On evaluating higher-order derivatives of the QR decomposition of tall matrices with full column rank in forward and reverse mode algorithmic differentiation. (English) Zbl 1242.65048
Summary: We address the task of higher-order derivative evaluation of computer programs that contain QR decompositions of tall matrices with full column rank. The approach is a combination of univariate Taylor polynomial arithmetic and matrix calculus in the (combined) forward/reverse mode of algorithmic differentiation. Explicit algorithms are derived and presented in an accessible form.
MSC:
65D25 | Numerical differentiation |
12E05 | Polynomials in general fields (irreducibility, etc.) |
58C15 | Implicit function theorems; global Newton methods on manifolds |
65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
65F25 | Orthogonalization in numerical linear algebra |
65F05 | Direct numerical methods for linear systems and matrix inversion |
Keywords:
automatic differentiation; null space; univariate Taylor polynomial arithmetic; higher-order derivative evaluation; QR decompositions; forward/reverse mode; algorithmic differentiation; algorithmsSoftware:
NumPyReferences:
[1] | Bernstein D. J., Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, Mathematical Sciences Research Institute Publications 44 pp 325– (2008) |
[2] | DOI: 10.1080/10556789308805543 · doi:10.1080/10556789308805543 |
[3] | DOI: 10.1080/10556789208805508 · doi:10.1080/10556789208805508 |
[4] | Giles, M. B. 2008.Collected matrix derivative results for forward and reverse mode algorithmic differentiation, in Advances in Automatic Differentiation, Edited by: Bischof, C. H., Martin Bücker, H., Hovland, P. D., Naumann, U. and Utke, J. 35–44. Berlin, Heidelberg: Springer. · Zbl 1154.65308 |
[5] | Golub G. H., Matrix Computations, 3. ed. (1996) · Zbl 0865.65009 |
[6] | DOI: 10.1080/10556789208805505 · doi:10.1080/10556789208805505 |
[7] | DOI: 10.1017/S0962492902000132 · Zbl 1047.65012 · doi:10.1017/S0962492902000132 |
[8] | DOI: 10.1090/S0025-5718-00-01120-0 · Zbl 0952.65028 · doi:10.1090/S0025-5718-00-01120-0 |
[9] | Griewank A., Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation 105, 2. ed. (2008) · Zbl 1159.65026 |
[10] | Healy M. J.R., Matrices for Statistics, 2. ed. (2000) · Zbl 0618.15004 |
[11] | DOI: 10.1016/j.compstruc.2009.10.001 · doi:10.1016/j.compstruc.2009.10.001 |
[12] | S. Körkel,Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen, Ph.D. thesis, Universitat Heidelberg, 2002 |
[13] | Körkel S., Reactive Flows, Diffusion and Transport pp 117– (2007) |
[14] | Magnus J. R., Matrix Differential Calculus with Applications in Statistics and Econometrics, 2. ed. (1999) · Zbl 0912.15003 |
[15] | Neidinger R. D., Math. Comput. 74 (249) pp 321– (2005) |
[16] | Oliphant T. E., Guide to NumPy (2006) |
[17] | E.T. Phipps,Taylor Series Integration of Differential-Algebraic Equations: Automatic Differentiation as a Tool For Simulating Rigid Body Mechanical Systems, Ph.D. thesis, Cornell University, February 2003 |
[18] | Schott J. R., Matrix Analysis for Statistics (1997) · Zbl 0872.15002 |
[19] | DOI: 10.1002/9780470226797 · doi:10.1002/9780470226797 |
[20] | DOI: 10.1080/10618600.1995.10474671 · doi:10.1080/10618600.1995.10474671 |
[21] | DOI: 10.1137/080718036 · Zbl 1194.65084 · doi:10.1137/080718036 |
[22] | A. Walther,Program Reversal Schedules for Single- and Multi-processor Machines, Ph.D. thesis, Institute of Scientific Computing, Technical University Dresden, Germany, 1999 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.