Inversion, matrix

The inverse of a matrix is defined as a matrix which, when multiplied by the original matrix, gives the identity matrix. This is a diagonal matrix (a square matrix with terms on the diagonal but zeros on all the olT-diagonal positions) with 1 terms on the diagonal. The 2x2 identity matrix is 1  

The inverse of a matrix is usually calculated numerically, particularly in realistically complex engineering applications. Standard computer library subroutines are readily available. We use the IMSL subroutine LEQ2C in this book to calculate the inverse of a complex matrix. 

Inverses of simple matrices can be calculated analytically from the equation 

The adjoint of a matrix is the transpose of the matrix which is formed by replacing each clement with its cofactor, A cofactor is the determinant formed by eliminating the row and column in which each element lies and using the 

Example 15l6. The inverse of the d matrix of Example 15,1 is found from the following steps  

We find the inverse of the matrix in (1.46). On input, the original matrix is stored in the array A, and its inverse will occupy the array on output. Performing LL) decomposition by the module M14, the original matrix will be destroyed. The program and the output are as follows  

You may find interesting to compare the results with the output in Example 

As you learned in the previous sections, LU decomposition with built-in partial pivoting, followed by backsubstitution is a good method to solve the matrix equation Ax = b. You can use, however, considerable simpler technics if the matrix A has some special structure. In this section we assume that A is symmetric (i.e., AT = A), and positive definite (i.e., x Ax 0 for all x 0 you will encounter the expression x Ax many times in this book, and hence we note that it is called quadratic form.) The problem considered here is special, but very important. In particular, estimating parameters in Chapter 3 you will have to invert matrices of the form A = X X many times, where X is an nxm matrix. The matrix X X is clearly symmetric, and it is positive definite if the columns of X are linearly independent. Indeed, x (x X)x = (Xx) (Xx) 0 for every x since it is a sum of squares. Thus (Xx) (Xx) = 0 implies Xx = 0 and also x = 0 if the columns of X are linearly independent. 

The method (ref. 2) is based an solving the matrix equation = Ax, where is not a fixed right-hand side, but a vector of variables 1 2 with completely free values. To solve the equation for x in terms of notice that an 0 due to positive definiteness of A, since an = (e ) Ae. We can therefore solve the first equation for x, and replace x by the resulting expression in the other equations  

To proceed we have to assume that a 22 can s lown that this 

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There are two matrix inverses that appear on the right-hand side of these equations. One of these is trivial the... 

I to model systems with fewer atoms than molecular mechanics some operations that integral to certain minimisation procedures (such as matrix inversion) are trivial for... 

In conjunction with the discrete penalty schemes elements belonging to the Crouzeix-Raviart group arc usually used. As explained in Chapter 2, these elements generate discontinuous pressure variation across the inter-element boundaries in a mesh and, hence, the required matrix inversion in the working equations of this seheme can be carried out at the elemental level with minimum computational cost. 

The attractive feature in matrix inversion is seen by premultiplying both sides of Ax = b by A , ... 

MOBAS was written by the author (Rogers, 1983) in BASIC to illustrate matrix inversion in molecular orbital calculations. It is modeled after a program in FORTRAN n given by Dickson (Dickson, 1968). 

Semiempiricals For very large molecules (limited by matrix inversion)... 

This completes tl foi ward steps to give AXv = —Fy. Remaining values of corr tions AX ar obtjined by successive backward substitution from AX = —Fj —(Fj — CjFj + i). Matrix inversions are best done by... 

There are various ways to obtain the solutions to this problem. The most straightforward method is to solve the full problem by first computing the Lagrange multipliers from the time-differentiated constraint equations and then using the values obtained to solve the equations of motion [7,8,37]. This method, however, is not computationally cheap because it requires a matrix inversion at every iteration. In practice, therefore, the problem is solved by a simple iterative scheme to satisfy the constraints. This scheme is called SHAKE [6,14] (see Section V.B). Note that the computational advantage has to be balanced against the additional work required to solve the constraint equations. This approach allows a modest increase in speed by a factor of 2 or 3 if all bonds are constrained. 

While it is possible to reeover the original distribution from the leading moments, e.g. by matrix inversion, it is often the ease that simply the mean eharaeteristies together with an estimate of the spread of the distribution are suffieient e.g. 

The left-hand side is the convolution of ViPQ(ri) with the familiar expression - i)IPa ) - < Qp(rii 2)], which has a matrix inverse given... 

A linear coordinate transformation may be illustrated by a simple two-dimensional example. The new coordinate system is defined in term of the old by means of a rotation matrix, U. In the general case the U matrix is unitary (complex elements), although for most applications it may be chosen to be orthogonal (real elements). This means that the matrix inverse is given by transposing the complex conjugate, or in the... 

If det C 0, C exists and can be found by matrix inversion (a modification of the Gauss-Jordan method), by writing C and 1 (the identity matrix) and then performing the same operations on each to transform C into I and, therefore, I into C". ... 

Recall that the matrix CT is formed by taking every row of C and placing it as a column in CT. Next, we eliminate the quantity [C CT] from the right-hand side of equation [31]. We can do this by post-multiplying each side of the equation by [C CT] the matrix inverse of [C CT]. 

Multiple Linear Regression (MLR), Classical Least-Squares (CLS, K-matrix), Inverse Least-Squares (ILS, P-matrix)... 

Now we are ready to formulate the basic idea of the correction algorithm in order to correct the four-indexed operator f1, it is enough to correct the two-indexed operators fc and f 1 in the supermatrix representation (7.100). The real advantage of this proposal is its compatibility with any definite way of f2 and P7 correction [61, 294], The matrix inversion demanded in (7.99) is divided into two stages. In the fi, v subspace it is possible to find the inverse matrix analytically with the help of the Frobenius formula that is well known in matrix algebra [295]. The... 

If we consider the relative merits of the two forms of the optimal reconstructor, Eq. s 16 and 17, we note that both require a matrix inversion. Computationally, the size of the matrix inversion is important. Eq. 16 inverts an M x M (measurements) matrix and Eq. 17 a P x P (parameters) matrix. In a traditional least squares system there are fewer parameters estimated than there are measurements, ie M > P, indicating Eq. 16 should be used. In a Bayesian framework we are hying to reconstruct more modes than we have measurements, ie P > M, so Eq. 17 is more convenient. 

MCT allows one to choose any conceivable error distribution for the variables, and to transform these into a result by any set of equations or algorithms, such as recursive (e.g., root-finding according to Newton) or matrix inversion (e.g., solving a set of simultaneous equations) procedures. Characteristic error distributions are obtained from experience or the literature, e.g.. Ref. 95. 

The PLS algorithm is relatively fast because it only involves simple matrix multiplications. Eigenvalue/eigenvector analysis or matrix inversions are not needed. The determination of how many factors to take is a major decision. Just as for the other methods the right number of components can be determined by assessing the predictive ability of models of increasing dimensionality. This is more fully discussed in Section 36.5 on validation. 

Figure 10.1 Schematic diagram of the sequential solution of model and sensitivity equations. The order is shown for a three parameter problem. Steps l, 5 and 9 involve iterative solution that requires a matrix inversion at each iteration of the fully implicit Euler s method. All other steps (i.e., the integration of the sensitivity equations) involve only one matrix multiplication each.

The above equations are developed using the theory of least squares and making use of the matrix inversion lemma... 

Equations 13.14 to 13.16 constitute the well known recursive least squares (RLS) algorithm. It is the simplest and most widely used recursive estimation method. It should be noted that it is computationally very efficient as it does not require a matrix inversion at each sampling interval. Several researchers have introduced a variable forgetting factor to allow a more precise estimation of 0 when the process is not "sensed" to change. 

Tan, T.B. and J.P. Letkeman, "Application of D4 Ordering and Minimization in an Effective Partial Matrix Inverse Iterative Method", paper SPE 10493 presented at the 1982 SPE Symposium on Reservoir Siumulation, San Antonio, TX (1982). 

Westlake, J. R. (1968) A handbook of numerical matrix inversion and solution of linear equations (Wiley). 

The above model was solved numerically by writing finite difference approximations for each term. The equations were decoupled by writing the reaction terms on the previous time steps where the concentrations are known. Similarly the equations were linearized by writing the diffusivities on the previous time step also. The model was solved numerically using a linear matrix inversion routine, updating the solution matrix between iterations to include the proper concentration dependent diffusivities and reactions. 

The diagonal elements of the matrix A are af1 and the off-diagonal elements of Aij are Ty. Equation (9-21) determines how the dipoles are coupled to the static electric field. There are three major methods to determine the dipoles matrix inversion, iterative methods and predictive methods. 

Although a direct comparison between the iterative and the extended Lagrangian methods has not been published, the two methods are inferred to have comparable computational speeds based on indirect evidence. The extended Lagrangian method was found to be approximately 20 times faster than the standard matrix inversion procedure [117] and according to the calculation of Bernardo et al. [208] using different polarizable water potentials, the iterative method is roughly 17 times faster than direct matrix inversion to achieve a convergence of 1.0 x 10-8 D in the induced dipole. 

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