Matrix upper triangular

In some applications the diagonal elements of the upper triangular matrix are not predetermined to be unity. The formula used for the LU decomposition procedure in these applications is slightly different from those given in Equations (6.10) to (6.12), (Press et al., 1987). 

This is eontinued n — 1 times until the entire eoeffieient maPix has been eonverted to an upper triangular matrix, that is, a maP ix with only zeros below the principal diagonal. The b veetor is operated on with exactly the same sequenee of operations as the eoeffieient matrix. The last equation at the very bottom of the triangle, aititXit = bn, is one equation in one unknown. It ean be solved for whieh is baek-substituted into the equation above it to obtain x i and so on, until the entire solution set has been generated. 

In the process of obtaining the upper triangular matrix, the nonhomogeneous vector has been transformed to (j). The bottom equation of Ax = b... 

The net result of all these operations is that in place of the system Ax = h, the system CAx = Ch is obtained, where CA is an upper triangular matrix. Such a matrix is easfiy inverted (the inverse will be exhibited below), and the triangular system is even more easily solved. With this explanation of the gaussian method, the basic theory of this and related methods will now be developed. 

Note that an upper triangular matrix can be partitioned and inverted as follows ... 

I — 2vnoH)A is a matrix whose first column has only its first element nonzero. The same principle can now be applied to the submatrix that remains after removing the first row and first column of the transformed matrix, and so on until there results, finally, an upper triangular matrix. Notice that interchange of rows is not necessary. 

A.8 A technique called LU decomposition can be used to solve sets of linear algebraic equations. L and U are lower and upper triangular matrices, respectively. A lower triangular matrix has zeros above the main diagonal an upper triangular matrix has zeros below the main diagonal. Any matrix A can be formed by the product of LU. 

Calculate L 1 and 6. Then solve for x using substitution from the upper triangular matrix U. 

The decomposition is "in place", i.e., all results are stored in the locations that matrix A used to occupy. The upper triangular matrix U will replace the diagonal elements of A and the ones above, whereas L is stored in the part below the diagonal, the unit elements of its diagonal being not stored. We will then say that the matrices are in "packed" form. The permutation matrix P is not stored at all. As discussed, the row interchanges can be described by n-l pairs (i,k ), and all information is contained in a... 

A symmetric positive definite matrix, A, can also be written as A = UL, where U is an upper triangular matrix and L U. This is not the Cholesky decomposition, however. Obtain this decomposition of the matrix in Exercise 9. 

In theory there is nothing wrong but for numerical computations, the above formulas are rather costly and inaccurate They involve inverting the upper triangular bidiagonal matrix Ax, whose inverse will be a dense upper triangular matrix. Moreover, formula (6.107) for Y involves the inverse of C which itself is given by a triple and complicated matrix product correction term subtracted from Ax in (6.105). 

Special orthogonal matrices such as Householder matrices H = Im — 2vv for a unit column vector v G Cm with v = v v = 1 can be used repeatedly to zero out the lower triangle of a matrix Amn much like the row reduction process that finds a REF of A in subsection (B). The result of this elimination process is the QR factorization of Am,n as A = QR for an upper triangular matrix Rm,n and a unitary matrix Qm,m that is the product of n — 1 Householder elimination matrices Hi. 

In other words, the modified moments fill in an upper triangular matrix fi = /ii in which all the elements below the main diagonal are equal to zero [45] ... 

The Gaussian algorithm described in Section A.4 transforms the matrix A into an upper triangular matrix U by operations equivalent to premultiplication of A by a nonsingular matrix. Denoting the latter matrix by one obtains the representation... 

Here U(k) and V( ) are orthogonal, % is diagonal, and r ) is an upper-triangular matrix whose diagonal elements are normalized to unity. 

The beauty of the REPSWA approach is that it can easily be employed to remove barriers in many degrees of freedom simultaneously without loss of efficiency. However, the method does require the evaluation of a Jacobian, which is not in general easy. Consider that proteins are polymers. It is, therefore, natural to consider growing in the barriers in the tjj angles one at a time. In this way, a lower/upper triangular matrix is formed. Since the Jacobian of an upper/lower triangular matrix is the product of its diagonal elements, REPSWA can be easily implemented. 

THIS PROGRAM FORMS THE LU EQUIVALENT OF THE SQUARE COEFFICIENT MATRIX A. THE LU, IS THEN RETURNED IN THE A MATRIX SPACE. THE UPPER TRIANGULAR MATRIX U HAS ONES ON ITS DIAGONAL. THESE VALUES ARE NOT INCLUDED IN THE RESULT. ... 

LU decomposition For a sqnare matrix A of order n, given that the determinants of the matrices Ap (p = l,2,...,n - formed by the elements at the intersection of the first p rows and columns of A are nonzeroes, then there exists a unique lower triangular matrix L and a unique upper triangular matrix U such that... 

An upper triangular matrix consists of a square matrix whose elements below the main diagonal are zero, e.g.,... 

First observe that the operations required to transform R into an upper triangular matrix are the same regardless of the particular set of elements appearing in the C matrix. Next, examine steps 1 and 2 of part (a) and observe that the elements of the C matrix are operated on by only the multipliers used to transform R into the upper triangular matrix R Thus, the final set of elements shown in the derived matrix C in Eqs. (B) and (C) can be obtained by commencing with the original set of elements in C and the multipliers stored in R and performing the operations shown below. Let the elements of C and C be denoted by Cu C2, C3 and C, C 2, C3, respectively. Examination of Eqs. (A) and (B) shows that... 

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