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Most tensor problems are NP hard
 CORR
, 2009
"... The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has ..."
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Cited by 44 (6 self)
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The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3tensor, determining a best rank1 approximation to a 3tensor, determining the rank of a 3tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.
From coinductive proofs to exact real arithmetic
"... Abstract. We give a coinductive characterisation of the set of continuous functions defined on a compact real interval, and extract certified programs that construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresp ..."
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Cited by 20 (7 self)
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Abstract. We give a coinductive characterisation of the set of continuous functions defined on a compact real interval, and extract certified programs that construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching nonwellfounded trees describing when the algorithm writes and reads digits. This is a pilot study in using prooftheoretic methods for obtaining certified algorithms in exact real arithmetic. 1
RZ: A tool for bringing constructive and computable mathematics closer to programming practice
 CiE 2007: Computation and Logic in the Real World, volume 4497 of LNCS
, 2007
"... Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Obje ..."
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Cited by 6 (3 self)
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Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. 1
INTERVAL ARITHMETIC USING SSE2
"... ABSTRACT. We present an implementation of double precision interval arithmetic using the singleinstructionmultipledata SSE2 instruction and register set extensions. The implementation is part of a package for exact real arithmetic, which defines the interval arithmetic variation that must be use ..."
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ABSTRACT. We present an implementation of double precision interval arithmetic using the singleinstructionmultipledata SSE2 instruction and register set extensions. The implementation is part of a package for exact real arithmetic, which defines the interval arithmetic variation that must be used: incorrect operations such as division by zero cause exceptions and loose evaluation of the operations is in effect. The SSE2 extensions are suitable for the job, because they can be used to operate on a pair of double precision numbers and include separate rounding mode control and detection of the exceptional conditions. The paper describes the ideas we use to fit interval arithmetic to this set of instructions, shows a performance comparison with other freely available interval arithmetic packages, and discusses possible very simple hardware extensions that can significantly increase the performance of interval arithmetic. 1.
Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability
, 2009
"... It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f: A → B, additional structural information about the input x ∈ A (that is any kind of promise that x belongs to a certain subset A ′ ⊆ A) should be taken advantage of. Some examples from ..."
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It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f: A → B, additional structural information about the input x ∈ A (that is any kind of promise that x belongs to a certain subset A ′ ⊆ A) should be taken advantage of. Some examples from real number computation show that such discrete advice can even make the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problems how much advice is necessary and sufficient to render them computable. Specifically, finding a nontrivial solution to a homogeneous linear equation A · x = 0 for a given singular real n × nmatrix A is possible when knowing rank(A) ∈ {0, 1,..., n−1}; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors of) a given real symmetric n × nmatrix A is possible when knowing the number of distinct eigenvalues: an integer between 1 and n (the latter corresponding to the nondegenerate case). And again we show that n–fold (i.e. roughly log n bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas finding some single eigenvector of A requires and suffices with Θ(log n)–fold advice.
M.: Time complexity and convergence analysis of domain theoretic picard method. Extended Version available from http://wwwusers.aston.ac.uk/ ˜farjudia/AuxFiles/2008Picard.pdf
, 2008
"... Abstract. We present an implementation of the domaintheoretic Picard method for solving initial value problems (IVPs) introduced by Edalat and Pattinson [1]. Compared to Edalat and Pattinson’s implementation, our algorithm uses a more efficient arithmetic based on an arbitrary precision floatingpo ..."
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Abstract. We present an implementation of the domaintheoretic Picard method for solving initial value problems (IVPs) introduced by Edalat and Pattinson [1]. Compared to Edalat and Pattinson’s implementation, our algorithm uses a more efficient arithmetic based on an arbitrary precision floatingpoint library. Despite the additional overestimations due to floatingpoint rounding, we obtain a similar bound on the convergence rate of the produced approximations. Moreover, our convergence analysis is detailed enough to allow a static optimisation in the growth of the precision used in successive Picard iterations. Such optimisation greatly improves the efficiency of the solving process. Although a similar optimisation could be performed dynamically without our analysis, a static one gives us a significant advantage: we are able to predict the time it will take the solver to obtain an approximation of a certain (arbitrarily high) quality. 1
Semantics of QueryDriven Communication of Exact Values1
"... Abstract: We address the question of how to communicate among distributed processes values such as real numbers, continuous functions and geometrical solids with arbitrary precision, yet efficiently. We extend the established concept of lazy communication using streams of approximants by introducin ..."
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Abstract: We address the question of how to communicate among distributed processes values such as real numbers, continuous functions and geometrical solids with arbitrary precision, yet efficiently. We extend the established concept of lazy communication using streams of approximants by introducing explicit queries. We formalise this approach using protocols of a queryanswer nature. Such protocols enable processes to provide valid approximations with certain accuracy and focusing on certain locality as demanded by the receiving processes through queries. A latticetheoretic denotational semantics of channel and process behaviour is developed. The query space is modelled as a continuous lattice in which the top element denotes the query demanding all the information, whereas other elements denote queries demanding partial and/or local information. Answers are interpreted as elements of lattices constructed over suitable domains of approximations to the exact objects. An unanswered query is treated as an error and denoted using the top element. The major novel characteristic of our semantic model is that it reflects the dependency of answers on queries. This enables the definition and analysis of an appropriate concept of convergence rate, by assigning an effort indicator to each query and a measure of information content to each answer. Thus we capture not only what function a process computes, but also how a process transforms the convergence rates from its inputs to its outputs. In future work these indicators can be used to capture further computational complexity measures. A robust prototype implementation of our model is available.
Automatic Numerical Analysis Based on Infiniteprecision Arithmetic
"... Abstract—Numerical analysis is an important process for creating reliable numerical software. However, traditional analysis methods rely on manual estimation by numerical analysts, which is restricted by the problem size. Although some stateofart software packages can check whether a program is n ..."
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Abstract—Numerical analysis is an important process for creating reliable numerical software. However, traditional analysis methods rely on manual estimation by numerical analysts, which is restricted by the problem size. Although some stateofart software packages can check whether a program is numerical unstable, they cannot tell whether it is caused by illposed problem itself or by some improper implementation practices, while these packages work on the floating point values in the program. In this paper, we introduce an automatic framework that utilizes infiniteprecision arithmetic to analyze largescale numerical problems by computer. To eliminate rounding errors, the computing process iterates itself to increase intermediate precision until the calculation reaches the desired final precision. Then the framework perturbs the inputs and intermediate values of a certain numerical problem. By checking the gaps among different program outputs, the framework helps us understand whether the problem is wellconditioned or illconditioned. The framework also compares the infiniteprecision arithmetic with fixedprecision arithmetic. The evaluation of a bunch of classical problems shows that our framework is able to detect the illconditioning in largescale problems effectively.
Semantics of Distributed Numerical Computation with Lazy Communication of Exact Values1
"... Abstract: We adjust the concept of dataflow process networks as used for example by Kahn to suit exact computation over data types related to real numbers, such as continuous functions and geometrical solids. Furthermore, we consider communicating these exact objects among processes using protocols ..."
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Abstract: We adjust the concept of dataflow process networks as used for example by Kahn to suit exact computation over data types related to real numbers, such as continuous functions and geometrical solids. Furthermore, we consider communicating these exact objects among processes using protocols of a queryanswer nature. This enables processes to provide valid approximations with certain precision and focusing on certain locality as demanded by the receiving processes through queries. A latticetheoretic denotational semantics of channel and process behaviour is developed. The query space is modelled as a continuous lattice in which the top element denotes the query demanding all the information, whereas other elements denote queries demanding partial and/or local information. Answers are interpreted as elements of lattices constructed over suitable domains of approximations to the exact objects. An unanswered query is treated as an error and denoted using the top element. The major characteristic of our semantic model is that it reflects the dependency of answers on queries. This enables the definition and analysis of an appropriate concept of convergence rate, by assigning an effort indicator to each query and a measure of information content to each answer. Thus we capture not only what function a process computes, but also how a process transforms the convergence rates from its inputs to its outputs. In future work these indicators can be used to capture also various computational complexity measures. We provide several theorems that validate the compositional derivation of semantics for nested and recursively defined networks and define several process properties that help maintain compositionality. A robust prototype implementation of our model is available. Key Words: exact real computation, distributed computation, dataflow networks, denotational semantics, domain theory