algorithms for the matrix exponential and frechet derivative

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Algorithms for the Matrix Exponential andits Frechet DerivativeAwad H. Al-MohyJuly 2010MIMS EPrint: 2010.63Manchester Institute for Mathematical SciencesSchool of MathematicsThe University of ManchesterReports available from: http://www.manchester.ac.uk/mims/eprintsAnd by contacting: The MIMS SecretarySchool of MathematicsThe University of ManchesterManchester, M13 9PL, UKISSN 1749-9097ALGORITHMS FOR THE MATRIXEXPONENTIAL AND ITS FRECHETDERIVATIVEA thesis submitted to the University of Manchesterfor the degree of Doctor of Philosophyin the Faculty of Engineering and Physical Sciences2010Awad H. Al-MohySchool of MathematicsContentsAbstract 7Declaration 9Copyright Statement 10Publications 11Acknowledgements 12Dedication 131 Background 141.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4 Floating point arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 Forward and backward errors . . . . . . . . . . . . . . . . . . . . . . 201.6 Frechet derivative and conditioning of a matrix function . . . . . . . 211.6.1 Kronecker form of the Frechet derivative . . . . . . . . . . . . 221.6.2 Computing or estimating the condition number . . . . . . . . 232 A New Scaling and Squaring algorithm for the Matrix Exponential 262.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Squaring phase for triangular matrices . . . . . . . . . . . . . . . . . 312.3 The existing scaling and squaring algorithm . . . . . . . . . . . . . . 332.4 Practical bounds for norm of matrix power series . . . . . . . . . . . 352.5 New algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 422.7 Cost of Pade versus Taylor approximants . . . . . . . . . . . . . . . . 462.7.1 Single precision . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Computing the Action of the Matrix Exponential, with an Applica-tion to Exponential Integrators 513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Exponential integrators: avoiding the functions . . . . . . . . . . . 523.3 Computing eAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.1 Preprocessing and termination criterion . . . . . . . . . . . . . 583.4 Rounding error analysis and conditioning . . . . . . . . . . . . . . . . 6023.5 Computing etAB over a time interval . . . . . . . . . . . . . . . . . . 633.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 654 Computing the Frechet Derivative of the Matrix Exponential, withan Application to Condition Number Estimation 734.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Frechet derivative via function of block triangular matrix . . . . . . . 744.3 Frechet derivative via power series . . . . . . . . . . . . . . . . . . . . 764.4 Cost analysis for polynomials . . . . . . . . . . . . . . . . . . . . . . 774.5 Computational framework . . . . . . . . . . . . . . . . . . . . . . . . 784.6 Scaling and squaring algorithm for the exponential and its Frechetderivative (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.7 Condition number estimation . . . . . . . . . . . . . . . . . . . . . . 874.8 Scaling and squaring algorithm for the exponential and its Frechetderivative (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 The Complex Step Approximation to the Frechet Derivative of aMatrix Function 975.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Complex step approximation: scalar case . . . . . . . . . . . . . . . . 985.3 Complex step approximation: matrix case . . . . . . . . . . . . . . . 985.4 Cost analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.5 Sign function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.6 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.7 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 Conclusions 108Bibliography 1113List of Tables2.1 Errors and condition numbers for A in (2.3) and B = QAQ. Thecolumns headed s show the values of s used by expm to produce theresults in the previous column. The superscripts and denote that aparticular choice of s was forced: s = 0 for and the s [0, 25] givingthe most accurate result for . . . . . . . . . . . . . . . . . . . . . . . 272.2 Parameters m needed in Algorithm 2.3.1 and Algorithm 2.5.1 andupper bounds for A(qm(A)). . . . . . . . . . . . . . . . . . . . . . . 342.3 The number of matrix products m in (2.33) needed for the Paterson-Stockmeyer scheme, m dened by (2.34), and Cm from (2.36). . . . . 483.1 Selected constants m for tol = 224(single precision) and tol = 253(double). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.2 Experiment 4: speed is time for method divided by time for Algo-rithm 3.5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.3 Experiment 5: speed is time for method divided by time for Algo-rithm 3.3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4 Experiment 7: cost is the number of matrixvector products, exceptfor ode15s for which it number of matrixvector products/number ofLU factorizations/number of linear system solves. The subscript onpoisson denotes the value of . . . . . . . . . . . . . . . . . . . . . . 714.1 Maximal values m of |2sA| such that the backward error bound(4.19) does not exceed u = 253, along with maximal values m suchthat a bound for |A|/|A| does not exceed u. . . . . . . . . . . . . 804.2 Number of matrix multiplications, m, required to evaluate rm(A) andLrm(A, E), and measure of overall cost Cm in (4.24). . . . . . . . . . . 834.3 Matrices that must be computed and stored during the initial eAeval-uation, to be reused during the Frechet derivative evaluations. LUfact stands for LU factorization of um +vm, and B = A/2s. . . . . 884List of Figures2.1 |Ak|1/k2 20k=1 for 54 16 16 matrices A with |A|2 = 1. . . . . . . . . 292.2 For the 22 matrix A in (2.10), |Ak|2 and various bounds for k = 1: 20. 302.3 Relative errors from Code Fragment 2.2.1 and expm mod for a single8 8 matrix with s = 0: 53. . . . . . . . . . . . . . . . . . . . . . . . 322.4 Results for test matrix Set 1. . . . . . . . . . . . . . . . . . . . . . . . 432.5 Results for test matrix Set 2. . . . . . . . . . . . . . . . . . . . . . . . 432.6 Results for test matrix Set 3. . . . . . . . . . . . . . . . . . . . . . . . 442.7 Results for test matrix Set 4. . . . . . . . . . . . . . . . . . . . . . . . 442.8 Quantities associated with the computed rm rm(2sA) for

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