Petroleum Reservoir Simulation

This paper surveys the state of the art in sparse matrix research in January 1976. Much of the survey deals with the solution of sparse simultaneous linear equations, including the storage of such matrices and the effect of paging on sparse matrix algorithms. In the symmetric case, relevant terms from graph theory are defined. Band systems and matrices arising from the discretization of partial differential equations are treated as separate cases. Preordering techniques are surveyed with particular emphasis on partitioning (to block triangular form) and tearing (to bordered block triangular form). Methods for solving the least squares problem and for sparse linear programming are also reviewed. The sparse eigenproblem is discussed with particular reference to some fairly recent iterative methods. There is a short discussion of general iterative techniques, and reference is made to good standard texts in this field. Design considerations when implementing sparse matrix algorithms are examined and finally comments are made concerning the availability of codes in this area.

A survey of sparse matrix research

The purpose of this note is to describe the simple extension of a popular method of solving second-order ordinary differential equations with two end-point boundary conditions to nth order ordinary differential equations and to partial differential equations that are second order in one direction. The orginal method is simply a version of Gaussian elimination; and the extension (to be described) has, we have discovered, been published, sliihtly differently, before. We feel, however, that the present note will be of value, since the extension has proven very useful to both of us, and is seldom used among numerical analysts and meteorologists. We begin by reviewing second-order ordinary differential equations. Consider 8x is the grid interval used in finite-difference approximation to equation (1),

A Reliable Method for the Numerical Integration of a Large Class of Ordinary and Partial Differential Equations

Several new fast direct methods for solving constant coefficient elliptic boundary value problems are presented. The methods make extensive use of the algebraic properties of the modified Chebyshev polynomials $S_n (x)$ and $C_n (x)$, which allow us to obtain operation counts of $O(n^2 )$ or $O(n^2 \log _2 ({n / k}))$ for solving problems on an $n \times n$ grid. The algorithms are shown to be numerically stable by giving a Wilkinson-style error analysis. Previously studied fast direct methods, and shooting and multiple shooting techniques are related to the algorithms.

Marching Algorithms for Elliptic Boundary Value Problems. I: The Constant Coefficient Case

A factoring and block elimination method for the fast numerical solution of block five diagonal linear algebraic equations is described. Applications of the method are given for the numerical solution of several boundary-value problems involving the bi- harmonic operator. In particular, 22 eigenvalues and eigenfunctions of the clamped square plate are computed and sketched.

/pdf/block-five-diagonal-matrices-and-the-fast-numerical-solution-48at9qpna7.pdf

Block five diagonal matrices and the fast numerical solution of the biharmonic equation

/pdf/quasi-tridiagonal-matrices-and-type-insensitive-difference-3z0ulphb0l.pdf

Quasi-tridiagonal matrices and type-insensitive difference equations

Various discretizations of boundary problems lead to such matrices. That this is so for Poisson's and the biharmonic equations is well known, and has been shown by 0. Karlqvist, A. F. Cornook and others. These authors prcpose the use of @irect, rather than iterative, methods to solve oroblems for these equations° Their methods are extended here to the general matrix (i.I) and it is shown that the positive symmetric difference systems of K. O. Friedrichs also fall into this c!ass~ These inc!ude~ in addition to pure elllptio

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Samuel Schechter

Author Tools

Chat about Author

Papers

Quasi-tridiagonal matrices and type-insensitive difference equations

Quasi-tridiagonal matrices and type-insensitive difference equations