Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 8C (Numerical Analysis)

Table of contents
- Solution 8C

Solution

Gaussian elimination without row exchanges gives the LU decomposition

A = LU, L = 121 - 2 011 - 1 001 - 1 0001, U = 3000 2 - 1 00 - 3 - 1 - 2 0 - 3 - 2 1 - 1 .

Because

L

has unit diagonal and

U

is upper triangular,

det A = det L det U = 3 (- 1) (- 2) (- 1) = - 6 .

Solving the two triangular systems gives

L y = b ⟹ y = (3, - 3, - 1, - 1)^{T},

and

Ux = y ⟹ x = (3, 0, 1, 1)^{T} .

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