Matrix inverse numerical calculation

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. 

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

Note that the most expensive part of the numerical calculations is the determination of the matrix Q using the Lanczos method. This matrix depends only on the coefficients of the matrix and the vector fic. Therefore, due to the fact that matrix does not depend on frequency, we should apply this decomposition only once for all frequency ranges (if also vector c does not depend on frequency, which is typical for many practical problems). The calculation of the inverse of the matrix (T+iujfj,(T) is computationally a much simpler problem, because T is a tri-diagonal matrix, and jj, and diagonal matrices. As a result, one application of SLDM allows us to solve forward problems for the entire frequency range. That is why SLDM increases the speed of solution of the forward problem by an order for multifrequency data. This is the main advantage of this method over any other approach. 

Another matrix, called the N matrix, is obtained by inverting the 1 - Q matrix. Matrix inversion can be done analytically for small matrices, but it is impractical to do for large realistic matrices that represent more complicated systems. A reasonable solution is available, however. Many spreadsheet programs in common use have the ability to numerically invert a matrix. This tool can be used to make quick work of previously time-consuming MTTF calculations. A numerical matrix inversion of the (1 - Q) matrix using a spreadsheet provides the following ... 

Note that this formula involves the inversion of XjX, the covariance matrix of X. If this inverse does not exist, then A cannot be calculated by means of MLR. Singularity of XjX corresponds to the linear dependence of a subset of the X variables. So, if X contains such relationships, MLR cannot be applied. In practice, it is rare to have exact linear dependence, since the measurement errors will tend to preclude this. However, near linearity will tend to make the inverse numerically unstable and subject to large errors, so much the same effect occurs. Another formulation of the... 

It is instructive to compare the approximate weak-coupling theory to essential exact, numerical (density matrix renormalization group) calculations on the same model (namely the Pariser-Parr-Pople model). The numerical calculations are performed on polymer chains with the polyacetylene geometry. Since these chains posses inversion symmetry the many-body eigenstates are either even (Ag) or odd By). As discussed previously, the singlet exciton wave function has either even or odd parity when the particle-hole eigenvalue is odd or even. Conversely, the triplet exciton wavefunction has either even or odd parity when the particle-hole eigenvalue is even or odd. As a consequence, we can express a B state as... 

The matrix inversion J is performed by the Gauss elimination method applied to linear equations. The derivatives d//dy, are calculated from analytical expressions or numerical approximations. The approximation of the derivatives d//dy, with forward differences is... 

The bound-state contribution to matrix P was calculated by the two alternative methods presented in Section 4.6.1. Without too much numerical effort five significant digits (relative error 0.1%) are easily obtained. It turns out that the second method is more advantageous, because it enables the derivative of the cofactor matrix to be calculated at a point where the determinant passes through zero (via calculation of the inverse near this singular point and multiplication by the determinant), while in the first method the derivative of G , which is usually a smooth function near this point, must be calculated. 

To write down explicitly the elements of the inverse of a matrix for larger matrices we need the definitions of cofactors and minors given above in connection with the evaluation of determinants. Each position in A is occupied by the signed minor of the cofactor for that position, divided by the determinant of the matrix and then transposed. The explicit form rapidly becomes very cumbersome, and accurate numerical calculation of the inverse of matrices larger than, say 3x3, requires special techniques that are available in computer programs such as Maple (see above). Problems arise in numerical work if a matrix is ill conditioned, that is a matrix with a determinant which is very small compared to its elements. 

Remember that a numerical calculation on a computer can give its inverse matrix. 

Discuss how to obtain the inverse matrix of A when the transformation matrix A is singular. Remind that a numerical calculation by a computer can give its inverse matrix. 

Adjugate Matrix of a Matrix Let Ay denote the cofactor of the element Oy in the determinant of the matrix A. The matrix B where B = (Ay) is called the adjugate matrix of A written adj A = B. The elements by are calculated by taking the matrix A, deleting the ith row and Jth. column, and calculating the determinant of the remaining matrix times (—1) Then A" = adj A/lAl. This definition may be used to calculate A"h However, it is very laborious and the inversion is usually accomplished by numerical techniques shown under Numerical Analysis and Approximate Methods. ... 

So, the calculation of the shape of an IR spectrum in the case of anticorrelated jumps of the orienting field in a complete vibrational-rotational basis reduces to inversion of matrix (7.38). This may be done with routine numerical methods, but it is impossible to carry out this procedure analytically. To elucidate qualitatively the nature of this phenomenon, one should consider a simplified energy scheme, containing only the states with j = 0,1. In [18] this scheme had four levels, because the authors neglected degeneracy of states with j = 1. Solution (7.39) [275] is free of this drawback and allows one to get a complete notion of the spectrum of such a system. 

Explicit calculation of the electronic coupling matrix element, Hah, is performed by modeling the transition state (Fig. 3) as a supermolecule, [M(H20)6]2+, and optimizing its geometry under the constraint of having an inversion center of symmetry The numerical value of Hab is then obtained from the energy gap between the appropriate molecular orbitals of the supermolecule. 

In a strictly mathematical sense this matrix is not singular but numerically it is rank deficient and has effectively a rank of only 4. Calculation of its pseudo-inverse consequently is impossible, or at least numerically unsafe. What can we do about that ... 

Because the approximation described by Eq. (3.47) fails if the inversion and vibrational wave functions are strongly mixed, the Coriolis operator defined by Eq. (5.8) cannot be treated by the numerical methods described in Sections 5.1 and 5.2. Instead of the perturbation treatment described in Section 5.1, we must use a variational approach in which the energy levels are calculated as eigenvalues of an energy matrix the off-diagonal elements of this matrix are the matrix elements of the Coriolis operator ) 2,4 ... 

We study the dielectric and energy loss properties of diamond via first-principles calculation of the (0,0)-element ( head element) of the frequency and wave-vector-dependent dielectric matrix eg.g CQ, The calculation uses all-electron Kohn-Sham states in the integral of the irreducihle polarizahility in the random phase approximation. We approximate the head element of the inverse matrix hy the inverse of the calculated head element, and integrate over frequencies and momenta to obtain the electronic energy loss of protons at low velocities. Numerical evaluation for diamond targets predicts that the band gap causes a strong nonlinear reduction of the electronic stopping power at ion velocities below 0.2 a.u. 

This form of the general Jacobian element allows for the straightforward solution of the SSOZ equation for molecules of arbitrary symmetry. However, in the numerical solution using Gillan s methods, most of the computation time is involved in calculating the elements of the Jacobian matrix, rather than in the calculation of its inverse or in the calculation of transforms. Indeed, as the forward and backward Fourier transforms can be carried out using a fast Fourier transform routine, the time-limiting step is the double summation over / and j in Eq. (4.3.36). With this restriction in mind, it is... 

Instead of applying Householder s formula, the calculation of an inverse of the jacobian may be avoided altogether by use of the algorithm proposed by Bennett for updating the LU factors of the jacobian matrix. Example 4-9 will show that fewer numerical operations are required to compute the LU factors than are required to compute the inverse of a matrix. Bennett s algorithm is applied to the Broyden equations as follows. 

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