Lecture 11

Question 1

Undo a linear operation

Answer 1

Ax = y, then
x = A^-1.y

Question 2

Solve lower triangular systems

Answer 2

Forward substitution:
Lx = b
xi = (bi - sum(j=1>{i-1}) Lij xj) / Lii (derivate…)
If any Lii=0, singular and cannot be solved, otherwise unique solution

Question 3

Solve upper triangular systems

Answer 3

Backward substitution:
Ux = b
xi = (bi - sum(j=i+1 > n) Uij Xj) / Uii

If any Uii=0, singular and cannot be solved, otherwise unique solution

Derivation: x1 [U11 0 … 0] + … + xi [U1i U2i … Uii 0 … 0] + … + xn [U1n … Unn] = [b1 .. bn]
so xn = bn/Unn and
xn-1 = (b{n-1} - xn U{n-1,n}) / U{n-1,n-1}
xi Uii = bi - sum… xj Uij

Question 4

Complexity Forward/backward substitution?

Answer 4

n^2

Question 5

Solve Ax=b when A non triangular?

Answer 5

A=LU factorization, with L diagonal = 1 (because unicity –we could have chosen U diagonal = 1 instead). If the LU decomposition exists then it is unique (factorization does not always exist).
Ax = b > LUx = b
2 problems:
 - solve Ly = b (forward)
 - solve Ux = y (backward)

Question 6

2x2 example
A11 A12
A21 A22
find L, U

Answer 6

L =
1 0
A21/A11 1

U =
A11 A12
0 A22-(A21/A11)A12

Question 7

A in invertible, U?

Answer 7

U is invertible