Numerical methods Assignment

1.The kth principal submatrix of A Cⁿⁿ is the upper left-most k k portion of A. Show that A has an LU factorization of the form A = LU where L is unit lower triangular with ones along the diagonal, and U is invertible and upper triangular if and only if all principal submatrices of A are invertible.

2. Show both of the computations below can be performed in a backward stable manner. Assume all of the computations are performed in IEEE floating point arithmetic. For simplicity also assume, in both parts, x and A are representable in IEEE floating point arithmetic.

(a) The summation f(x) = x₁ + x₂ + x₃ as a function of x R ³

(b) The product f(A) = Ax as a function of A R ⁿⁿ only, and for a fixed x R ⁿ .