Important operations in numerical computing are vector orthogonalization. One of the well-known algorithms for vector orthogonalisation is Gram–Schmidt algorithm. This is a method for constructing a set of orthogonal vectors in an inner product space, most commonly the Euclidean space Rn. This process takes a finite, linearly independent set S = {b1, b2, ..., bk} vectors for k ≤ n and generates...

We invert the Fredholm equation representing the light scattered by a single spherical particle or a distribution of spherical particles to obtain the particle size distribution function and refractive index. We obtain the solution by expanding the distribution function as a linear combination of a set of orthonormal basis functions. The set of orthonormal basis functions is composed of Schmidt...

K e y w o r d s N u m e r i c a l linear algebra, QR factorization, Gram-Schmidt orthogonalization, Reorthogonalization, Rounding error analysis. 1. I N T R O D U C T I O N Scientific comput ing and ma themat i ca l models in engineering are becoming increasingly dependent upon development and implementa t ion of efficient paral le l a lgor i thms on modern high performance computers . Numerica...

A new inverse iteration algorithm that can be used to compute all the eigenvectors of a real symmetric tri-diagonal matrix on parallel computers is developed. The modified Gram-Schmidt orthogonalization is used in the classical inverse iteration. This algorithm is sequential and causes a bottleneck in parallel computing. In this paper, the use of the compact WY representation is proposed in the...

It is well-known that orthogonalization of column vectors in a rectangular matrix B with respect to the bilinear form induced by a nonsingular symmetric indefinite matrix A can be seen as its factorization B = QR that is equivalent to the Cholesky-like factorization in the form BTAB = RTΩR, where Ω is some signature matrix. Under the assumption of nonzero principal minors of the matrix M = BTAB...

In this paper, we compare with the inverse iteration algorithms on PowerXCell 8i processor, which has been known as a heterogeneous environment. When some of all the eigenvalues are close together or there are clusters of eigenvalues, reorthogonalization must be adopted to all the eigenvectors associated with such eigenvalues. Reorthogonalization algorithms need a lot of computational cost. The...

Convergence acceleration by preconditioning is usually essential when solving the standard least squares problems by an iterative method. IMGS, is an incomplete modiied version of Gram-Schmidt orthogonalization to obtain an incomplete orthogonal factorization pre-conditioner M = R, where A = Q R + E is an approximation of a QR factorization, Q is an orthogonal matrix and R is upper triangular m...

This report presents a faster version of the polynomial recursive orthogonalization (PRO) algorithm, which is used in segmented image coding to generate orthonormal base functions on a non-rectangular discrete image. It shows that up to 25% of the oating point operations can be eliminated by using previously computed results; also, some orthogonalizations can be avoided altogether, because they...