Adapt the program $UWHPSC/codes/lapack/randomsys3.f90 to use a specific matrix A in place of the random matrix used in the original code. The matrix to use is the Hilbert matrix defined by

\(a_{i,j} = \frac{1}{i+j-1}\)

This is a notorious matrix since it is always nonsingular but is very ill-conditioned even for moderately small values of n.

For more discussion of this matrix, and a formula for how the condition number grows with n, see this Cleve’s Corner blog post.

Note that in order to create an executable for your program, in the linking step you will need to make sure gfortran also links in the BLAS and LAPACK library. See the LFLAGS set in `$UWHPSC/codes/lapack/Makefile for the arguments you need to add to the linking step.

Instead of using the random_number subroutine to generate a random x for checking the relative error, as is done in $UWHPSC/codes/lapack/randomsys3.f90, try taking x to be a vector of all 1’s. (And as in the original code compute \(b = Ax\) usint matmul and then solve the system to recover x.) Print out the computed x as well as computing the relative error in the 1-norm as in the original code. How well does it do? How does the accuracy relate to the condition number?

You might want to look at the dgecon documentation.

Try different values of n with your program to see if it gives the expected behavior. Note that the LAPACK function dgecon does not compute the exact condition number but only estimates it. Also note that the program estimates the 1-norm condition number, while the approximate formula is for the 2-norm condition number (but they grow in a similar exponential fashion).

**If you have time to do more...**

Modify your code by creating a Fortran function hilbert_condition that returns the condition number estimate for a given value of n.

Then write a main program that loops over n from 1 to 20, computes the estimate for each n, and writes a text file with two columns n and the estimate. These statements might be useful:

open(21, file='cond.txt',status='unknown') do n=1,20 cond = hilbert_condition(n) ! print *, "cond = ",cond write(21, 210) n,cond 210 format(i4,e16.6) enddo

The text file produced should be readable by the Python script $UWHPSC/labs/lab16/plot_cond.py, which plots the results on a logarithmic scale, along with what the formula predicts.

For the function version you do not need to solve a linear system, so you don’t need to call dgesv, but you do need to compute the LU factorization of A before calling dgecon. The could be done by calling dgetrf instead of dgesv. You might want to look at the dgetrf documentation.

**There is quiz for Lab 16**