Solve Lower and Upper Triangular

[rachka]

With these solvers, is it safe to assume that the arguments L and U both contain the main diagonal with the test suites used for grading?

[cswiercz]

Yes. A lower (resp. upper) triangular matrix contains non-zero element at and below (resp. above) the main diagonal.

[rachka]

Thanks for the clarification. I figured that's how it generally is but wanted to be extra sure.

[bplilley]

Just want to make sure I have this correct: in L (and U) from HW#1 the main diagonal contains all zeros but in L (and U) from HW#2 (I'm looking at solve_lower_triangular and solve_upper_triangular) the main diagonal is non-zero?

[rachka]

Yeah the back and forward substitution algorithms rely on the diagonals being non zero. That is the default definition of lower and upper triangular. HW 1 had it differently because Jacobi and Gauss Seidel use special cases of U and L to solve Ax = b iteratively.

[cswiercz]

@rachka is correct.

The matrix names L and U are overloaded in numerical analysis and scientific computing. Here is a partial list of their uses:

• for iterative methods: write A = L + U + D where D is diagonal and L and U are triangular with zero diagonals
• for triangular linear systems: Lx = b where L is a lower triangular matrix. (The definition of triangular allows for non-zero diagonal entries.)
• for matrix factorization: the "LU-factorization of a matrix" is A = LU where L and U are triangular. Note that these L and U are very different from those used in the iterative methods additive decomposition. (This is a multiplicative decomposition.)