Lecture 13

Question 1

sensitivity of solutions linear system (perturbation to b)

Answer 1

||Δx|| / ||x|| <= cond(A) ||Δb||/||b||

Question 2

condition number

Answer 2

output relerr / input relerr

= (||Δx||/||x||)/(||Δb||/||b||) <= = ||A^-1|| ||A|| = cond(A)

Question 3

ill conditioned system

Answer 3

Amplifies error of the input (large cond(A))

Question 4

well conditioned system

Answer 4

small cond(A)

Question 5

sensitivity of solutions linear system (perturbation to A)

Answer 5

||Δx|| / ||x|| <= cond(A) ||ΔA||/||A||

Question 6

norm condition number

Answer 6

||A^-1|| ||A|| = cond(A)

cons relative to a given norm, cond_2(A), cond_infty(A)…

Question 7

What’s the best condition number

Answer 7

cond(A) = 1, indeed:
cond(A) = ||A||||A^-1|| >= ||AA^-1|| = ||Id||
||Id|| = max_{||x||=1} ||Id.x|| = max_{||x||=1} ||x|| = 1

Question 8

Matrix with optimal conditioning

Answer 8

Orthogonal matrices have optimal conditioning (A-1 = A.T) and are very well-behaved in computation

Question 9

Condition number of
100 0 0
0 13 0
0 0 0.5

Answer 9

cond(A) = ||A||||A^-1|| = 100.(1/0.5) = 200

Question 10

Condition number properties (4)

Answer 10

cond(A) >= 1
cond(Id) = 1
cond(aA) = cond(A)
for any diagonal matrix D, cond(D)=max|di|/min|di|

Question 11

Pivoting and singularity of the matrix relation

Answer 11

not related things, pivoting for numerical issues only (little c…)

Question 12

condition number indicator

Answer 12

Measure of how close a matrix is to being singular (det(A) close to zero doesn’t mean anything!) large number is nearly singular, a cond close to 1 is far from being singular

Question 13

residual error, error vector

Answer 13

||r|| = ||A ^x - b|| (no knowledge of x!)
||e|| = ||x - ^x||

Question 14

Guaranteed small if LU with pivot?

Answer 14

regardless of the condition number, the relative residual is always small when LU with pivot (not the rel err)
||r|| / (||A|| ||^x||) <= ||ΔA||/||A|| <= c.em

Question 15

Assuming IEEE double precision, how many digits of accuracy Ax=b when k(A)=1000?

Answer 15

10^3 10^-16 > 10^-13 digits of accuracy

Question 16

cond(A) if A-1 doesn’t exist

Answer 16

\infty by convention (A singular, det A = 0)

Question 17

Rule of the thumb for conditioning

Answer 17

||Δx|| / ||x|| <= cond(A) c.em

when solving with LU/pivot, loss of W decimal digits if cond(A)=10^w (from log10(em) digits = log10(2.2e-16))

Question 18

ABx=b, ||Δb||/||b||=1e-4, cond(A)=6, cond(B)=14, worst case rel err ||Δx||/||x||=?

Answer 18

84e-4 (because relerrx=rellerrb.condA.condB)