मान्य संख्याएँ
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मान्य संख्यात्मकता, या कठोर गणना, सत्यापित गणना, विश्वसनीय गणना, संख्यात्मक सत्यापन (German: Zuverlässiges Rechnen) गणितीय रूप से सख्त त्रुटि (राउंडिंग त्रुटि, ट्रंकेशन त्रुटि, विवेकाधीन त्रुटि) मूल्यांकन सहित संख्यात्मक है, और यह संख्यात्मक विश्लेषण का एक क्षेत्र है। गणना के लिए, अंतराल अंकगणित का उपयोग किया जाता है, और सभी परिणाम अंतराल द्वारा दर्शाए जाते हैं। स्माले की 14वीं समस्याओं को हल करने के लिए वारविक टकर द्वारा मान्य संख्याओं का उपयोग किया गया था,[1]और आज इसे गतिशील प्रणालियों के अध्ययन के लिए एक शक्तिशाली उपकरण के रूप में मान्यता प्राप्त है।[2]
महत्व
सत्यापन के बिना गणना से दुर्भाग्यपूर्ण परिणाम हो सकते हैं। नीचे कुछ उदाहरण हैं.
दुम का उदाहरण
1980 के दशक में रम्प ने एक उदाहरण बनाया।[3][4]उन्होंने एक जटिल फ़ंक्शन बनाया और उसका मूल्य प्राप्त करने का प्रयास किया। एकल परिशुद्धता, दोहरी परिशुद्धता, विस्तारित परिशुद्धता परिणाम सही प्रतीत होते थे, लेकिन इसका प्लस-माइनस चिह्न वास्तविक मान से भिन्न था।
प्रेत समाधान
ब्रेउर-प्लम-मैककेना ने एम्डेन समीकरण की सीमा मूल्य समस्या को हल करने के लिए स्पेक्ट्रम विधि का उपयोग किया, और बताया कि एक असममित समाधान प्राप्त किया गया था।[5]अध्ययन का यह परिणाम गिडास-नी-निरेनबर्ग के सैद्धांतिक अध्ययन से विरोधाभासी है जिसमें दावा किया गया था कि कोई असममित समाधान नहीं है।[6]ब्रेउर-प्लम-मैककेना द्वारा प्राप्त समाधान विवेकाधीन त्रुटि के कारण उत्पन्न एक प्रेत समाधान था। यह एक दुर्लभ मामला है, लेकिन यह हमें बताता है कि जब हम अंतर समीकरणों पर सख्ती से चर्चा करना चाहते हैं, तो संख्यात्मक समाधानों को सत्यापित किया जाना चाहिए।
संख्यात्मक त्रुटियों के कारण दुर्घटनाएँ
निम्नलिखित उदाहरण संख्यात्मक त्रुटियों के कारण होने वाली दुर्घटनाओं के रूप में जाने जाते हैं:
- खाड़ी युद्ध में मिसाइलों को रोकने में विफलता (1991)[7]* एरियन 5 रॉकेट की विफलता (1996)[8]*चुनाव परिणाम के समग्रीकरण में गलतियाँ[9]
मुख्य विषय
मान्य संख्याओं के अध्ययन को निम्नलिखित क्षेत्रों में विभाजित किया गया है:
- Verification in numerical linear algebra
- Verification of special functions:
- Verification of numerical quadrature[31][32][33]
- Verification of nonlinear equations (The Kantorovich theorem,[34] Krawczyk method, interval Newton method, and the Durand–Kerner–Aberth method are studied.)
- Verification for solutions of ODEs, PDEs[35] (For PDEs, knowledge of functional analysis are used.[34])
- Verification of linear programming[36]
- Verification of computational geometry
- Verification at high-performance computing environment
उपकरण
- INTLAB Library made by MATLAB/GNU Octave
- kv Library made by C++. This library can obtain multiple precision outputs by using GNU MPFR.
- Arb Library made by C. It is capable to rigorously compute various special functions.
- CAPD A collection of flexible C++ modules which are mainly designed to computation of homology of sets, maps and validated numerics for dynamical systems.
- JuliaIntervals on GitHub (Library made by Julia)
- [1] Boost Safe Numerics - C++ header only library of validated replacements for all builtin integer types]].
यह भी देखें
संदर्भ
- ↑ Tucker, Warwick. (1999). "The Lorenz attractor exists." Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 328(12), 1197–1202.
- ↑ Zin Arai, Hiroshi Kokubu, Paweãl Pilarczyk. Recent Development In Rigorous Computational Methods In Dynamical Systems.
- ↑ Rump, Siegfried M. (1988). "Algorithms for verified inclusions: Theory and practice." In Reliability in computing (pp. 109–126). Academic Press.
- ↑ Loh, Eugene; Walster, G. William (2002). Rump's example revisited. Reliable Computing, 8(3), 245-248.
- ↑ Breuer, B.; Plum, Michael; McKenna, Patrick J. (2001). "Inclusions and existence proofs for solutions of a nonlinear boundary value problem by spectral numerical methods." In Topics in Numerical Analysis (pp. 61–77). Springer, Vienna.
- ↑ Gidas, B.; Ni, Wei-Ming; Nirenberg, Louis (1979). "Symmetry and related properties via the maximum principle." Communications in Mathematical Physics, 68(3), 209–243.
- ↑ "The Patriot Missile Failure".
- ↑ ARIANE 5 Flight 501 Failure, http://sunnyday.mit.edu/nasa-class/Ariane5-report.html
- ↑ Rounding error changes Parliament makeup
- ↑ Yamamoto, T. (1984). Error bounds for approximate solutions of systems of equations. Japan Journal of Applied Mathematics, 1(1), 157.
- ↑ Oishi, S., & Rump, S. M. (2002). Fast verification of solutions of matrix equations. Numerische Mathematik, 90(4), 755-773.
- ↑ Yamamoto, T. (1980). Error bounds for computed eigenvalues and eigenvectors. Numerische Mathematik, 34(2), 189-199.
- ↑ Yamamoto, T. (1982). Error bounds for computed eigenvalues and eigenvectors. II. Numerische Mathematik, 40(2), 201-206.
- ↑ Mayer, G. (1994). Result verification for eigenvectors and eigenvalues. Topics in Validated Computations, Elsevier, Amsterdam, 209-276.
- ↑ Ogita, T. (2008). Verified Numerical Computation of Matrix Determinant. SCAN’2008 El Paso, Texas September 29–October 3, 2008, 86.
- ↑ Shinya Miyajima, Verified computation for the Hermitian positive definite solution of the conjugate discrete-time algebraic Riccati equation, Journal of Computational and Applied Mathematics, Volume 350, Pages 80-86, April 2019.
- ↑ Shinya Miyajima, Fast verified computation for the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation, Computational and Applied Mathematics, Volume 37, Issue 4, Pages 4599-4610, September 2018.
- ↑ Shinya Miyajima, Fast verified computation for the solution of the T-congruence Sylvester equation, Japan Journal of Industrial and Applied Mathematics, Volume 35, Issue 2, Pages 541-551, July 2018.
- ↑ Shinya Miyajima, Fast verified computation for the solvent of the quadratic matrix equation, The Electronic Journal of Linear Algebra, Volume 34, Pages 137-151, March 2018
- ↑ Shinya Miyajima, Fast verified computation for solutions of algebraic Riccati equations arising in transport theory, Numerical Linear Algebra with Applications, Volume 24, Issue 5, Pages 1-12, October 2017.
- ↑ Shinya Miyajima, Fast verified computation for stabilizing solutions of discrete-time algebraic Riccati equations, Journal of Computational and Applied Mathematics, Volume 319, Pages 352-364, August 2017.
- ↑ Shinya Miyajima, Fast verified computation for solutions of continuous-time algebraic Riccati equations, Japan Journal of Industrial and Applied Mathematics, Volume 32, Issue 2, Pages 529-544, July 2015.
- ↑ Rump, Siegfried M. (2014). Verified sharp bounds for the real gamma function over the entire floating-point range. Nonlinear Theory and Its Applications, IEICE, 5(3), 339-348.
- ↑ Yamanaka, Naoya; Okayama, Tomoaki; Oishi, Shin’ichi (2015, November). Verified Error Bounds for the Real Gamma Function Using Double Exponential Formula over Semi-infinite Interval. In International Conference on Mathematical Aspects of Computer and Information Sciences (pp. 224-228). Springer.
- ↑ Johansson, Fredrik (2019). Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms. In Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory (pp. 269-293). Springer, Cham.
- ↑ Johansson, Fredrik (2019). Computing Hypergeometric Functions Rigorously. ACM Transactions on Mathematical Software (TOMS), 45(3), 30.
- ↑ Johansson, Fredrik (2015). Rigorous high-precision computation of the Hurwitz zeta function and its derivatives. Numerical Algorithms, 69(2), 253-270.
- ↑ Miyajima, S. (2018). Fast verified computation for the matrix principal pth root. en:Journal of Computational and Applied Mathematics, 330, 276-288.
- ↑ Miyajima, S. (2019). Verified computation for the matrix principal logarithm. Linear Algebra and its Applications, 569, 38-61.
- ↑ Miyajima, S. (2019). Verified computation of the matrix exponential. Advances in Computational Mathematics, 45(1), 137-152.
- ↑ Johansson, Fredrik (2017). Arb: efficient arbitrary-precision midpoint-radius interval arithmetic. IEEE Transactions on Computers, 66(8), 1281-1292.
- ↑ Johansson, Fredrik (2018, July). Numerical integration in arbitrary-precision ball arithmetic. In International Congress on Mathematical Software (pp. 255-263). Springer, Cham.
- ↑ Johansson, Fredrik; Mezzarobba, Marc (2018). Fast and Rigorous Arbitrary-Precision Computation of Gauss--Legendre Quadrature Nodes and Weights. SIAM Journal on Scientific Computing, 40(6), C726-C747.
- ↑ 34.0 34.1 Eberhard Zeidler, Nonlinear Functional Analysis and Its Applications I-V. Springer Science & Business Media.
- ↑ Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe (2019) Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics).
- ↑ Oishi, Shin’ichi; Tanabe, Kunio (2009). Numerical Inclusion of Optimum Point for Linear Programming. JSIAM Letters, 1, 5-8.
अग्रिम पठन
- Tucker, Warwick (2011). Validated Numerics: A Short Introduction to Rigorous Computations. Princeton University Press.
- Moore, Ramon Edgar, Kearfott, R. Baker., Cloud, Michael J. (2009). Introduction to Interval Analysis. Society for Industrial and Applied Mathematics.
- Rump, Siegfried M. (2010). Verification methods: Rigorous results using floating-point arithmetic. Acta Numerica, 19, 287–449.
बाहरी संबंध
- Validated Numerics for Pedestrians
- Reliable Computing, An open electronic journal devoted to numerical computations with guaranteed accuracy, bounding of ranges, mathematical proofs based on floating-point arithmetic, and other theory and applications of interval arithmetic and directed rounding.
