Sparse and 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. 

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 Antarctic coal beds are apparently less persistent, and locally may be thicker, than many of the beds in Paleozoic coal fields of North America. It is hazardous to generalize about petrographic composition from hand specimens that are available from many of the coal beds, but one obtains the impression that dull, moderately dull, and midlustrous attrital layers are more prevalent than in Paleozoic coal of the Northern Hemisphere. Vitrain bands tend to be relatively sparse and thin fusain chips and partings generally are present and may be abundant. Many coal specimens are relatively impure, apparently owing to well-dispersed detrital mineral matter. 

Generally, the uncertainty in determining the AH stretching frequency increases with the width and complexity of the band for strong hydrogen bonds it can amount to 50 cm-1 ( 3%) and even more. The low temperature study of crystals in the infrared and in the Raman can improve the accuracy of the band position considerably. However, crystallographic data about R(A.. B) distances at low temperature are sparse, and the comparison of frequency vs. distance is thus less reliable. 

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. 

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. 

Irrespective of boundary conditions, it is easy to see that this matrix is sparse and highly banded. It connects each (field)... 

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... 

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. 

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