Homework 5

Question One

See LIZFCM → Matrix Routines → lu decomp & bsubst.

The test UTEST(matrix, lu_decomp) is a unit test for the lu_decomp routine, and UTEST(matrix, bsubst) verifies back substitution on an upper triangular 3 × 3 matrix with a known solution that can be verified manually.

Both can be found in tests/matrix.t.c.

Question Two

Unless the following are met, the resulting solution will be garbage.

The matrix $U$ must be not be singular.
$U$ must be square (or it will fail the assert).
The system created by $Ux = b$ must be consistent.
$U$ is (quite obviously) in upper-triangular form.

Thus, the actual calculation performing the $LU$ decomposition (in lu_decomp) does a sanity check for 1-3 will fail an assert, should a point along the diagonal (pivot) be zero, or the matrix be non-factorable.

Question Three

See LIZFCM → Matrix Routines → fsubst.

UTEST(matrix, fsubst) verifies forward substitution on a lower triangular 3 × 3 matrix with a known solution that can be verified manually.

Question Four

See LIZFCM → Matrix Routines → gaussian_elimination and solve_gaussian_elimination.

Question Five

See LIZFCM → Matrix Routines → m_dot_v, and the UTEST(matrix, m_dot_v) in tests/matrix.t.c.

Question Six

See UTEST(matrix, solve_gaussian_elimination) in tests/matrix.t.c, which generates a diagonally dominant 10 × 10 matrix and shows that the solution is consistent with the initial matrix, according to the steps given. Then, we do a dot product between each row of the diagonally dominant matrix and the solution vector to ensure it is near equivalent to the input vector.

Question Seven

See UTEST(matrix, solve_matrix_lu_bsubst) which does the same test in Question Six with the solution according to solve_matrix_lu_bsubst as shown in the Software Manual.

Question Eight

No, since the time complexity for Gaussian Elimination is always less than that of the LU factorization solution by $O(n^2)$ operations (in LU factorization we perform both backwards and forwards substitutions proceeding the LU decomp, in Gaussian Elimination we only need back substitution).

Question Nine, Ten

See LIZFCM Software manual and shared library in dist after compiling.

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