Sparse direct method

Duff, I. S. Direct Methods for Sparse Matrices, Oxford, Charendon Press (1986). 

To avoid such small time steps, which become smaller as Ax decreases, an implicit method could be used. This leads to large, sparse matrices rather than convenient tridiagonal matrices. These can be solved, but the alternating direction method is also useful (Ref. 221). This reduces a problem on an /i X n grid to a series of 2n one-dimensional problems on an n grid. 

More often, however, s.a. methods are applied at the outset, with more or less arbitrary initial approximations, and without previous application of a direct method. Usually this is done if the matrix is extremely large and extremely sparse, which is most often true of the matrices that arise in the algebraic representation of a functional equation, especially a partial differential equation. The advantage is that storage requirements are much more modest for s.a. methods. 

Sparse matrices are ones in which the majority of the elements are zero. If the structure of the matrix is exploited, the solution time on a computer is greatly reduced. See Duff, I. S., J. K. Reid, and A. M. Erisman (eds.), Direct Methods for Sparse Matrices, Clarendon Press, Oxford (1986) Saad, Y., Iterative Methods for Sparse Linear Systems, 2d ed., Society for Industrial and Applied Mathematics, Philadelphia (2003). The conjugate gradient method is one method for solving sparse matrix problems, since it only involves multiplication of a matrix times a vector. Thus the sparseness of the matrix is easy to exploit. The conjugate gradient method is an iterative method that converges for sure in n iterations where the matrix is an n x n matrix. 

The most valuable information one should be able to obtain from the carbon spectrum of a coal derivative is the ratio of aromatic to nonaromatic carbon atoms. We know of no other direct method by which this value can be obtained. Since the literature pertinent to C13 NMR is sparse, we wish to... 

The round-off error propagation associated with the use of Shacham and Kehat s direct method for the solution of large sparse systems of linear equations is investigated. A reordering scheme for reducing error propagation is proposed as well as a method for iterative refinement of the solution. Accurate solutions for linear systems, which contain up to 500 equations, have been obtained using the proposed method, in very short computer times. 

Shacham, M. Kehat, E., "A Direct Method for the Solution of Large Sparse Systems of Linear Equations", Comp. J., 1976, 1 ( 0, 353. 

There are four methods for solving systems of linear equations. Cramer s rule and computing the inverse matrix of A are inefficient and produce inaccurate solutions. These methods must be absolutely avoided. Direct methods are convenient for stored matrices, i.e. matrices having only a few zero elements, whereas iterative methods generally work better for sparse matrices, i.e. matrices having only a few non-zero elements (e.g. band matrices). Special procedures are used to store and fetch sparse matrices, in order to save memory allocations and computer time. 

The matrices derived from partial differential equations are always sparse, i.e. most of their elements are zero. For one-dimensional systems the discretization process leads to tri-diagonal systems, a system with only three non-zero coefficients per equation. Since the systems are often very large we find that iterative methods are generally much more economical than direct methods. 

Duff, I. S., Erisman, A. M., and Reid, J. K. (1987), Direct Methods for Sparse Matrices, Oxford University Press, New York. 

The strategy of use of sparseness of the coefficient matrix by direct methods can be divided into three groups ... 

George J.A. (1977) Solutions of Linear Systems of Equations Direct Methods for Finite Element Probelms. In Barker V.A. (ed) Sparse Matrix Techniques. Lecture notes in mathematic, Vol. 572, Springer Berlin, Heidelberg, New York, pp 52-101. 

There are a number of methods available to solve for the solution of a given set of linear algebraic equations. One class is the direct method (i.e., requires no iteration) and the other is the iterative method, which requires iteration as the name indicates. For the second class of method, an initial guess must be provided. We will first discuss the direct methods in Section B.5 and the iterative methods will be dealt with in Section B.6. The iterative methods are preferable when the number of equations to be solved is large, the coefficient matrix is sparse and the matrix is diagonally dominant (Eqs. B.8 and B.9). 

Iterative algorithms are recommended for some linear systems Ax = b as an alternative to direct algorithms. An iteration usually amounts to one or two multiplications of the matrix A by a vector and to a few linear operations with vectors. If A is sparse, small storage space suffices. This is a major advantage of iterative methods where the direct methods have large fill-in. Furthermore, with appropriate data structures, arithmetic operations are actually performed only where both operands are nonzeros then, D A) or 2D A) flops per iteration and D(A) + 2n units of storage space suffice, where D(A) denotes the number of nonzeros in A. Finally, iterative methods allow implicit symmetrization, when the iteration applies to the symmetrized system A Ax = A b without explicit evaluation of A A, which would have replaced A by less sparse matrix A A. 

The inner loop starts at this step. In this loop the variables Sp RpRj, and Qj are calculated to satisfy Equations 13.49,13.50,13.51, and 13.52, using the simple thermodynamic models. Begin by computing from Equation set 13.49. Eor each component i, these are N linear equations represented by a tridiagonal matrix which can be solved by a special sparse matrix method, the Thomas algorithm, described further along in this section. Next, VJ, are calculated from Equation set 13.50, and Lj, Vj, and Xj are calculated directly ... 

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