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20cs700 [2020-09-03] – [Synopsis] Martin Ziegler | 20cs700 [2020-09-17] – [Synopsis] Martin Ziegler | ||
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IEEE 754 / Floatingpoint Discretization Properties and Reliability, | IEEE 754 / Floatingpoint Discretization Properties and Reliability, | ||
Numerical Folklore and Myths \\ | Numerical Folklore and Myths \\ | ||
- | **I. Recap on discrete Theory of Computation** ({{20cs700a.pdf|pdf}}, | + | **I. Recap on discrete Theory of Computation** ({{20cs700a.pdf|pdf}}, |
- | Un-/ | + | Un-/ |
- | **II. Computability theory over the Reals**: \\ | + | **II. Computability theory over the Reals**: |
- | real numbers: binary vs. approximation | + | a) |
- | sequences, limits, rate of convergence | + | b) Computing Real Sequences: semi-decidability / strong undecidability |
- | qualitative stability: computability | + | c) Computing Real Functions: closure under composition |
- | real arithmetic, join, maximum, integral | + | d) Advanced Un/ |
- | root finding, argmax, derivative | + | e) Multi-Functions & Enrichment: generalized restriction, |
- | uncomputable | + | f) Computing Real Operators: encoding continuous functions, encoding compact subsets |
- | analytic functions, discrete enrichment | + | |
- | multivaluedness, computability | + | |
**III. Computability on T0 spaces**: \\ | **III. Computability on T0 spaces**: \\ | ||
C[0;1] and computable Weierstrass Theorem | C[0;1] and computable Weierstrass Theorem | ||
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2nd-order polynomial complexity theory | 2nd-order polynomial complexity theory | ||
Weihrauch reductions | Weihrauch reductions | ||
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==== Test questions for self-assessment ==== | ==== Test questions for self-assessment ==== | ||
Mathematical background: | Mathematical background: |