comphy (version 1.0.5)

LUdeco: LU decomposition

Description

Transform an \(n\times n\) matrix into a product of a lower-triangular and upper-triangular matrices, using the Crout (method="crout" - default) or Doolittle (method= "doolittle") method.

Usage

LUdeco(A, method = "crout")

Value

A named list with the lower triangular, L, upper triangular, U, matrices, and with a vector, ord, containing the permutation needed to achieve the LU factorisation.

Arguments

A

An \(n\times n\) matrix.

method

A character string. This calls two different procedures for the decomposition. Only "crout" (default) and "doolittle" are recognised methods. A different character string forces the function to return NULL.

Details

The "crout" method returns the upper triangular matrix, U, with ones on its diagonal. The "doolittle" method returns the lower triangular matrix, L, with ones on its diagonal.

Some matrices do not have an LU decomposition unless a row permutation is done to the matrix. In this function, the order of such a permutation is included in the named vector ord, returned as part of the output. When the vector is equal to 1,2,...,n (first n numbers, naturally ordained), this means that there was no need of permuting the original matrix to carry out the LU decomposition.

Examples

# 3X3 matrix
#
# [ 3  1  1
#   1 -1  2
#  -1  1  1]

# Input matrix
A <- matrix(c(3,1,-1,1,-1,1,1,2,1),ncol=3)

# LU decomposition
ltmp <- LUdeco(A)
print(ltmp$L)
print(ltmp$U)
print(ltmp$ord) # No permutation needed

# The product is the original matrix, A
print(ltmp$L%*%ltmp$U)

# Singular matrix with LU decomposition
A <- matrix(c(1,0,0,0,1,1,1,0,0),ncol=3)
print(det(A))
ltmp <- LUdeco(A,"doolittle")
print(ltmp$L)
print(ltmp$U)
print(ltmp$ord) # No permutation needed

# The product is the original matrix, A
print(ltmp$L%*%ltmp$U)

# Singular matrix without LU decomposition
A <- matrix(c(1,0,0,0,0,0,0,0,0),ncol=3)
ltmp <- LUdeco(A)
print(ltmp)
#