01 May 2011 11:28:16 AM
HERMITE_CUBIC_PRB
C++ version
Test the HERMITE_CUBIC library.
TEST01:
HERMITE_CUBIC_VALUE evaluates a Hermite cubic polynomial.
Try out four sets of data:
(F1,D1,F2,D2) = (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)
on [0,1] and [1.0,-2.0] (interval reversed)
J X F D
-3 -0.3 0.676 2.34
-2 -0.2 0.864 1.44
-1 -0.1 0.968 0.66
*Data 0 1 0
0 0 1 0
1 0.1 0.972 -0.54
2 0.2 0.896 -0.96
3 0.3 0.784 -1.26
4 0.4 0.648 -1.44
5 0.5 0.5 -1.5
6 0.6 0.352 -1.44
7 0.7 0.216 -1.26
8 0.8 0.104 -0.96
9 0.9 0.028 -0.54
10 1 0 0
*Data 1 0 0
11 1.1 0.032 0.66
12 1.2 0.136 1.44
J X F D
-3 -0.3 -0.507 2.47
-2 -0.2 -0.288 1.92
-1 -0.1 -0.121 1.43
*Data 0 0 1
0 0 0 1
1 0.1 0.081 0.63
2 0.2 0.128 0.32
3 0.3 0.147 0.07
4 0.4 0.144 -0.12
5 0.5 0.125 -0.25
6 0.6 0.096 -0.32
7 0.7 0.063 -0.33
8 0.8 0.032 -0.28
9 0.9 0.009 -0.17
10 1 0 0
*Data 1 0 0
11 1.1 0.011 0.23
12 1.2 0.048 0.52
J X F D
-3 -0.3 0.324 -2.34
-2 -0.2 0.136 -1.44
-1 -0.1 0.032 -0.66
*Data 0 0 0
0 0 0 0
1 0.1 0.028 0.54
2 0.2 0.104 0.96
3 0.3 0.216 1.26
4 0.4 0.352 1.44
5 0.5 0.5 1.5
6 0.6 0.648 1.44
7 0.7 0.784 1.26
8 0.8 0.896 0.96
9 0.9 0.972 0.54
10 1 1 0
*Data 1 1 0
11 1.1 0.968 -0.66
12 1.2 0.864 -1.44
J X F D
-3 -0.3 -0.117 0.87
-2 -0.2 -0.048 0.52
-1 -0.1 -0.011 0.23
*Data 0 0 0
0 0 0 0
1 0.1 -0.009 -0.17
2 0.2 -0.032 -0.28
3 0.3 -0.063 -0.33
4 0.4 -0.096 -0.32
5 0.5 -0.125 -0.25
6 0.6 -0.144 -0.12
7 0.7 -0.147 0.07
8 0.8 -0.128 0.32
9 0.9 -0.081 0.63
10 1 0 1
*Data 1 0 1
11 1.1 0.121 1.43
12 1.2 0.288 1.92
J X F D
-3 1.9 0.676 -0.78
-2 1.6 0.864 -0.48
-1 1.3 0.968 -0.22
*Data 1 1 0
0 1 1 0
1 0.7 0.972 0.18
2 0.4 0.896 0.32
3 0.1 0.784 0.42
4 -0.2 0.648 0.48
5 -0.5 0.5 0.5
6 -0.8 0.352 0.48
7 -1.1 0.216 0.42
8 -1.4 0.104 0.32
9 -1.7 0.028 0.18
10 -2 0 0
*Data -2 0 0
11 -2.3 0.032 -0.22
12 -2.6 0.136 -0.48
J X F D
-3 1.9 1.521 2.47
-2 1.6 0.864 1.92
-1 1.3 0.363 1.43
*Data 1 0 1
0 1 0 1
1 0.7 -0.243 0.63
2 0.4 -0.384 0.32
3 0.1 -0.441 0.07
4 -0.2 -0.432 -0.12
5 -0.5 -0.375 -0.25
6 -0.8 -0.288 -0.32
7 -1.1 -0.189 -0.33
8 -1.4 -0.096 -0.28
9 -1.7 -0.027 -0.17
10 -2 0 2.22045e-16
*Data -2 0 0
11 -2.3 -0.033 0.23
12 -2.6 -0.144 0.52
J X F D
-3 1.9 0.324 0.78
-2 1.6 0.136 0.48
-1 1.3 0.032 0.22
*Data 1 0 0
0 1 0 0
1 0.7 0.028 -0.18
2 0.4 0.104 -0.32
3 0.1 0.216 -0.42
4 -0.2 0.352 -0.48
5 -0.5 0.5 -0.5
6 -0.8 0.648 -0.48
7 -1.1 0.784 -0.42
8 -1.4 0.896 -0.32
9 -1.7 0.972 -0.18
10 -2 1 0
*Data -2 1 0
11 -2.3 0.968 0.22
12 -2.6 0.864 0.48
J X F D
-3 1.9 0.351 0.87
-2 1.6 0.144 0.52
-1 1.3 0.033 0.23
*Data 1 0 0
0 1 0 0
1 0.7 0.027 -0.17
2 0.4 0.096 -0.28
3 0.1 0.189 -0.33
4 -0.2 0.288 -0.32
5 -0.5 0.375 -0.25
6 -0.8 0.432 -0.12
7 -1.1 0.441 0.07
8 -1.4 0.384 0.32
9 -1.7 0.243 0.63
10 -2 0 1
*Data -2 0 1
11 -2.3 -0.363 1.43
12 -2.6 -0.864 1.92
TEST02:
HERMITE_CUBIC_VALUE evaluates a Hermite cubic polynomial.
Try out data from a cubic function:
on [0,10] and [-1.0,1.0] and [0.5,0.75]
J X F D S T
Exact -3 -120 79 -32 6
-3 -3 -120 79 -32 6
Exact -2 -56 50 -26 6
-2 -2 -56 50 -26 6
Exact -1 -18 27 -20 6
-1 -1 -18 27 -20 6
*Data 0 0 10
Exact 0 0 10 -14 6
0 0 0 10 -14 6
Exact 1 4 -1 -8 6
1 1 4 -1 -8 6
Exact 2 0 -6 -2 6
2 2 0 -6 -2 6
Exact 3 -6 -5 4 6
3 3 -6 -5 4 6
Exact 4 -8 2 10 6
4 4 -8 2 10 6
Exact 5 0 15 16 6
5 5 0 15 16 6
Exact 6 24 34 22 6
6 6 24 34 22 6
Exact 7 70 59 28 6
7 7 70 59 28 6
Exact 8 144 90 34 6
8 8 144 90 34 6
Exact 9 252 127 40 6
9 9 252 127 40 6
Exact 10 400 170 46 6
10 10 400 170 46 6
*Data 10 400 170
Exact 11 594 219 52 6
11 11 594 219 52 6
Exact 12 840 274 58 6
12 12 840 274 58 6
J X F D S T
Exact -1.6 -38.016 40.08 -23.6 6
-3 -1.6 -38.016 40.08 -23.6 6
Exact -1.4 -30.464 35.48 -22.4 6
-2 -1.4 -30.464 35.48 -22.4 6
Exact -1.2 -23.808 31.12 -21.2 6
-1 -1.2 -23.808 31.12 -21.2 6
*Data -1 -18 27
Exact -1 -18 27 -20 6
0 -1 -18 27 -20 6
Exact -0.8 -12.992 23.12 -18.8 6
1 -0.8 -12.992 23.12 -18.8 6
Exact -0.6 -8.736 19.48 -17.6 6
2 -0.6 -8.736 19.48 -17.6 6
Exact -0.4 -5.184 16.08 -16.4 6
3 -0.4 -5.184 16.08 -16.4 6
Exact -0.2 -2.288 12.92 -15.2 6
4 -0.2 -2.288 12.92 -15.2 6
Exact 0 0 10 -14 6
5 0 0 10 -14 6
Exact 0.2 1.728 7.32 -12.8 6
6 0.2 1.728 7.32 -12.8 6
Exact 0.4 2.944 4.88 -11.6 6
7 0.4 2.944 4.88 -11.6 6
Exact 0.6 3.696 2.68 -10.4 6
8 0.6 3.696 2.68 -10.4 6
Exact 0.8 4.032 0.72 -9.2 6
9 0.8 4.032 0.72 -9.2 6
Exact 1 4 -1 -8 6
10 1 4 -1 -8 6
*Data 1 4 -1
Exact 1.2 3.648 -2.48 -6.8 6
11 1.2 3.648 -2.48 -6.8 6
Exact 1.4 3.024 -3.72 -5.6 6
12 1.4 3.024 -3.72 -5.6 6
J X F D S T
Exact 0.425 3.06239 4.59187 -11.45 6
-3 0.425 3.06239 4.59187 -11.45 6
Exact 0.45 3.17363 4.3075 -11.3 6
-2 0.45 3.17362 4.3075 -11.3 6
Exact 0.475 3.2778 4.02688 -11.15 6
-1 0.475 3.2778 4.02688 -11.15 6
*Data 0.5 3.375 3.75
Exact 0.5 3.375 3.75 -11 6
0 0.5 3.375 3.75 -11 6
Exact 0.525 3.46533 3.47687 -10.85 6
1 0.525 3.46533 3.47687 -10.85 6
Exact 0.55 3.54888 3.2075 -10.7 6
2 0.55 3.54888 3.2075 -10.7 6
Exact 0.575 3.62573 2.94188 -10.55 6
3 0.575 3.62573 2.94188 -10.55 6
Exact 0.6 3.696 2.68 -10.4 6
4 0.6 3.696 2.68 -10.4 6
Exact 0.625 3.75977 2.42188 -10.25 6
5 0.625 3.75977 2.42188 -10.25 6
Exact 0.65 3.81713 2.1675 -10.1 6
6 0.65 3.81712 2.1675 -10.1 6
Exact 0.675 3.86817 1.91687 -9.95 6
7 0.675 3.86817 1.91687 -9.95 6
Exact 0.7 3.913 1.67 -9.8 6
8 0.7 3.913 1.67 -9.8 6
Exact 0.725 3.9517 1.42688 -9.65 6
9 0.725 3.9517 1.42688 -9.65 6
Exact 0.75 3.98438 1.1875 -9.5 6
10 0.75 3.98438 1.1875 -9.5 6
*Data 0.75 3.98438 1.1875
Exact 0.775 4.01111 0.951875 -9.35 6
11 0.775 4.01111 0.951875 -9.35 6
Exact 0.8 4.032 0.72 -9.2 6
12 0.8 4.032 0.72 -9.2 6
TEST03:
HERMITE_CUBIC_INTEGRATE integrates a Hermite cubic
polynomial from A to B.
Exact Computed
J A B Integral Integral
-3 -1 -3 120.667 120.667
-2 -1 -2 35.0833 35.0833
-1 -1 -1 0 0
0 -1 0 -7.58333 -7.58333
1 -1 1 -4.66667 -4.66667
2 -1 2 -2.25 -2.25
3 -1 3 -5.33333 -5.33333
4 -1 4 -12.9167 -12.9167
5 -1 5 -18 -18
6 -1 6 -7.58333 -7.58333
7 -1 7 37.3333 37.3333
8 -1 8 141.75 141.75
9 -1 9 336.667 336.667
10 -1 10 659.083 659.083
11 -1 11 1152 1152
12 -1 12 1864.42 1864.42
Exact Computed
J A B Integral Integral
-3 -2 -1.6 -18.6709 -18.6709
-2 -2 -1.4 -25.5036 -25.5036
-1 -2 -1.2 -30.9163 -30.9163
0 -2 -1 -35.0833 -35.0833
1 -2 -0.8 -38.1696 -38.1696
2 -2 -0.6 -40.3303 -40.3303
3 -2 -0.4 -41.7109 -41.7109
4 -2 -0.2 -42.4476 -42.4476
5 -2 0 -42.6667 -42.6667
6 -2 0.2 -42.4849 -42.4849
7 -2 0.4 -42.0096 -42.0096
8 -2 0.6 -41.3383 -41.3383
9 -2 0.8 -40.5589 -40.5589
10 -2 1 -39.75 -39.75
11 -2 1.2 -38.9803 -38.9803
12 -2 1.4 -38.3089 -38.3089
Exact Computed
J A B Integral Integral
-3 -0.5 0.425 -0.82513 -0.82513
-2 -0.5 0.45 -0.747165 -0.747165
-1 -0.5 0.475 -0.666508 -0.666508
0 -0.5 0.5 -0.583333 -0.583333
1 -0.5 0.525 -0.497815 -0.497815
2 -0.5 0.55 -0.410123 -0.410123
3 -0.5 0.575 -0.320427 -0.320427
4 -0.5 0.6 -0.228892 -0.228892
5 -0.5 0.625 -0.135681 -0.135681
6 -0.5 0.65 -0.0409568 -0.0409568
7 -0.5 0.675 0.0551225 0.0551225
8 -0.5 0.7 0.1524 0.1524
9 -0.5 0.725 0.250721 0.250721
10 -0.5 0.75 0.349935 0.349935
11 -0.5 0.775 0.449891 0.449891
12 -0.5 0.8 0.550442 0.550442
TEST04:
HERMITE_CUBIC_SPLINE_VALUE evaluates a Hermite cubic spline.
I X F computed F exact Error
0 0 0 0 0
1 0.2 0.198223 0.198669 0.000446022
2 0.4 0.388329 0.389418 0.00108958
3 0.6 0.56347 0.564642 0.00117281
4 0.8 0.716799 0.717356 0.000556784
5 1 0.841471 0.841471 2.22045e-16
6 1.2 0.931 0.932039 0.00103876
7 1.4 0.9831 0.98545 0.00235021
8 1.6 0.997217 0.999574 0.00235692
9 1.8 0.9728 0.973848 0.00104769
10 2 0.909297 0.909297 0
11 2.2 0.80782 0.808496 0.000676464
12 2.4 0.674013 0.675463 0.00145007
13 2.6 0.514127 0.515501 0.00137409
14 2.8 0.334413 0.334988 0.00057535
15 3 0.14112 0.14112 1.38778e-16
16 3.2 -0.0580664 -0.0583741 0.000307767
17 3.4 -0.254758 -0.255541 0.000783257
18 3.6 -0.441648 -0.44252 0.000872071
19 3.8 -0.611432 -0.611858 0.00042596
20 4 -0.756802 -0.756802 1.11022e-16
21 4.2 -0.870567 -0.871576 0.00100904
22 4.4 -0.949306 -0.951602 0.00229646
23 4.6 -0.991375 -0.993691 0.00231645
24 4.8 -0.995129 -0.996165 0.00103564
25 5 -0.958924 -0.958924 0
26 5.2 -0.882672 -0.883455 0.000782604
27 5.4 -0.771066 -0.772764 0.00169831
28 5.6 -0.629636 -0.631267 0.0016311
29 5.8 -0.463909 -0.464602 0.000693162
30 6 -0.279415 -0.279415 0
31 6.2 -0.0832528 -0.0830894 0.000163353
32 6.4 0.116088 0.116549 0.00046126
33 6.6 0.310987 0.311541 0.00055388
34 6.8 0.493827 0.494113 0.00028661
35 7 0.656987 0.656987 0
36 7.2 0.792709 0.793668 0.000959124
37 7.4 0.896511 0.898708 0.00219675
38 7.6 0.96569 0.96792 0.00222962
39 7.8 0.99754 0.998543 0.00100287
40 8 0.989358 0.989358 0
41 8.2 0.939857 0.940731 0.000873081
42 8.4 0.852686 0.854599 0.00191256
43 8.6 0.732542 0.734397 0.00185546
44 8.8 0.58412 0.584917 0.0007971
45 9 0.412118 0.412118 1.66533e-16
46 9.2 0.222906 0.22289 1.56686e-05
47 9.4 0.0249055 0.0247754 0.000130031
48 9.6 -0.174102 -0.174327 0.000224603
49 9.8 -0.366338 -0.366479 0.000141524
50 10 -0.544021 -0.544021 1.11022e-16
I X D computed D exact Error
0 0 1 1 0
1 0.2 0.976527 0.980067 0.00353908
2 0.4 0.918821 0.921061 0.00223951
3 0.6 0.826882 0.825336 0.00154633
4 0.8 0.700709 0.696707 0.00400217
5 1 0.540302 0.540302 4.44089e-16
6 1.2 0.354531 0.362358 0.00782652
7 1.4 0.166001 0.169967 0.00396636
8 1.6 -0.025289 -0.0291995 0.00391048
9 1.8 -0.219338 -0.227202 0.00786385
10 2 -0.416147 -0.416147 5.55112e-17
11 2.2 -0.593419 -0.588501 0.00491829
12 2.4 -0.73944 -0.737394 0.00204655
13 2.6 -0.854209 -0.856889 0.00267936
14 2.8 -0.937727 -0.942222 0.00449553
15 3 -0.989992 -0.989992 2.22045e-16
16 3.2 -0.995783 -0.998295 0.00251179
17 3.4 -0.965043 -0.966798 0.00175485
18 3.6 -0.897774 -0.896758 0.00101516
19 3.8 -0.793974 -0.790968 0.00300595
20 4 -0.653644 -0.653644 2.22045e-16
21 4.2 -0.482628 -0.490261 0.00763254
22 4.4 -0.30339 -0.307333 0.00394284
23 4.6 -0.115929 -0.112153 0.00377634
24 4.8 0.0797552 0.087499 0.00774378
25 5 0.283662 0.283662 0
26 5.2 0.474253 0.468517 0.00573597
27 5.4 0.637199 0.634693 0.00250581
28 5.6 0.7725 0.775566 0.00306557
29 5.8 0.880158 0.88552 0.00536201
30 6 0.96017 0.96017 1.11022e-16
31 6.2 0.995108 0.996542 0.00143422
32 6.4 0.99195 0.993185 0.00123505
33 6.6 0.950696 0.950233 0.000463665
34 6.8 0.871347 0.869397 0.00194956
35 7 0.753902 0.753902 0
36 7.2 0.601066 0.608351 0.0072858
37 7.4 0.434707 0.438547 0.00384041
38 7.6 0.254826 0.25126 0.00356661
39 7.8 0.0614241 0.0539554 0.00746872
40 8 -0.1455 -0.1455 0
41 8.2 -0.345594 -0.339155 0.00643885
42 8.4 -0.522204 -0.519289 0.00291491
43 8.6 -0.67533 -0.67872 0.00339043
44 8.8 -0.804972 -0.811093 0.00612117
45 9 -0.91113 -0.91113 3.33067e-16
46 9.2 -0.974516 -0.974844 0.000327953
47 9.4 -0.999002 -0.999693 0.000690544
48 9.6 -0.984591 -0.984688 9.71047e-05
49 9.8 -0.93128 -0.930426 0.000854156
50 10 -0.839072 -0.839072 3.33067e-16
TEST05:
HERMITE_CUBIC_TO_POWER_CUBIC converts the Hermite data
to the coefficients of the power form of the polynomial
POWER_CUBIC_TO_HERMITE_CUBIC converts the power form
to Hermite form
Hermite data:
X1, F1, D1: -1 -18 27
X2, F2, D2: 1 4 -1
Power form:
p(x) = 0 + 10 * x + -7 * x^2 + 1 * x^3
X F (Hermite) F (power)
-1.6 -38.016 -38.016
-1.4 -30.464 -30.464
-1.2 -23.808 -23.808
-1 -18 -18
-0.8 -12.992 -12.992
-0.6 -8.736 -8.736
-0.4 -5.184 -5.184
-0.2 -2.288 -2.288
0 0 0
0.2 1.728 1.728
0.4 2.944 2.944
0.6 3.696 3.696
0.8 4.032 4.032
1 4 4
1.2 3.648 3.648
1.4 3.024 3.024
Use POWER_CUBIC_TO_HERMITE_CUBIC to recover the
original Hermite data:
Original Recovered
F1: -18 -18
D1: 27 27
F2: 4 4
D2: -1 -1
TEST06:
HERMITE_CUBIC_INTEGRATE integrates a Hermite cubic
polynomial from A to B.
Use A, B vectors for the calculation.
Exact Computed
J A B Integral Integral
-3 -1 -3 120.667 120.667
-2 -1 -2 35.0833 35.0833
-1 -1 -1 0 0
0 -1 0 -7.58333 -7.58333
1 -1 1 -4.66667 -4.66667
2 -1 2 -2.25 -2.25
3 -1 3 -5.33333 -5.33333
4 -1 4 -12.9167 -12.9167
5 -1 5 -18 -18
6 -1 6 -7.58333 -7.58333
7 -1 7 37.3333 37.3333
8 -1 8 141.75 141.75
9 -1 9 336.667 336.667
10 -1 10 659.083 659.083
11 -1 11 1152 1152
12 -1 12 1864.42 1864.42
TEST07:
HERMITE_CUBIC_INTEGRAL integrates a Hermite cubic
polynomial over the definition interval [X1,X2].
Exact Computed
X1 X2 Integral Integral
0 10 666.667 666.667
-1 1 -4.66667 -4.66667
0.5 0.75 0.933268 0.933268
TEST08:
HERMITE_CUBIC_SPLINE_INTEGRAL integrates a Hermite
cubic spline over the definition interval [X1,XNN].
Exact Computed
X1 XNN Integral Integral
0 1 -0.166667 -0.166667
0 1 -0.166667 -0.166667
0 3.14159 2 1.99997
TEST09:
HERMITE_CUBIC_SPLINE_INTEGRATE integrates a Hermite
cubic spline from A to B.
Exact Computed
I A B Integral Integral
0 2.5 -1 3.02604 3.02604
1 2.5 -0.5 -3 -3
2 2.5 0 -4.55729 -4.55729
3 2.5 0.5 -3.58333 -3.58333
4 2.5 1 -1.64062 -1.64062
5 2.5 1.5 0.0833333 0.0833333
6 2.5 2 0.776042 0.776042
7 2.5 2.5 0 0
8 2.5 3 -2.30729 -2.30729
9 2.5 3.5 -5.83333 -5.83333
10 2.5 4 -9.89063 -9.89062
11 2.5 4.5 -13.4167 -13.4167
12 2.5 5 -14.974 -14.974
13 2.5 5.5 -12.75 -12.75
14 2.5 6 -4.55729 -4.55729
15 2.5 6.5 12.1667 12.1667
16 2.5 7 40.3594 40.3594
17 2.5 7.5 83.3333 83.3333
18 2.5 8 144.776 144.776
19 2.5 8.5 228.75 228.75
20 2.5 9 339.693 339.693
21 2.5 9.5 482.417 482.417
22 2.5 10 662.109 662.109
23 2.5 10.5 884.333 884.333
24 2.5 11 1155.03 1155.03
TEST10:
HERMITE_CUBIC_SPLINE_INTEGRAL integrates a Hermite
cubic spline over the definition interval [X1,XNN].
If the subintervals are equally spaced, the derivative
information has no effect on the result, except for
the first and last values, DN(1) and DN(NN).
Exact Computed
X1 XNN Integral Integral Comment
0 3.14159 2 1.99997 Equal spacing, correct DN
0 3.14159 2 1.99997 Equal spacing, DN(2:N-1) random
0 3.14159 2 7.08569 Equal spacing, DN(1:N) random
0 3.14159 2 1.99935 Variable spacing, correct DN
0 3.14159 2 1.2796 Variable spacing, a single internal DN randomized.
TEST11:
HERMITE_CUBIC_LAGRANGE_VALUE evaluates the four
Lagrange basis functions associated with F1, D1,
F2 and D2 such that
P(X) = F1 * LF1(X) + D1 * LD1(X)
+ F2 * LF2(X) + D2 * LD2(X).
The first, second and third derivatives of these four
Lagrange basis functions are also computed.
The Lagrange basis functions:
I X LF1 LD1 LF2 LD2
0 0 -4 -4 5 -2
1 0.25 -1.53125 -2.29688 2.53125 -0.984375
2 0.5 0 -1.125 1 -0.375
3 0.75 0.78125 -0.390625 0.21875 -0.078125
4 1 1 0 0 -0
5 1.25 0.84375 0.140625 0.15625 -0.046875
6 1.5 0.5 0.125 0.5 -0.125
7 1.75 0.15625 0.046875 0.84375 -0.140625
8 2 0 0 1 0
9 2.25 0.21875 0.078125 0.78125 0.390625
10 2.5 1 0.375 0 1.125
The derivative of the Lagrange basis functions:
I X LF1 LD1 LF2 LD2
0 0 12 8 -12 5
1 0.25 7.875 5.6875 -7.875 3.1875
2 0.5 4.5 3.75 -4.5 1.75
3 0.75 1.875 2.1875 -1.875 0.6875
4 1 -0 1 0 -0
5 1.25 -1.125 0.1875 1.125 -0.3125
6 1.5 -1.5 -0.25 1.5 -0.25
7 1.75 -1.125 -0.3125 1.125 0.1875
8 2 0 0 0 1
9 2.25 1.875 0.6875 -1.875 2.1875
10 2.5 4.5 1.75 -4.5 3.75
TEST12:
HERMITE_CUBIC_LAGRANGE_INTEGRAL returns the integrals
of the four Lagrange basis functions associated
with F1, D1, F2 and D2 such that
P(X) = F1 * LF1(X) + D1 * LD1(X)
+ F2 * LF2(X) + D2 * LD2(X).
The Lagrange basis function integrals:
X1 X2 LF1 LD1 LF2 LD2
-6 1 3.5 4.08333 3.5 -4.08333
-5 1 3 3 3 -3
-4 1 2.5 2.08333 2.5 -2.08333
-3 1 2 1.33333 2 -1.33333
-2 1 1.5 0.75 1.5 -0.75
-1 1 1 0.333333 1 -0.333333
0 1 0.5 0.0833333 0.5 -0.0833333
1 1 0 0 0 -0
2 1 -0.5 0.0833333 -0.5 -0.0833333
TEST13:
HERMITE_CUBIC_LAGRANGE_INTEGRATE integrates a Hermite cubic
Lagrange polynomial from A to B.
Compute each result TWICE:
First row computed using HERMITE_CUBIC_INTEGRATE.
Second row computed using HERMITE_CUBIC_LAGRANGE_INTEGRATE.
A B LF1 LD1 LF2 LD2
-1 -3 -1.7 5.93333 -0.3 1.06667
-1.7 5.93333 -0.3 1.06667
-1 -2 -0.9225 2.00417 -0.0775 0.270833
-0.9225 2.00417 -0.0775 0.270833
-1 -1 0 0 0 0
0 0 0 0
-1 0 0.9895 -0.569167 0.0105 -0.0358333
0.9895 -0.569167 0.0105 -0.0358333
-1 1 1.98 -0.133333 0.02 -0.0666667
1.98 -0.133333 0.02 -0.0666667
-1 2 2.9175 0.9375 0.0825 -0.2625
2.9175 0.9375 0.0825 -0.2625
-1 3 3.76 2.33333 0.24 -0.733333
3.76 2.33333 0.24 -0.733333
-1 4 4.4775 3.80417 0.5225 -1.52917
4.4775 3.80417 0.5225 -1.52917
-1 5 5.052 5.16 0.948 -2.64
5.052 5.16 0.948 -2.64
-1 6 5.4775 6.27083 1.5225 -3.99583
5.4775 6.27083 1.5225 -3.99583
-1 7 5.76 7.06667 2.24 -5.46667
5.76 7.06667 2.24 -5.46667
-1 8 5.9175 7.5375 3.0825 -6.8625
5.9175 7.5375 3.0825 -6.8625
-1 9 5.98 7.73333 4.02 -7.93333
5.98 7.73333 4.02 -7.93333
-1 10 5.9895 7.76417 5.0105 -8.36917
5.9895 7.76417 5.0105 -8.36917
-1 11 6 7.8 6 -7.8
6 7.8 6 -7.8
-1 12 6.0775 8.07083 6.9225 -5.79583
6.0775 8.07083 6.9225 -5.79583
TEST14:
HERMITE_CUBIC_SPLINE_QUAD_RULE returns a quadrature rule
for Hermite cubic splines.
Case 1: Random spacing
I J X W Q
0 0 0.218418 0.478159 0.478159
0 1 1.17474 0.892913 0.892913
0 2 2.00425 0.695602 0.695602
0 3 2.56594 0.488501 0.488501
0 4 2.98125 0.240713 0.240713
0 5 3.04737 0.161848 0.161848
0 6 3.30494 0.183767 0.183767
0 7 3.4149 0.0768929 0.0768929
0 8 3.45873 0.338897 0.338897
0 9 4.0927 0.347846 0.347846
0 10 4.15442 0.0308636 0.0308636
1 0 0.218418 0.0762119 0.0762119
1 1 1.17474 -0.0188715 -0.0188715
1 2 2.00425 -0.0310486 -0.0310486
1 3 2.56594 -0.0119185 -0.0119185
1 4 2.98125 -0.014009 -0.014009
1 5 3.04737 0.00516455 0.00516455
1 6 3.30494 -0.00452132 -0.00452132
1 7 3.4149 -0.00084746 -0.00084746
1 8 3.45873 0.0333326 0.0333326
1 9 4.0927 -0.0331752 -0.0331752
1 10 4.15442 -0.000317521 -0.000317521
Case 2: Uniform spacing
F(2:N-1) have equal weight.
D(2:N-1) have zero weight.
I J X W Q
0 0 0.5 0.025 0.025
0 1 0.55 0.05 0.05
0 2 0.6 0.05 0.05
0 3 0.65 0.05 0.05
0 4 0.7 0.05 0.05
0 5 0.75 0.05 0.05
0 6 0.8 0.05 0.05
0 7 0.85 0.05 0.05
0 8 0.9 0.05 0.05
0 9 0.95 0.05 0.05
0 10 1 0.025 0.025
1 0 0.5 0.000208333 0.000208333
1 1 0.55 -8.94467e-19 -9.25186e-19
1 2 0.6 8.94467e-19 9.25186e-19
1 3 0.65 -8.94467e-19 -9.25186e-19
1 4 0.7 8.94467e-19 9.25186e-19
1 5 0.75 0 0
1 6 0.8 -8.94467e-19 -9.25186e-19
1 7 0.85 8.94467e-19 9.25186e-19
1 8 0.9 -8.94467e-19 -9.25186e-19
1 9 0.95 8.94467e-19 9.25186e-19
1 10 1 -0.000208333 -0.000208333
TEST15:
HERMITE_CUBIC_SPLINE_QUAD_RULE returns a quadrature rule
for Hermite cubic splines.
Random spacing
Number of points N = 11
Interval = [0.218418, 4.15442]
Q = -6.75338
Q (exact) = -6.75338
HERMITE_CUBIC_PRB
Normal end of execution.
01 May 2011 11:28:16 AM