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Computational Characteristics of High Performance Embedded Algorithms and Applications
Published in David R. Martinez, Robert A. Bond, Vai M. Michael, High Performance Embedded Computing Handbook, 2018
Arakawa Masahiro, A. Bond Robert
The QR decomposition (or equivalently the LQ) is used extensively in adaptive signal and image processing as part of the weight computation algorithm. The complexity derivations for the complex and real cases are similar. The complex case is examined below. A complex-data matrix A∈Cmxn can be factored into an unitary matrix Q∈Cmxn and an upper triangular matrix R∈Cnxn so that A = QR. The QR decomposition can be accomplished using three algorithms: Modified Gram-Schmidt, Givens, and Householder. The complexity of the Householder algorithm is derived, but all of the algorithms are based on the idea of iteratively constructing the upper triangular matrix R by applying a series of unitary transformation matrices to A: Pn∗…∗P1∗A=R.
Linear Algebra
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Fatemeh Hamidi Sepehr, Erchin Serpedin
Let A be an m × n matrix of rank n. The QR decomposition consists in the factorization of matrix A into a product QR, in which Q is an m × n matrix consisting of orthonormal column vectors and R is a non-singular upper triangular n × n matrix. Let a1, a2,…, an be the columns of A such that the column space of A coincides with span (a1, a2,…, an). Consider also the matrix: () Q=[q1,q2,…,qn],
State Estimation
Published in James A. Momoh, Electric Power System Applications of Optimization, 2017
Other methods called the orthogonal transformation method and the hybrid method directly perform the QR decomposition of the Jacobian matrix. The QR decomposition, also known as the QR factorization of a matrix, is a decomposition of the matrix into an orthogonal and triangular matrix. It is a method often used to solve the least squares problem. As a result, these are more stable and preferable to the NE method. There exist, however, some disadvantages. Namely, with regard to the orthogonal transformation method, Q needs to be stored which would require costly memory to store the nonspare and high dimension matrix. With the hybrid method is comparatively less stable than the orthogonal transformation method.
Spectral Efficient Precoding Design for Multi-cell Large MU-MIMO System
Published in IETE Journal of Research, 2022
The multi-cell BD precoding scheme represented by Equation (33) is the basis for the design of our proposed precoding. To design this precoding scheme, we have utilized SVD technique to determine the null space. Now, we utilize QR decomposition instead of SVD in the first step of multi-cell BD precoding to lower the computational complexity of the proposed precoding design which eliminates intercell interference from adjacent cells [46,49]. The QR decomposition of matrix produces two output matrices, the upper triangular matrix Rj,all and orthogonal matrix Qj,all. The QR operation performed on intercell interference matrix can be Since ,
An estimation scheme of road friction coefficient based on novel tyre model and improved SCKF
Published in Vehicle System Dynamics, 2022
Zhida Zhang, Ling Zheng, Hang Wu, Ziwei Zhang, Yinong Li, Yixiao Liang
Step 2: Calculate the predicted value of state and the square-root factor of its error covariance matrix The weighted central matrix is defined as follows: where the operation of is defined as follows: let be the upper triangular matrix of the QR decomposition of matrix , then the QR decomposition algorithm of matrix can be expressed as , and is the lower triangular matrix. Measurement update
Recursive orthogonal least squares based adaptive control of a polymerisation reactor
Published in Indian Chemical Engineer, 2019
Sarthak Tiwari, Purushottam Sawant, Imran Rahman
A recursively updated RBF network-based model was applied in an IMC framework to control the non linear polymerisation process. ROLS algorithm was used to update the weight coefficients of the RBF network in real time. Open loop simulations show excellent tracking ability of ROLS based RBF. One advantage of ROLS algorithm, over other learning algorithms, is that the number of tuning parameters required are less, in our case only the centre vectors. Furthermore, it’s algorithmic complexity is low as it is based on a simple QR decomposition. Hence the time required in the training and testing phases is also less. The fact that ROLS algorithm can handle a large data set, is time efficient and has low computational demands is an interesting feature. Also, the proposed algorithm fits well in the IMC framework. Closed loop simulations of the continuous polymerisation process, under a variety of setpoint changes and disturbances, show that ROLS based IMC controller outperforms fixed model IMC and conventional PID for both setpoint tracking and disturbance rejection. Furthermore, ROLS based IMC, which is based on an adaptive algorithm, is expected to be more robust to changes in system dynamics and variations in time delays. Whereas in order to attain comparable results, a linear PID controller would have to be tuned separately for each case, which is not possible in continuous operation.