Math 460, Spring 1999: Computational Methods

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The midterm exam will be on Friday, March 16.
Problem 1(d): Actually, you need the hint for (ii), not (i). For (i), you should show that for any matrix B there holds ||B||_infinity = ||B||₁.
Important: Do not compute A^-1 in (d)!!. The whole point of (d) is to show that one can get a good estimate for ||A^-1|| without computing A^-1.
Problem 2 (c): Note that for vectors u, v the relative error ||v-u||/||u|| with respect to the norm || || is different from the componentwise relative errors |v_j - u_j|/|u_j|. E.g., consider the vectors u = (5, .1, 10)^T and v = (5, -.1, 10)^T.
In (c), we have that |ttilde_j - t_j|/|t_j| < 0.01. Use this to derive a bound for the corresponding right hand side vectors of the linear system: ||btilde - b||/||b|| < ??. Then derive an upper bound for ||utilde - u||/||u||. Then use this to get an upper bound for the componentwise relative errors |utilde_j - u_j|/|u_j|.

For cond_infinity(A) use the approximations for cond₁(A) which you computed in (b).
Play with interpolating polynomials
If you are not familiar with Matlab:
- Read chapter 1 of the textbook and try out the Matlab commands on a computer
- Read the Matlab tutorial (Postscript) and try out the Matlab commands on a computer
- Look at the other Matlab information below