QR decomposition

About: QR decomposition is a research topic. Over the lifetime, 3504 publications have been published within this topic receiving 100599 citations. The topic is also known as: QR factorization.

A Constrained Least Squares Approach to Interactive Mesh Deformation

Abstract: In this paper, we propose a constrained least squares approach for stably computing Laplacian deformation with strict positional constraints. In the existing work on Laplacian deformation, strict positional constraints are described using large values of least squares weights, which often cause numerical problems when Laplacians are described using cotangent weights. In our method, we describe strict positional constraints as hard constraints. We solve the combination of hard and soft constraints by constructing a typical least squares matrix form using QR decomposition. In addition, our method can manage shape deformation under over-constraints, such as redundant and conflicting constraints. Our framework achieves excellent performance for interactive deformation of mesh models.

Algorithmic engineering in adaptive signal processing: worked examples

Abstract: Algorithmic engineering provides a rigorous framework for describing and manipulating the type of building blocks commonly used to define parallel algorithms and architectures for digital signal processing. So far, the concept has only been illustrated by means of relatively simple examples relating to the use of QR decomposition (QRD) by Givens rotations for the purposes of adaptive filtering and beamforming. Two more challenging examples are presented that illustrate the use of simple diagrammatic transformations to develop novel algorithms and architectures, and demonstrate the potential power of algorithmic engineering as a formal design technique. The first example constitutes the only known derivation of a modular processing architecture for generalised sidelobe cancellation based on QR decomposition. The second provides a simple derivation of the QRD-based lattice algorithm for multichannel least-squares linear prediction.

Computing the R of the QR factorization of tall and skinny matrices using MPI_Reduce

Abstract: A QR factorization of a tall and skinny matrix with n columns can be represented as a reduction. The operation used along the reduction tree has in input two n-by-n upper triangular matrices and in output an n-by-n upper triangular matrix which is defined as the R factor of the two input matrices stacked the one on top of the other. This operation is binary, associative, and commutative. We can therefore leverage the MPI library capabilities by using user-defined MPI operations and MPI_Reduce to perform this reduction. The resulting code is compact and portable. In this context, the user relies on the MPI library to select a reduction tree appropriate for the underlying architecture.

Dynamic Selection and Estimation of the Digital Predistorter Parameters for Power Amplifier Linearization

Abstract: This paper presents a new technique that dynamically estimates and updates the coefficients of a digital predistorter (DPD) for power amplifier (PA) linearization. The proposed technique is dynamic in the sense of estimating, at every iteration of the coefficient’s update, only the minimum necessary parameters according to a criterion based on the residual estimation error. At the first step, the original basis functions defining the DPD in the forward path are orthonormalized for DPD adaptation in the feedback path by means of a precalculated principal component analysis (PCA) transformation. The robustness and reliability of the precalculated PCA transformation (i.e., PCA transformation matrix obtained off line and only once) is tested and verified. Then, at the second step, a properly modified partial least squares (PLS) method, named dynamic partial least squares (DPLS), is applied to obtain the minimum and most relevant transformed components required for updating the coefficients of the DPD linearizer. The combination of the PCA transformation with the DPLS extraction of components is equivalent to a canonical correlation analysis (CCA) updating solution, which is optimum in the sense of generating components with maximum correlation (instead of maximum covariance as in the case of the DPLS extraction alone). The proposed dynamic extraction technique is evaluated and compared in terms of computational cost and performance with the commonly used QR decomposition approach for solving the least squares (LS) problem. Experimental results show that the proposed method (i.e., combining PCA with DPLS) drastically reduces the amount of DPD coefficients to be estimated while maintaining the same linearization performance.

Hardware-efficient matrix inversion algorithm for complex adaptive systems

Abstract: This work shows an FPGA implementation for the matrix inversion algebra operation. Usually, large matrix dimension is required for real-time signal processing applications, especially in case of complex adaptive systems. A hardware efficient matrix inversion procedure is described using QR decomposition of the original matrix and modified Gram-Schmidt method. This works attempts a direct VHDL description using few predefined packages and fixed point arithmetic for better optimization. New proposals for intermediate calculations are described, leading to efficient logic occupation together with better performance and accuracy in the vector space algebra. Results show that, for a relatively small device as Xilinx Spartan3 XC3S1000, a matrix size up to 23 × 23 can be implemented, having a matrix inversion computation time of 253μs. Accuracy results compared to floating point computation and an estimation of required clock cycles as a function of matrix size are analyzed.