Matrices triangular

After obtaining the described decomposition the set of equations can be readily solved. This is because all of the information required for transfonnation of the coefficient matrix to an upper triangular fonn is essentially recorded in the lower triangle. Therefore modification of the right-hand side is quite straightforward and can be achieved using the lower triangular matrix as... 

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. 

The triangular matrix A resulting from Gaussian elimination is... 

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

A triangular matrix is a matrix all of whose elements above or below the main diagonal (set of elements an,. . . , a j) are zero. 

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... 

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. 

Provided that the m variances Vj - sj are roughly equal (Bartlett s test, see Section 1.7.3), the m means are ordered (cf. subroutine SORT, Table 5.17). The smallest mean has index 1, the largest has index m. A triangular matrix (see Tables 4.9, 4.10) is then printed that gives the m (m- l)/2 differences AXmean.wi = Xmean.M Xmean.i for all possible pairings. Every element of the... 

Data Analysis Because of the danger of false conclusions if only one or two parameters were evaluated, it was deemed better to correlate every parameter with all the others, and to assemble the results in a triangular matrix, so that trends would become more apparent. The program CORREL described in Section 5.2 retains the sign of the correlation coefficient (positive or negative slope) and combines this with a confidence level (probability p of obtaining such a correlation by chance alone). 

Calculate the correlation coefficient r for every combination of columns, and display the results in a triangular matrix (an absolute value just under 1.00 indicates a strong correlation between the measurements in columns i and j a minus sign indicates that the slope is negative). 

Multiple Range Test) yields a triangular matrix of differences Axmean.y... 

The triangular matrix of differences AXmean, ij is converted into a triangu-... 

RND-l-15.dat A triangular matrix of random numbers that serves the same purpose as ND 160.dat, but introduces the vector length as a factor. Use with MSD or HUBER. 

Let Ao = A - -A2, where Ai and A2 are adjoint or triangular (with a triangular matrix) operators, so that... 

QR Factorization of a Matrix If A is an m x n matrix with m > n, there exists inmxji unitary matrix = [qij q2,..., q ] and an m X n right triangular matrix R such that A = QR. The OR factorization is frequently used in the actual computations when the other transformations are unstable. 

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. 

Using the normalized overlap formula, Eq. (68), the derivative with respect to the nonzero terms of the lower triangular matrix Lk is... 

Here, L is a lower triangular matrix (not to be confused with L, the Cholesky factor of the matrix of nonlinear parameters A ), and D is a diagonal matrix. The scheme of the solution of the generalized symmetric eigenvalue problem above has proven to be very efficient and accurate in numerous calculations. But the main advantage of this scheme is revealed when one has to routinely solve the secular equation with only one row and one column of matrices H and S changed. In this case, the update of factorization (117) requires only oc arithmetic operations while, in general, the solution from scratch needs oc operations. 

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