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List of Figures
3.1 (a) x-values (b) y-values of solution paths of H(x, y, t) =0(0≤ t ≤ 1) . . 31
4.1 Example of a polynomial with multiple locations of the optimum ..... 44
4.2 Example polynomial p 1 (x 1 ,x 2 ) ........................ 48
5.1 Relationship between time series in (5.16) and the initial state w 0,0 .... 56
5.2 Time instants to compute the action of A T p 1
................. 57
5.3 Constructing and translating a minimal stable pattern ........... 64
5.4 A shortest path to compute y 13,12 ...................... 65
5.5 (a) The initial state w 0,0,0 (b) The initial state and the full recursions . . 67
5.6 A stable pattern (green) for shifts in all three directions .......... 67
5.7 (a) Points with total time 4. (b) Connections of the points ........ 68
5.8 Two different viewpoints of Figure 5.7(b) .................. 69
5.9 Computation of points with total time 4 ................... 70
5.10 A non-minimal stable pattern allowing shifts in three time directions . . . 71
5.11 Non-minimal stable patterns allowing for shifts along one time axis .... 72
5.12 Points required by the Linear method .................... 74
5.13 Points required by the Diagonal method ................... 75
5.14 Points required by the Equalizing method .................. 75
5.15 Points required by the Axis method . . . .................. 76
5.16 Points required by the linear, diagonal, equalizing and axis method .... 77
5.17 Points required by the linear, diagonal, equalizing and axis method .... 78
5.18 Points required by the least-increments method ............... 78
5.19 Region of points that occur in (a) 2-D paths and (b) 3-D paths ...... 80
7.1 Sparsity structure of the matrices A x1 and A p1 ............... 109
7.2 Residuals before and after projection ..................... 116
7.3 Sparsity structure of the matrices A p1 and A xi ,i=1,...,4 ........ 119
7.4 Eigenvalue spectra of the matrices A p1 and A xi ,i=1,...,4........ 120
7.5 Residual norms against MVs with A p1 for the JD and JDCOMM methods . . 121
7.6 Sparsity structure of the matrices A p1 and A xi ,i=1,...,4 ........ 122
270