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.

Algorithm 933: Reliable calculation of numerical rank, null space bases, pseudoinverse solutions, and basic solutions using suitesparseQR

Abstract: The SPQR_RANK package contains routines that calculate the numerical rank of large, sparse, numerically rank-deficient matrices. The routines can also calculate orthonormal bases for numerical null spaces, approximate pseudoinverse solutions to least squares problems involving rank-deficient matrices, and basic solutions to these problems. The algorithms are based on SPQR from SuiteSparseQR (ACM Transactions on Mathematical Software 38, Article 8, 2011). SPQR is a high-performance routine for forming QR factorizations of large, sparse matrices. It returns an estimate for the numerical rank that is usually, but not always, correct. The new routines improve the accuracy of the numerical rank calculated by SPQR and reliably determine the numerical rank in the sense that, based on extensive testing with matrices from applications, the numerical rank is almost always accurately determined when our methods report that the numerical rank should be correct. Reliable determination of numerical rank is critical to the other calculations in the package. The routines work well for matrices with either small or large null space dimensions.

Efficient detection algorithm for 2Nx2N MIMO systems using alamouti code and QR decomposition

Abstract: We propose an efficient 2Ntimes2N MIMO detection algorithm where the transmit signals are grouped in pairs and separately coded using the standard Alamouti space-time code. At the receiver, one or more QR decompositions are performed and the upper triangular property of the R matrices so obtained is exploited in order to successively decode the transmitted symbols starting with those interference-free symbols corresponding to the last two rows and columns of R. Bit-error-rate simulation results, for a 4times4 MIMO system and a bandwidth efficiency of 8 bits/s/Hz, show that the proposed technique, while less complex than ordered MMSE V-BLAST, outperforms the latter by 2-6 dB at a BER of 10-4

Matrix Shapes Invariant under the Symmetric QR Algorithm

Abstract: The QR algorithm is a basic algorithm for computing the eigenvalues of dense matrices. For efficiency reasons it is prerequisite that the algorithm is applied only after the original matrix has been reduced to a matrix of a particular shape, most notably Hessenberg and tridiagonal, which is preserved during the iterative process. In certain circumstances a reduction to another matrix shape may be advantageous. In this paper, we identify which zero patterns of symmetric matrices are preserved under the QR algorithm.

Fault tolerant matrix operations using checksum and reverse computation

Abstract: In this paper, we present a technique, based on checksum and reverse computation, that enables high-performance matrix operations to be fault-tolerant with low overhead. We have implemented this technique on five matrix operations: matrix multiplication, Cholesky factorization, LU factorization, QR factorization and Hessenberg reduction. The overhead of checkpointing and recovery is analyzed both theoretically and experimentally. These analyses confirm that our technique can provide fault tolerance for these high-performance matrix operations with low overhead.

Wavelets Based Neural Network for Function Approximation

Abstract: In this paper, a new type of WNN is proposed to enhance the function approximation capability. In the proposed WNN, the nonlinear activation function is a linear combination of wavelets, that can be updated during the networks training process. As a result the approximate error is significantly decreased. The BP algorithm and the QR decomposition based training method for the proposed WNN is derived. The obtained results indicate that this new type of WNN exhibits excellent learning ability compared to the conventional ones.