Sparse matrix banded

Note that in equation system (2.64) the coefficients matrix is symmetric, sparse (i.e. a significant number of its members are zero) and banded. The symmetry of the coefficients matrix in the global finite element equations is not guaranteed for all applications (in particular, in most fluid flow problems this matrix will not be symmetric). However, the finite element method always yields sparse and banded sets of equations. This property should be utilized to minimize computing costs in complex problems. 

Although this algorithm is clear and simple, it presents the most ineffective way of storing the global stiffness matrix since it results in a full sparse matrix. Later in this section we will discuss how the storage space and computation time is minimized by using alternative storing schemes such as banded matrices. 

The equations will still form the same block-banded sparse matrix as in the Naphtali-Sandholm method. No matter what size the time step, the same matrix solution technique can be used to calculate the next set of independent variables. 

There are several general characteristics of a matrix that are particularly useful for analysis of minimization algorithms. Density of a matrix is a measurement given by the ratio of the nonzero to zero matrix components. A matrix is said to be dense when this ratio is large and sparse when it is small. A sparse matrix may be structured (e.g., block diagonal, band) or unstructured (Figure 2). 

After the Broyden correction for the independent variables has been computed, Broyden proposed that the inverse of the jacobian matrix of the Newton-Raphson equations be updated by use of Householder s formula. Herein lies the difficulty with Broyden s method. For Newton-Raphson formulations such as the Almost Band Algorithm for problems involving highly nonideal solutions, the corresponding jacobian matrices are exceedingly sparse, and the inverse of a sparse matrix is not necessarily sparse. The sparse characteristic of these jacobian matrices makes the application of Broyden s method (wherein the inverse of the jacobian matrix is updated by use of Householder s formula) impractical. 

In general, we choose compact wavelets (i.e. only a finite number of coefficients for the dilation and wavelet equations are non-zero) and therefore only a finite number of Aj blocks are non-zero. In this case the matrix W is sparse and banded (see Section 4.2.2). Compactly supported wavelets have good localisation properties but may not always have a high degree of smoothness (e.g. the Haar wavelet). 

The Jacobian for this problem is a sparse constant matrix (banded with bandwidth 5). The property... 

Figure 8.10 gives MATLAB m-file Band.m for implementing the band algorithm. An alternative formulation of the problem is possible in MAT-LAB. Rather than using the band algorithm directly, we can use the sparse matrix capability of MATLAB. The matrix to be inverted can be defined and stored in the sparse or band format (see Chapter 2, section 2.2.3.3). This option is particularly efficient in MATLAB. 

If Equation 15-34 is to be written for each (i,j,k) node and solved at the new time step (n+1), we obtain a complicated system of algebraic equations that is costly to invert computationally. When it cannot be locally linearized, the full but sparse matrix is solved using even more expensive Newton-Raphson iterations. Thus, we employ approximate factorization techniques to resolve the system into three simpler, but sequential banded ones. In this approach. 

A matrix is termed sparse if it has a large percentage of zero entries otherwise it is dense. A sparse matrix can be structured (as in a banded matrix where there are zeros for I — j > p) or unstructured, as shown in Figure 3(a). This... 

Even though it is possible to store a sparse matrix efficiently, can we efficiently solve a system by Gaussian elimination using this notation We can if the matrix is banded i.e., if the nonzero values are clustered in the vicinity of the principal diagonal. 

The MATLAB elimination solver , also known as the midivide function, can handle matrices stored in sparse-matrix format. If the matrix is banded, the bandwidth is determined and the elimination algorithm modified accordingly. If the matrix is not banded, the solver attempts to reduce the bandwidth as much as possible by applying a hemistic algorithm that interchanges rows and columns. 

As the number of elements in the mesh increases the sparse banded nature of the global set of equations becomes increasingly more apparent. However, as Equation (6,4) shows, unlike the one-dimensional examples given in Chapter 2, the bandwidth in the coefficient matrix in multi-dimensional problems is not constant and the main band may include zeros in its interior terms. It is of course desirable to minimize the bandwidth and, as far as possible, prevent the appearance of zeros inside the band. The order of node numbering during... 

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. 

It can be seen as four different submatrices An, A12, A21, A22, where An is band matrix with lower band size 3 and upper band size 1, A12 is extremely sparse as A21, and the smallest matrix A22 is dense. 

---