Validated numerics

Validated numerics, or rigorous computation, verified computation, reliable computation, numerical verification (German: Zuverlässiges Rechnen) is numerics including mathematically strict error (rounding error, truncation error, discretization error) evaluation, and it is one field of numerical analysis. For computation, interval arithmetic is used, and all results are represented by intervals. Validated numerics were used by Warwick Tucker in order to solve the 14th of Smale's problems,[1] and today it is recognized as a powerful tool for the study of dynamical systems.[2]

See also: Numerical analysis and Interval arithmetic

Importance

Computation without verification may cause unfortunate results. Below are some examples.

Rump's example

In the 1980s, Rump made an example.[3][4] He made a complicated function and tried to obtain its value. Single precision, double precision, extended precision results seemed to be correct, but its plus-minus sign was different from the true value.

Phantom solution

Breuer–Plum–McKenna used the spectrum method to solve the boundary value problem of the Emden equation, and reported that an asymmetric solution was obtained.[5] This result to the study conflicted to the theoretical study by Gidas–Ni–Nirenberg which claimed that there is no asymmetric solution.[6] The solution obtained by Breuer–Plum–McKenna was a phantom solution caused by discretization error. This is a rare case, but it tells us that when we want to strictly discuss differential equations, numerical solutions must be verified.

Accidents caused by numerical errors

The following examples are known as accidents caused by numerical errors:

주요 주제

검증된 숫자의 연구는 다음과 같은 분야로 나뉜다.

참고 항목: 일반 미분 방정식, 수치 선형 대수, 수치 사분법 및 계산 기하학에 대한 숫자 방법

도구들

참고 항목

References

  1. ^ Tucker, Warwick. (1999). "The Lorenz attractor exists." Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 328(12), 1197–1202.
  2. ^ Zin Arai, Hiroshi Kokubu, Paweãl Pilarczyk. Recent Development In Rigorous Computational Methods In Dynamical Systems.
  3. ^ Rump, Siegfried M. (1988). "Algorithms for verified inclusions: Theory and practice." In Reliability in computing (pp. 109–126). Academic Press.
  4. ^ Loh, Eugene; Walster, G. William (2002). Rump's example revisited. Reliable Computing, 8(3), 245-248.
  5. ^ 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.
  6. ^ Gidas, B.; Ni, Wei-Ming; Nirenberg, Louis (1979). "Symmetry and related properties via the maximum principle." Communications in Mathematical Physics, 68(3), 209–243.
  7. ^ http://www-users.math.umn.edu/~arnold//disasters/patriot.html
  8. ^ ARIANE 5 Flight 501 Failure, http://sunnyday.mit.edu/nasa-class/Ariane5-report.html
  9. ^ Rounding error changes Parliament makeup
  10. ^ Yamamoto, T. (1984). Error bounds for approximate solutions of systems of equations. Japan Journal of Applied Mathematics, 1(1), 157.
  11. ^ Oishi, S., & Rump, S. M. (2002). Fast verification of solutions of matrix equations. Numerische Mathematik, 90(4), 755-773.
  12. ^ Yamamoto, T. (1980). Error bounds for computed eigenvalues and eigenvectors. Numerische Mathematik, 34(2), 189-199.
  13. ^ Yamamoto, T. (1982). Error bounds for computed eigenvalues and eigenvectors. II. Numerische Mathematik, 40(2), 201-206.
  14. ^ Mayer, G. (1994). Result verification for eigenvectors and eigenvalues. Topics in Validated Computations, Elsevier, Amsterdam, 209-276.
  15. ^ Ogita, T. (2008). Verified Numerical Computation of Matrix Determinant. SCAN’2008 El Paso, Texas September 29–October 3, 2008, 86.
  16. ^ 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.
  17. ^ 미야지마 신야, Fast는 비대칭 대수 리카티 방정식, 연산 및 응용 수학, 37권, 이슈 4, 4599-4610페이지, 2018년 9월 최소 비음성 해법에 대한 연산을 검증했다.
  18. ^ 미야지마 신야, Fast는 T-concurnence Sylvester 방정식, 일본 산업 및 응용 수학 저널, 제35권, 제2권, 제541-551쪽, 2018년 7월 해답을 검증했다.
  19. ^ 미야지마 신야, 2차 행렬 방정식의 용매에 대한 고속 검증 연산, 선형 대수 전자 저널, 제34권 137-151, 2018년 3월
  20. ^ 미야지마 신야, Fast는 2017년 10월, 응용을 이용한 수치 선형대수학, 제24권, 제5권, 제1-12페이지에서 발생하는 대수 리카티 방정식의 해법에 대한 계산을 검증했다.
  21. ^ 미야지마 신야는 2017년 8월 이산 시간 대수 리카티 방정식, 연산 및 응용 수학 저널, 제319권 352-364페이지의 솔루션을 안정화하기 위한 연산을 검증했다.
  22. ^ 미야지마 신야, Fast는 연속시간 대수 리카티 방정식, 일본 산업 및 응용 수학 저널, 32권, 이슈 2, 529-544페이지, 2015년 7월, 연속시간 대수 리카티 방정식의 해법에 대한 연산을 검증했다.
  23. ^ 럼프, 지그프리드 M. (2014). 전체 부동 소수점 범위에 걸쳐 실제 감마 함수에 대해 확인된 예리한 한계. 비선형 이론과 그 응용, IEICE, 5(3), 339-348.
  24. ^ 야마나카, 나오야; 오카야마, 도모아키, 오이시, 신이치(2015년 11월). 반무한 구간에 대한 이중 지수 공식을 사용하여 실제 감마 함수에 대한 확인된 오차 한계. 컴퓨터와 정보과학의 수학적 측면에 관한 국제회의 (pp. 224-228)에서. 스프링거.
  25. ^ 요한슨, 프레드릭(2019년). 타원함수, 타원형 적분 및 모듈형식의 수치적 평가 Elliptic Integals, Elliptic Functions and Modular Forms in Quantum 필드 이론(pp. 269-293)에서. 스프링거, 참.
  26. ^ 요한슨, 프레드릭(2019년). 초지하학적 기능을 엄격하게 계산하는 중. ACM ToMS(수학적 소프트웨어에 관한 거래), 45(3), 30.
  27. ^ Johansson, Fredrik (2015). Rigorous high-precision computation of the Hurwitz zeta function and its derivatives. Numerical Algorithms, 69(2), 253-270.
  28. ^ Miyajima, S. (2018). Fast verified computation for the matrix principal pth root. en:Journal of Computational and Applied Mathematics, 330, 276-288.
  29. ^ Miyajima, S. (2019). Verified computation for the matrix principal logarithm. Linear Algebra and its Applications, 569, 38-61.
  30. ^ Miyajima, S. (2019). Verified computation of the matrix exponential. Advances in Computational Mathematics, 45(1), 137-152.
  31. ^ Johansson, Fredrik (2017). Arb: efficient arbitrary-precision midpoint-radius interval arithmetic. IEEE Transactions on Computers, 66(8), 1281-1292.
  32. ^ Johansson, Fredrik (2018, July). Numerical integration in arbitrary-precision ball arithmetic. In International Congress on Mathematical Software (pp. 255-263). Springer, Cham.
  33. ^ 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. ^ a b Eberhard Zeidler [de], Nonlinear Functional Analysis and Its Applications I-V. Springer Science & Business Media.
  35. ^ Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe (2019) Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics).
  36. ^ Oishi, Shin’ichi; Tanabe, Kunio (2009). Numerical Inclusion of Optimum Point for Linear Programming. JSIAM Letters, 1, 5-8.

Further reading

External links

  • 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.