Task: Test the code developed above (K2_cond) to compute the condition number of the Hilbert matrix as the size of the matrix is increased. Tabulate and/or graph the results from your work.
This process can be executed in Fortran with the following code.
INTEGER :: n, maxiter, i, printit, j, k
REAL*8, ALLOCATABLE :: A(:, :), v0(:), v(:)
REAL*8 :: tol, lam, alpha, cond
INTEGER :: s(1:4)
tol = 10.D-18
maxiter = 10000
printit = 1
s = (/4, 8, 16, 32/)
DO k = 1, SIZE(s)
IF(ALLOCATED(A)) THEN
DEALLOCATE(A, v0, v)
END IF
n = s(k)
ALLOCATE(A(n, n), v(n), v0(n))
WRITE(*,*) "Hilbert", n, "x", n
DO i = 1, n
DO j = 1, n
A(i, j) = 1.0D0/(REAL(i) + REAL(j) - 1.0D0)
END DO
END DO
v0 = 1.D0
CALL K2_cond(A, n, v0, tol, maxiter, printit, cond)
WRITE(*,*) cond
END DO
The results of running this code are shown here
| Size of Hilbert Matrix | Condition Number |
|---|---|
| 4x4 | 15513.738738930397 |
| 8x8 | 15257575817.032135 |
| 16x16 | 3.6910873798176121E+019 |
| 32x32 | 1.1010765858137861E+018 |
Hilbert matrices are terribly conditioned. This means that a small perturbation in the right hand side of a system of equations with these matrices will yield drastically different results.