Solving Systems of Linear Equations

Matrix multiplication and inversion provide very useful means of representing and solving systems of linear equations. Consider the following matrix equation  

According to the rule for matrix multiplication introduced earlier, each element of y is calculated as the scalar product between c and the corresponding column of A. These linear operations are represented exactly by the following system of inhomogeneous linear equations  

Let s assume the elements ci, a and cz of vector c are the unknowns. Thus, the system is comprised of three equations with three unknowns. Such systems of n equations with n unknowns have exactly one solution if none of the individual equations can be expressed by linear combinations of the remaining ones, i.e. if they are linearly independent. Then, the coefficient matrix A is of full rank and non-singular and its inverse, A1, exists such that right multiplication of equation (2.20) with A 1 allows the determination of the unknowns. 

A typical example arises from Beer-Lambert s law. In spectrophotometry, it describes the linear relationship between the concentration of a chemical species and the measured absorbance at a particular wavelength. The corresponding coefficients are called molar absorptivities. They are specific for each species and wavelength. We refer to Chapter 3.1, Beer-Lambert s Law, for a more detailed introduction of Beer-Lambert s law. 

Consider a mixture of three species of unknown concentrations ci, C2 and C3 for which the absorbances yi, 1/2 and yz have been measured at three 

---

Farkas O and Schlegel H B 1998 Methods for geometry optimization In large molecules. I. An O(N ) algorithm for solving systems of linear equations for the transformation of coordinates and forces J. Chem. Phys. 109 7100... 

To solve the set of linear equations introduced in our previous chapter referenced as [1], we will now use elementary matrix operations. These matrix operations have a set of rules which parallel the rules used for elementary algebraic operations used for solving systems of linear equations. The rules for elementary matrix operations are as follows [2] ... 

In Chapters 2 and 3, we discussed the rules related to solving systems of linear equations using elementary algebraic manipulation, including simple matrix operations. The past chapters have described the inverse and transpose of a matrix in at least an introductory fashion. In this installment we would like to introduce the concepts of matrix algebra and their relationship to multiple linear regression (MLR). Let us start with the basic spectroscopic calibration relationship ... 

Within the Matlab s numerical precision X is singular, i.e. the two rows (and columns) are identical, and this represents the simplest form of linear dependence. In this context, it is convenient to introduce the rank of a matrix as the number of linearly independent rows (and columns). If the rank of a square matrix is less than its dimensions then the matrix is call rank-deficient and singular. In the latter example, rank(X)=l, and less than the dimensions of X. Thus, matrix inversion is impossible due to singularity, while, in the former example, matrix X must have had full rank. Matlab provides the function rank in order to test for the rank of a matrix. For more information on this topic see Chapter 2.2, Solving Systems of Linear Equations, the Matlab manuals or any textbook on linear algebra. 

The task is to determine the shift vector 5c for which the new vector of differences d(c+8c) is minimal or within the Taylor approximation for which this difference is zero. In Chapter 2.2, Solving Systems of Linear Equations, we have given the solution ... 

Excel solves systems of linear equations with a single statement. If the following matrix mathematics is unfamiliar to you, disregard it. The important result is the template in Figure 19-5 for solving simultaneous equations. You can use this template by following the instructions in the last paragraph of this section, even if the math is not familiar. 

Riley, J.D., Solving systems of linear equations with a positive definite symmetric but possibly ill-conditioned matrix, Math. Table Aids Comput., 9, 96-101, 1955. 

There are four methods for solving systems of linear equations. Cramer s rule and computing the inverse matrix of A are inefficient and produce inaccurate solutions. These methods must be absolutely avoided. Direct methods are convenient for stored matrices, i.e. matrices having only a few zero elements, whereas iterative methods generally work better for sparse matrices, i.e. matrices having only a few non-zero elements (e.g. band matrices). Special procedures are used to store and fetch sparse matrices, in order to save memory allocations and computer time. 

A matrix can be used to solve systems of linear equations. If... 

From the retention volume for some compounds consisting of different number of the same groups it is possible to determine the contribution of groups to the retention by solving system of linear equations (1). 

This method of solving systems of linear equations (or even of inverting a matrix) is NOT recommended for systems of order greater than 2. Better methods are discussed in texts on numerical methods (e.g.. Press et al., 1992) (see, also. Appendix C). 

To illustrate the use of an inverse for solving systems of linear equations, consider two equations in two unknowns. 

A. K. Jainandunsing. Parallel algorithms for solving systems of linear equations and their mapping on systolic arrays. PhD thesis. Delft University of Technology, January 1989. 

L, U] = LU(A) LU decomposition of matrix A. Typically used within MATLAB to solve systems of linear equations. You need to provide [L, U] so that both matrices are stored. 

Whereas the procedures for solving systems of linear equations are straightforward, those for solving sets of nonlinear equations are not nearly so well formulated. The method that will be used for the solution... 

Linear algebra is so named because it grew out of methods for solving systems of linear equations. For our purposes, it is the branch of mathematics that describes how to perform arithmetic and algebra using vectors and matrices. 

Crout, P. D., A Short Method for Evaluating Determinants and Solving Systems of Linear Equations with Real or Complex CoetTicients, Marchant Ckculating Machine Co., Bulletins MM-I82 and 183, ASTBA, September 1941. Dwyer, P. S., Psychomeiria, Vol 6, 1941, p. 101. Hotting, H., Am. Math. Stat., Vol 14, 1943, p. 1. 

Here we do not aim at presenting the standard methods of solving the Newton vibrational equations. It should be emphasized that essential results of these calculations are the transfonnation coefficients Lij that define the relative contribution of each internal coordinate to the respective normal vibrations in the molecule. As underlined, the availability of accurate vibrational form coefficients are needed in intensity analysis. This is detennined simply by the fact that vibrational intensities in the infrared spectra of molecules in the gas-phase (at low pressure so that no considerable intermolecular interaction is present) are governed by two principal factors (1) the intramolecular charge rearrangements accompanying vibrational distortions and (2) the form of the normal vibrations as expressed in the coefficients of the normal coordinate transfonnation matrix L. The elements of L are determined by solving systems of linear equations of the type [4,6]... 

THIS IS A STANDARD LIBRARY SUBROUTINE FOR SOLVING SYSTEMS OF LINEAR EQUATIONS. SEE YOUR COMPUTER CENTER LIBRARY FOR AN EXPLANATION OF HOW IT WORKS. 

---