Solution of linear equation systems with matrix operations

1 A shows the proposed feedback circuit for solving a system of linear equations in one step, and the hardware circuit on a printed circuit board is shown in SI Appendix, Fig. Results Cross-Point Circuits for Solving a System of Linear Equations.ДЇig. However, iteration raises a fundamental limit toward achieving high computing performance in terms of energy and latency. The higher the desired accuracy of the solution, the more iterations are needed to complete the operation. It has also been shown recently that iterated MVM operations with resistive cross-point arrays can solve systems of linear equations, in combination with digital floating-point computers ( 22).

Implemented in the cross-point array architecture, resistive memories can naturally accelerate data-intensive operations with enhanced time/energy efficiencies compared with classical digital computing ( 5, 6, 17). 20), and spin-transfer torque magnetic memory ( 21). Resistive memories include various device concepts, such as resistive switching memory (RRAM, refs. Owing to their nonvolatile and reconfigurable behavior, resistive memories have been widely investigated and developed for storage-class memory ( 11, 12), stateful logic ( 13вЂ“ 15), in-memory computing ( 5, 6, 16, 17), and neuromorphic computing applications ( 7, 8, 18, 19). Resistive memories are two-terminal elements that can change their conductance in response to applied voltage stimuli ( 9, 10). Here, we show that a feedback circuit including a reconfigurable cross-point resistive array can provide the solution to algebraic problems such as systems of linear equations, matrix eigenvectors, and differential equations in just one step. However, solving matrix equations, such as a linear system Ax = b, in a single operation remains an open challenge. In-memory MVM has been adopted for several tasks, including image compression ( 5), sparse coding ( 6), and the training of deep neural networks ( 7, 8). One example is computing within cross-point arrays, which can accelerate matrix-vector multiplication (MVM) by OhmвЂ™s law and KirchhoffвЂ™s law with analog and reconfigurable resistive memories ( 5вЂ“ 8). To get around these fundamental limits, in-memory computing has recently emerged as a promising technique to conduct computing in situ, i.e., within the memory unit ( 5). As conventional computers are increasingly challenged by the scaling limits of the complementary metal-oxide-semiconductor (CMOS) technology ( 3), and by the energy and latency burdens of moving data between the memory and the computing units ( 4), improving the computing performance with increasing hardware resources becomes difficult and noneconomic. Traditionally, these problems in forms of matrix equations are solved by matrix factorizations or iterative matrix multiplications ( 1, 2), which are computationally expensive with polynomial time complexity, e.g., O( N 3) where N is the size of the problem. Linear algebra problems, such as solving systems of linear equations and computing matrix eigenvectors, lie at the heart of modern scientific computing and data-intensive tasks. Algebraic problems are demonstrated in hardware and applied to classical computing tasks, such as ranking webpages and solving the SchrГ¶dinger equation in one step. These operations are completed in just one single step, thanks to the physical computing with OhmвЂ™s and KirchhoffвЂ™s laws, and thanks to the negative feedback connection in the cross-point circuit. Here, we show that a cross-point array of resistive memory devices can directly solve a system of linear equations, or find the matrix eigenvectors. To accelerate the execution of such advanced tasks, in-memory computing with resistive memories provides a promising avenue, thanks to analog data storage and physical computation in the memory. As a result, complicated tasks such as solving a linear system or solving a differential equation require a large number of computing steps and an extensive use of memory units to store individual bits. Conventional digital computers can execute advanced operations by a sequence of elementary Boolean functions of 2 or more bits.

Solution of linear equation systems with matrix operations