1987 Fiscal Year Final Research Report Summary
Research on Programming Languages and Softwares for Automatic Calculation of Partial Derivatives and Rounding Error Estimates
| Project/Area Number |
60460130
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
計算機工学
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| Research Institution | The University of Tokyo |
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Principal Investigator |
IRI Masao Department of Mathematical Engineering and Information Physics, Faculty of Engineering, University of Tokyo., 工学部, 教授 (40010722)
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| Co-Investigator(Kenkyū-buntansha) |
KUBOTA Koichi Department of Mathematical Engineering and Information Physics, Fuculty of Engin, 工学部, 助手 (90178046)
MUROTA Kazuo Department of Mathematical Engineering and Information Physics, Faculty of Engin, 工学部, 助教授 (50134466)
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| Project Period (FY) |
1985 – 1987
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| Keywords | Fast Automatic Differentiation / Computational Graph / Automatic Differentiation / Partial Derivatives / Rounding Error / プリプロセッサ |
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| Research Abstract |
The aim of this research is to investigate a new method proposed by M. Iri, head investigator, which can rapidly and accurately evaluate partial derivatives of a function and rounding estimates in the computed values of the function. The new method solved difficulties in symbolic differentiation, numeric differentiation and rounding error estimation when we compute partial derivatives and norms in circuit and system analysis. We call the method "FAD: Fast Automatic Differentiation" because we combine the method with the conventional sutomatic differentiation. The results obtained by this research project are the following:
1. We confirmed that partial derivatives can actually be computed rapidly and accurately, and introduced a new concept of weighted norm with respect to rounding error estimates;
2. We proposed a new way to fast calculate the product of the Jacobian matrix of a vector-valued function and a given vector;
3. We proposed a new way to calculate the product of the Hessian matrix of a scalar-valued function and a given vector using the computational graph;
4. We described rigorously the algorithms of the new method in terms of a new concept of "computational subgraph" and gave interpretation to the quantities appearing in the algorithm;
5. We proposed a new algorithm for calculating rigorous and sharp estimates of rounding errors and compared it with the conventional;
6. We defined a new concept of "reducible subgraph" as well as a method to extract reducible subgraphs;
7. We completed the implementation of a FORTRAN preprocessor for FAD, and confirmed its performance through numerical experiments for large-scale problems.
In the field of numerical computation, those results will make FAD a widely usable basic practical technique, and techniques as well as theoretical properties concerning FAD should be worth further defailed investigation.
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