T-matrix Programs

A Fortran computer program has been written to solve various scattering problems in the framework of the null-field method. This section gives a short description of the code, while more details concerning the significance of the input and output parameters are given in the documentation on CD-ROM. 

The main program TMATRIX.fQO calls a T-matrix routine for solving a specific scattering problem. These routines computes the T-matrix of  

Three other routine are invoked by the main program 

Essentially, the code performs a convergence test and computes the T-matrix and the scattering characteristics of particles with uniform orientation distribution functions. 

An important part of the T-matrix calculation is the convergence procedure over the maximum expansion order A rank, maximum azimuthal order Afrank and the number of integration points Nmt- In fact, Afrank Afrank and A int are input parameters and their optimal values must be found by the user. This is accomplished by repeated convergence tests based on the analysis of the differential scattering cross-section as discussed in Sect. 2.1. 

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T-matrix Program 185 3.1.1 Complete Uniform Distribution Function... 

The main drawback of the code is that the computer time requirements might by higher in comparison to other more optimized T-matrix programs. Our intention was to cover a large class of electromagnetic scattering problem and therefore we sacrifice the speed in favor of the flexibihty and code modularization. The code shares several modules which are of general use and are not devoted to a specific application. 

TAXSYM routine and the program developed by Bohren [16]. This program was coded by Ute Comberg and is available from www.T-matrix.de. The scattering characteristics are computed in the azimuthal plane y> = 0° and for two polarizations of the incident wave. 

This section discusses numerical and practical aspects of T-matrix calculations for inhomogeneous particles. The flow diagram of the TINHOM routine is shown in Fig. 3.34. The program supports scattering computation for axisymmetric and dielectric host particles with real refractive index. The main feature of this routine is that the T-matrix of the inclusion is provided as input parameter. The inclusion can be a homogeneous, axisymmetric or... 

T. Wriedt, Electromagnetic Scattering Programs, http //www.t-matrix.de (2006)... 

The text form for parameters uses white space or commas to separate the fields (columns) of the parameter tiles. They can be read by ordinary text editors, w ord processors, etc. In the text form, param eters are easy to m odify but not easy to com pare, stn dy, etc. Many database program s are capable of reading column s of text as a database, h owever. Wh ile spreadsheets are n ot. per se, databases, they can be useful for examining parameter sets. Microsoft Excel, for example, can read the text form of a param eter file and pn t the data in a form easily manipulated as a matrix or a database. The text form of parameters are stored, by default only, in Tart files. 

A. M. Morshedi, C. R. Cutier, and T. A. Skrovanek, "Optimal Solution of Dynamic Matrix Control with Linear Programming Techniques,"... 

The 21 equations (given as Equation 6.68) should be solved simultaneously with the three state equations (Equation 6.64). Integration of these 24 equations yields x(t) and G(t) which are used in setting up matrix A and vector b at each iteration of the Gauss-Newton method. Given the complexity of the ODEs when the dimensionality of the problem increases, it is quite helpful to have a general purpose computer program that sets up the sensitivity equations automatically. 

In the above expressions, the subscript F refers to the full relaxation and exchange matrices that include the P2 and P2 protons since their magnetizations do not experience rf saturation during in the on-resonance saturation experiment and during ti + t in the off-resonance irradiation experiment (control spectrum). Hence, experience coupled recovery with the rest of the protons during these periods. The subscript r refers to the reduced matrix containing elements for I, I, PI, and PI extracted from the full matrix. We have implemented the above expressions as an option in the CORCEMA-ST program. 

To solve a diffusion equation, one needs to diagonalize the D matrix. This is best done with a computer program. For a ternary system, one can find the two eigenvalues by solving the quadratic Equation 3-lOOe. The two vectors of matrix T can then be found by solving... 

In solving the underlying model problem, the Jacobian matrix is an iteration matrix used in a modified Newton iteration. Thus it usually doesn t need to be computed too accurately or updated frequently. The Jacobian s role in sensitivity analysis is quite different. Here it is a coefficient in the definition of the sensitivity equations, as is 3f/9a matrix. Thus accurate computation of the sensitivity coefficients depends on accurate evaluation of these coefficient matrices. In general, for chemically reacting flow problems, it is usually difficult and often impractical to derive and program analytic expressions for the derivative matrices. However, advances in automatic-differentiation software are proving valuable for this task [36]. 

Principal computer programs used in this study T. Ottersen, COMPARE data reduction program, University of Hawaii, 1973 full-matrix least-squares, P. K. Gantzel, R. A. Sparks, and... 

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