Matrix.83g is a group file containing CHOLESKY.83P, LUFACT.83P and QRFACT.83P.







CHOLESKY.83P computes Cholesky decomposition of a symmetric positive

definite matrix.  The result is returned as [A].

 

Cholesky Decomposition decomposes a symmetric positive definite real matrix,

A into a lower triangular matrix L, such that A = L*transpose(L).  Note that

a matrix is positive definite if and only if, transpose(x)*A*x > 0, for any

real non-zero vector x.  If the matrix does not satisfy the positive

definite requirement, INVALID MATRIX, will be displayed.  An reference

for this algorithm is: Fundamentals of Matrix Computations, by David S.

Watkins, Section 1.5.  The algorithm is based on the "outer-product"

formulation.

 

LUFACT.83P computes LU decomposition of a general matrix using partial

pivoting.  The lower triangular portion, L, is returned as [A], the upper

triangular portion, U, is returned as [B] and the pivot list is returned as

piv.

 

LU Decomposition decomposes a general matrix A into two matrices, L and U, such

that L*U = A(piv,:).  Here, A(piv,:) indicates a re-ordering of the rows of A,

so that they match the produce L*U.  Following the decomposition, L is a lower

triangular matrix with 1's along the main diagonal and U is an upper triangular

matrix.  One of the primary uses of LU factorization is to allow one to solve a

system of simultaneous equations using back substitution.

 

QRFACT.83P computes the QR decomposition of a general matrix using the modified

Gram-Schmidt procedure.  The unitary matrix Q is returned in [A] and the upper

triangular matrix R is returned in [B].  The number of rows of the input matrix

must be at least as large a the number of columns.

 

QR Decomposition decomposes a general matrix A into two matrices, Q and R, such

that Q*R = A.  The Q matrix is unitary and the R matrix is upper triangular.

If A is m-by-n, then Q is m-by-n, transpose(Q)*Q = identity(m) and R is n-by-n.

 Some of the uses of QR decomposition is in the solution to least squares

problems and as a building block for the QR algorithm, which is used to compute

the eigenvalues of a matrix.