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Pipelined RLS FOR VLSI: STAR-RLS Filters
Published in Keshab K. Parhi, Takao Nishitani, Digital Signal Processing for Multimedia Systems, 2018
K. J. Raghunath, Keshab K. Parhi
The orthogonal matrix Q(n) can be determined as a product of Givens rotation matrices. A single Givens rotation matrix can nullify one element in the matrix on which it is applied. Suppose, we want to null the (n,m) element ynm using the (m,m) element ymm of a matrix Y. The Givens rotation matrix G can be defined as G(m,m)=cosϕG(m,n)=sinϕG(n,m)=−sinϕG(n,n)=cosϕG(i,i)=1,i≠n,m () G(i,j)=0,otherwise,
Linear Systems
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
A matrix such as the Qi above, consisting of Im with a plane rotation matrix inserted in its (k,k), (k,p), (p,k), and (p,p) entries in the natural way, is called a Givens matrix (or Givens rotation); the 2×2 matrix of Example 1 is a special case. For an m×n matrix with m≥n, we might in principle need (m−1)+(m−2)+…+1=m(m−1)/2 such matrices (working west to east across the matrix) to introduce all the zeroes needed to drive A to triangular form. This will give QNQN−1···Q1A=R (N=m(m−1)/2), or A=QR with Q=Q1TQ2T···QNT orthogonal and R upper triangular. This is the method of QR decomposition using Givens rotations.
QR decomposition for the least squares method: theory and practice
Published in International Journal of Mathematical Education in Science and Technology, 2022
The proposed examples of the QR decomposition implementation can be used not only for explanation of the concept but also as the source of exercises. They are mainly connected with computer programming and may be integrated into courses about numerical methods: Translation of the suggested MATLAB code to Python (with NumPy), Julia, R, Scilab or other programming language that supports matrix arithmetic.Implementation of the least squares method (based on Givens rotations or Householder reflections) using lower-level programming languages such as C, Pascal or Fortran.Implementation of alternative method of QR decomposition, e.g. the modified Gram-Schmidt method or fast Givens rotations.Exploring the performance issues and code profiling.Usage of the QR decomposition for transformation of a design matrix X to an orthogonal matrix, i.e. for replacement of ordinary polynomials to orthogonal ones.Implementation of parallelized versions of Givens or Householder methods, e.g. for GPU.
Sliced Rotated Sphere Packing Designs
Published in Technometrics, 2019
In He (2017b), G is recommended to be M*p in (3). A Givens rotation Rp(i, j, α) is the p × p identity matrix with the (i, i)th, (i, j)th, (j, i)th, and (j, j)th elements being replaced by cos (α), − sin (α), sin (α) and cos (α), respectively. For p = 2, R = I2 and w = 1 was recommended. For p > 2, it was recommended to use w = 100 and generate R's randomly by multiplying p(p − 1)/2 sequential Givens rotations with α sampled independently and uniformly from [0, 2π].