LU Matrix Factorization
LU decomposition of a positive definite matrix as Gaussian factorization.
lu(A, scheme = c("kji", "jki", "ijk"))
lufact(A)
lusys(A, b)A |
square positive definite numeric matrix (will not be checked). |
scheme |
order of row and column operations. |
b |
right hand side of a linear system of equations. |
For a given matrix A, the LU decomposition exists and is unique iff
its principal submatrices of order i=1,...,n-1 are nonsingular. The
procedure here is a simple Gauss elimination with or without pivoting.
The scheme abbreviations refer to the order in which the cycles of row- and column-oriented operations are processed. The “ijk” scheme is one of the two compact forms, here the Doolite factorization (the Crout factorization would be similar).
lufact applies partial pivoting (along the rows).
lusys uses LU factorization to solve the linear system A*x=b.
lu returns a list with components L and U, the two
lower and upper triangular matrices such that A=L%*%U.
lufact returns a list with L and U combined into one
matrix LU, the rows used in partial pivoting, and det
representing the determinant of A. See the examples how to extract
matrices L and U from LU.
lusys returns the solution of the system as a column vector.
This function is not meant to process huge matrices or linear systems of equations. Without pivoting it may also be harmed by considerable inaccuracies.
Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics. Second edition, Springer-Verlag, Berlin Heidelberg.
J.H. Mathews and K.D. Fink (2003). Numerical Methods Using MATLAB. Fourth Edition, Pearson (Prentice-Hall), updated 2006.
A <- magic(5) D <- lu(A, scheme = "ijk") # Doolittle scheme D$L %*% D$U ## [,1] [,2] [,3] [,4] [,5] ## [1,] 17 24 1 8 15 ## [2,] 23 5 7 14 16 ## [3,] 4 6 13 20 22 ## [4,] 10 12 19 21 3 ## [5,] 11 18 25 2 9 H4 <- hilb(4) lufact(H4)$det ## [1] 0.0000001653439 x0 <- c(1.0, 4/3, 5/3, 2.0) b <- H4 %*% x0 lusys(H4, b) ## [,1] ## [1,] 1.000000 ## [2,] 1.333333 ## [3,] 1.666667 ## [4,] 2.000000
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