lu | R Documentation |
LU decomposition of a positive definite matrix as Gaussian factorization.
lu(A, scheme = c("kji", "jki", "ijk"))
lu_crout(A)
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).
lu_crout
implements the Crout algorithm. For the Doolite algorithm,
the L
matrix has ones on its diagonal, for the Crout algorithm, the
diagonal of the U
matrix only has ones.
lufact
applies partial pivoting (along the rows).
lusys
uses LU factorization to solve the linear system A*x=b
.
These function are not meant to process huge matrices or linear systems of equations. Without pivoting they may also be harmed by considerable inaccuracies.
lu
and lu_crout
return a list with components L
and U
, the 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.
To get the Crout decomposition of a matrix A
do
Z <- lu(t(A)); L <- t(Z$U); U <- t(Z$L)
.
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.
qr
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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