modify the example in the text by changing from to , and find the factorization of this new trdiagonal matrix.
Write down the 3 by 3 finite difference matrix () for
Find the 5 by 5 matrix that approximates , replacing the boundary conditions by and . Check that your matrix applied to the constant vector , yields zero; A is singular. Analogously, show that if is a solution of the continuous problem, then so is . The two boundary conditions do not remove the uncertainty in the term , and the solution is not unique.
With and , the difference equation (5) is Solve for and find their error in comparison with the true solution at and .
What 5 by 5 system replaces (6) if the boundary conditions are changed to ?
(recommended) Compute the inverse of the 3 by 3 Hilbert matrix in two ways using the ordinary Gauss-Jordan elimination sequence: (1) by exact computation and (ii) by rounding off each number to three figures. Note: this is a cas where pivoting doesn not help; is ill-donditioned and incurable.
For the same matrix comapre the rightsides of when the solutions are and
Solve for the 10 by 10 Hilbert matrix with , using any computer code for linear equations. Then make a small change in all entry of or and compare the solutions.
Compare the pivots in direct elimination to those with partial pivoting for , (This is actually an example that needs rescaling before elimination.)
Exlain why partial pivoting all the multiplers in satify . Deduce that if the original entries of satisfy , then afte rproducing zeros in the first column all entries are bounded by 2; after stages they are bounded by . Can you contruct a 3 by 3 example with all and whose last pivot is 4?