The Inversion Problem for Computable Linear Operators
- Vasco BrattkaAffiliated withTheoretische Informatik I, Informatikzentrum FernUniversität
Given a program of a linear bounded and bijective operator T, does there exist a program for the inverse operator T -1? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T -1? This is the inversion problem for computable linear operators on Banach spaces in its non-uniform and uniform formulation, respectively. We study this problem from the point of view of computable analysis which is the Turing machine based theory of computability on Euclidean space and other topological spaces. Using a computable version of Banach’s Inverse Mapping Theorem we can answer the first question positively. Hence, the non-uniform version of the inversion problem is solvable, while a topological argument shows that the uniform version is not. Thus, we are in the striking situation that any computable linear operator has a computable inverse while there exists no general algorithmic procedure to transfer a program of the operator into a program of its inverse. As a consequence, the computable version of Banach’s Inverse Mapping Theorem is a powerful tool which can be used to produce highly non-constructive existence proofs of algorithms. We apply this method to prove that a certain initial value problem admits a computable solution.
Keywordscomputable analysis linear operators inversion problem.
- The Inversion Problem for Computable Linear Operators
- Book Title
- STACS 2003
- Book Subtitle
- 20th Annual Symposium on Theoretical Aspects of Computer Science Berlin, Germany, February 27 – March 1, 2003 Proceedings
- pp 391-402
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
- Additional Links
- computable analysis
- linear operators
- inversion problem.
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