"File: Lower-Upper decomposition"

D_(a):=DIMENSION(a)

I_(n):=IDENTITY_MATRIX(n)

E_(a,i):=ELEMENT(a,i)

EE_(a,i,j):=ELEMENT(a,i,j)

E_(a,k)-EE_(a,k,j)*E_(a,i)/EE_(a,i,j)

E_(a,i)/EE_(a,i,j)

IF(k=i,E_(a,i)/EE_(a,i,j),E_(a,k)-EE_(a,k,j)*E_(a,i)/EE_(a,i,j))

IF(k<i,E_(a,k),IF(k=i,E_(a,i)/EE_(a,i,j),E_(a,k)-EE_(a,k,j)*E_(a,i)/EE_(a,i,j)~
))

DOWN(a,i,j):=VECTOR(IF(k<i,E_(a,k),IF(k=i,E_(a,i)/EE_(a,i,j),E_(a,k)-EE_(a,k,j~
)*E_(a,i)/EE_(a,i,j))),k,D_(a))

i>D_(a) OR j>D_(a`)

SUM(ABS(EE_(a,s,j)),s,i,D_(a))

NEXT(a,i,j):=

H_(a,i,j):=

H_(DOWN(a,i,j),i+1,j+1)

IF(EE_(a,i,j)=0,NEXT(a,i,j),H_(DOWN(a,i,j),i+1,j+1),H_(DOWN(a,i,j),i+1,j+1))

H_(a,i,j):=IF(i>D_(a) OR j>D_(a`),a,IF(EE_(a,i,j)=0,NEXT(a,i,j),H_(DOWN(a,i,j)~
,i+1,j+1),H_(DOWN(a,i,j),i+1,j+1)))

[["Row Interchange Required"]]

NEXT(a,i,j):=IF(SUM(ABS(EE_(a,s,j)),s,i,D_(a))=0,H_(a,i,j+1),[["Row Interchang~
e Required"]],[["Row Interchange Required"]])

APPEND(a`,I_(D_(a)))`

VECTOR(E_(a,k),k,D_(a)-n+1,D_(a))`

L_(a,n):=IF(D_(a)=1,a^(-1),VECTOR(E_(a,k),k,D_(a)-n+1,D_(a))`)

"*****************************"

"USER FUNCTIONS"

UPPER(a):=H_(a,1,1)

LOWER(a):=L_(H_(APPEND(a`,I_(D_(a)))`,1,1)`,D_(a))^(-1)

"************************"

"INSTRUCTIONS"

"If A is a square matrix that can be put in lower triangular"

"form without row interchanges, then..."

"UPPER(A) gives an upper triangular matrix"

"LOWER(A) gives a lower triangular matrix"

"LOWER(A) UPPER(A) =A"

"#################"

"Example"

a:=[[1,3,2,4],[3,5,6,7],[4,5,3,4],[1,2,3,3]]

"Simplify the following to see the lower-upper decomposition of A"

[LOWER(a),UPPER(a)]