Homework 5 Task 8: QR factorization Using Gram-Schmidt on Hilbert Matrices

Task: Try out your QR-factorization method from the previous task on the Hilbert matrices of sizes n=4,6,8,10. Test to see if the orthogonal matrix is really orthogonal by multiplying QtQ and comparing the result to the identity matrix. Explain the results you obtain.

This process can be executed in Fotran with the following code.

INTEGER :: m, n, i, j
REAL*8, ALLOCATABLE :: A(:, :), Q(:, :), R(:, :), QTQ(:, :)

m = 8
n = 8
ALLOCATE(A(1:m, 1:n), Q(1:m, 1:n), R(1:n, 1:n), QTQ(1:n, 1:n))
DO i = 1, m
	DO j = 1, n
		A(i, j) = 1.0D0/(REAL(i) + REAL(j) - 1.0D0)
	END DO
END DO
CALL qr_factor_gs(A, m, n, Q, R)
WRITE(*,*) "A"
DO i = 1, m
	WRITE(*,*) A(i, :), "/"
END DO
WRITE(*,*) "Q"
DO i = 1, m
	WRITE(*,*) Q(i, :), "/"
END DO
WRITE(*,*) "R"
DO i = 1, n
	WRITE(*,*) R(i, :), "/"
END DO
CALL mat_prod(TRANSPOSE(Q), Q, n, m, n, QTQ)
WRITE(*,*) "QTQ"
DO i = 1, n
	WRITE(*,*) QTQ(i, :), "/"
END DO
CALL mat_prod(Q, R, m, n, n, A)
WRITE(*,*) "A = QR"
DO i = 1, m
	WRITE(*,*) A(i, :), "/"
END DO

The output of running the Hilbert matrices can be seen here. As can be seen, the QTQ section of each of the successive matrices becomes less and less similar to the identity matrix as the size of the matrix increases. This essentially means that Q is becoming less and less orthogonal due to round off error. The entry in the software manual can be seen here.