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ERROR AND SENSITIVITY ANALYSIS FOR SYSTEMS OF LINEAR EQUATIONS
ERROR AND SENSITIVITY ANALYSIS FOR SYSTEMS OF LINEAR EQUATIONS
• Read Sections 2.7, 3.3, 3.4, 3.5. • Conditioning of linear systems . • Estimating accuracy • Error analysis 
Perturbation analysis Consider a linear system Ax = b. The question addressed by perturbation analysis is to determine the variation of the solution x when the data, namely A and b, undergoes small variations. A problem is illconditioned if small variations in the data lead to very large variation in the solution. Let E, be an n × n matrix and e be an nvector. “Perturb” A into A(ε) = A + εE and b into b + εe. Note: A + εE is nonsingular for ε small enough. Why? The solution x (ε) of the perturbed system is s.t.


Let . Then, x(ε) is differentiable at ε = 0 and its derivative is
A small variation [εE, εe] will cause the solution
to vary by Since : 
The quantity is called the
condition number of the linear system with respect to the norm . When using the standard norms , we label κ(A) with the same label as the associated norm. Thus, Note: ratio of largest to smallest singular values of A . Allows to define when A is not square . Determinant *is not* a good indication of sensitivity Small eigenvalues *do not* always give a good indication of poor conditioning.


Example: Consider matrices of the form for large α. The inverse of is and for the ∞norm we have so that For a large α, this can give a very large condition number, whereas all the eigenvalues of are equal to unity. 
Rigorous normbased error bounds First need to show that A+E is nonsingular if A is nonsingular and E is small: LEMMA: If A is nonsingular and then A + E is nonsingular and Proof is based on expansion THEOREM 1: Assume that (A + E)y = b + e and Ax = b and that . Then A + E is nonsingular and


roof: From (A + E)y = b + e and Ax = b we get A(y − x) = e − Ey = e − Ex − E(y − x). Hence:
Note: stated in a slightly weaker form in text.
Assume that 
Another common form: THEOREM 2: Let (A + ΔA)y = b + Δb and Ax = b where , and assume that . Then


Normwise backward error Question: How much do we need to perturb data for an approxi mate solution y to be the exact solution of the perturbed system? Normwise backward error for y is defined as smallest ε for which Denoted by . y is given (some computed solution). E and e are to be selected (most likely ’directions of perturbation for A and b’). Typical choice: E = A, e = b Explain why this is not unreasonable

Let r = b − Ax. Then we have: THEOREM 3: Normwise backward error is for case E = A, e = b: Show how this can be used in practice as a means to stop some iterative method which computes a sequence of approximate solutions to Ax = b. Consider the 6×6 Vandermonde system Ax = b where . We perturb A by E, with E ≤ 10^{−10}A and b similarly and solve the system. Evaluate the backward error for this case. Evaluate the forward bound provided by Theorem 2. Comment on the results. 

Componentwise backward error A few more definitions on norms... A norm is absolute for all x. (satisfied by all pnorms). A norm is monotone if . It can be shown that these two properties are equivalent . ... and some notation: (where E ≥ 0, e ≥ 0) is the componentwise backward error. 
Show: a function which satisfies the first 2
requirements of vector norms (1. (==0, iff x = 0) and 2. satisfies the triangle inequality iff its unit ball is convex. Use the above to construct a norm in R^{2} that is *not* absolute. Define absolute *matrix* norms in the same way. Which of the usual norms , and are absolute? Recall that for any matrix fl(A) = A+E with . For an absolute matrix norm What does this imply? 

THEOREM 4 [OettliPrager] Let r = b − Ay
(residual). Then
Analogue of theorem 2:

is equal to


Example of illconditioning: The Hilbert Matrix Notorious example of ill conditioning.
For
. 
Estimating Condition numbers. Let A,B be two n ×
n matrices with A nonsingular and B singular. Then Proof: B singular →: x ≠ 0 such that Bx = 0. Divide both sides by result. QED. Example: let and Then . 

Estimating errors from residual norms Let an approximate solution to system Ax = b (e.g., computed from an iterative process). We can compute the residual norm: Question: How to estimate the error from ? One option is to use the inequality We must have an estimate of κ(A). 
Proof of inequality. First, note that . So: Also note that from the relation b = Ax, we get Therefore, Show that


THEOREM 6 Let A be a nonsingular matrix and
an
approximate solution to Ax = b. Then for any norm , In addition, we have the relation in which κ(A) is the condition number of A associated with the norm . It can be shown that

Iterative refinement Define residual vector: We have seen that: , i.e., we have Idea: Compute r accurately (double precision) then solve ... and correct by ... repeat if needed. 

ALGORITHM : 1 Iterative refinement Do { 1. Compute r = b − A 2. Solve 3. Compute } while Why does this work? Model: each solution gets m digits at most because of the conditioning: For example 3 digits. At the first iteration, the error is roughly . Second iteration: error in is roughly . (but now is much smaller than ). etc .. 
Iterative refinement  Analysis Assume residual is computed exactly. Backward error analysis: So: A previous result showed that if then So : process will converge if (suff. condition) 

Important: Iterative refinement won’t work when
the residual r consists only of noise. When b − Ax is already very small (≈ ε) it is likely to be just noise, so not much can be done because Heuristic: If ε = 10^{−d}, and then each iterative refinement step will gain about d − q digits. A matlab experiment [do operations indicated ] 
repeat a couple of times .. Observation: we gain about 3 digits per iteration. Read Section 3.5.4 on estimating accuracy. 

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