Matrix banded

MODIFY - addressing of members of the coefficient matrix are adjusted to allocate their row and column index in the banded matrix. 

Dense (few zero elements) and small. A banded matrix has all zero elements except for a band centered on the main diagonal, e.g.,... 

Finite element methods [20,21] have replaced finite difference methods in many fields, especially in the area of partial differential equations. With the finite element approach, the continuum is divided into a number of finite elements that are assumed to be joined by a discrete number of points along their boundaries. A function is chosen to represent the variation of the quantity over each element in terms of the value of the quantity at the boundary points. Therefore a set of simultaneous equations can be obtained that will produce a large, banded matrix. 

The matrix presented in the above equation is clearly a banded matrix, with a bandwidth of 4. Note that the bandwidth is the maximum difference between node numbers of the elements of a given mesh times the number of degrees of freedom per node. Algorithm 8 computes the bandwidth of any mesh with nelem nodes per element. 

Algorithm 9 Global stiffness matrix and force vector assembly for a banded matrix... 

This is a band-structured matrix or a band matrix which is in this particular tridiagonal form also called the Jacobi matrix. Projecting both sides of Eq. (53) onto Pn+ fn+ I and using Eq. (59), we find ... 

In addition to the usual array representation which stores A(i,j,k) as A (((i-l)n+j-l)o+k), where A is dimensioned (m,n,o) and A1, (mno), Multilin provides representations taking advantage of sparsity and symmetry. Patterned sparsity such as that exhibited by a band diagonal matrix can be treated through another matrix. Thus a band matrix, A, dimensioned (n,r), with k lower... 

The structure of this formula is interesting. The X s and fi s lying by definition between 0 and 1, is a band matrix with a dominant diagonal = 2). The same is approximately true for close to 0.5). This shows that the m, s are re-... 

A proper numbering of the equations and unknowns ensures that the matrix representing the linear part of the differential operator will be a band matrix with bandwidth proportional to the lesser of m and n. Standard methods for decomposing such matrices exist (19), which allow savings in both storage and time. 

DDAPLUS will perform better if it is told to use band-matrix algorithms w henever the lower and upper bandwddths of G satisfy 2ml - - mu < Nstvar. The matrix G (and 1 if nondiagonal) wall then be stored more... 

Set Info(6) = l if G(t) is to be treated as a band matrix. Set the lower and upper bandwidths, Iwork(l)=ml and Iwork(2)=mu. These bandwidths are nonnegative integers such that the nonzero elements of the matrix lie in a band extending from the mlth subdiagonal to the muth superdiagonal. The efficiency will be further enhanced if the equations and unknowns are arranged so that ml[Pg.195]

Iwork(l) is accessed only when the user has requested band matrix algorithms by specifying Info(6) = l. This register must then be set to the lower bandwidth ml, defined under Info(6), of the iteration matrix G t). 

If Info(13) = 2 and Info(6) = 1, then E must be a banded matrix. Its work space is dimensioned as Ework( ) and its elements are inserted as follows ... 

If Info(5)=l and Info(6)=l, then df /du is treated as a banded matrix. Then Esub must declare Pdwork as a vector and insert its nonzero elements as follows ... 

This implies that we could determine a control polygon whose limit curve would interpolate all the points of Q. All we have to do is invert E and multiply Q by it. This is not in fact practical for two reasons. The first is that we do have to worry about end conditions to make E finite. The second is that although E is a narrow-banded matrix, its inverse is typically completely full. It is therefore much cheaper to solve the system EP = Q for P than either to invert E or to multiply Q by it. 

Consider that the system of linear discretization equations for phase k can be expressed on the general form Ak k = b. The and bfc vectors contain the variable solution and source terms for the whole calculation domain, respectively. Ak is a banded matrix containing the coefficients of the discretized equation. In three-dimensional problems, the neighbor coefficients are arranged in two sub-diagonals located next to the main diagonal and four peripherical diagonals ... 

For a collinear, electronically adiabatic reaction in the variables xAB, xBC, the procedure is to digitate these variables and number the finite-difference mesh so that the differential operator becomes a banded matrix. Then, the Schrodinger equation becomes... 

Table 5-6 Statement of examples used in the comparison of the 2N Newton-Raphson and the Almost Band matrix methods...

Example 2N Newton-Raphson methodf Almost Band matrix method ... 

The simple implementation of the translation operator is a consequence of a general property of the Fourier transform that a convolution of two functions in coordinate space becomes a multiplication of the transform function in momentum space. This fact can be used to study local implementations of the differential operators. In all local methods the derivative matrix D is a banded matrix. For example, consider the mapping of the fourth-order finite difference (FD) kinetic energy operator ... 

A (p, q)-banded matrix B has all non-zero elements contained in a band consisting of the principal diagonal together with p diagonals above it and q diagonals below it. This is shown as follows ... 

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. 

The conjugate gradient method is slower and less accurate than direct factorization when the positive definite matrix is dense as well as when it has a particular structure (i.e., a band matrix). 

This version solves a linear system or the equivalent minimization of a quadratic function using the PCG method where the preconditioner is the band matrix with the band dimension provided by the integer in the argument. 

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