by the executor 66 of the initial block matrix multiplication unit 60, in a clock cycle in which the executor 66 of the initial block matrix multiplication unit 60 outputs a second intermediate stream B. The selector 68 of the initial block matrix multiplication unit 60 selects and outputs dummy data, in a clock cycle in which the executor 66 of the initial block matrix multiplication unit 60 does not output a second intermediate stream B.

The selector 68 of the kth block matrix multiplication unit 60 excluding the initial block matrix multiplication unit selects and outputs the second intermediate stream B output by the executor 66 of the kth block matrix multiplication unit 60, in a clock cycle in which the executor 66 of the kth block matrix multiplication unit 60 outputs a second intermediate stream B. In a clock cycle in which the executor 66 of the kth block matrix multiplication unit 60 does not output a second intermediate stream B, the second intermediate stream B output by the selector 68 of the block matrix multiplication unit 60 on the preceding stage is selected and output.

FIG. 11 is a diagram illustrating an implementation example of the matrix multiplication units 28 and the time evolution unit 30 according to the third embodiment.

P_r3 matrix multiplication units 28 and the time evolution unit 30 according to the third embodiment can be implemented, for example, on (P_r3+1) chips 70-1 to 70-(P_r3+1) which are individually independent semiconductor chips.

Each of the first to P_r3^rd chips 70-1 to 70-P_r3 includes a block matrix multiplication unit 60. The (P_r3+1)^th chip 70-(P_r3+1) includes the time evolution unit 30.

Each chip 70 includes a receiver 74 and a transmitter 76. The receiver 74 is a link port for receiving data. The transmitter 76 is a link port for transmitting data. The receiver 74 and the transmitter 76 may be a transmitter/receiver that performs full-duplex communication. Each chip 70 may include an additional communication port. For example, each chip 70 may include two independent receiving link ports and two independent transmitting link ports.

The transmitter 76 combines the first intermediate variable x′_i and the second intermediate variable b_i in one clock cycle and outputs a combined stream. When the bit width of the first intermediate stream X′ is Wx′ and the bit width of the second intermediate stream B is Wb, the combined stream has a bit width equal to or greater than Wx′+Wb.

Each of the receiver 74 and the transmitter 76 includes, for example, a FIFO memory. The transmitter 76 combines the first intermediate stream X′ and the second intermediate stream B and writes the combined stream into the FIFO. The transmitter 76 then successively transmits the combined stream stored in the FIFO. The receiver 74 writes the received combined stream into the FIFO. The receiver 74 then reads the combined stream in order from the FIFO and separates it into the first intermediate stream X′ and the second intermediate stream B.

Two chips 70 are connected through a communication link 72. The output terminal of the first chip 70-1 is connected to the input terminal of the second chip 70-2 through the first communication link 72-1. The output terminal of the kth chip 70-k is connected to the input terminal of the (k+1)^th chip 70-(k+1)^th rough the kth communication link 72-k. Then, the output terminal of the (P_r3+1)^th chip 70-(P_r3+1) is connected to the input terminal of the first chip 70-1 through the (P_r3+1)^th communication link 72-(P_r3+1). In this way, (P_r3+1) chips 70-1 to 70-(P_r3+1) are mutually connected in the form of ring topology.

For example, the input terminal and the output terminal of the chip 70 may be quad small form-factor pluggable (QSFP) ports. The communication link 72 may be a QSFP-adapted optical cable or a QSFP-adapted metal cable. The communication link 72 between two chips 70 may be a high-speed serial link, an Ethernet link, or a peer-to-peer link.

In the present embodiment, each chip 70 transmits/receives a combined stream. However, the first intermediate stream X′ and the second intermediate stream B may be each individually transmitted/received. In this case, (P_r3+1) chips 70-1 to 70-(P_r3+1) are mutually connected in the form of two-ring topology.

Such a calculation apparatus 10 according to the third embodiment can cope with change in number of first variables x_i to be determined by an optimization problem, by changing the number of block matrix multiplication units 60. Therefore, when the number of first variables x_i to be determined by an optimization problem significantly increases, the calculation apparatus 10 can cope, for example, by increasing the number of chips 70 having the same configuration. In addition, the calculation apparatus can complete the optimization problem in a short time because a plurality of block matrix multiplication units 60 perform matrix operations in parallel.

Fourth Embodiment

The calculation apparatus 10 according to a fourth embodiment will be described.

FIG. 12 is a diagram illustrating the relation between N first intermediate variables x′_i, the coefficient matrix J, and N second intermediate variables b₁ in the fourth embodiment. In the fourth embodiment, N first intermediate variables x′_i, the coefficient matrix J, and N second intermediate variables b₁ have the relation as illustrated in FIG. 12, in addition to the similar relation as in the third embodiment.

N first intermediate variables x′_i are divided into N_s data sets. Each of N_s data sets includes P_c first intermediate variables x′_i. N_s and P_c are divisors of N. For example, N first intermediate variables x′_i are divided into N_G data sets x′b0, x′b1, . . . , and x′bNs.

Each of P_r3 block matrices is divided into P_r2 submatrices. Each of P_r2 submatrices includes coefficients J_i,j in P_r3 rows and N columns. For example, the block matrix JG0 is divided into P_r2 submatrices jb0, jb1, . . . , jbs, . . . , and jbPr2-1.

P_r1 and P_r2 are divisors of N, and P_r1×P_r2×P_r3 are N. Here, s is any integer from zero to P_r2-1.

Each of P_r3 blocks included in N second intermediate variables b₁ is divided into P_r2 sub-blocks. Each of Pre sub-blocks includes P_r1 second intermediate variables b_i. For example, the block BG0 is divided into P_r2 sub-blocks bb0, bb1, . . . , bbs, . . . , and bbPr2-1.

P_r2 sub-blocks included in the kth block are associated with P_r2 submatrices included in the kth block matrix in one-to-one correspondence. For example, the s^th sub-block included in the kth block BGk corresponds to the s^th submatrix jbs included in the kth block matrix JGk.

In the fourth embodiment, the block matrix multiplication unit 60 performs matrix multiplication between N first intermediate variables x′_i for each submatrix included in the corresponding block matrix. For example, the kth block matrix multiplication unit 60 calculates P_r2 sub-blocks included in the kth block BGk by individually performing matrix multiplication between N first intermediate variables x′_i and each of P_r2 submatrices included in the kth block matrix JGk.

FIG. 13 is a diagram illustrating a configuration of the block matrix multiplication unit 60 according to the fourth embodiment. The executor 66 included in the block matrix multiplication unit 60 includes P_r2 submatrix multiplication units 80 and a multiplexer 82.

P_r2 submatrix multiplication units 80 are associated with P_r2 submatrices included in the corresponding block matrix in one-to-one correspondence. For example, P_r2 submatrix multiplication units 80 included in the block matrix multiplication unit 60 associated with the kth block matrix are associated with P_r2 submatrices included in the kth block matrix in one-to-one correspondence.

Each of P_r2 submatrix multiplication units 80 included in the kth block matrix multiplication unit 60 calculates P_r1 second intermediate variables b₁ included in the corresponding sub-block by performing matrix multiplication between N first intermediate variables x′_i and the corresponding submatrix.

Each of P_r2 submatrix multiplication units 80 outputs P_r1 second intermediate variables b_i included in the corresponding sub-block in parallel in one clock cycle. For example, the s^th submatrix multiplication unit 80 included in the kth block matrix multiplication unit 60 outputs P_r1 second intermediate variables b_i included in the s^th sub-block included in the kth block in the same clock cycle.

Further, each of P_r2 submatrix multiplication units 80 outputs P_r1 second intermediate variables b_i in a clock cycle different from other submatrix multiplication units 80. That is, the second intermediate variable b_i is not output from a plurality of submatrix multiplication units 80 in the same clock cycle.

The multiplexer 82 generates a second intermediate stream B including P_r1 second intermediate variables b_i in one clock cycle by multiplexing a set of P_r2 second intermediate variables b_i output from each of P_r2 submatrix multiplication units 80. For example, the multiplexer 82 included in the kth block matrix multiplication unit 60 outputs a second intermediate stream B including P_r2 sub-blocks included in the kth block in P_r2 clock cycles. The buffer 62 included in the block matrix multiplication unit 60 includes P_r2 stages of registers 84 functioning as a shift register, an in-buffer receiver 86, and an in-buffer transmitter 88.

P_r2 stages of registers 84 are associated with P_r2 submatrix multiplication units 80 in one-to-one correspondence. Each of P_r2 stages of registers 84 stores a data set including P(first intermediate variables x′1 in one clock cycle. Each of P_r2 stages of registers 84 transfers the stored data set including P_c first intermediate variables x′_i to the register 84 on the next stage in parallel in the next clock cycle.

The in-buffer receiver 86 receives a first intermediate stream X′ and converts the received first intermediate stream X′ into a stream with a word width of P_c. The in-buffer receiver 86 then writes a data set including P_c first intermediate variables x′_i into an initial register 84 for each clock cycle.

The in-buffer transmitter 88 reads a data set including P_c first intermediate variables x′_i from the register 84 on the last stage for each clock cycle and converts the read data set into a first intermediate stream X′. The in-buffer transmitter 88 then transmits the first intermediate stream X′.

Each of P_r2 submatrix multiplication units 80 reads a data set including P_c first intermediate variables x′i stored in the corresponding register 84, for each clock cycle. Each of P_r2 submatrix multiplication units 80 multiplies each of the read P_c first intermediate variables x′_i and the coefficient J_i,j in the corresponding column in the corresponding submatrix, for each clock cycle. Each of P_r2 submatrix multiplication units 80 then cumulatively adds the multiplication result of the first intermediate variable x′_i and the coefficient J_i,j, for each row included in the corresponding submatrix. Each of P_r2 submatrix multiplication units 80 thus can calculate P_r1 second intermediate variables b_i included in the corresponding sub-block.

Here, each of P_r2 submatrix multiplication units 80 performs multiplication and cumulative addition in a period before an initial first intermediate variable x′₀ to the last first intermediate variable x′_N-1 of N first intermediate variables x′_i are stored into the corresponding register 84. Each of P_r2 submatrix multiplication units 80 then outputs P_r1 second intermediate variables h_i included in the corresponding sub-block after passage of a predetermined number of clock cycles after the clock cycle in which the last first intermediate variable x′_N1 is stored into the corresponding register 84. Then, each of P_r2 submatrix multiplication units 80 can output Pit second intermediate variables b₁ included in the corresponding sub-block in clock cycles different from each other.

The calculation apparatus 10 according to the fourth embodiment having such a configuration can perform matrix multiplication with the degree of parallelism P_r2×P_r2. The calculation apparatus 10 according to the fourth embodiment thus can perform matrix multiplication at high speed.

Fifth Embodiment

The calculation apparatus 10 according to a fifth embodiment will be described.

FIG. 14 is a diagram illustrating the relation between N first intermediate variables x′_i, the coefficient matrix J, and N second intermediate variables b_i in the fifth embodiment. In the fifth embodiment, N first intermediate variables x′_i, the coefficient matrix J, and N second intermediate variables b_i further have the relation as illustrated in FIG. 14, in addition to the similar relation as in the fourth embodiment.

Each of P_r3 block matrices is divided into Pre submatrices. Each of P_r2 submatrices is further divided into N_s coefficient sets, in units of P_c columns. Each coefficient set includes a coefficient J_i,j with P_r1 rows and P_c columns. For example, the submatrix jb0 included in the block matrix JG0 is divided into N_s coefficient sets of jb0(0), jb0(1), . . . , and jb0(N_s−1).

N_s coefficient sets have one-to-one correspondence with N_c data sets obtained by dividing N first intermediate variables x′_i in units of P_c.

In the fifth embodiment, the submatrix multiplication unit 80 calculates a multiply-accumulate value of P_c coefficients J_i,j and the corresponding P_c first intermediate variables x′_i, for each of P_r1 rows included in one coefficient set. The submatrix multiplication unit 80 then adds all of the multiply-accumulate values calculated for each of N_s coefficient sets, for each of P_r1 rows. The submatrix multiplication unit 80 thus can multiply-accumulate N coefficients J_i,j and N first intermediate variables x′_i, for each of P_r1 rows.

FIG. 15 is a diagram illustrating a configuration of the submatrix multiplication unit 80 according to the fifth embodiment with the corresponding register 84. Each submatrix multiplication unit 80 includes P_r1 multiply-accumulators 90.

P_r1 multiply-accumulators 90 are associated with P_r1 rows included in the corresponding submatrix in one-to-one correspondence. For example, P_r1 multiply-accumulators 90 associated with the 5th submatrix included in the block matrix multiplication unit 60 associated with the k^th block matrix are associated with P_r1 rows included in the s^th submatrix included in the kth block matrix in one-to-one correspondence.

P_r1 multiply-accumulators 90 all have the same configuration. Each of P_r1 multiply-accumulators 90 calculates a second intermediate variable b_i at a corresponding position included in the corresponding sub-block by multiply-accumulating the corresponding row in the coefficient matrix J and N first intermediate variables x′_i. P_r1 multiply-accumulators 90 calculate second intermediate variables b_i in parallel and output the same in the same clock cycle.

More specifically, for each clock cycle, each of P_r1 multiply-accumulators 90 acquires an initial first intermediate variable x′₀ to the last N first intermediate variables of a first intermediate stream X′, successively stored in the corresponding register 84 of Pre stages of registers 84. In this case, each of P_r1 multiply-accumulators 90 acquires a data set including P_c first intermediate variables x′_i in one clock cycle. Then, each of P_r1 multiply-accumulators 90 performs acquisition of a data set over N_s (N_s=N/P_c) clock cycles to acquire N_s data sets. The p^th (p is any integer from zero to P_r1−1) multiply-accumulator 90 of P_r1 multiply-accumulators 90 acquires P_c coefficients ((jp, S0) to (jp, SPc−1)) included in the p^th corresponding row included in a coefficient set corresponding to the acquired data set, in one clock cycle. Each of P_r1 multiply-accumulators 90 then acquires P_c coefficients included in the coefficient set over N_s clock cycles. Each of P_r1 multiply-accumulators 90 multiply-accumulates P_c first intermediate variables x′_i and the corresponding P_c coefficients J_i,j included in the corresponding row, for each clock cycle. Each of P_r1 multiply-accumulators 90 calculates a second intermediate variable b_i by cumulatively adding the multiply-accumulate results over N_s clock cycles.

FIG. 16 is a diagram illustrating a block matrix stored in the block matrix memory 64. The block matrix memory 64 stores a block matrix divided into P_r2 submatrices. P_r2 submatrices are output in parallel to the corresponding submatrix multiplication units 80.

Each of P_r2 submatrices includes N_s coefficient sets. Each of N_c coefficient sets includes p_c×P_r1 coefficients. The block matrix memory 64 can output p_c×P_r1 coefficients included in one coefficient set in parallel in one clock cycle, for each of P_r2 submatrices.

FIG. 17 is a diagram illustrating P_r1 multiply-accumulators 90 included in one submatrix multiplication unit 80 and a submatrix transmitted to the submatrix multiplication unit 80.

To each of P_r1 multiply-accumulators 90 included in one submatrix multiplication unit 80, a data set including P_c first intermediate variables x′_i stored in the corresponding register 84 is broadcasted for each clock cycle.

The block matrix memory 64 includes N_s coefficient sets for each submatrix. In the block matrix memory 64, one coefficient set included in each submatrix is specified by a pointer for each clock cycle. More specifically, in the block matrix memory 64, a coefficient set corresponding to the data set stored in the register 84 corresponding to the submatrix is specified.

The block matrix memory 64 outputs P_c×P_r1 coefficients included in the coefficient set specified by a pointer in parallel, for each clock cycle. For each of P_c×P_r1 coefficients included in a coefficient set, its output destination multiply-accumulator 90 is determined in advance. The block matrix memory 64 outputs each of p_c×p_r1 coefficients included in one coefficient set to a predetermined multiply-accumulator 90. For example, an initial to the P_c^th P_c coefficients in a coefficient set are output to the first multiply-accumulator 90. For example, the (P_c+1)^th to (2×P_c)^th P_c coefficients are output to the second multiply-accumulator 90. The (P_r1-P_c)^th to P_r1^st P_c coefficients are output to the P_r1^st multiply-accumulator 90. FIG. 18 is a diagram illustrating a configuration of the multiply-accumulator 90. The multiply-accumulator 90 includes P_c multipliers 92, an adder 94, and an accumulator 96. For each clock cycle, the multiply-accumulator 90 acquires a data set including P_c first intermediate variables x′_i and a coefficient set including P_c coefficients J_i,j.

P_c first intermediate variables x′_i included in a data set have one-to-one correspondence with P_c coefficients J_i,j included in a coefficient set. P_c multipliers 92 have one-to-one correspondence with P_c first intermediate variables x′_i included in a data set and P_c coefficients J_i,j included in a coefficient set. Each of P_c multipliers 92 multiplies one corresponding first intermediate variable x′_i included in a data set and one corresponding coefficient J_i,j included in a coefficient set and outputs a multiplication value. Each of P_c multipliers 92 outputs the multiplication value for each clock cycle.

The adder 94 calculates a multiply-accumulate value by adding P_c multiplication values output from P_c multipliers 92, for each clock cycle. The accumulator 96 cumulatively adds the multiply-accumulate value output from the adder 94, for each clock cycle. The accumulator 96 resets the cumulatively added value to zero at a timing (S0=0) when the initial first intermediate variable x′₀ in the first intermediate stream X′ is stored in the corresponding register 84.

The multiply-accumulator 90 having such a configuration acquires N first intermediate variables x′i from the initial first intermediate variable x′₀ to the last first intermediate variable of the first intermediate stream X′ successively stored in the corresponding register 84 of P_r2 stages of registers 84, for each clock cycle. Further, the multiply-accumulator 90 multiplies each of the acquired first intermediate variables x′_i and the coefficient J_i,j on the column corresponding to the acquired first intermediate variable x′_i that is included in the corresponding row in the coefficient matrix J. The multiply-accumulator 90 then cumulatively adds the multiplication result of the first intermediate variable x′_i and the coefficient J_i,j from the initial first intermediate variable x′₀ to the last first intermediate variable x′_N-1 of the first intermediate stream X′. With such processing, the multiply-accumulator 90 can calculate the second intermediate variable b_i at the corresponding position included in the corresponding sub-block.

FIG. 19 is a diagram illustrating an example of specific values of parameters and an example of processing timing in the fifth embodiment.

In the example in FIG. 19, N=16384. In the example in FIG. 19, P_c=4, N_s=4096. In the example in FIG. 19, P_r1=8, P_r2=256, P_r3=8. In FIGS. 19, t₀, t₁, t₂, . . . , and t6143 represent the order of clock cycles.

The initial data set in N first intermediate variables x′_i is stored in the initial register 84 in the initial matrix multiplication unit 28, at t₀. Therefore, the initial coefficient set in the submatrix jb0 in the block matrix JG0 is read out at t₀.

The last data set in N first intermediate variables x′_i is stored in the initial register 84 in the initial matrix multiplication unit 28, at t4095. Therefore, the last coefficient set in the submatrix jb0 in the block matrix JG0 is read out at t4095.

Then, the initial block in N second intermediate variables b_i can be output after the clock cycle (t4095) in which multiply-accumulation of the last coefficient set in the submatrix jb0 in the block matrix JG0 is completed. Therefore, the initial block in the second intermediate variable b_i is output at t4095+α (α is a predetermined delay time).

The initial data set in N first intermediate variables x′_i is delayed for (P_r1×P_r3=2048) clock cycles after t₀ and thereafter stored in the last register 84 in the last matrix multiplication unit 28. Therefore, the initial coefficient set in the submatrix jbPr2-1 in the block matrix JGPr3-1 is read out at t2047.

The last data set in N first intermediate variables x′_i is delayed for (P_r1×P_r3=2048) clock cycles after t4095 and thereafter stored in the last register 84 in the last matrix multiplication unit 28. Therefore, the last coefficient set in the submatrix jbPr2-1 in the block matrix JGPr3-1 is read out at t6143.

Then, the last block in N second intermediate variables b_i can be output after the clock cycle (t6143) in which the multiply accumulation of the last coefficient set in the submatrix jbPr2-1 in the block matrix JGPr3-1 is completed. Therefore, the last block in the second intermediate variable b_i is output at t6143+α.

The calculation apparatus 10 according to the fifth embodiment having such a configuration can perform matrix multiplication with the degree of parallelism P_r3×P_r2×P_r1.

The calculation apparatus 10 according to the fifth embodiment thus can perform matrix multiplication at high speed.

Sixth Embodiment

The calculation apparatus 10 according to a sixth embodiment will be described.

FIG. 20 is a diagram illustrating a configuration of the time evolution unit 30 according to the sixth embodiment with a plurality of matrix multiplication units 28. In the sixth embodiment, each matrix multiplication unit 28 outputs a second intermediate stream B including P_r1 second intermediate variables b_i in one clock cycle. In the sixth embodiment, the time evolution unit 30 receives a second intermediate stream B including P_r1 second intermediate variables b_i in one clock cycle.

The time evolution unit 30 according to the sixth embodiment includes P_r1 first variable memories 40, P_r1 second variable memories 42, P_r1 first addition units 44, P_r1 function computing units 46, P_r1 first multiplication units 48, and one first intermediate variables memory 50.

P_r1 first variable memories 40, P_r1 second variable memories 42, P_r1 first addition units 44, P_r1 function computing units 46, and P_r1 first multiplication units 48 have one-to-one correspondence with P_r1 second intermediate variables b_i included in one clock cycle.

Each of P_r1 first addition units 44 and each of P_r1 function computing units 46 perform a computation process for corresponding one second intermediate variable b_i of P_r1 second intermediate variables b_i included in one clock cycle. Each of P_r1 first variable memories 40 and each of P_r1 second variable memories 42 store the first variable x_i and the second variable y_i calculated using corresponding one second intermediate variable b_i of P_r1 second intermediate variables b_i included in one clock cycle. Each of P_r1 first multiplication units 48 performs a computation process for the first variable x_i calculated using corresponding one second intermediate variable b_i of P_r1 second intermediate variables b_i included in one clock cycle.

In this way, the time evolution unit 30 according to the sixth embodiment performs the calculation process of N first variables x_i and N second variables y_i for P_r1 second intermediate variables b_i included in one clock cycle, with the degree of parallelism P_r1. The calculation apparatus according to the sixth embodiment thus can perform the calculation process of N first variables x_i and N second variables y_i at high speed.

FIG. 21 is a diagram illustrating the output timing of the first intermediate variable x′_i and the second intermediate variable b₁. More specifically, (A) in FIG. 21 illustrates the timing when the first intermediate variable x′_i is output from the time evolution unit 30 to an initial matrix multiplication unit 28. (B) in FIG. 21 illustrates the timing when the second intermediate variable b_i is output from the last matrix multiplication unit 28 to the time evolution unit 30. (C) in FIG. 21 illustrates the timing of calculating the first intermediate variable x′_i by the time evolution unit 30. The first intermediate variable x′_i is transferred with the degree of parallelism P_c. Therefore, as illustrated by (A) in FIG. 21, the transfer period of the first intermediate variables (x′b0 to x′b4095(t₁)) at the first time point is N(=N/P_c) clock cycles.

The second intermediate variable b_i is transferred with the degree of parallelism P_r2. Therefore, as illustrated by (B) in FIG. 21, the transfer period of the second intermediate variables (bb0 to bb2047(t₁)) at the first time point is N/P_r1 clock cycles.

In the time evolution unit 30, processing is executed with the degree of parallelism P_r1. Therefore, as illustrated by (C) in FIG. 21, the computation period of the second intermediate variables (bb0 to bb2047(t₂)) at the second time point is N/P_r1 clock cycles.

Here, L_a is a delay time from when output of the last first intermediate variable (x′b4095(t₂)) at the first time point is completed to when output of the initial second intermediate variable (bb0(t₁)) at the first time point is started. L_b is a delay time from when output of the initial second intermediate variable (bb0(t₁)) at the first time point is started to when the initial first intermediate variable (x′b0(t₂)) at the second time point is calculated. L_c is a delay time from when the initial first intermediate variable (x′b0(t₂)) at the second time point is calculated to when the initial first intermediate variable (x′b0(t₂)) at the second time point is output. Here, a period from the start of a certain loop process to the start of the next loop process (one loop period) is N_s+L_a+L_b L_c. The calculation apparatus 10 can make (L_b+L_c) shorter than the transfer period (N/P_r1) of second intermediate variables (bb0 to bb2047(t₁)) at the first time point. Therefore, in the calculation apparatus 10, output of the first intermediate variables (x′b0 to x′b4095(t₂)) at the second time point is started before output of the second intermediate variables (bb0 to bb2047(t₁)) at the first time point is completed. That is, the calculation apparatus 10 implements overlapping of loop processes. The calculation apparatus 10 can perform processing at high speed by implementing such overlapping.

When N is large, N_s>>L_a+L_b L_c. Therefore, when N is large, L_a+L_b L can be shortened to a negligible level with respect to the entire execution time. In this case, the matrix multiplication time in the calculation apparatus 10 can be considered as N_s (=N/P_c) clock cycles.

For example, the matrix operation time of a matrix with a size N by a single-core processor is (N×N) clock cycles. The matrix multiplication time in the calculation apparatus 10 relative to the matrix multiplication time by a single-core processor is (N/P_c)/N×N. Therefore, the calculation apparatus 10 can perform matrix multiplication in a time 1/(P_c×N), compared with a single-core processor.

While certain embodiments have been described, these embodiments have been presented by way of example only, and are not intended to limit the scope of the inventions.

Indeed, the novel embodiments described herein may be embodied in a variety of other forms; furthermore, various omissions, substitutions and changes in the form of the embodiments described herein may be made without departing from the spirit of the inventions. The accompanying claims and their equivalents are intended to cover such forms or modifications as would fall within the scope and spirit of the inventions.

Claims

1. A calculation apparatus comprising: matrix multiplication circuitry configured to calculate N second intermediate variables at a first time point by performing matrix multiplication between N (N being an integer equal to or greater than two) first intermediate variables at the first time point and a coefficient matrix including preset coefficients in N rows and N columns;time evolution circuitry configured to calculate, based on the N second intermediate variables at the first time point, N first variables at a second time point, N second variables at the second time point, and N first intermediate variables at the second time point, the second time point being a time point following one sampling period after the first time point;management circuitry configured to increment a time point from a start time point to an end time point by adding a sampling period for each time point and to control the matrix multiplication circuitry and the time evolution circuitry such that at each time point, the matrix multiplication circuitry calculates N second intermediate variables at the time point, and the time evolution circuitry calculates N first variables and N first intermediate variables at a subsequent incremented time point; andoutput circuitry configured to output N first variables at a preset end time point, wherein the time evolution circuitry comprises:first addition circuitry configured to update N second variables at the first time point by adding the N second intermediate variables at the first time point to the N second variables at the first time point;first FX computing circuitry configured to calculate N second derivative values by performing a first function operation on each of N first variables at the first time point;first FX addition circuitry configured to calculate N second update values by adding the N second derivative values calculated by the first FX computing circuitry and the updated N second variables calculated by the first addition circuitry;first FY computing circuitry configured to calculate N first derivative values by performing a second function operation on each of the N second update values calculated by the first FX addition circuitry; andfirst FY addition circuitry configured to calculate N first update values by adding the N first derivative values calculated by the first FY computing circuitry and the N first variables at the first time point.