By Vasile Cirtoaje

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N, to produce the symmetric n x n matrix A and solution vector xexact. Then the corresponding discrete right-hand side is produced as frexact = A xexact. 4. A Problem with a Discontinuous Solution (wing) This test problem appears as problem VI. 10 in [373], and it is different from the above two problems in that the solution is discontinuous. , in image processing (across edges in the image) and in seismic deconvolution (across layer boundaries), so it makes sense to include such a test problem here.

21), many RRQR algorithms also provide a matrix WT€ £ ]Rnx(n~r«) whose columns span an approximation to the numerical null space of R. Another approximate basis for MTf (A) can be computed as the columns of the matrix The existence of RRQR decompositions was proved in [213] and the necessity of the column permutations is treated in [131]. 22) stay small. The key to computing a rank-revealing decomposition is to choose the appropriate column permutation matrix II. We can illustrate this with a simple example.

Four large singular values and n — 4 small singular values of order crnoise. 2. Computation of the Second Derivative (deriv2) This is a classical example of an ill-posed problem, and it is used in numerous papers on regularization algorithms. 4. FOUR TEST PROBLEMS 15 and both integration intervals are [0,1]. The symmetric kernel K is not differentiable across the line s = t. The singular values and functions for i = 1,2,... 13]) Since the singular values are proportional to i 2, the problem is moderately ill posed.