Matrix Gauss elimination

Figure 12.57. Growth of storage requirements with number of spatial intervals for sparse matrix Gauss elimination.

LV Factorization of a Matrix To eveiy m X n matrix A there exists a permutation matrix P, a lower triangular matrix L with unit diagonal elements, and a.nm X n (upper triangular) echelon matrix U such that PA = LU. The Gauss elimination is in essence an algorithm to determine U, P, and L. The permutation matrix P may be needed since it may be necessaiy in carrying out the Gauss elimination to... 

Many methods exist for solving the basic form AU = b for the potential U = A-1 b. The methods depend on various features exhibited by the matrices themselves, immediate byproducts of how the problem was set up in the previous stage of specification. A general method, assuming that A is nonsingular (determinant is nonzero), is to find the inverse matrix A-1, using techniques such as Gauss elimination. However, in practice this approach is not computationally viable. Typically, one looks for features of the problem that simplify A. 

This is a system of equations of the form Ax = B. There are several numeral algorithms to solve this equation including Gauss elimination, Gauss-Jacobi method, Cholesky method, and the LU decomposition method, which are direct methods to solve equations of this type. For a general matrix A, with no special properties such as symmetric, band diagonal, and the like, the LU decomposition is a well-established and frequently used algorithm. 

This procedure completes the Gauss elimination. We can carry out the elimination process by writing only the coefficients and the matrix vector in an array as... 

There is a similar procedure known as Gauss elimination, in which row operations are carried out until the left part of the augmented matrix is in upper triangular form. The bottom row of the augmented matrix then provides the root for one variable. This is substituted into the equation represented by the next-to-bottom row, and it is solved to give the root for the second variable. The two values are substituted into the next equation, and so on. 

Addition Subtraction Multiplication Transpose Inverse Determinant Eigenvalues Matrix left division (uses Gauss elimination to solve a set of linear equations)... 

Gauss-Jordan elimination is a variation of the Gauss elimination scheme. Instead of obtaining the triangular matrix at the end of the elimination, the Gauss-Jordan has one extra step to reduce the matrix A to an identity matrix. In this way the augmented vector b is simply the solution vector x. 

In the Doolittle method, the upper triangular matrix U is determined by the Gauss elimination process, while the matrix L is the lower triangular matrix containing the multipliers employed in the Gauss process as the elements below the unity diagonal line. More details on the Doolittle and Grout methods can be found in Hoffman (1992). 

Here, the matrices H and V are symmetric and positive definite matrices, which are each, after suitable permutation of indices, tridiagonal matrices. The matrix S is a non-negative diagonal matrix. Recalling that tridiagonal matrix equations are efficiently solved by the Gauss elimination method, we consider now the Peaceman-Rachford iterative method [27], a particular variant of the lAD methods, which is defined by... 

Let A [M, yV] be arbitrary, let A be a matrix obtained from A by Gauss elimination. A that by Gauss-Jordan elimination. There then exists regular matrix L [M, M] resp. L [M, M such that... 

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 equation implies that a linear equation system always has to be solved when the parameter k, is calculated. The matrix I — ]aijh is inverted using the Gauss elimination. 

The matrix algebra used for calculation of the coefficient matrix k is particularly favourable for programming linear regression. If the number of independent variables Xk exceed three or four, it is in many cases better to solve the equation system (3) by Gauss elimination instead of inverting the matrix X X in the traditional way. 

Tlie overall Gauss elimination procedure applied on the fnl x (n + 1) augmented matrix is condensed into a three-part mathematical formula for initialization, elimination, and back substitution as shown below ... 

The Gauss elimination procedure, which was described above in formula form, can also be accomplished by series of matrix multiplications. Two types of special matrices are involved in this operation. Both of these matrices are modifications of the identity matrix. The first type, which we designate as P,, is the identity matrix with the following changes The unity at position ii switches places with the zero at position ij, and the unity at position jj switches places with the zero at position ji. For example, for a fifth-order system is... 

Therefore, the entire Gauss elimination method, which reduces a nonsingular matrix A to an upper triangular matrix U, can be represented by the following series of matrix multiplications ... 

Example 2.1 demonstrates the Gauss elimination method with complete pivoting strategy in solving a set of simultaneous linear algebraic equations and in calculating the determinant of the matrix of coefficients. 

Method of Solution The function is written based on Gauss elimination in matrix form. It applies complete pivoting strategy by searching rows and columns for the maximum pivot element. It keeps track of column interchanges, which affect the positions of the unknown variables. The function applies the back-substimtion formula [Eq. (2.110)] to calculate the unknown variables and interchanges their order to correct for column pivoting. 

The Gauss-Jordan reduction method applies the same series of elementary operations that are used by the Gauss elimination method. It applies these operations both below and above the diagonal in order to reduce all the off-diagonal elements of the matrix to zero. In addition, it converts the elements on the diagonal to unity. 

In Sec. 2.5.2, we showed that the Gauss elimination method can be represented in matrix form as... 

We, therefore, conclude that if the Gauss elimination method is extended so that matrix A is postmultiplied by L , at each step of the operation, in addition to being premulliplied by L, the resulting matrix B is similar to A. Tliis operation is called the elementary similarity... 

Apply the Gauss elimination method with complete pivoting to the matrix (A - XT) to evaluate the eigenvectors corresponding to each eigenvalue. Several different possibilities exist when the eigenvalues are real ... 

---