This is an old revision of the document!


Alan Coelho 2017/08/28 16:43

TOPAS uses the Conjugate Gradient (CG) routine as described at:

Coelho, A. A. (2005). J. Appl. Cryst. 38, 455-461. "A bound constrained conjugate gradient solution method as applied to crystallographic refinement problems"

The routine uses min/max limits during the solution to the normal equations. Newton methods such as LU-Decomposition, SVD or Cholsky Decomposition cannot use min/max limits and hence struggle to solve the normal equations due to ill-conditioning. The following are outputs comparing CG with SVD:

Use of TOPAS Conjugate Gradient routine on PAWLEY1.INP:

TOPAS-32 Version 6 (c) 1992-2017 Alan A. Coelho
   Maximum number of threads 8
Time   0.01, INP file pre-processed
Loading xyz's for p-1 from file C:\c\t5\sg\p-1.sg
Number of hkls generated for C:\c\t5\sg\p-1.sg 530
Number of independent parameters : 551
  0  Time   0.02  Rwp   92.483    0.000 MC   0.00 0
Sparse matrix methods invoked - 83.0% of the A matrix elements are zero
  1  Time   0.03  Rwp   67.173  -25.310 MC   1.00 2
  2  Time   0.05  Rwp   37.184  -29.990 MC   1.15 1
  3  Time   0.06  Rwp   21.833  -15.351 MC   0.47 1
  4  Time   0.07  Rwp   14.298   -7.535 MC   0.16 1
  5  Time   0.08  Rwp   10.961   -3.337 MC   0.06 1
  6  Time   0.10  Rwp    8.309   -2.653 MC   0.02 1
  7  Time   0.11  Rwp    5.847   -2.462 MC   0.00 1
  8  Time   0.13  Rwp    4.554   -1.292 MC   0.00 1
  9  Time   0.14  Rwp    4.136   -0.418 MC   0.00 1
 10  Time   0.16  Rwp    3.966   -0.170 MC   0.00 1
 11  Time   0.17  Rwp    3.908   -0.059 MC   0.00 1
 12  Time   0.18  Rwp    3.894   -0.014 MC   0.00 1
 13  Time   0.19  Rwp    3.891   -0.002 MC   0.00 1
 14  Time   0.20  Rwp    3.888   -0.004 MC   0.00 1
 15  Time   0.21  Rwp    3.888   -0.000 MC   0.00 1
--- 0.213 seconds ---

*** Parameter(s) close to limit(s).
    Check for LIMIT_MIN and LIMIT_MAX in Grid/Text
Errors calculated
    
File C:\c\t5\test_examples\peak-intensity-extraction\pawley1.out updated
    with parameters from last iteration

Process Times (secs)
    0.01 = Peak buffer derivatives
    0.01 = Ycalc calculation and Penalties
    0.10 = A and Y matrix dot products and derivatives
    0.06 = Ycalc derivatives
    0.04 = A and Y matrix dot products
    0.07 = Solution to the normal equations
Using Cholsky Decomposition on PAWLEY1.INP:

TOPAS-32 Version 6 (c) 1992-2017 Alan A. Coelho
   Maximum number of threads 8
Loading C:\c\t5\topas.inc
Time   0.02, INP file pre-processed
Loading xyz's for p-1 from file C:\c\t5\sg\p-1.sg
Number of hkls generated for C:\c\t5\sg\p-1.sg 530
Number of independent parameters : 551
  0  Time   0.04  Rwp   92.483    0.000 MC   0.00 0
Loading C:\c\t5\topas.inc
Loading C:\c\t5\interface.inc
  1  Time   0.18  Rwp   61.400  -31.083 MC   1.00 2
  2  Time   0.26  Rwp   31.414  -29.986 MC   0.47 1
  3  Time   0.33  Rwp   18.986  -12.428 MC   0.16 1
  4  Time   0.40  Rwp   16.034   -2.952 MC   0.06 1
  5  Time   0.47  Rwp   15.974   -0.060 MC   0.03 1
  6  Time   0.54  Rwp   10.716   -5.258 MC   1.60 1
  7  Time   0.61  Rwp    8.807   -1.909 MC   0.56 1
  8  Time   0.68  Rwp    7.804   -1.003 MC   0.17 1
  9  Time   0.74  Rwp    7.694   -0.109 MC   0.05 1
 10  Time   0.82  Rwp    6.206   -1.488 MC   1.61 1
 11  Time   0.88  Rwp    5.753   -0.453 MC   0.52 1
 12  Time   0.99  Rwp    5.414   -0.339 MC   2.50 2
 13  Time   1.06  Rwp    5.303   -0.111 MC   0.74 1
 14  Time   1.16  Rwp    5.144   -0.159 MC   3.09 2
 15  Time   1.23  Rwp    5.097   -0.046 MC   0.89 1
 16  Time   1.33  Rwp    4.998   -0.100 MC   3.72 2
 17  Time   1.40  Rwp    4.967   -0.030 MC   1.05 1
 18  Time   1.51  Rwp    4.900   -0.068 MC   4.15 2
 19  Time   1.58  Rwp    4.879   -0.021 MC   1.18 1
 20  Time   1.69  Rwp    4.828   -0.050 MC   4.78 2
 21  Time   1.76  Rwp    4.811   -0.017 MC   1.35 1
 22  Time   1.87  Rwp    4.774   -0.038 MC   5.30 2
 23  Time   1.93  Rwp    4.760   -0.014 MC   1.49 1
 24  Time   2.03  Rwp    4.730   -0.029 MC   5.80 2
 25  Time   2.09  Rwp    4.718   -0.012 MC   1.64 1
 26  Time   2.20  Rwp    4.695   -0.023 MC   6.20 2
 27  Time   2.26  Rwp    4.685   -0.010 MC   1.76 1
 28  Time   2.36  Rwp    4.665   -0.019 MC   6.68 2
 29  Time   2.43  Rwp    4.656   -0.009 MC   1.86 1
 30  Time   2.52  Rwp    4.639   -0.016 MC   7.02 2
 31  Time   2.59  Rwp    4.631   -0.008 MC   1.96 1
 32  Time   2.69  Rwp    4.617   -0.014 MC   7.37 2
 33  Time   2.75  Rwp    4.609   -0.008 MC   2.08 1
 34  Time   2.85  Rwp    4.597   -0.012 MC   7.86 2
 35  Time   2.92  Rwp    4.590   -0.007 MC   2.20 1
 36  Time   3.03  Rwp    4.580   -0.011 MC   8.18 2
 37  Time   3.10  Rwp    4.573   -0.006 MC   2.29 1
 38  Time   3.20  Rwp    4.564   -0.009 MC   8.47 2
 39  Time   3.26  Rwp    4.558   -0.006 MC   2.38 1
 40  Time   3.36  Rwp    4.550   -0.008 MC   8.78 2
 41  Time   3.42  Rwp    4.544   -0.006 MC   2.54 1
 42  Time   3.52  Rwp    4.537   -0.007 MC   9.03 2
 43  Time   3.58  Rwp    4.532   -0.005 MC   2.57 1
 44  Time   3.68  Rwp    4.525   -0.007 MC   9.24 2
 45  Time   3.75  Rwp    4.520   -0.005 MC   2.58 1
 46  Time   3.85  Rwp    4.513   -0.006 MC   9.23 2
 47  Time   3.91  Rwp    4.509   -0.005 MC   2.57 1
 48  Time   4.01  Rwp    4.502   -0.006 MC   9.06 2
 49  Time   4.08  Rwp    4.498   -0.004 MC   2.53 1
 50  Time   4.18  Rwp    4.492   -0.006 MC   9.08 2
 51  Time   4.24  Rwp    4.488   -0.004 MC   2.51 1
 52  Time   4.34  Rwp    4.482   -0.006 MC   8.92 2
 53  Time   4.41  Rwp    4.478   -0.004 MC   2.46 1
 54  Time   4.50  Rwp    4.473   -0.005 MC   8.92 2
 55  Time   4.57  Rwp    4.469   -0.004 MC   2.45 1
 56  Time   4.67  Rwp    4.464   -0.005 MC   8.94 2
 57  Time   4.73  Rwp    4.460   -0.004 MC   2.46 1
 58  Time   4.84  Rwp    4.455   -0.005 MC   8.81 2
 59  Time   4.90  Rwp    4.452   -0.003 MC   2.34 1
 60  Time   5.00  Rwp    4.446   -0.005 MC   8.74 2
 61  Time   5.06  Rwp    4.443   -0.003 MC   2.40 1
 62  Time   5.16  Rwp    4.438   -0.005 MC   8.75 2
 63  Time   5.22  Rwp    4.435   -0.003 MC   2.39 1
 64  Time   5.32  Rwp    4.430   -0.005 MC   8.82 2
 65  Time   5.39  Rwp    4.427   -0.003 MC   2.53 1
 66  Time   5.49  Rwp    4.423   -0.005 MC   8.48 2
 67  Time   5.57  Rwp    4.420   -0.002 MC   2.38 1
 68  Time   5.67  Rwp    4.415   -0.005 MC   8.95 2
 69  Time   5.73  Rwp    4.413   -0.002 MC   2.38 1
 70  Time   5.83  Rwp    4.408   -0.005 MC   9.00 2
 71  Time   5.89  Rwp    4.406   -0.002 MC   2.46 1
 72  Time   5.99  Rwp    4.401   -0.004 MC   9.36 2
 73  Time   6.05  Rwp    4.399   -0.002 MC   2.54 1
 74  Time   6.16  Rwp    4.395   -0.004 MC   9.29 2
 75  Time   6.22  Rwp    4.393   -0.002 MC   2.55 1
 76  Time   6.32  Rwp    4.389   -0.004 MC   9.13 2
 77  Time   6.38  Rwp    4.387   -0.002 MC   2.54 1
 78  Time   6.48  Rwp    4.383   -0.004 MC   9.21 2
 79  Time   6.55  Rwp    4.381   -0.002 MC   2.57 1
 80  Time   6.65  Rwp    4.377   -0.004 MC   9.36 2
 81  Time   6.72  Rwp    4.375   -0.002 MC   2.65 1
 82  Time   6.83  Rwp    4.371   -0.004 MC   9.50 2
 83  Time   6.90  Rwp    4.369   -0.002 MC   2.65 1
 84  Time   7.00  Rwp    4.366   -0.004 MC   9.63 2
 85  Time   7.06  Rwp    4.364   -0.002 MC   2.62 1
 86  Time   7.16  Rwp    4.361   -0.003 MC   9.65 2
 87  Time   7.22  Rwp    4.359   -0.002 MC   2.63 1
 88  Time   7.32  Rwp    4.355   -0.003 MC   9.58 2
 89  Time   7.38  Rwp    4.354   -0.002 MC   2.63 1
 90  Time   7.48  Rwp    4.350   -0.003 MC   9.55 2
 91  Time   7.56  Rwp    4.349   -0.002 MC   2.64 1
 92  Time   7.66  Rwp    4.346   -0.003 MC   9.44 2
 93  Time   7.74  Rwp    4.344   -0.002 MC   2.65 1
 94  Time   7.84  Rwp    4.341   -0.003 MC   9.85 2
 95  Time   7.92  Rwp    4.339   -0.001 MC   2.64 1
 96  Time   8.02  Rwp    4.336   -0.003 MC  10.16 2
 97  Time   8.09  Rwp    4.335   -0.002 MC   2.72 1
 98  Time   8.19  Rwp    4.332   -0.003 MC   9.83 2
 99  Time   8.25  Rwp    4.330   -0.001 MC   2.70 1
100  Time   8.38  Rwp    4.327   -0.003 MC   9.96 2
101  Time   8.45  Rwp    4.326   -0.002 MC   2.86 1
102  Time   8.56  Rwp    4.323   -0.003 MC   9.99 2
103  Time   8.64  Rwp    4.322   -0.001 MC   2.72 1
104  Time   8.74  Rwp    4.319   -0.003 MC  10.20 2
105  Time   8.81  Rwp    4.317   -0.002 MC   2.91 1
106  Time   8.90  Rwp    4.315   -0.003 MC  10.09 2
107  Time   8.97  Rwp    4.314   -0.001 MC   2.73 1
108  Time   9.07  Rwp    4.311   -0.003 MC  10.33 2
109  Time   9.13  Rwp    4.310   -0.001 MC   2.80 1
110  Time   9.25  Rwp    4.307   -0.003 MC  10.90 2
111  Time   9.31  Rwp    4.306   -0.001 MC   2.71 1
112  Time   9.41  Rwp    4.303   -0.003 MC  11.08 2
113  Time   9.48  Rwp    4.302   -0.001 MC   3.04 1
114  Time   9.58  Rwp    4.299   -0.002 MC  10.32 2
115  Time   9.64  Rwp    4.298   -0.001 MC   2.80 1
116  Time   9.74  Rwp    4.296   -0.002 MC  10.59 2
117  Time   9.80  Rwp    4.295   -0.001 MC   2.90 1
118  Time   9.90  Rwp    4.292   -0.002 MC  10.43 2
119  Time   9.96  Rwp    4.291   -0.001 MC   2.82 1
120  Time  10.06  Rwp    4.289   -0.002 MC  10.48 2
121  Time  10.12  Rwp    4.288   -0.001 MC   2.82 1
122  Time  10.22  Rwp    4.286   -0.002 MC  10.43 2
123  Time  10.29  Rwp    4.285   -0.001 MC   2.79 1
124  Time  10.38  Rwp    4.282   -0.002 MC  10.64 2
125  Time  10.47  Rwp    4.281   -0.001 MC   2.86 1
126  Time  10.57  Rwp    4.279   -0.002 MC  10.38 2
127  Time  10.63  Rwp    4.278   -0.001 MC   2.82 1
--- 10.649 seconds ---

*** Parameter(s) close to limit(s).
    Check for LIMIT_MIN and LIMIT_MAX in Grid/Text
Errors calculated
    
File C:\c\t5\test_examples\peak-intensity-extraction\pawley1.out updated
    with parameters from last iteration

Process Times (secs)
    0.07 = Peak buffer derivatives
    0.20 = Ycalc calculation and Penalties
    1.37 = A and Y matrix dot products and derivatives
    0.58 = Ycalc derivatives
    0.79 = A and Y matrix dot products
    6.37 = Solution to the normal equations
SVD convergence behavior on PAWLEY1.INP:

  0  Time   0.02  Rwp   92.483    0.000 MC   0.00 0
  1  Time   2.77  Rwp   61.400  -31.083 MC   1.00 2
  2  Time   4.14  Rwp   31.414  -29.986 MC   0.47 1
  3  Time   5.50  Rwp   18.986  -12.428 MC   0.16 1
  4  Time   6.78  Rwp   16.034   -2.952 MC   0.06 1
  5  Time   8.07  Rwp   15.974   -0.060 MC   0.03 1
  6  Time   9.48  Rwp   10.716   -5.258 MC   1.60 1
  7  Time  10.84  Rwp    8.807   -1.909 MC   0.56 1
  8  Time  12.16  Rwp    7.804   -1.003 MC   0.17 1
  9  Time  13.46  Rwp    7.694   -0.109 MC   0.05 1
 10  Time  14.84  Rwp    6.206   -1.488 MC   1.61 1
 11  Time  16.19  Rwp    5.753   -0.453 MC   0.52 1
 12  Time  18.93  Rwp    5.414   -0.339 MC   2.50 2
 13  Time  20.37  Rwp    5.303   -0.111 MC   0.74 1
 14  Time  23.12  Rwp    5.144   -0.159 MC   3.09 2
 15  Time  24.54  Rwp    5.097   -0.046 MC   0.89 1
 16  Time  27.31  Rwp    4.998   -0.100 MC   3.72 2
 17  Time  28.73  Rwp    4.967   -0.030 MC   1.05 1
 18  Time  31.48  Rwp    4.900   -0.068 MC   4.15 2
 19  Time  32.87  Rwp    4.879   -0.021 MC   1.18 1
 20  Time  35.67  Rwp    4.828   -0.050 MC   4.78 2
 21  Time  37.10  Rwp    4.811   -0.017 MC   1.35 1
 22  Time  39.94  Rwp    4.774   -0.038 MC   5.30 2
 23  Time  41.34  Rwp    4.760   -0.014 MC   1.49 1
 24  Time  44.12  Rwp    4.730   -0.029 MC   5.80 2
 25  Time  45.57  Rwp    4.718   -0.012 MC   1.64 1
 26  Time  48.36  Rwp    4.695   -0.023 MC   6.20 2
 27  Time  49.82  Rwp    4.685   -0.010 MC   1.76 1
 28  Time  52.63  Rwp    4.665   -0.019 MC   6.68 2
 29  Time  54.04  Rwp    4.656   -0.009 MC   1.86 1
 30  Time  56.93  Rwp    4.639   -0.016 MC   7.02 2
 31  Time  58.33  Rwp    4.631   -0.008 MC   1.96 1
 32  Time  61.20  Rwp    4.617   -0.014 MC   7.37 2
 33  Time  62.63  Rwp    4.609   -0.008 MC   2.08 1
 34  Time  65.53  Rwp    4.597   -0.012 MC   7.86 2
 35  Time  66.96  Rwp    4.590   -0.007 MC   2.20 1
 36  Time  69.81  Rwp    4.580   -0.011 MC   8.18 2
 37  Time  71.28  Rwp    4.573   -0.006 MC   2.29 1
 38  Time  74.16  Rwp    4.564   -0.009 MC   8.47 2
 39  Time  75.58  Rwp    4.558   -0.006 MC   2.38 1
 40  Time  78.43  Rwp    4.550   -0.008 MC   8.78 2
 41  Time  79.86  Rwp    4.544   -0.006 MC   2.54 1
 42  Time  82.68  Rwp    4.537   -0.007 MC   9.03 2
 43  Time  84.13  Rwp    4.532   -0.005 MC   2.57 1
 44  Time  87.03  Rwp    4.525   -0.007 MC   9.24 2
 45  Time  88.46  Rwp    4.520   -0.005 MC   2.58 1
 46  Time  91.31  Rwp    4.513   -0.006 MC   9.23 2
 47  Time  92.73  Rwp    4.509   -0.005 MC   2.57 1
 48  Time  95.57  Rwp    4.502   -0.006 MC   9.06 2
 49  Time  97.04  Rwp    4.498   -0.004 MC   2.53 1

A few points to note:

- Convergence of the CG routine is fast.

- The SVD routine does try and minimize on the residual (A x – b)^2

- The SVD refinement was truncated at 50 iterations to minimize on waiting time.

- On a problem with 551 independent parameters, SVD is 100s of times slower over the whole refinement. Note, the CG routine is threaded.

- The Solution to the normal equations for CG took 0.07s; for LU decomposition its 6.37s.

Using the CG routine without limits

The following INP file is used:

r_wp  0
process_times

xdd alvo4.xdd

   CuKa1(0.001)
   Radius(200.5)
   Simple_Axial_Model(@, 1)
   LP_Factor(27)
   Zero_Error(@, 0)
   bkg @ 0 0 0 0 0 0 0 0
   
   hkl_Is
      a   @   6.54
      b   @   7.75
      c   @   9.12
      al  @   96.1
      be  @  107.2
      ga  @  101.4
      
      PV_Peak_Type(, 0, @, 0.05, @, 0.05, @, 0.05, @, 0.05, @, .05)
      
       
      macro VV(v) 
         {
            #m_unique i v min = -1e40; max = 1e40; update = Max(Val+Change,0);
            = i;
         }
      space_group "p-1"
      
      load hkl_m_d_th2 prm I
      {
            0   0   1   2    8.58454       10.29625 VV(1)
            0   1   0   2    7.48537       11.81320 VV(1)
            0   1  -1   2    6.21484       14.23967 VV(1)
            1   0   0   2    6.06127       14.60239 VV(1)
            1   0  -1   2    5.94891       14.87973 VV(1)
            1  -1   0   2    5.39088       16.43013 VV(1)
            0   1   1   2    5.20274       17.02859 VV(1)
            1  -1  -1   2    4.94022       17.94077 VV(1)
            1   1  -1   2    4.41793       20.08252 VV(1)
            1   0   1   2    4.33066       20.49155 VV(1)
            0   0   2   2    4.29227       20.67682 VV(1)
            1  -1   1   2    4.26461       20.81240 VV(1)
            1   1   0   2    4.23626       20.95326 VV(1)
            1   0  -2   2    4.20949       21.08801 VV(1)
            0   1  -2   2    4.04647       21.94796 VV(1)
            0   2   0   2    3.74269       23.75433 VV(1)
            1   1  -2   2    3.67893       24.17217 VV(1)
            0   2  -1   2    3.67841       24.17564 VV(1)
            1  -1  -2   2    3.65936       24.30337 VV(1)
            1  -2   0   2    3.59687       24.73228 VV(1)
            0   1   2   2    3.46733       25.67175 VV(1)
            1   1   1   2    3.38351       26.31901 VV(1)
            1  -2  -1   2    3.34477       26.62939 VV(1)
            1  -2   1   2    3.29076       27.07465 VV(1)
            0   2   1   2    3.22729       27.61759 VV(1)
            2   0  -1   2    3.20450       27.81798 VV(1)
            2  -1  -1   2    3.17113       28.11671 VV(1)
            1  -1   2   2    3.12045       28.58292 VV(1)
            0   2  -2   2    3.10742       28.70539 VV(1)
            2  -1   0   2    3.08006       28.96593 VV(1)
            1   0   2   2    3.06338       29.12706 VV(1)
            2   0   0   2    3.03063       29.44889 VV(1)
            1   2  -1   2    3.01639       29.59110 VV(1)
            1   0  -3   2    2.99141       29.84393 VV(1)
            2   0  -2   2    2.97446       30.01803 VV(1)
            1   2   0   2    2.88769       30.94216 VV(1)
            2  -1  -2   2    2.87939       31.03365 VV(1)
            0   0   3   2    2.86151       31.23245 VV(1)
            0   1  -3   2    2.84677       31.39836 VV(1)
            1   1  -3   2    2.84354       31.43496 VV(1)
            1   2  -2   2    2.80572       31.86981 VV(1)
            1  -2  -2   2    2.78839       32.07327 VV(1)
            2   1  -1   2    2.76272       32.37944 VV(1)
            1  -2   2   2    2.72639       32.82297 VV(1)
            1  -1  -3   2    2.71643       32.94678 VV(1)
            2  -2  -1   2    2.69977       33.15596 VV(1)
            2  -2   0   2    2.69544       33.21076 VV(1)
            2  -1   1   2    2.68688       33.31959 VV(1)
            2   1  -2   2    2.66184       33.64236 VV(1)
            1   1   2   2    2.61607       34.24898 VV(1)
            2   0   1   2    2.60370       34.41676 VV(1)
            0   2   2   2    2.60137       34.44849 VV(1)
            2   1   0   2    2.59905       34.48019 VV(1)
            2   0  -3   2    2.53337       35.40325 VV(1)
            1  -3   0   2    2.53286       35.41063 VV(1)
            0   1   3   2    2.52738       35.49006 VV(1)
            1   2   1   2    2.52316       35.55128 VV(1)
            0   3  -1   2    2.51883       35.61446 VV(1)
            0   3   0   2    2.49512       35.96437 VV(1)
            0   2  -3   2    2.49721       35.93325 VV(1)
            2  -2  -2   2    2.47011       36.34125 VV(1)
            2  -2   1   2    2.46020       36.49280 VV(1)
            1  -3   1   2    2.45887       36.51316 VV(1)
            2  -1  -3   2    2.43306       36.91444 VV(1)
            1   2  -3   2    2.41617       37.18181 VV(1)
            1  -3  -1   2    2.40082       37.42835 VV(1)
            1  -1   3   2    2.37322       37.88022 VV(1)
            2   1  -3   2    2.36761       37.97335 VV(1)
            0   3  -2   2    2.34567       38.34228 VV(1)
            1   0   3   2    2.31286       38.90808 VV(1)
            0   3   1   2    2.28950       39.32133 VV(1)
            2   1   1   2    2.28101       39.47363 VV(1)
            1   0  -4   2    2.26918       39.68803 VV(1)
            1  -2  -3   2    2.26465       39.77089 VV(1)
            2  -1   2   2    2.24302       40.17080 VV(1)
            2   2  -1   2    2.23414       40.33737 VV(1)
            1   1  -4   2    2.23270       40.36449 VV(1)
            1  -3   2   2    2.22465       40.51696 VV(1)
            1  -2   3   2    2.21481       40.70495 VV(1)
            1   3  -1   2    2.21255       40.74827 VV(1)
            2   2  -2   2    2.20896       40.81743 VV(1)
            2  -3   0   2    2.20561       40.88230 VV(1)
            2  -3  -1   2    2.17882       41.40781 VV(1)
            3  -1  -1   2    2.17007       41.58250 VV(1)
            2   0   2   2    2.16533       41.67785 VV(1)
            0   1  -4   2    2.16750       41.63424 VV(1)
            1   3  -2   2    2.15230       41.94204 VV(1)
            0   0   4   2    2.14613       42.06819 VV(1)
            2  -2  -3   2    2.14308       42.13106 VV(1)
            1  -3  -2   2    2.14055       42.18324 VV(1)
            1   3   0   2    2.13290       42.34165 VV(1)
            2  -2   2   2    2.13231       42.35412 VV(1)
            3   0  -1   2    2.12744       42.45570 VV(1)
            3  -1  -2   2    2.12723       42.46010 VV(1)
            1   2   2   2    2.12541       42.49828 VV(1)
            2   2   0   2    2.11813       42.65137 VV(1)
            1  -1  -4   2    2.11524       42.71249 VV(1)
            3   0  -2   2    2.11260       42.76840 VV(1)
            2   0  -4   2    2.10475       42.93597 VV(1)
            0   2   3   2    2.10039       43.02952 VV(1)
            2  -3   1   2    2.09604       43.12333 VV(1)
            3  -1   0   2    2.08131       43.44394 VV(1)
            1   1   3   2    2.07603       43.55996 VV(1)
            0   3  -3   2    2.07161       43.65768 VV(1)
            2   2  -3   2    2.05555       44.01651 VV(1)
            3  -2  -1   2    2.04359       44.28769 VV(1)
            2  -3  -2   2    2.02914       44.62011 VV(1)
            2   1  -4   2    2.02948       44.61227 VV(1)
            1   2  -4   2    2.02987       44.60318 VV(1)
            3   0   0   2    2.02042       44.82299 VV(1)
            2  -1  -4   2    2.02289       44.76545 VV(1)
            0   2  -4   2    2.02323       44.75736 VV(1)
            0   3   2   2    2.00780       45.12036 VV(1)
            3  -2   0   2    1.99039       45.53706 VV(1)
            3  -2  -2   2    1.98570       45.65067 VV(1)
            3   0  -3   2    1.98297       45.71699 VV(1)
            1   3  -3   2    1.98182       45.74513 VV(1)
            3  -1  -3   2    1.97346       45.94984 VV(1)
            0   1   4   2    1.97232       45.97795 VV(1)
            1   3   1   2    1.95182       46.48914 VV(1)
            3   1  -2   2    1.95060       46.51995 VV(1)
            2   1   2   2    1.94811       46.58276 VV(1)
            3   1  -1   2    1.94170       46.74562 VV(1)
            1  -3   3   2    1.93943       46.80366 VV(1)
            1  -4   0   2    1.92397       47.20255 VV(1)
            2   2   1   2    1.91538       47.42716 VV(1)
            1  -4   1   2    1.90990       47.57172 VV(1)
            3  -1   1   2    1.90179       47.78707 VV(1)
            2  -3   2   2    1.90068       47.81661 VV(1)
            0   4  -1   2    1.89996       47.83609 VV(1)
            1  -1   4   2    1.89136       48.06709 VV(1)
            0   4   0   2    1.87134       48.61434 VV(1)
            2  -1   3   2    1.86941       48.66780 VV(1)
            3   1  -3   2    1.86485       48.79473 VV(1)
            1  -2  -4   2    1.86180       48.87980 VV(1)
            1  -3  -3   2    1.85649       49.02865 VV(1)
            3  -2   1   2    1.84890       49.24346 VV(1)
            1  -4  -1   2    1.84650       49.31167 VV(1)
            3   1   0   2    1.84181       49.44567 VV(1)
            1   0   4   2    1.84284       49.41615 VV(1)
            3  -2  -3   2    1.84140       49.45740 VV(1)
            3   0   1   2    1.83761       49.56613 VV(1)
            2   2  -4   2    1.83946       49.51292 VV(1)
            0   4  -2   2    1.83920       49.52041 VV(1)
            2  -2  -4   2    1.82968       49.79561 VV(1)
            1  -2   4   2    1.82476       49.93920 VV(1)
            2  -2   3   2    1.82030       50.06993 VV(1)
            3  -3  -1   2    1.81896       50.10930 VV(1)
            2  -3  -3   2    1.81898       50.10880 VV(1)
            1   0  -5   2    1.81429       50.24730 VV(1)
            2   3  -2   2    1.81239       50.30348 VV(1)
            2   3  -1   2    1.80962       50.38595 VV(1)
            1   1  -5   2    1.81169       50.32438 VV(1)
            2   0   3   2    1.80719       50.45859 VV(1)
            1  -4   2   2    1.80988       50.37810 VV(1)
            2  -4   0   2    1.79843       50.72150 VV(1)
            3  -3   0   2    1.79696       50.76605 VV(1)
            3   0  -4   2    1.79114       50.94287 VV(1)
            0   3  -4   2    1.79195       50.91820 VV(1)
            1   2   3   2    1.78799       51.03886 VV(1)
            3  -1  -4   2    1.76863       51.63845 VV(1)
            2  -4  -1   2    1.76834       51.64754 VV(1)
            0   4   1   2    1.76437       51.77233 VV(1)
            1   3  -4   2    1.76643       51.70760 VV(1)
            3  -3  -2   2    1.76245       51.83302 VV(1)
            2   0  -5   2    1.76009       51.90776 VV(1)
            2  -4   1   2    1.75221       52.15866 VV(1)
            0   1  -5   2    1.74222       52.48059 VV(1)
            2   3  -3   2    1.73909       52.58219 VV(1)
            1   3   2   2    1.73470       52.72538 VV(1)
            0   3   3   2    1.73425       52.74030 VV(1)
            0   2   4   2    1.73366       52.75943 VV(1)
            2   3   0   2    1.73177       52.82164 VV(1)
            1   4  -1   2    1.72945       52.89782 VV(1)
            2   1  -5   2    1.72947       52.89714 VV(1)
            3   2  -2   2    1.72188       53.14873 VV(1)
            1  -1  -5   2    1.71847       53.26243 VV(1)
            0   0   5   2    1.71691       53.31475 VV(1)
            3   1  -4   2    1.71646       53.32974 VV(1)
            1   4  -2   2    1.71396       53.41368 VV(1)
            1   2  -5   2    1.71187       53.48409 VV(1)
            0   4  -3   2    1.71155       53.49493 VV(1)
            1  -4  -2   2    1.70604       53.68154 VV(1)
            3  -3   1   2    1.70431       53.74038 VV(1)
            1   1   4   2    1.70234       53.80743 VV(1)
            3   2  -1   2    1.70196       53.82040 VV(1)
            2  -1  -5   2    1.69769       53.96700 VV(1)
            3  -1   2   2    1.69209       54.16016 VV(1)
            2   2   2   2    1.69175       54.17175 VV(1)
            3   1   1   2    1.68675       54.34575 VV(1)
            2  -3   3   2    1.68277       54.48465 VV(1)
            1   4   0   2    1.67741       54.67336 VV(1)
            0   2  -5   2    1.67750       54.67019 VV(1)
            3   2  -3   2    1.67487       54.76335 VV(1)
            1  -3   4   2    1.67413       54.78958 VV(1)
            2  -4  -2   2    1.67238       54.85149 VV(1)
            3  -2   2   2    1.66717       55.03764 VV(1)
            2   1   3   2    1.66221       55.21571 VV(1)
            3  -2  -4   2    1.65893       55.33419 VV(1)
            1  -4   3   2    1.65850       55.34988 VV(1)
            3  -3  -3   2    1.64674       55.77951 VV(1)
            2  -4   2   2    1.64538       55.82953 VV(1)
            3   0   2   2    1.63421       56.24482 VV(1)
            1   4  -3   2    1.63601       56.17753 VV(1)
            4  -1  -2   2    1.63125       56.35621 VV(1)
            4  -1  -1   2    1.62287       56.67349 VV(1)
            3   2   0   2    1.62157       56.72279 VV(1)
            2   2  -5   2    1.61891       56.82439 VV(1)
            0   4   2   2    1.61365       57.02691 VV(1)
            0   1   5   2    1.61224       57.08135 VV(1)
            2   3  -4   2    1.61377       57.02193 VV(1)
            2   3   1   2    1.60405       57.39957 VV(1)
            2  -3  -4   2    1.60388       57.40635 VV(1)
            1  -3  -4   2    1.60297       57.44183 VV(1)
            4   0  -2   2    1.60225       57.47019 VV(1)
            4  -2  -1   2    1.58883       58.00145 VV(1)
            3   0  -5   2    1.58863       58.00937 VV(1)
            4  -2  -2   2    1.58556       58.13241 VV(1)
            4   0  -1   2    1.58322       58.22656 VV(1)
            4  -1  -3   2    1.58302       58.23478 VV(1)
            2  -1   4   2    1.57981       58.36433 VV(1)
            3  -4  -1   2    1.57970       58.36891 VV(1)
            3  -4   0   2    1.57593       58.52211 VV(1)
            3   2  -4   2    1.57559       58.53596 VV(1)
            1   4   1   2    1.57405       58.59869 VV(1)
            3  -3   2   2    1.56942       58.78873 VV(1)
            2  -2  -5   2    1.56782       58.85447 VV(1)
            4   0  -3   2    1.56700       58.88828 VV(1)
            1  -1   5   2    1.56427       59.00139 VV(1)
            1  -2  -5   2    1.56338       59.03818 VV(1)
            4  -1   0   2    1.56036       59.16385 VV(1)
            3  -1  -5   2    1.56222       59.08613 VV(1)
            2  -2   4   2    1.56023       59.16932 VV(1)
            1   3  -5   2    1.55511       59.38332 VV(1)
            0   4  -4   2    1.55371       59.44242 VV(1)
            0   3  -5   2    1.54863       59.65705 VV(1)
            3   1  -5   2    1.54594       59.77131 VV(1)
            1  -5   1   2    1.54563       59.78450 VV(1)
            1  -5   0   2    1.54282       59.90486 VV(1)
            1  -4  -3   2    1.54121       59.97386 VV(1)
            4  -2   0   2    1.54003       60.02439 VV(1)
            2  -4  -3   2    1.53858       60.08652 VV(1)
            1  -2   5   2    1.53587       60.20384 VV(1)
            2   0   4   2    1.53169       60.38498 VV(1)
            3  -4  -2   2    1.53222       60.36189 VV(1)
            4  -2  -3   2    1.53115       60.40858 VV(1)
            1   3   3   2    1.52628       60.62173 VV(1)
            1   0   5   2    1.52655       60.60950 VV(1)
            1   2   4   2    1.52256       60.78538 VV(1)
            3  -4   1   2    1.52197       60.81150 VV(1)
            0   5  -1   2    1.52124       60.84365 VV(1)
            3   1   2   2    1.51566       61.09166 VV(1)
            4   0   0   2    1.51532       61.10679 VV(1)
            1   4  -4   2    1.51800       60.98731 VV(1)
            1   1  -6   2    1.51467       61.13586 VV(1)
            4   1  -2   2    1.50936       61.37406 VV(1)
            2   4  -2   2    1.50820       61.42645 VV(1)
            2  -4   3   2    1.50725       61.46924 VV(1)
            1   0  -6   2    1.50664       61.49659 VV(1)
            3   2   1   2    1.50326       61.65004 VV(1)
            3  -3  -4   2    1.50289       61.66692 VV(1)
            0   3   4   2    1.50066       61.76876 VV(1)
            1  -5   2   2    1.50042       61.77965 VV(1)
            0   5  -2   2    1.49873       61.85686 VV(1)
            2   4  -1   2    1.49724       61.92545 VV(1)
            0   5   0   2    1.49707       61.93282 VV(1)
            3   3  -2   2    1.49498       62.02941 VV(1)
            2   0  -6   2    1.49571       61.99578 VV(1)
            4  -3  -1   2    1.49358       62.09372 VV(1)
            3  -1   3   2    1.49207       62.16359 VV(1)
            2  -5   0   2    1.49321       62.11064 VV(1)
            1  -5  -1   2    1.49272       62.13340 VV(1)
            4  -1  -4   2    1.49165       62.18326 VV(1)
            1  -4   4   2    1.49243       62.14716 VV(1)
            4   1  -3   2    1.48882       62.31452 VV(1)
            4   0  -4   2    1.48723       62.38862 VV(1)
            2   2   3   2    1.48517       62.48496 VV(1)
            4   1  -1   2    1.48430       62.52546 VV(1)
            3  -2   3   2    1.48385       62.54682 VV(1)
            2   1  -6   2    1.48576       62.45704 VV(1)
            4  -3  -2   2    1.48179       62.64331 VV(1)
            2  -3   4   2    1.47988       62.73350 VV(1)
            3  -2  -5   2    1.47611       62.91204 VV(1)
            2  -5   1   2    1.47523       62.95369 VV(1)
            3   3  -1   2    1.47298       63.06120 VV(1)
            2   4  -3   2    1.47393       63.01569 VV(1)
            3   3  -3   2    1.47264       63.07707 VV(1)
            2  -5  -1   2    1.46705       63.34518 VV(1)
            2   3  -5   2    1.46731       63.33286 VV(1)
            0   2   5   2    1.46507       63.44112 VV(1)
            4  -3   0   2    1.46137       63.62026 VV(1)
            1   2  -6   2    1.46345       63.51918 VV(1)
            4  -1   1   2    1.46037       63.66919 VV(1)
            2   3   2   2    1.45709       63.82915 VV(1)
            0   4   3   2    1.45248       64.05578 VV(1)
            4  -2   1   2    1.45205       64.07730 VV(1)
            0   1  -6   2    1.45372       63.99480 VV(1)
            1  -3   5   2    1.45114       64.12210 VV(1)
            3   2  -5   2    1.44897       64.22964 VV(1)
            2  -1  -6   2    1.44837       64.25922 VV(1)
            3  -4  -3   2    1.44593       64.38095 VV(1)
            1   4   2   2    1.44495       64.42992 VV(1)
            3   0   3   2    1.44355       64.49980 VV(1)
            2   4   0   2    1.44385       64.48501 VV(1)
            1  -1  -6   2    1.44205       64.57506 VV(1)
            4  -2  -4   2    1.43969       64.69361 VV(1)
            1   1   5   2    1.43554       64.90364 VV(1)
            0   5  -3   2    1.43530       64.91572 VV(1)
            2   1   4   2    1.43222       65.07294 VV(1)
            0   5   1   2    1.43240       65.06360 VV(1)
            3  -4   2   2    1.43163       65.10271 VV(1)
            0   0   6   2    1.43076       65.14755 VV(1)
            4  -3  -3   2    1.42896       65.23964 VV(1)
            4   1  -4   2    1.42787       65.29543 VV(1)
            0   2  -6   2    1.42339       65.52709 VV(1)
            3  -3   3   2    1.42154       65.62299 VV(1)
            4   1   0   2    1.41993       65.70637 VV(1)
            2   2  -6   2    1.42177       65.61092 VV(1)
            1  -5   3   2    1.41855       65.77839 VV(1)
            4   0   1   2    1.41534       65.94654 VV(1)
            2  -5   2   2    1.41757       65.82979 VV(1)
            1   5  -1   2    1.41408       66.01281 VV(1)
            3   3   0   2    1.41209       66.11813 VV(1)
            1   5  -2   2    1.41331       66.05337 VV(1)
            2  -3  -5   2    1.41086       66.18301 VV(1)
            3   3  -4   2    1.41150       66.14908 VV(1)
            1  -5  -2   2    1.40776       66.34772 VV(1)
            2  -5  -2   2    1.40316       66.59322 VV(1)
            3   0  -6   2    1.40316       66.59303 VV(1)
            2   4  -4   2    1.40286       66.60924 VV(1)
            2  -4  -4   2    1.39419       67.07791 VV(1)
            4  -3   1   2    1.39297       67.14481 VV(1)
            1  -3  -5   2    1.39299       67.14358 VV(1)
            0   4  -5   2    1.39446       67.06368 VV(1)
            1   4  -5   2    1.38481       67.59329 VV(1)
            4   2  -2   2    1.38136       67.78515 VV(1)
            4   0  -5   2    1.38130       67.78834 VV(1)
            1  -4  -4   2    1.37940       67.89464 VV(1)
            3   1  -6   2    1.38070       67.82171 VV(1)
            1   5   0   2    1.37792       67.97737 VV(1)
            4  -1  -5   2    1.37756       67.99769 VV(1)
            3  -1  -6   2    1.37759       67.99610 VV(1)
            1   5  -3   2    1.37579       68.09726 VV(1)
            4   2  -3   2    1.37265       68.27406 VV(1)
            3   2   2   2    1.37118       68.35747 VV(1)
            3  -5   0   2    1.37192       68.31576 VV(1)
            1   3  -6   2    1.36889       68.48760 VV(1)
            3  -5  -1   2    1.36729       68.57940 VV(1)
            4  -4  -1   2    1.36576       68.66649 VV(1)
            2  -4   4   2    1.36320       68.81395 VV(1)
            0   1   6   2    1.36145       68.91493 VV(1)
            2   4   1   2    1.36029       68.98160 VV(1)
            2  -1   5   2    1.35790       69.12062 VV(1)
            2  -2  -6   2    1.35821       69.10205 VV(1)
            4   2  -1   2    1.35515       69.28084 VV(1)
            3  -3  -5   2    1.35601       69.23063 VV(1)
            3   1   3   2    1.35298       69.40742 VV(1)
            2  -2   5   2    1.35218       69.45477 VV(1)
            4  -4  -2   2    1.34988       69.58975 VV(1)
            4  -4   0   2    1.34772       69.71770 VV(1)
            0   3  -6   2    1.34882       69.65246 VV(1)
            4  -3  -4   2    1.34696       69.76247 VV(1)
            1   3   4   2    1.34429       69.92169 VV(1)
            4  -1   2   2    1.34327       69.98200 VV(1)
            4  -2   2   2    1.34344       69.97205 VV(1)
            0   5  -4   2    1.34502       69.87805 VV(1)
            3  -5   1   2    1.34251       70.02778 VV(1)
            0   5   2   2    1.34144       70.09199 VV(1)
            1  -2  -6   2    1.33999       70.17880 VV(1)
            4   1  -5   2    1.33996       70.18070 VV(1)
            3  -4  -4   2    1.33893       70.24221 VV(1)
            1  -4   5   2    1.33450       70.51049 VV(1)
            2  -5   3   2    1.33284       70.61097 VV(1)
            4   1   1   2    1.33014       70.77582 VV(1)
            4   2  -4   2    1.33092       70.72834 VV(1)
            1  -1   6   2    1.33044       70.75743 VV(1)
            4  -2  -5   2    1.32978       70.79802 VV(1)
            3  -5  -2   2    1.32961       70.80858 VV(1)
            3   3   1   2    1.32540       71.06717 VV(1)
            3   3  -5   2    1.32468       71.11204 VV(1)
            3  -4   3   2    1.32309       71.21034 VV(1)
            2   0   5   2    1.32056       71.36747 VV(1)
            2   3  -6   2    1.32211       71.27122 VV(1)
            1  -2   6   2    1.31914       71.45648 VV(1)
            3  -2   4   2    1.31796       71.53034 VV(1)
            3  -1   4   2    1.31734       71.56858 VV(1)
            1   2   5   2    1.31638       71.62923 VV(1)
            1  -5   4   2    1.31667       71.61117 VV(1)
            3   2  -6   2    1.31657       71.61740 VV(1)
            2  -5  -3   2    1.31497       71.71812 VV(1)
            2   3   3   2    1.31250       71.87355 VV(1)
            1   5   1   2    1.31259       71.86803 VV(1)
            1   4   3   2    1.31144       71.94114 VV(1)
            3  -2  -6   2    1.31118       71.95763 VV(1)
            0   3   5   2    1.30995       72.03553 VV(1)
            2   2   4   2    1.30803       72.15765 VV(1)
            5  -1  -2   2    1.30738       72.19939 VV(1)
            2   4  -5   2    1.30959       72.05844 VV(1)
            1   5  -4   2    1.30952       72.06293 VV(1)
            1  -5  -3   2    1.30461       72.37720 VV(1)
            2  -3   5   2    1.30496       72.35418 VV(1)
            4  -4  -3   2    1.30342       72.45367 VV(1)
            4   0   2   2    1.30185       72.55484 VV(1)
            4  -3   2   2    1.30230       72.52545 VV(1)
            0   4   4   2    1.30069       72.63006 VV(1)
            1   0   6   2    1.30088       72.61734 VV(1)
            4   2   0   2    1.29953       72.70522 VV(1)
            4  -4   1   2    1.29953       72.70513 VV(1)
            3   4  -2   2    1.29828       72.78626 VV(1)
            1   1  -7   2    1.29725       72.85314 VV(1)
            5  -1  -3   2    1.29557       72.96301 VV(1)
            5  -2  -2   2    1.29492       73.00537 VV(1)
            2   0  -7   2    1.29278       73.14578 VV(1)
            1  -6   1   2    1.29237       73.17309 VV(1)
            2   1  -7   2    1.29227       73.17960 VV(1)
            5  -1  -1   2    1.28941       73.36848 VV(1)
            3   4  -3   2    1.28946       73.36530 VV(1)
            1   0  -7   2    1.28628       73.57617 VV(1)
            3  -5   2   2    1.28511       73.65436 VV(1)
            1  -6   0   2    1.28482       73.67373 VV(1)
            5  -2  -1   2    1.28325       73.77857 VV(1)
            5   0  -2   2    1.28096       73.93251 VV(1)
            3  -3   4   2    1.27946       74.03345 VV(1)
            2   5  -2   2    1.27983       74.00864 VV(1)
            3   0   4   2    1.27779       74.14661 VV(1)
            3   4  -1   2    1.27799       74.13336 VV(1)
            5  -2  -3   2    1.27764       74.15671 VV(1)
            5   0  -3   2    1.27554       74.29936 VV(1)
            1  -6   2   2    1.27122       74.59445 VV(1)
            1  -3   6   2    1.26990       74.68520 VV(1)
            1   2  -7   2    1.27027       74.66014 VV(1)
            2   5  -1   2    1.26745       74.85470 VV(1)
            0   6  -1   2    1.26703       74.88391 VV(1)
            3  -5  -3   2    1.26639       74.92801 VV(1)
            4   0  -6   2    1.26669       74.90740 VV(1)
            2  -6   0   2    1.26643       74.92511 VV(1)
            0   2   6   2    1.26369       75.11588 VV(1)
            4   2  -5   2    1.26442       75.06512 VV(1)
            2   5  -3   2    1.26424       75.07760 VV(1)
            2   4   2   2    1.26158       75.26305 VV(1)
            2  -6   1   2    1.26091       75.31028 VV(1)
            5   0  -1   2    1.25850       75.47980 VV(1)
            0   6  -2   2    1.25942       75.41504 VV(1)
            4  -1  -6   2    1.25826       75.49667 VV(1)
            5  -1  -4   2    1.25627       75.63691 VV(1)
            2  -4  -5   2    1.25627       75.63726 VV(1)
            2  -1  -7   2    1.25634       75.63235 VV(1)
            2   2  -7   2    1.25492       75.73256 VV(1)
            3   4  -4   2    1.25322       75.85399 VV(1)
            1   4  -6   2    1.25384       75.80945 VV(1)
            2   1   5   2    1.24978       76.09968 VV(1)
            4  -3  -5   2    1.25024       76.06652 VV(1)
            1  -6  -1   2    1.25002       76.08244 VV(1)
            0   6   0   2    1.24756       76.25924 VV(1)
            5  -3  -2   2    1.24672       76.31999 VV(1)
            0   4  -6   2    1.24861       76.18398 VV(1)
            2  -3  -6   2    1.24692       76.30525 VV(1)
            5  -1   0   2    1.24511       76.43672 VV(1)
            5  -2   0   2    1.24486       76.45482 VV(1)
            4   3  -2   2    1.24526       76.42531 VV(1)
            0   1  -7   2    1.24604       76.36896 VV(1)
            2  -6  -1   2    1.24499       76.44540 VV(1)
            4   3  -3   2    1.24416       76.50517 VV(1)
            5   0  -4   2    1.24326       76.57064 VV(1)
            3   2   3   2    1.24191       76.66954 VV(1)
            3   0  -7   2    1.24349       76.55381 VV(1)
            5  -3  -1   2    1.24155       76.69553 VV(1)
            0   5  -5   2    1.24297       76.59214 VV(1)
            1  -1  -7   2    1.24008       76.80303 VV(1)
            0   5   3   2    1.23920       76.86752 VV(1)
            4   1  -6   2    1.23981       76.82279 VV(1)
            1   1   6   2    1.23790       76.96323 VV(1)
            4  -4  -4   2    1.23505       77.17303 VV(1)
            5  -2  -4   2    1.23467       77.20133 VV(1)
            2  -5   4   2    1.23529       77.15550 VV(1)
            1  -4  -5   2    1.23348       77.28978 VV(1)
            3   4   0   2    1.23240       77.36997 VV(1)
            3   1  -7   2    1.23293       77.33086 VV(1)
            4  -5  -1   2    1.23126       77.45502 VV(1)
            0   2  -7   2    1.23191       77.40623 VV(1)
            4   1   2   2    1.22946       77.58971 VV(1)
            4  -2   3   2    1.22999       77.54972 VV(1)
            4  -4   2   2    1.23010       77.54187 VV(1)
            1   5   2   2    1.22971       77.57074 VV(1)
            2   5   0   2    1.22940       77.59406 VV(1)
            2  -6   2   2    1.22944       77.59145 VV(1)
            3   3   2   2    1.22708       77.76862 VV(1)
            2  -4   5   2    1.22793       77.70424 VV(1)
            3  -4  -5   2    1.22680       77.78990 VV(1)
            5  -3  -3   2    1.22615       77.83861 VV(1)
            0   0   7   2    1.22636       77.82252 VV(1)
            4   2   1   2    1.22453       77.96074 VV(1)
            4  -1   3   2    1.22476       77.94380 VV(1)
            3   3  -6   2    1.22631       77.82653 VV(1)
            0   6  -3   2    1.22614       77.83964 VV(1)
            1   5  -5   2    1.22618       77.83628 VV(1)
            1  -6   3   2    1.22533       77.90044 VV(1)
            1  -3  -6   2    1.22291       78.08360 VV(1)
            4  -5   0   2    1.22301       78.07600 VV(1)
            5   1  -2   2    1.22214       78.14275 VV(1)
            5   1  -3   2    1.22244       78.11934 VV(1)
            2   5  -4   2    1.22356       78.03452 VV(1)
            4   3  -1   2    1.22090       78.23715 VV(1)
            3  -1  -7   2    1.22053       78.26509 VV(1)
            3  -3  -6   2    1.21979       78.32199 VV(1)
            4   3  -4   2    1.21779       78.47497 VV(1)
            2  -5  -4   2    1.21637       78.58457 VV(1)
            4  -2  -6   2    1.21653       78.57217 VV(1)
            4  -5  -2   2    1.21461       78.72073 VV(1)
            5   0   0   2    1.21225       78.90324 VV(1)
            5  -3   0   2    1.21156       78.95700 VV(1)
            3  -4   4   2    1.21115       78.98941 VV(1)
            1   3  -7   2    1.21183       78.93593 VV(1)
            3   1   4   2    1.20878       79.17414 VV(1)
            3  -5   3   2    1.20973       79.10024 VV(1)
            1  -5   5   2    1.20925       79.13741 VV(1)
            2   4  -6   2    1.20809       79.22882 VV(1)
            0   6   1   2    1.20453       79.50926 VV(1)
            4  -3   3   2    1.20297       79.63291 VV(1)
            2  -6  -2   2    1.20041       79.83661 VV(1)
            5   1  -1   2    1.19781       80.04542 VV(1)
            5   1  -4   2    1.19868       79.97561 VV(1)
            3  -6   0   2    1.19896       79.95315 VV(1)
            1  -5  -4   2    1.19725       80.08967 VV(1)
            1   6  -2   2    1.19818       80.01569 VV(1)
            5  -1  -5   2    1.19641       80.15774 VV(1)
            3   4  -5   2    1.19596       80.19403 VV(1)
            1  -6  -2   2    1.19411       80.34368 VV(1)
            1   6  -1   2    1.19391       80.35942 VV(1)
            1  -4   6   2    1.19420       80.33659 VV(1)
      }                                                

The CG routine in this case is used to solve the normal equations but the limits on the intensity parameter is set to the very large values of -1e40 and 1e40. The macro VV limits the intensity to a positive value after the solution to the normal equations; this is performed using:

#m_unique i v min = -1e40; max = 1e40; update = Max(Val+Change,0);

Thus convergence output is as follows:

TOPAS-32 Version 6 (c) 1992-2017 Alan A. Coelho
   Maximum number of threads 8
Time   0.01, INP file pre-processed
Number of independent parameters : 551
  0  Time   0.03  Rwp   92.483    0.000 MC   0.00 0
Sparse matrix methods invoked - 83.0% of the A matrix elements are zero
  1  Time   0.04  Rwp   37.725  -54.758 MC   0.00 1
  2  Time   0.05  Rwp   27.737   -9.988 MC   0.05 1
  3  Time   0.07  Rwp   19.533   -8.204 MC   1.27 2
  4  Time   0.08  Rwp   14.885   -4.647 MC   0.48 1
  5  Time   0.09  Rwp   13.575   -1.311 MC   0.15 1
  6  Time   0.10  Rwp   12.636   -0.939 MC   1.63 2
  7  Time   0.11  Rwp   12.187   -0.449 MC   0.48 1
  8  Time   0.12  Rwp   11.871   -0.316 MC   2.34 2
  9  Time   0.13  Rwp   11.660   -0.211 MC   0.66 1
 10  Time   0.14  Rwp   11.484   -0.176 MC   2.73 2
 11  Time   0.15  Rwp   11.352   -0.132 MC   0.75 1
 12  Time   0.16  Rwp   11.228   -0.124 MC   3.00 2
 13  Time   0.17  Rwp   11.121   -0.107 MC   0.82 1
 14  Time   0.18  Rwp   11.022   -0.099 MC   3.14 2
 15  Time   0.19  Rwp   10.915   -0.107 MC   0.85 1
 16  Time   0.20  Rwp   10.871   -0.044 MC   0.22 1
 17  Time   0.21  Rwp   10.455   -0.416 MC   0.83 1
 18  Time   0.21  Rwp   10.133   -0.322 MC   0.22 1
 19  Time   0.22  Rwp    9.835   -0.297 MC   0.06 1
 20  Time   0.24  Rwp    8.753   -1.082 MC   1.24 2
 21  Time   0.24  Rwp    8.026   -0.727 MC   0.38 1
 22  Time   0.25  Rwp    7.176   -0.850 MC   0.10 1
 23  Time   0.27  Rwp    6.769   -0.407 MC   0.03 1
 24  Time   0.27  Rwp    5.443   -1.326 MC   0.28 1
 25  Time   0.29  Rwp    5.159   -0.284 MC   0.09 1
 26  Time   0.30  Rwp    4.742   -0.417 MC   1.31 2
 27  Time   0.31  Rwp    4.645   -0.097 MC   0.39 1
 28  Time   0.32  Rwp    4.482   -0.164 MC   2.08 2
 29  Time   0.33  Rwp    4.455   -0.027 MC   0.59 1
 30  Time   0.34  Rwp    4.362   -0.093 MC   2.71 2
 31  Time   0.35  Rwp    4.351   -0.011 MC   0.76 1
 32  Time   0.36  Rwp    4.290   -0.061 MC   3.45 2
 33  Time   0.36  Rwp    4.282   -0.008 MC   0.96 1
 34  Time   0.37  Rwp    4.240   -0.042 MC   3.99 2
 35  Time   0.38  Rwp    4.235   -0.005 MC   1.11 1
 36  Time   0.39  Rwp    4.205   -0.031 MC   4.71 2
 37  Time   0.41  Rwp    4.200   -0.004 MC   1.30 1
 38  Time   0.42  Rwp    4.178   -0.023 MC   5.14 2
 39  Time   0.42  Rwp    4.175   -0.003 MC   1.40 1
 40  Time   0.43  Rwp    4.157   -0.018 MC   5.73 2
 41  Time   0.44  Rwp    4.154   -0.003 MC   1.57 1
 42  Time   0.45  Rwp    4.139   -0.014 MC   5.86 2
 43  Time   0.46  Rwp    4.138   -0.001 MC   1.59 1
 44  Time   0.47  Rwp    4.130   -0.009 MC   2.12 1
 45  Time   0.47  Rwp    4.118   -0.012 MC   6.46 2
 46  Time   0.48  Rwp    4.118   -0.000 MC   1.65 1
--- 0.483 seconds ---

*** Parameter(s) close to limit(s).
    Check for LIMIT_MIN and LIMIT_MAX in Grid/Text
File C:\c\t5\test_examples\peak-intensity-extraction\pawley1.out updated
    with parameters from last iteration

Process Times (secs)
    0.02 = Peak buffer derivatives
    0.04 = Ycalc calculation and Penalties
    0.22 = A and Y matrix dot products and derivatives
    0.15 = Ycalc derivatives
    0.07 = A and Y matrix dot products
    0.13 = Solution to the normal equations

As can be seen, not having min/max limits on the intensity parameters leads to slow convergence.

Removing the 'update' attribute by changing the VV macro to:

          #m_unique i v min = -1e40; max = 1e40; 

results in fast convergence but many of the intensities become negative.


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