(a) Solve the integral equation on the interval [0, 1] by discretizing the integral using the…

(a) Solve the integral equation

(a) Solve the integral equation on the interval [0, 1] by discretizing the integral using the...

on the interval [0, 1] by discretizing the integral using the composite Simpson quadrature rule with n equally spaced points tj , and also using the same n points for the si. Solve the resulting linear system Ax = y using a library routine for Gaussian elimination with partial pivoting. Experiment with various values for n in the range from 3 to 15, comparing your results with the known unique solution, u(t) = t. Which value of n gives the best results? Can you explain why?

(b) For each value of n in part a, compute the condition number of the matrix A. How does it behave as a function of n?

(c) Repeat part a, this time solving the linear system using the singular value decomposition, but omit any “small” singular values. Try various thresholds for truncating the singular values, and again compare your results with the known true solution.

(d) Repeat part a, this time using the method of regularization. Experiment with various values for the regularization parameter µ to determine which value yields the best results for a given value of n. For each value of µ, plot a point on a twodimensional graph whose axes are the norm of the solution and the norm of the residual. What is the shape of the curve traced out as µ varies? Does this shape suggest an optimal value for µ?

(e) Repeat part a, this time using an optimization routine to minimize ||y−Ax||22 subject to the constraint that the components of the solution must be nonnegative. Again, compare your results with the known true solution.

(f ) Repeat part e, this time imposing the additional constraint that the solution be monotonically increasing, i.e., x1 ≥ 0 and xi − xi−1 ≥ 0, i = 2, . . . , n. How much difference does this make in approximating the true solution?

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