scholarly journals A parallel Cholesky algorithm for the solution of symmetric linear systems

2004 ◽  
Vol 2004 (25) ◽  
pp. 1315-1327
Author(s):  
R. R. Khazal ◽  
M. M. Chawla
Keyword(s):  
Parallel Algorithm ◽  
Linear Systems ◽  
Triangular Matrix ◽  
Coefficient Matrix ◽  
Positive Definite ◽  
Identity Matrix ◽  
Lower Triangular Matrix ◽  
Elimination Method ◽  
Lower Triangular ◽  
Positive Definite System

For the solution of symmetric linear systems, the classical Cholesky method has proved to be difficult to parallelize. In the present paper, we first describe an elimination variant of Cholesky method to produce a lower triangular matrix which reduces the coefficient matrix of the system to an identity matrix. Then, this elimination method is combined with the partitioning method to obtain a parallel Cholesky algorithm. The total serial arithmetical operations count for the parallel algorithm is of the same order as that for the serial Cholesky method. The present parallel algorithm could thus perform withefficiencyclose to 1 if implemented on a multiprocessor machine. We also discuss theexistenceof the parallel algorithm; it is shown that for a symmetric and positive definite system, the presented parallel Cholesky algorithm is well defined and will run to completion.

scholarly journals Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Luis Verde-Star
Keyword(s):  
Triangular Matrix ◽  
Identity Matrix ◽  
Lower Triangular Matrix ◽  
Triangular Matrices ◽  
Hessenberg Matrices ◽  
Banded Matrices ◽  
Lower Triangular ◽  
And Inversion ◽  
Block Hessenberg Matrices ◽  
Explicit Expressions

AbstractWe use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.

scholarly journals CONNECTEDNESS OF SELF-AFFINE SETS WITH PRODUCT DIGIT SETS

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750053 ◽  
Author(s):  
JING-CHENG LIU ◽  
JUN JASON LUO ◽  
KE TANG
Keyword(s):  
Triangular Matrix ◽  
Sufficient Condition ◽  
Lower Triangular Matrix ◽  
Necessary And Sufficient Condition ◽  
Lower Triangular ◽  
Necessary And Sufficient ◽  
Affine Set

Let [Formula: see text] be an expanding lower triangular matrix and [Formula: see text]. Let [Formula: see text] be the associated self-affine set. In the paper, we generalize some connectedness results on self-affine tiles to self-affine sets and provide a necessary and sufficient condition for [Formula: see text] to be connected.

scholarly journals Matrix transformations of starshaped sequences

2001 ◽  
Vol 28 (4) ◽  
pp. 189-200
Author(s):  
Chikkanna R. Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Lower Triangular Matrix ◽  
Lower Triangular ◽  
Necessary And Sufficient

We deal with matrix transformations preserving the starshape of sequences. The main result gives the necessary and sufficient conditions for a lower triangular matrixAto preserve the starshape of sequences. Also, we discuss the nature of the mappings of starshaped sequences by some classical matrices.

scholarly journals The inverse along a lower triangular matrix

2012 ◽  
Vol 219 (3) ◽  
pp. 886-891 ◽  
Author(s):  
Xavier Mary ◽  
Pedro Patrício
Keyword(s):  
Triangular Matrix ◽  
Lower Triangular Matrix ◽  
Lower Triangular

scholarly journals On solving classes of positive-definite quantum linear systems with quadratically improved runtime in the condition number

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 573
Author(s):  
Davide Orsucci ◽  
Vedran Dunjko
Keyword(s):  
Linear Systems ◽  
Condition Number ◽  
Quantum Algorithms ◽  
Coefficient Matrix ◽  
Positive Definite ◽  
Fundamental Parameter ◽  
Worst Case ◽  
Complete Problems ◽  
Matrix Block ◽  
Potential Applications

Quantum algorithms for solving the Quantum Linear System (QLS) problem are among the most investigated quantum algorithms of recent times, with potential applications including the solution of computationally intractable differential equations and speed-ups in machine learning. A fundamental parameter governing the efficiency of QLS solvers is κ, the condition number of the coefficient matrix A, as it has been known since the inception of the QLS problem that for worst-case instances the runtime scales at least linearly in κ [Harrow, Hassidim and Lloyd, PRL 103, 150502 (2009)]. However, for the case of positive-definite matrices classical algorithms can solve linear systems with a runtime scaling as κ, a quadratic improvement compared to the the indefinite case. It is then natural to ask whether QLS solvers may hold an analogous improvement. In this work we answer the question in the negative, showing that solving a QLS entails a runtime linear in κ also when A is positive definite. We then identify broad classes of positive-definite QLS where this lower bound can be circumvented and present two new quantum algorithms featuring a quadratic speed-up in κ: the first is based on efficiently implementing a matrix-block-encoding of A−1, the second constructs a decomposition of the form A=LL† to precondition the system. These methods are widely applicable and both allow to efficiently solve BQP-complete problems.

An Inclusion Theorem for Bohr-Hardy Summability Factors

1971 ◽  
Vol 23 (4) ◽  
pp. 653-658 ◽  
Author(s):  
B. Thorpe
Keyword(s):  
Convergent Sequence ◽  
Triangular Matrix ◽  
Partial Sums ◽  
Its Sequence ◽  
Summability Factors ◽  
Lower Triangular Matrix ◽  
Inclusion Theorem ◽  
Sequence Transformation ◽  
Lower Triangular ◽  
Regular Transformation

1. Let A denote a sequence to sequence transformation given by the normal matrix A = (ank)(n, k = 0, 1, 2, …), i.e., a lower triangular matrix with ann ≠ 0 for all n. For B = (bnk) we write B ⇒ A if every B limitable sequence is A limitable to the same limit, and say that B is equivalent to A if B ⇒ A and A ⇒ B. If B is normal, then it is well known that the inverse of B exists (we denote it by B-l) and that B ⇒ A if and only if F = AB-1 is a regular transformation, i.e., transforms every convergent sequence into a sequence converging to the same limit. We say that a series ∑ an† is summable A if its sequence of partial sums is A-limitable.

A Hardy-Davies-Petersen Inequality for a Class of Matrices

1978 ◽  
Vol 30 (03) ◽  
pp. 458-465 ◽  
Author(s):  
P. D. Johnson ◽  
R. N. Mohapatra
Keyword(s):  
Triangular Matrix ◽  
Diagonal Matrix ◽  
Lower Triangular Matrix ◽  
Lower Triangular ◽  
Class Of Matrices

Let ω be the set of all real sequences a = ﹛an﹜ n ≧0. Unless otherwise indicated operations on sequences will be coordinatewise. If any component of a has the entry oo the corresponding component of a-1 has entry zero. The convolution of two sequences s and q is given by s * q . The Toeplitz martix associated with sequence s is the lower triangular matrix defined by tnk = sn-k (n ≧ k), tnk = 0 (n < k). It can be seen that Ts(q) = s * q for each sequence q and that Ts is invertible if and only if s0 ≠ 0. We shall denote a diagonal matrix with diagonal sequence s by Ds.

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