# Special Topics in Computer Science I (CS493) in Summer 2018: Algorithmic Foundation of Numerics

This course offers a hands-on introduction to the rigorous foundations of computing with continuous data (real numbers, sequences, functions): from in/computability via complexity to programming practice.

### Purpose and Goals

We emphasize the difference between a recipe/heuristic and an algorithm. You will be reminded of aspects from the classical (i.e. discrete) Theory of Computation, from computability to complexity. We then apply it to numerical problems, yielding a rigorous perspective on the (i) computability, (ii) complexity, and (iii) semantics of: real tests, root finding, differentiation, maximization, integration, ODE and PDE solving. The course culminates in a programming assignment in iRRAM: a C++ library providing via object-oriented overloading an abstract data type REAL. In addition, each week will conclude with a theoretical homework assignment.

### Administrative

Lecturer: Martin Ziegler

Language: English only

Prerequisites: Introduction to Computer Science, Discrete Mathematics, Calculus I, C++, root on some Linux computer

Grading: 10% attendance, 50% homework/programming assignments, 40% final exam

### Schedule

- Tue, July 3, 12:10-13:00
- Wed, July 4, 12:10-13:00
- Thu, July 5, 12:10-13:00
- Fri, July 6, 9:00-12:00
- Tue, July 10, 12:10-13:00
- Wed, July 11, 12:10-13:00
- Thu, July 12, 12:10-13:00
- Tue, July 17, 12:10-13:00
- Wed, July 18, 12:10-13:00
- Thu, July 19, 12:10-13:00
- Fri, July 20, 9:00-12:00
- Tue, July 24, 12:10-13:00
- Wed, July 25, 12:10-13:00
- Thu, July 26, 12:10-13:00

### Synopsis

- Recap on Discrete Computability:
- Halting Problem
- Un/Semi/Decidability/Enumerability
- Reductions
- WHILE Programs
- Oracles

- Computing over the Reals
- Computable Real numbers: non/equivalent notions
- Equality, real sequences and limits
- Computing real functions, Effective Weierstrass Theorem
- Computational cost, compactness, and continuity
- Root finding and uncomputable argmax
- Uncomputable derivative/wave equation
- Non-extensionality, discrete enrichment

- Recap on Discrete Complexity:
- Runtime and memory, asymptotically
- Complexity classes P, NP, #P, PSPACE, EXP
- Polynomial-time reductions
- Cook-Levin Theorem (w/o proof)

- Real Complexity Theory
- Real arithmetic, join, maximum, integral
- Polynomial-time un/computable reals
- Polynomial-time un/computable functions
- Maximizing smooth functions and P/NP
- Riemann Integration and #P, ODEs and PSPACE
- Complexity Phase Transition and Gevrey's Hierarchy

- Practice of Exact Real Computation
- iRRAM library
- Semantics of tests and choose()
- Example Algorithms

### Reading List

- Mark Braverman, Stephen Cook: Computing over the Reals: Foundations for Scientific Computing, Notices of the AMS (2006).
- A. Kawamura, M. Ziegler: “Invitation to Real Complexity Theory: Algorithmic Foundations to Reliable Numerics with Bit-Costs”, 18th Korea-Japan Joint Workshop on Algorithms and Computation (2015).
- Norbert Müller: The iRRAM: Exact Arithmetic in C++ (2000).
- Norbert Müller: iRRAM C++ Library.

### Suggested Literature

- Ker-I Ko: Complexity Theory of Real Functions, Birkhäuser (1991).
- Klaus Weihrauch: Computable Analysis: An Introduction, Springer (2000).
- Akitoshi Kawamura, Stephen Cook: Complexity Theory for Operators in Analysis, ACM Transactions on Computation Theory 4 (2012).
- Akitoshi Kawamura, Hiroyuki Ota, Carsten Rösnick, Martin Ziegler: Computational Complexity of Smooth Differential Equations, Logical Methods in Computer Science (2014).
- Akitoshi Kawamura, Norbert Müller, Carsten Rösnick, Martin Ziegler: Computational Benefit of Smoothness: Parameterized Bit-Complexity of Numerical Operators on Analytic Functions and Gevrey's Hierarchy, pp.689-714 in the Journal of Complexity vol.31:5 (2015).
- Akitoshi Kawamura, Florian Steinberg, Martin Ziegler: On the Computational Complexity of the Dirichlet Problem for Poisson's Equation, Mathematical Structures in Computer Science (2017).