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Backward Thinking

Backward Thinking. Confessions of a Numerical Analyst Keith Evan Schubert. Simple Problem. Consider the problem ax=b The resulting x value is. Simple Problem 2. Consider the problem ax=b The resulting x value is. What’s Up?.

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Backward Thinking

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  1. Backward Thinking Confessions of a Numerical Analyst Keith Evan Schubert

  2. Simple Problem • Consider the problem ax=b • The resulting x value is

  3. Simple Problem 2 • Consider the problem ax=b • The resulting x value is

  4. What’s Up? • The condition number (sensitivity to perturbations) is about 400. • A condition number of 1 is perfect. • Perturbation is 0.01, so 0.01*400=4. • Components of x can vary by this much!

  5. What Can We Do? • Rather than solve it the standard way • X=a\b • X=(ATA)-1atb • Consider the following: • X=(ATA+i)-1atb •  =.01 • Then:

  6. Lucky Guess?

  7. Does It Always Work? • No • Consider  X0 • Consider si2(si is singular value of A) X± • Picking the wrong value can get junk

  8. Skyline • Consider a 1 dimensional picture • Use height instead of color • Result looks like the silhouette of a city’s skyline • Have smog which blurs and softens • Don’t know exactly how much blur • Want to get sharp edges

  9. Getting Garbage

  10. Getting Improvement

  11. Why Backward? • Forward errors • Explicitly account for each error source • (X+d1)(y+d2)=xy+(yd1+xd2+d1d2) • Backward errors • Check that my algorithm acting on data will give me a solution that is “near” to the actual system acting on a nearby set of data • I.E. My algorithm with good data should do about as well as a perfect calculation on ok data

  12. Picture Please! Inherent errors in A b Perfect Calculations b* Errors due to algorithm best My Algorithm Actual Data (x) Nearby Data (x*)

  13. Least Squares • Usually we don’t have an invertible matrix • Need to find an estimated solution • Criterion: minimize ||ax-b|| • Normal equation • ATA x = ATb • Solution • X = (ATA)-1atb

  14. Backward Error • Criterion: minimize ||Ax-b||/(||A|| ||x||+||b||) • Normal Equations • Solution:

  15. Non Convex

  16. Finding The Root

  17. Look At Critical Region

  18. Informal Algorithm • Get (A,b) • svd(A)  [u1 u2],,v • U1b  b1 • Use rootfinder (bisection, Newton, etc.) to get  in [-sn2,0] • vT(2- I)-1  b1  x

  19. What You Get

  20. Least Squares

  21. Total Least Squares

  22. Tikhonov

  23. Backward Error

  24. Original

  25. Comparison

  26. Final Thoughts • BE is always optimistic in that it presumes that the real system is “better” • Even with this it is “robust” • There is a perturbed version of this algorithm which can be either optimistic or pessimistic • That version is not fully proven

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