Automatic differentiation through the use of hyper-dual numbers for second derivatives. Automatic differentiation techniques are typically derived based on the chain rule of differentiation. Other methods can be derived based on the inherent mathematical properties of generalized complex numbers that enable first-derivative information to be carried in the non-real part of the number. These methods are capable of producing effectively exact derivative values. However, when second-derivative information is desired, generalized complex numbers are not sufficient. Higher-dimensional extensions of generalized complex numbers, with multiple non-real parts, can produce accurate second-derivative information provided that multiplication is commutative. One particular number system is developed, termed hyper-dual numbers, which produces exact first- and second-derivative information. The accuracy of these calculations is demonstrated on an unstructured, parallel, unsteady Reynolds-averaged Navier-Stokes solver.

References in zbMATH (referenced in 487 articles , 1 standard article )

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  1. Wallcraft, Alan J.: SPMD OpenMP versus MPI for ocean models (2000)
  2. Winstead, Carl; McKoy, Vincent: Parallel computational studies of electron-molecule collisions (2000)
  3. Hammond, Kevin (ed.); Michaelson, Greg (ed.): Research directions in parallel functional programming (1999)
  4. Huber, Gary A.; McCammon, J. Andrew: OOMPAA -- Object-oriented model for probing assemblages of atoms (1999)
  5. Lappa, M.; Savino, R.: Parallel solution of three-dimensional Marangoni flow in liquid bridges (1999)
  6. Murphy, K.; Clint, M.; Perrott, R. H.: Re-engineering statistical software for efficient parallel execution. (1999)
  7. Murty, Ravi; Okunbor, Daniel: Efficient parallel algorithms for molecular dynamics simulations (1999)
  8. Nakajima, Kengo; Okuda, Hiroshi: Parallel iterative solvers with localized ILU preconditioning for unstructured grids on workstation clusters (1999)
  9. Nakano, Aiichiro: Multiresolution load balancing in curved space: The wavelet representation (1999)
  10. Reuther, J.; Alonso, J. J.; Rimlinger, M. J.; Jameson, A.: Aerodynamic shape optimization of supersonic aircraft configurations via an adjoint formulation on distributed memory parallel computers (1999)
  11. Rossi, Tuomo; Toivanen, Jari: Parallel fictitious domain method for a non-linear elliptic Neumann boundary value problem (1999)
  12. Rossi, Tuomo; Toivanen, Jari: A parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension (1999)
  13. Borges, Leonardo; Oliveira, Suely: A parallel Davidson-type algorithm for several eigenvalues (1998)
  14. Dongarra, J. J.; Hammarling, S.; Walker, D. W.: Key concepts for parallel out-of-core LU factorization (1998)
  15. Elman, Howard C.; O’Leary, Dianne P.: Efficient iterative solution of the three-dimensional Helmholtz equation (1998)
  16. Kshemkalyani, Ajay D.: A framework for viewing atomic events in distributed computations (1998)
  17. Meyer, Ulrich; Sanders, Peter: (\Delta)-stepping: A parallel single source shortest path algorithm (1998)
  18. Nicholls, David P.: Traveling water waves: Spectral continuation methods with parallel implementation (1998)
  19. Shock, Carter T.; Chang, Chialin; Moon, Bongki; Acharya, Anurag; Davis, Larry: The design and evaluation of a high-performance earth science database (1998)
  20. Balay, Satish; Gropp, William D.; McInnes, Lois Curfman; Smith, Barry F.: Efficient management of parallelism in object-oriented numerical software libraries (1997)

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