SIAM Digital Library
 
 
 

SIAM J. on Scientific Computing

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue

1981

Volume 2, Issue 4, pp. 375-489


Solution of Homogeneous Systems of Linear Equations Arising from Compartmental Models

R. E. Funderlic and J. B. Mankin

SIAM J. Sci. and Stat. Comput. 2, pp. 375-383 (9 pages) | Cited 26 times

Full Text: | Download PDF

Show Abstract
Systems of linear differential equations with constant coefficients, $Ax = \dot x$, with the matrix $A$ having nonnegative off-diagonal elements and zero column sums, occur in compartmental analysis. The steady-state solution leads to the homogeneous system of linear equations $Ax(\infty ) = \dot x(\infty ) = 0$. $LU$-factorization, the Crout algorithm, error analysis and solution of a modified system are treated.

Condition Number Estimation for Sparse Matrices

Roger G. Grimes and John G. Lewis

SIAM J. Sci. and Stat. Comput. 2, pp. 384-388 (5 pages) | Cited 5 times

Full Text: | Download PDF

Show Abstract
The LINPACK package of linear equation solving software provides a reliable and inexpensive algorithm for estimating the condition number of a dense matrix. The direct generalization to banded or sparse matrices is reliable, but not necessarily inexpensive. The simple modification described in this note can bring the cost of the algorithm back to a reasonable level.

Numerical Technique to Trace the Loci of the Complex Roots of Characteristic Equations

E. Bahar and M. Fitzwater

SIAM J. Sci. and Stat. Comput. 2, pp. 389-403 (15 pages) | Cited 1 time

Full Text: | Download PDF

Show Abstract
A numerical technique is presented to compute the complex roots of characteristic equations for a wide class of engineering problems. Approximate values for the roots need not be known a priori and closed form analytical expressions for the derivative of the modal equations are not required. A method has also been developed to trace the loci of the complex roots as one or several parameters of the characteristic equation vary.
The key to the method described in this paper is the determination of the covering space on which all the complex roots lie. As a result the complex roots can be found using standard numerical techniques developed for the real zero problem.
As an illustrative example, the solution of the modal equation for the vertically polarized waves in an irregular spheroidal model of the earth-ionosphere waveguide is presented. Since it is not necessary to use several pole-free forms of the modal equation to overcome overflow and underflow problems in the numerical computations, one can avoid the inadvertent search for “phantom” roots and the possibility of losing a root due to the necessity to switch functions over the complex

Multiple Solutions and Bifurcation of Finite Difference Approximations to Some Steady Problems of Fluid Dynamics

A. B. Stephens and G. R. Shubin

SIAM J. Sci. and Stat. Comput. 2, pp. 404-415 (12 pages) | Cited 10 times

Full Text: | Download PDF

Show Abstract
We review and extend an earlier study of the behavior of multiple finite difference solutions for a centered difference approximation of the steady Burgers’ equation. Using the fact that all of the inviscid (viscosity = 0) solutions can be found, we numerically continue these solutions with respect to viscosity and thereby uncover turning points and bifurcation points. In addition, we demonstrate analogous behavior for a model of one-dimensional duct flow and for a particular discretization of the supersonic blunt body problem.

Solution of Large-Scale Sparse Least Squares Problems Using Auxiliary Storage

J. A. George, M. T. Heath, and R. J. Plemmons

SIAM J. Sci. and Stat. Comput. 2, pp. 416-429 (14 pages) | Cited 9 times

Full Text: | Download PDF

Show Abstract
Very large sparse linear least squares problems arise in a variety of applications, such as geodetic network adjustments, photogrammetry, earthquake studies, and certain types of finite element analysis. Many of these problems are so large that it is impossible to solve them without using auxiliary storage devices. Some problems are so massive that the storage needed for their solution exceeds the virtual address space of the largest machines. In this paper we describe a method for solving such problems on a typical (large) computer and provide the results of some experiments illustrating the effectiveness of our approach. The method includes an automatic partitioning scheme which is essential to the efficient management of the data on auxiliary files.

The Multi-Grid Method for the Diffusion Equation with Strongly Discontinuous Coefficients

R. E. Alcouffe, Achi Brandt, J. E. Dendy, Jr., and J. W. Painter

SIAM J. Sci. and Stat. Comput. 2, pp. 430-454 (25 pages) | Cited 121 times

Full Text: | Download PDF

Show Abstract
The subject of this paper is the application of the multi-grid method to the solution of \[ - \nabla \cdot (D(x,y)\nabla U(x,y)) + \sigma (x,y)U(x,y) = f(x,y) \] in a bounded region $\Omega $ of $R^2 $ where $D$ is positive and $D$, $\sigma $, and $f$ are allowed to be discontinuous across internal boundaries $\Gamma $ of $\Omega $. The emphasis is on discontinuities of orders of magnitude in $D$, when special techniques must be applied to restore the multi-grid method to good efficiency. These techniques are based on the continuity of $D\nabla U$ across $\Gamma $. Two basic methods are derived, one in which the approximating finite difference operators on coarser grids are five point operators (assuming the finite difference operator on the finest grid is a five point one) and one in which they are nine point operators.

A Two-Dimensional Mesh Verification Algorithm

R. B. Simpson

SIAM J. Sci. and Stat. Comput. 2, pp. 455-473 (19 pages) | Cited 4 times

Full Text: | Download PDF

Show Abstract
A finite element mesh is usually represented in a program by lists of data, i.e., vertex coordinates, element incidences, boundary data. This paper is concerned with conditions on the list data which ensure that the lists describe a “tiling” of some planar region without overlap or gaps. For a particular format of lists, a set of such conditions is given which is proven to be sufficient to guarantee such a “tiling”. These conditions have been chosen so as to be verifiable by the algorithm referred to in the title, which is described in detail and is claimed to be of reasonable efficiency.

A Bidiagonalization-Regularization Procedure for Large Scale Discretizations of Ill-Posed Problems

Dianne P. O’Leary and John A. Simmons

SIAM J. Sci. and Stat. Comput. 2, pp. 474-489 (16 pages) | Cited 30 times

Full Text: | Download PDF

Show Abstract
In this paper, we consider ill-posed problems which discretize to linear least squares problems with matrices $K$ of high dimensions. The algorithm proposed uses $K$ only as an operator and does not need to explicitly store or modify it. A method related to one of Lanczos is used to project the problem onto a subspace for which $K$ is bidiagonal. It is then an easy matter to solve the projected problem by standard regularization techniques. These ideas are illustrated with some integral equations of the first kind with convolution kernels, and sample numerical results are given.
Close

Your Recent History Recent History Help

Recently Viewed

  1. Iterative Solution of Cyclically Reduced Systems Arising from Discretization of the Three-Dimensiona...
  2. Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws
  3. A Fast Multilevel Algorithm for the Solution of Nonlinear Systems of Conductive-Radiative Heat Trans...
  4. Parallel Newton--Krylov--Schwarz Algorithms for the Transonic Full Potential Equation
  5. Numerical Solution of the Riemann Problem for Two-Dimensional Gas Dynamics
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close