"QR DECOMPOSITION AND QR ALGORITHM"

U_(a):=a/ABS(a)

D_(a):=DIMENSION(a)

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

I_(n):=IDENTITY_MATRIX(n)

(E_(w,k) . E_(z,i))*E_(z,i)

SUM((E_(w,k) . E_(z,i))*E_(z,i),i,1,k-1)

IF(i<k,E_(z,i),U_(E_(w,k)-SUM((E_(w,k) . E_(z,i))*E_(z,i),i,1,k-1)))

VECTOR(IF(i<k,E_(z,i),U_(E_(w,k)-SUM((E_(w,k) . E_(z,i))*E_(z,i),i,1,k-1))),i,~
k)

GRAM_H(z,w,k):=IF(k>D_(w),z,GRAM_H(VECTOR(IF(i<k,E_(z,i),U_(E_(w,k)-SUM((E_(w,~
k) . E_(z,i))*E_(z,i),i,1,k-1))),i,k),w,k+1))

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

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

"USER FUNCTIONS"

Q(a):=GRAM_H([U_(E_(a`,1))],a`,2)`

R(a):=Q(a)` . a

QR(a,n):=ITERATE(R(z) . Q(z),z,a,n)

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

"INSTRUCTIONS"

"If A is a square, nonsingular matrix,"

"Q(A) is orthogonal, R(A) is upper triangular"

"A=Q(A)R(A)"

"If A has real eigenvalues, then"

"QR(A,n) executes n iterations of the QR algorithm"

"to produce a matrix in approximate upper triangular form"

"that is similar to A"

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

"Example 1"

a:=[[1,2,3],[2,3,1],[4,1,1]]

"Simplify the following to get the Q-R decomposition of A"

[Q(a),R(a)]

"Example 2"

"approX the following to see 20 iterations of the QR algorithm"

"... applied to A"

QR(a,20)