25 March 2017 10:04:27.609 AM PPPACK_PRB FORTRAN90 version Test the PPPACK library. TEST01 Demonstrate the failure of polynomial interpolation when applied to the Runge function, using equally spaced nodes. Use polynomial interpolation of order N. N Max error Decay exponent 2 0.9615E+00 0.00 4 0.7070E+00 -0.44 6 0.4327E+00 -1.21 8 0.2474E+00 -1.94 10 0.2994E+00 0.86 12 0.5567E+00 3.40 14 0.1069E+01 4.23 16 0.2099E+01 5.05 18 0.4214E+01 5.92 20 0.8573E+01 6.74 TEST02 Cubic Hermite interpolation appplied to Runge function, equally spaced nodes. Use polynomial interpolation of order N. N Max error Decay exponent 2 0.9246E+00 0.00 4 0.5407E+00 -0.77 6 0.2500E+00 -1.90 8 0.1141E+00 -2.73 10 0.5562E-01 -3.22 12 0.2932E-01 -3.51 14 0.1661E-01 -3.69 16 0.1000E-01 -3.80 18 0.6339E-02 -3.87 20 0.4195E-02 -3.92 TEST03 Compare B-spline coefficients to values at knot averages. I Tave(I) F at Tave(I) BCOEF(I) 1 0.00000 -162.00000 -162.00000 2 0.33333 -130.96296 -129.00000 3 1.00000 -80.00000 -75.00000 4 2.00000 -28.00000 -24.00000 5 3.00000 0.00000 3.00000 6 4.00000 10.00000 12.00000 7 5.00000 8.00000 9.00000 8 6.00000 0.00000 0.00000 9 7.00000 -8.00000 -9.00000 10 8.00000 -10.00000 -12.00000 11 9.00000 0.00000 -3.00000 12 9.66667 16.29630 15.00000 13 10.00000 28.00000 28.00000 TEST04 INTERV locates a point X within a knot sequence; BSPLVB evaluates a B-spline. Use these routines to plot some B splines. B-spline order K = 3 N = 7 Knot sequence T: 1 0.0000000 2 0.0000000 3 0.0000000 4 1.0000000 5 1.0000000 6 3.0000000 7 4.0000000 8 6.0000000 9 6.0000000 10 6.0000000 X LEFT B1(X) B2(X) B3(X) B4(X) B5(X) 0.000 3 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.200 3 0.0400000 0.0000000 0.0000000 0.0000000 0.0000000 0.400 3 0.1600000 0.0000000 0.0000000 0.0000000 0.0000000 0.600 3 0.3600000 0.0000000 0.0000000 0.0000000 0.0000000 0.800 3 0.6400000 0.0000000 0.0000000 0.0000000 0.0000000 1.000 5 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 1.200 5 0.8100000 0.1833333 0.0066667 0.0000000 0.0000000 1.400 5 0.6400000 0.3333333 0.0266667 0.0000000 0.0000000 1.600 5 0.4900000 0.4500000 0.0600000 0.0000000 0.0000000 1.800 5 0.3600000 0.5333333 0.1066667 0.0000000 0.0000000 2.000 5 0.2500000 0.5833333 0.1666667 0.0000000 0.0000000 2.200 5 0.1600000 0.6000000 0.2400000 0.0000000 0.0000000 2.400 5 0.0900000 0.5833333 0.3266667 0.0000000 0.0000000 2.600 5 0.0400000 0.5333333 0.4266667 0.0000000 0.0000000 2.800 5 0.0100000 0.4500000 0.5400000 0.0000000 0.0000000 3.000 6 0.0000000 0.3333333 0.6666667 0.0000000 0.0000000 3.200 6 0.0000000 0.2133333 0.7733333 0.0133333 0.0000000 3.400 6 0.0000000 0.1200000 0.8266667 0.0533333 0.0000000 3.600 6 0.0000000 0.0533333 0.8266667 0.1200000 0.0000000 3.800 6 0.0000000 0.0133333 0.7733333 0.2133333 0.0000000 4.000 7 0.0000000 0.0000000 0.6666667 0.3333333 0.0000000 4.200 7 0.0000000 0.0000000 0.5400000 0.4500000 0.0100000 4.400 7 0.0000000 0.0000000 0.4266667 0.5333333 0.0400000 4.600 7 0.0000000 0.0000000 0.3266667 0.5833333 0.0900000 4.800 7 0.0000000 0.0000000 0.2400000 0.6000000 0.1600000 5.000 7 0.0000000 0.0000000 0.1666667 0.5833333 0.2500000 5.200 7 0.0000000 0.0000000 0.1066667 0.5333333 0.3600000 5.400 7 0.0000000 0.0000000 0.0600000 0.4500000 0.4900000 5.600 7 0.0000000 0.0000000 0.0266667 0.3333333 0.6400000 5.800 7 0.0000000 0.0000000 0.0066667 0.1833333 0.8100000 6.000 7 0.0000000 0.0000000 0.0000000 0.0000000 1.0000000 TEST05 Polynomials that build splines. -0.8 -0.04266667 1.12373333 -17.16693333 10.48106667 -0.6 -0.01800000 0.80120000 -14.75080000 9.58320000 -0.4 -0.00533333 0.54346667 -12.55786667 8.73813333 -0.2 -0.00066667 0.34493333 -10.57773333 7.94426667 0.0 0.00000000 0.20000000 -8.80000000 7.20000000 0.2 0.00066667 0.10306667 -7.21426667 6.50373333 0.4 0.00533333 0.04853333 -5.81013333 5.85386667 0.6 0.01800000 0.03080000 -4.57720000 5.24880000 0.8 0.04266667 0.04426667 -3.50506667 4.68693333 1.0 0.08333333 0.08333333 -2.58333333 4.16666667 1.2 0.14400000 0.14240000 -1.80160000 3.68640000 1.4 0.22866667 0.21586667 -1.14946667 3.24453333 1.6 0.34133333 0.29813333 -0.61653333 2.83946667 1.8 0.48600000 0.38360000 -0.19240000 2.46960000 2.0 0.66666667 0.46666667 0.13333333 2.13333333 2.2 0.88733333 0.54173333 0.37106667 1.82906667 2.4 1.15200000 0.60320000 0.53120000 1.55520000 2.6 1.46466667 0.64546667 0.62413333 1.31013333 2.8 1.82933333 0.66293333 0.66026667 1.09226667 3.0 2.25000000 0.65000000 0.65000000 0.90000000 3.2 2.73066667 0.60106667 0.60373333 0.73173333 3.4 3.27533333 0.51053333 0.53186667 0.58586667 3.6 3.88800000 0.37280000 0.44480000 0.46080000 3.8 4.57266667 0.18226667 0.35293333 0.35493333 4.0 5.33333333 -0.06666667 0.26666667 0.26666667 4.2 6.17400000 -0.37960000 0.19640000 0.19440000 4.4 7.09866667 -0.76213333 0.15253333 0.13653333 4.6 8.11133333 -1.21986667 0.14546667 0.09146667 4.8 9.21600000 -1.75840000 0.18560000 0.05760000 5.0 10.41666667 -2.38333333 0.28333333 0.03333333 5.2 11.71733333 -3.10026667 0.44906667 0.01706667 5.4 13.12200000 -3.91480000 0.69320000 0.00720000 5.6 14.63466667 -4.83253333 1.02613333 0.00213333 5.8 16.25933333 -5.85906667 1.45826667 0.00026667 6.0 18.00000000 -7.00000000 2.00000000 0.00000000 6.2 19.86066667 -8.26093333 2.66173333 -0.00026667 6.4 21.84533333 -9.64746667 3.45386667 -0.00213333 6.6 23.95800000 -11.16520000 4.38680000 -0.00720000 6.8 26.20266667 -12.81973333 5.47093333 -0.01706667 7.0 28.58333333 -14.61666667 6.71666667 -0.03333333 TEST06 Piecewise representation -0.800000 -0.426667E-01 -0.600000 -0.180000E-01 -0.400000 -0.533333E-02 -0.200000 -0.666667E-03 0.00000 0.00000 0.200000 0.666667E-03 0.400000 0.533333E-02 0.600000 0.180000E-01 0.800000 0.426667E-01 1.00000 0.833333E-01 1.20000 0.142400 1.40000 0.215867 1.60000 0.298133 1.80000 0.383600 2.00000 0.466667 2.20000 0.541733 2.40000 0.603200 2.60000 0.645467 2.80000 0.662933 3.00000 0.650000 3.20000 0.603733 3.40000 0.531867 3.60000 0.444800 3.80000 0.352933 4.00000 0.266667 4.20000 0.194400 4.40000 0.136533 4.60000 0.914667E-01 4.80000 0.576000E-01 5.00000 0.333333E-01 5.20000 0.170667E-01 5.40000 0.720000E-02 5.60000 0.213333E-02 5.80000 0.266667E-03 6.00000 0.555112E-16 6.20000 -0.266667E-03 6.40000 -0.213333E-02 6.60000 -0.720000E-02 6.80000 -0.170667E-01 7.00000 -0.333333E-01 TEST07 Construct a spline via BVALUE -0.800000 0.00000 -0.600000 0.00000 -0.400000 0.00000 -0.200000 0.00000 0.00000 0.00000 0.200000 0.666667E-03 0.400000 0.533333E-02 0.600000 0.180000E-01 0.800000 0.426667E-01 1.00000 0.833333E-01 1.20000 0.142400 1.40000 0.215867 1.60000 0.298133 1.80000 0.383600 2.00000 0.466667 2.20000 0.541733 2.40000 0.603200 2.60000 0.645467 2.80000 0.662933 3.00000 0.650000 3.20000 0.603733 3.40000 0.531867 3.60000 0.444800 3.80000 0.352933 4.00000 0.266667 4.20000 0.194400 4.40000 0.136533 4.60000 0.914667E-01 4.80000 0.576000E-01 5.00000 0.333333E-01 5.20000 0.170667E-01 5.40000 0.720000E-02 5.60000 0.213333E-02 5.80000 0.266667E-03 6.00000 0.00000 6.20000 0.00000 6.40000 0.00000 6.60000 0.00000 6.80000 0.00000 7.00000 0.00000 TEST08 Cubic spline with good knots Run number 1 N Max error Decay exponent 4 0.3834E+03 0.00 6 0.1285E+03 -2.70 8 0.4653E+02 -3.53 10 0.1316E+02 -5.66 Run number 2 N Max error Decay exponent 4 0.4882E+02 0.00 6 0.1179E+02 -3.50 8 0.2897E+01 -4.88 10 0.5740E+00 -7.25 12 0.1384E+00 -7.80 14 0.8384E-01 -3.25 16 0.5457E-01 -3.21 18 0.3749E-01 -3.19 20 0.2685E-01 -3.17 Run number 3 N Max error Decay exponent 4 0.5001E+01 0.00 6 0.7842E+00 -4.57 8 0.1217E+00 -6.48 10 0.5025E-01 -3.96 12 0.3364E-01 -2.20 14 0.2408E-01 -2.17 16 0.1809E-01 -2.14 18 0.1408E-01 -2.13 20 0.1127E-01 -2.11 TEST09 Cubic spline, good knots Run 1 ITERMX = 0 NLOW = 4 NHIGH = 20 Take 0 cycles through NEWNOT N Max error Decay exponent 4 0.1476E+00 0.00 6 0.1114E+00 -0.69 8 0.9414E-01 -0.59 10 0.8303E-01 -0.56 12 0.7510E-01 -0.55 14 0.6908E-01 -0.54 16 0.6431E-01 -0.54 18 0.6041E-01 -0.53 20 0.5714E-01 -0.53 Run 2 ITERMX = 3 NLOW = 4 NHIGH = 20 Take 3 cycles through NEWNOT N Max error Decay exponent 4 0.1344E+00 0.00 6 0.1293E+00 -0.10 8 0.1097E+00 -0.57 10 0.9437E-01 -0.68 12 0.8363E-01 -0.66 14 0.7535E-01 -0.68 16 0.6880E-01 -0.68 18 0.6349E-01 -0.68 20 0.5910E-01 -0.68 TEST10 Quasi-interpolant Run 1 N Max error Decay exponent 4 0.7240E+00 0.00 6 0.9952E-01 -4.89 8 0.3031E-01 -4.13 10 0.1197E-01 -4.17 Run 2 N Max error Decay exponent 4 0.4128E+00 0.00 6 0.8937E-01 -3.77 8 0.3257E-01 -3.51 10 0.1532E-01 -3.38 12 0.8393E-02 -3.30 14 0.5085E-02 -3.25 16 0.3310E-02 -3.21 18 0.2274E-02 -3.19 20 0.1629E-02 -3.17 TEST11 A large norm amplifies noise. Run 1 Size of noise = 0.100000E-05 H Max error 0.17E+00 0.145E-05 0.83E-01 0.251E-05 0.42E-01 0.676E-05 0.21E-01 0.158E-04 0.10E-01 0.342E-04 0.52E-02 0.712E-04 0.26E-02 0.145E-03 0.13E-02 0.294E-03 0.65E-03 0.591E-03 0.33E-03 0.119E-02 Run 2 Size of noise = 0.00000 H Max error 0.17E+00 0.888E-15 0.83E-01 0.888E-15 0.42E-01 0.111E-14 0.21E-01 0.888E-15 0.10E-01 0.466E-14 0.52E-02 0.355E-14 0.26E-02 0.264E-13 0.13E-02 0.442E-13 0.65E-03 0.113E-12 0.33E-03 0.444E-13 TEST12 Interpolation at knot averages Run 1 We will take 0 cycles through NEWNOT. NLOW is 4 NHIGH is 20 N Max error Decay exponent 4 0.1476E+00 0.00 6 0.9126E-01 -1.19 8 0.7070E-01 -0.89 10 0.5975E-01 -0.75 12 0.5270E-01 -0.69 14 0.4767E-01 -0.65 16 0.4385E-01 -0.63 18 0.4082E-01 -0.61 20 0.3834E-01 -0.59 Run 2 We will take 3 cycles through NEWNOT. NLOW is 4 NHIGH is 20 N Max error Decay exponent 4 0.1476E+00 0.00 6 0.7753E-01 -1.59 8 0.4044E-01 -2.26 10 0.2593E-01 -1.99 12 0.1822E-01 -1.94 14 0.1364E-01 -1.88 16 0.1065E-01 -1.85 18 0.8637E-02 -1.78 20 0.7159E-02 -1.78 Run 3 We will take 6 cycles through NEWNOT. NLOW is 4 NHIGH is 20 N Max error Decay exponent 4 0.1476E+00 0.00 6 0.7545E-01 -1.65 8 0.3413E-01 -2.76 10 0.1870E-01 -2.70 12 0.1139E-01 -2.72 14 0.7548E-02 -2.67 16 0.5330E-02 -2.61 18 0.3926E-02 -2.60 20 0.3009E-02 -2.52 TEST13 Modified example 2 Run 1 Take 0 cycles through NEWNOT N Max error Decay exponent 4 0.1476E+00 0.00 6 0.9126E-01 -1.19 8 0.6436E-01 -1.21 10 0.4382E-01 -1.72 12 0.2896E-01 -2.27 14 0.1919E-01 -2.67 16 0.1278E-01 -3.04 18 0.8604E-02 -3.36 20 0.5865E-02 -3.64 Run 2 Take 1 cycles through NEWNOT N Max error Decay exponent 4 0.1476E+00 0.00 6 0.8335E-01 -1.41 8 0.5004E-01 -1.77 10 0.2833E-01 -2.55 12 0.1647E-01 -2.97 14 0.9871E-02 -3.32 16 0.6104E-02 -3.60 18 0.3895E-02 -3.81 20 0.2567E-02 -3.96 Run 3 Take 2 cycles through NEWNOT N Max error Decay exponent 4 0.1476E+00 0.00 6 0.7954E-01 -1.52 8 0.4188E-01 -2.23 10 0.2240E-01 -2.81 12 0.1259E-01 -3.16 14 0.7394E-02 -3.45 16 0.4526E-02 -3.68 18 0.2877E-02 -3.85 20 0.1891E-02 -3.98 TEST14 Optimal spline test TITAND returns some test data for a material property of titanium. I Data point Data Interpolant Error 1 595. 0.6440 0.6440 0.000E+00 2 605. 0.6220 0.1043 0.518E+00 3 615. 0.6380 0.0436 0.594E+00 4 625. 0.6490 0.2781 0.371E+00 5 635. 0.6520 0.6520 0.111E-15 6 645. 0.6390 1.0378 -0.399E+00 7 655. 0.6460 1.3359 -0.690E+00 8 665. 0.6570 1.4749 -0.818E+00 9 675. 0.6520 1.4114 -0.759E+00 10 685. 0.6550 1.1302 -0.475E+00 11 695. 0.6440 0.6440 0.111E-15 12 705. 0.6630 -0.0062 0.669E+00 13 715. 0.6630 -0.7513 0.141E+01 14 725. 0.6680 -1.4943 0.216E+01 15 735. 0.6760 -2.1102 0.279E+01 16 745. 0.6760 -2.4607 0.314E+01 17 755. 0.6860 -2.4516 0.314E+01 18 765. 0.6790 -2.0563 0.274E+01 19 775. 0.6780 -1.3160 0.199E+01 20 785. 0.6830 -0.3401 0.102E+01 21 795. 0.6940 0.6940 0.000E+00 22 805. 0.6990 1.5515 -0.853E+00 23 815. 0.7100 2.0381 -0.133E+01 24 825. 0.7300 2.0863 -0.136E+01 25 835. 0.7630 1.7620 -0.999E+00 26 845. 0.8120 1.2646 -0.453E+00 27 855. 0.9070 0.9070 -0.111E-15 28 865. 1.0440 0.9300 0.114E+00 29 875. 1.3360 1.3360 -0.222E-15 30 885. 1.8810 1.8775 0.354E-02 31 895. 2.1690 2.1690 0.444E-15 32 905. 2.0750 2.0308 0.442E-01 33 915. 1.5980 1.5980 -0.222E-15 34 925. 1.2110 1.1682 0.428E-01 35 935. 0.9160 0.9160 -0.222E-15 36 945. 0.7460 0.8395 -0.935E-01 37 955. 0.6720 0.8294 -0.157E+00 38 965. 0.6270 0.8007 -0.174E+00 39 975. 0.6150 0.7207 -0.106E+00 40 985. 0.6070 0.6070 0.222E-15 41 995. 0.6060 0.5031 0.103E+00 42 1005. 0.6090 0.4419 0.167E+00 43 1015. 0.6030 0.4403 0.163E+00 44 1025. 0.6010 0.4990 0.102E+00 45 1035. 0.6030 0.6030 -0.111E-15 46 1045. 0.6010 0.7211 -0.120E+00 47 1055. 0.6110 0.8060 -0.195E+00 48 1065. 0.6010 0.7947 -0.194E+00 49 1075. 0.6080 0.6080 0.000E+00 Optimal knots: 1 730.985 2 794.414 3 844.476 4 880.060 5 907.814 6 938.001 7 976.752 TEST15 SMOOTH defines a cubic smoothing spline The exact data values were rounded to 2 digits after the decimal point. Value and derivatives of noisefree function at some points: 1 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.5000000E+00 6 0.1041667E-01 0.6250000E-01 0.2500000E+00 0.5000000E+00 11 0.8333333E-01 0.2500000E+00 0.5000000E+00 -0.7000000E+00 16 0.2562500E+00 0.4125000E+00 0.1500000E+00 -0.7000000E+00 21 0.4666667E+00 0.4000000E+00 -0.2000000E+00 -0.7000000E+00 26 0.6270833E+00 0.2125000E+00 -0.5500000E+00 -0.7000000E+00 31 0.6500000E+00 -0.1500000E+00 -0.9000000E+00 0.1300000E+01 36 0.4895833E+00 -0.4375000E+00 -0.2500000E+00 0.1300000E+01 41 0.2666667E+00 -0.4000000E+00 0.4000000E+00 -0.2000000E+00 46 0.1125000E+00 -0.2250000E+00 0.3000000E+00 -0.2000000E+00 51 0.3333333E-01 -0.1000000E+00 0.2000000E+00 -0.2000000E+00 56 0.4166667E-02 -0.2500000E-01 0.1000000E+00 -0.2000000E+00 61 0.5551115E-16 0.5551115E-16 0.0000000E+00 -0.2000000E+00 Prescribed S = 600000. S(Smoothing spline) = 136817. Value and derivatives of smoothing spline at corresponding points: 1 0.2941036E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 6 0.2860153E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 11 0.2779270E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 16 0.2698387E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 21 0.2617504E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 26 0.2536621E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 31 0.2455738E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 36 0.2374855E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 41 0.2293971E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 46 0.2213088E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 51 0.2132205E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 56 0.2051322E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 61 0.1970439E+00 -0.1617663E-01 0.0000000E+00 0.0000000E+00 Prescribed S = 60000.0 S(Smoothing spline) = 60594.2 Value and derivatives of smoothing spline at corresponding points: 1 0.1147453E+00 0.1248896E+00 0.0000000E+00 -0.3144359E-02 6 0.1770639E+00 0.1239407E+00 -0.5373302E-02 -0.2344597E-01 11 0.2377963E+00 0.1176221E+00 -0.2177277E-01 -0.4643046E-01 16 0.2928591E+00 0.1004183E+00 -0.4818482E-01 -0.5869598E-01 21 0.3358277E+00 0.6906420E-01 -0.7672704E-01 -0.4991272E-01 26 0.3598108E+00 0.2521816E-01 -0.9641122E-01 -0.2004025E-01 31 0.3600840E+00 -0.2429080E-01 -0.9838592E-01 0.2024727E-01 36 0.3361813E+00 -0.6990344E-01 -0.8151629E-01 0.4990104E-01 41 0.2921234E+00 -0.1040603E+00 -0.5446361E-01 0.5577702E-01 46 0.2344183E+00 -0.1246005E+00 -0.2863262E-01 0.4312783E-01 51 0.1693762E+00 -0.1340802E+00 -0.1079232E-01 0.2425146E-01 56 0.1014326E+00 -0.1369733E+00 -0.2101441E-02 0.8258699E-02 61 0.3281814E-01 -0.1373147E+00 0.0000000E+00 0.8993137E-03 Prescribed S = 6000.00 S(Smoothing spline) = 6006.40 Value and derivatives of smoothing spline at corresponding points: 1 -0.7700496E-01 0.2016251E+00 0.0000000E+00 0.8348940E-01 6 0.2640664E-01 0.2191499E+00 0.8231865E-01 0.1899203E+00 11 0.1495967E+00 0.2779315E+00 0.1320411E+00 -0.9156231E-01 16 0.3019495E+00 0.3217662E+00 0.1666312E-01 -0.4016292E+00 21 0.4560014E+00 0.2752966E+00 -0.2109408E+00 -0.4678783E+00 26 0.5580908E+00 0.1166162E+00 -0.4065841E+00 -0.2165549E+00 31 0.5625813E+00 -0.1000095E+00 -0.4224959E+00 0.2521956E+00 36 0.4663443E+00 -0.2682390E+00 -0.2245256E+00 0.5361252E+00 41 0.3152549E+00 -0.3146008E+00 0.3119113E-01 0.4085792E+00 46 0.1693283E+00 -0.2574074E+00 0.1714570E+00 0.8681967E-01 51 0.6291959E-01 -0.1691475E+00 0.1614270E+00 -0.1512137E+00 56 -0.5006900E-02 -0.1104196E+00 0.6876211E-01 -0.1935220E+00 61 -0.5528434E-01 -0.9657137E-01 0.0000000E+00 -0.5993972E-01 Prescribed S = 600.000 S(Smoothing spline) = 600.601 Value and derivatives of smoothing spline at corresponding points: 1 -0.2963400E-01 0.7784390E-01 0.0000000E+00 0.1881186E+00 6 0.1512881E-01 0.1172257E+00 0.1860182E+00 0.4513100E+00 11 0.1052990E+00 0.2556813E+00 0.3274437E+00 -0.1074509E+00 16 0.2695089E+00 0.3856242E+00 0.1453313E+00 -0.6319346E+00 21 0.4664125E+00 0.3713366E+00 -0.2193693E+00 -0.7647145E+00 26 0.6094399E+00 0.1723324E+00 -0.5596879E+00 -0.4544145E+00 31 0.6189989E+00 -0.1388067E+00 -0.6152950E+00 0.4415859E+00 36 0.4845194E+00 -0.3686379E+00 -0.2554097E+00 0.9618103E+00 41 0.2875115E+00 -0.3840732E+00 0.1638807E+00 0.5533942E+00 46 0.1252297E+00 -0.2533486E+00 0.3071757E+00 -0.4324249E-01 51 0.3494297E-01 -0.1145944E+00 0.2279962E+00 -0.2802139E+00 56 0.1160666E-03 -0.3662131E-01 0.8773735E-01 -0.2464744E+00 61 -0.1190576E-01 -0.1897948E-01 0.0000000E+00 -0.7557853E-01 Prescribed S = 60.0000 S(Smoothing spline) = 60.0312 Value and derivatives of smoothing spline at corresponding points: 1 -0.8394129E-02 0.2581047E-01 0.0000000E+00 0.2518230E+00 6 0.1198464E-01 0.7524758E-01 0.2300879E+00 0.5644270E+00 11 0.8923731E-01 0.2504282E+00 0.4184971E+00 -0.2174806E+00 16 0.2591358E+00 0.4062645E+00 0.1575786E+00 -0.6429156E+00 21 0.4678743E+00 0.3973185E+00 -0.2160043E+00 -0.7442283E+00 26 0.6247285E+00 0.1993882E+00 -0.5940418E+00 -0.7224763E+00 31 0.6391664E+00 -0.1522376E+00 -0.7223112E+00 0.5305745E+00 36 0.4878095E+00 -0.4132885E+00 -0.2530386E+00 0.1289184E+01 41 0.2740688E+00 -0.3996796E+00 0.2468044E+00 0.5556576E+00 46 0.1133045E+00 -0.2373620E+00 0.3303067E+00 -0.1861492E+00 51 0.3150310E-01 -0.9930279E-01 0.2192157E+00 -0.2914548E+00 56 0.2946336E-02 -0.2602364E-01 0.9223091E-01 -0.2136055E+00 61 -0.3058048E-02 -0.6123582E-02 0.0000000E+00 -0.9174112E-01 Prescribed S = 6.00000 S(Smoothing spline) = 6.02024 Value and derivatives of smoothing spline at corresponding points: 1 -0.7761942E-04 -0.3258983E-02 0.0000000E+00 0.2902726E+00 6 0.1196397E-01 0.5910886E-01 0.3700503E+00 -0.1852071E+01 11 0.8113751E-01 0.2499691E+00 0.6298765E+00 -0.2068756E+01 16 0.2597290E+00 0.4002764E+00 -0.4646067E-01 0.7199026E+00 21 0.4704100E+00 0.4034641E+00 -0.4440256E+00 -0.1965103E+01 26 0.6287927E+00 0.2457430E+00 -0.8246287E+00 -0.2005511E+01 31 0.6490559E+00 -0.1504916E+00 -0.9944092E+00 0.1616665E+01 36 0.4882313E+00 -0.4450990E+00 -0.4212975E+00 0.6255360E+01 41 0.2690242E+00 -0.4038389E+00 0.1370717E-01 0.2163886E+01 46 0.1116081E+00 -0.2373397E+00 0.3946836E+00 -0.3037040E+00 51 0.2979722E-01 -0.9931821E-01 0.3889785E+00 0.4965903E+00 56 0.2233020E-02 -0.4057103E-01 0.4159226E+00 -0.2142740E+01 61 0.2673157E-04 0.1638648E-03 0.0000000E+00 0.9996781E-01 Prescribed S = 0.600000 S(Smoothing spline) = 0.602252 Value and derivatives of smoothing spline at corresponding points: 1 -0.1199480E-05 0.1596481E-03 0.0000000E+00 0.9792433E-01 6 0.1066315E-01 0.3943016E-01 0.1660023E+01 -0.1882883E+02 11 0.8046364E-01 0.2499976E+00 0.1445672E+01 -0.1892289E+02 16 0.2599986E+00 0.4000181E+00 0.4376973E-02 0.4982062E-01 21 0.4704912E+00 0.3992611E+00 0.4603068E+00 -0.1891737E+02 26 0.6297978E+00 0.2591464E+00 -0.1208571E+01 0.2336277E+01 31 0.6500599E+00 -0.1500205E+00 -0.9768544E+00 -0.2434928E+01 36 0.4893052E+00 -0.4598538E+00 -0.1667521E+01 0.3547889E+02 41 0.2700603E+00 -0.3998548E+00 0.2365105E-01 -0.2360405E+01 46 0.1106938E+00 -0.2600053E+00 0.1669383E+01 -0.2138201E+02 51 0.2947728E-01 -0.9997954E-01 -0.4677549E+00 0.2132809E+02 56 0.6645464E-03 -0.6055177E-01 0.1655647E+01 -0.1889271E+02 61 0.6698039E-07 -0.4084469E-05 0.0000000E+00 0.5468210E-02 TEST16 Solution of second order two point boundary value problem. Carrier's nonlinear perturbation problem EPS = 0.500000E-02 Approximation from a space of splines of order 6 on 4 intervals of dimension 18 Breakpoints: 0.250000 0.500000 0.750000 Results on interation 11 The piecewise polynomial representation of the approximation: 0.000 -0.1000E+01 0.222E-14 -0.195E-05 0.305E-03 -0.874E-02 0.132E+00 0.250 -0.1000E+01 0.769E-05 -0.436E-03 0.467E-01 -0.130E+01 0.193E+02 0.500 -0.9999E+00 0.112E-02 -0.629E-01 0.676E+01 -0.188E+03 0.280E+04 0.750 -0.9918E+00 0.163E+00 0.452E+01 -0.139E+02 0.215E+04 0.609E+05 X G(X) G(X)-F(X): 0.750000 -0.991843 -0.437119E-04 0.781250 -0.984779 -0.327376E-03 0.812500 -0.971626 -0.166146E-03 0.843750 -0.947207 0.693448E-03 0.875000 -0.902123 0.104774E-02 0.906250 -0.819725 -0.556448E-03 0.937500 -0.671950 -0.346509E-02 0.968750 -0.415996 -0.341015E-02 1.00000 -0.111022E-15 -0.111022E-15 Approximation from a space of splines of order 6 on 4 intervals of dimension 18 Breakpoints: 0.441428 0.652766 0.831348 Results on interation 2 The piecewise polynomial representation of the approximation: 0.000 -0.1000E+01 0.126E-14 -0.343E-03 0.138E-01 -0.227E+00 0.151E+01 0.441 -0.1000E+01 0.338E-03 -0.208E-02 0.936E+00 -0.273E+02 0.548E+03 0.653 -0.9988E+00 0.234E-01 0.260E+00 0.315E+02 -0.819E+03 0.242E+05 0.831 -0.9587E+00 0.820E+00 0.199E+02 -0.971E+02 0.235E+05 -0.155E+06 X G(X) G(X)-F(X): 0.750000 -0.991843 -0.383648E-04 0.781250 -0.984779 0.150282E-04 0.812500 -0.971626 0.188176E-04 0.843750 -0.947207 -0.188183E-03 0.875000 -0.902123 -0.182631E-03 0.906250 -0.819725 0.755006E-03 0.937500 -0.671950 0.916999E-04 0.968750 -0.415996 -0.126197E-02 1.00000 -0.111022E-15 -0.111022E-15 Approximation from a space of splines of order 6 on 4 intervals of dimension 18 Breakpoints: 0.445031 0.678895 0.846496 Results on interation 1 The piecewise polynomial representation of the approximation: 0.000 -0.1000E+01 -0.125E-14 -0.373E-03 0.149E-01 -0.242E+00 0.160E+01 0.445 -0.1000E+01 0.365E-03 -0.109E-01 0.161E+01 -0.461E+02 0.769E+03 0.679 -0.9980E+00 0.394E-01 0.554E+00 0.424E+02 -0.989E+03 0.351E+05 0.846 -0.9442E+00 0.110E+01 0.253E+02 -0.250E+02 0.290E+05 -0.254E+06 X G(X) G(X)-F(X): 0.750000 -0.991843 -0.335560E-04 0.781250 -0.984779 -0.221474E-04 0.812500 -0.971626 0.312361E-04 0.843750 -0.947207 -0.702137E-05 0.875000 -0.902123 -0.303323E-03 0.906250 -0.819725 0.414622E-03 0.937500 -0.671950 0.332633E-03 0.968750 -0.415996 -0.840483E-03 1.00000 -0.111022E-15 -0.222045E-15 TEST17 Taut spline interpolation Cubic spline versus taut spline GAMMA = 2.50000 595.000 0.644000 0.644000 597.400 0.645367 0.645206 599.800 0.646593 0.646300 602.200 0.647685 0.647286 604.600 0.648648 0.648168 607.000 0.649487 0.648949 609.400 0.650208 0.649634 611.800 0.650817 0.650226 614.200 0.651318 0.650731 616.600 0.651718 0.651151 619.000 0.652022 0.651490 621.400 0.652235 0.651754 623.800 0.652363 0.651945 626.200 0.652412 0.652068 628.600 0.652387 0.652127 631.000 0.652293 0.652125 633.400 0.652137 0.652068 635.800 0.651923 0.651958 638.200 0.651657 0.651800 640.600 0.651345 0.651598 643.000 0.650992 0.651355 645.400 0.650603 0.651076 647.800 0.650185 0.650766 650.200 0.649742 0.650427 652.600 0.649281 0.650064 655.000 0.648807 0.649680 657.400 0.648324 0.649281 659.800 0.647840 0.648870 662.200 0.647359 0.648450 664.600 0.646886 0.648026 667.000 0.646428 0.647603 669.400 0.645990 0.647183 671.800 0.645577 0.646771 674.200 0.645195 0.646371 676.600 0.644849 0.645987 679.000 0.644546 0.645622 681.400 0.644290 0.645282 683.800 0.644086 0.644970 686.200 0.643942 0.644690 688.600 0.643861 0.644445 691.000 0.643849 0.644241 693.400 0.643913 0.644080 695.800 0.644057 0.643968 698.200 0.644285 0.643906 700.600 0.644593 0.643894 703.000 0.644979 0.643931 705.400 0.645442 0.644016 707.800 0.645978 0.644150 710.200 0.646586 0.644331 712.600 0.647261 0.644559 715.000 0.648004 0.644833 717.400 0.648809 0.645152 719.800 0.649676 0.645517 722.200 0.650602 0.645926 724.600 0.651584 0.646379 727.000 0.652620 0.646875 729.400 0.653708 0.647413 731.800 0.654844 0.647993 734.200 0.656026 0.648615 736.600 0.657253 0.649278 739.000 0.658521 0.649980 741.400 0.659829 0.650722 743.800 0.661173 0.651503 746.200 0.662551 0.652322 748.600 0.663960 0.653179 751.000 0.665399 0.654073 753.400 0.666865 0.655004 755.800 0.668355 0.655982 758.200 0.669867 0.657024 760.600 0.671398 0.658148 763.000 0.672946 0.659373 765.400 0.674509 0.660715 767.800 0.676083 0.662194 770.200 0.677667 0.663826 772.600 0.679258 0.665630 775.000 0.680854 0.667623 777.400 0.682452 0.669824 779.800 0.684049 0.672250 782.200 0.685644 0.674919 784.600 0.687233 0.677850 787.000 0.688815 0.681059 789.400 0.690387 0.684565 791.800 0.691945 0.688385 794.200 0.693489 0.692539 796.600 0.695019 0.697041 799.000 0.696577 0.701892 801.400 0.698232 0.707080 803.800 0.700053 0.712593 806.200 0.702111 0.718421 808.600 0.704473 0.724552 811.000 0.707210 0.730975 813.400 0.710390 0.737677 815.800 0.714084 0.744648 818.200 0.718360 0.751875 820.600 0.723287 0.759348 823.000 0.728936 0.767056 825.400 0.735375 0.774985 827.800 0.742673 0.783126 830.200 0.750901 0.791467 832.600 0.760127 0.799995 835.000 0.770420 0.808713 837.400 0.781850 0.817720 839.800 0.794487 0.827168 842.200 0.808399 0.837209 844.600 0.823655 0.847993 847.000 0.840326 0.859671 849.400 0.858481 0.872395 851.800 0.878188 0.886316 854.200 0.899518 0.901585 856.600 0.922627 0.918467 859.000 0.948707 0.938556 861.400 0.979613 0.964302 863.800 1.01721 0.998168 866.200 1.06337 1.04262 868.600 1.11995 1.10012 871.000 1.18882 1.17314 873.400 1.27186 1.26413 875.800 1.37086 1.37550 878.200 1.48461 1.50529 880.600 1.60712 1.64483 883.000 1.73201 1.78488 885.400 1.85291 1.91618 887.800 1.96344 2.02949 890.200 2.05722 2.11557 892.600 2.12786 2.16515 895.000 2.16900 2.16900 897.400 2.17613 2.12528 899.800 2.15227 2.06119 902.200 2.10233 1.99483 904.600 2.03122 1.92660 907.000 1.94384 1.85604 909.400 1.84508 1.78268 911.800 1.73987 1.70605 914.200 1.63309 1.62570 916.600 1.52943 1.54125 919.000 1.43082 1.45362 921.400 1.33748 1.36453 923.800 1.24957 1.27573 926.200 1.16725 1.18893 928.600 1.09069 1.10589 931.000 1.02007 1.02834 933.400 0.955544 0.958012 935.800 0.897285 0.896632 938.200 0.845271 0.844901 940.600 0.799193 0.801954 943.000 0.758719 0.766785 945.400 0.723514 0.738392 947.800 0.693246 0.715771 950.200 0.667582 0.697918 952.600 0.646190 0.683830 955.000 0.628735 0.672503 957.400 0.614885 0.662934 959.800 0.604307 0.654282 962.200 0.596668 0.646373 964.600 0.591634 0.639199 967.000 0.588874 0.632755 969.400 0.588053 0.627033 971.800 0.588839 0.622027 974.200 0.590899 0.617730 976.600 0.593899 0.614137 979.000 0.597507 0.611239 981.400 0.601390 0.609032 983.800 0.605214 0.607508 986.200 0.608654 0.606652 988.600 0.611559 0.606264 991.000 0.613951 0.605991 993.400 0.615861 0.605722 995.800 0.617317 0.605458 998.200 0.618351 0.605201 1000.60 0.618992 0.604952 1003.00 0.619270 0.604712 1005.40 0.619216 0.604482 1007.80 0.618859 0.604264 1010.20 0.618230 0.604059 1012.60 0.617358 0.603868 1015.00 0.616273 0.603693 1017.40 0.615007 0.603534 1019.80 0.613587 0.603394 1022.20 0.612046 0.603272 1024.60 0.610412 0.603171 1027.00 0.608717 0.603092 1029.40 0.606989 0.603035 1031.80 0.605258 0.603003 1034.20 0.603556 0.602996 1036.60 0.601912 0.603016 1039.00 0.600355 0.603064 1041.40 0.598917 0.603142 1043.80 0.597627 0.603250 1046.20 0.596515 0.603389 1048.60 0.595611 0.603562 1051.00 0.594945 0.603769 1053.40 0.594548 0.604011 1055.80 0.594449 0.604291 1058.20 0.594678 0.604608 1060.60 0.595266 0.604964 1063.00 0.596242 0.605362 1065.40 0.597636 0.605801 1067.80 0.599479 0.606283 1070.20 0.601801 0.606809 1072.60 0.604631 0.607381 1075.00 0.608000 0.608000 TEST18 Two parameterizations Using the "natural" parameterization. X Y 0.200000 0.100000E-01 0.210000 0.810000E-02 0.220000 0.640000E-02 0.230000 0.490000E-02 0.240000 0.360000E-02 0.250000 0.250000E-02 0.260000 0.160000E-02 0.270000 0.900000E-03 0.280000 0.400000E-03 0.290000 0.100000E-03 0.300000 0.123982E-31 0.310000 0.100000E-03 0.320000 0.400000E-03 0.330000 0.900000E-03 0.340000 0.160000E-02 0.350000 0.250000E-02 0.360000 0.360000E-02 0.370000 0.490000E-02 0.380000 0.640000E-02 0.390000 0.810000E-02 0.400000 0.100000E-01 Using the "uniform" parameterization. X Y 0.200000 0.100000E-01 0.218147 0.740950E-02 0.236597 0.528676E-02 0.254442 0.357850E-02 0.270771 0.223144E-02 0.284677 0.119231E-02 0.295251 0.407828E-03 0.301598 -0.174957E-03 0.303697 -0.589267E-03 0.302881 -0.837758E-03 0.300599 -0.920454E-03 0.298298 -0.837378E-03 0.297429 -0.588554E-03 0.299439 -0.174006E-03 0.305663 0.408876E-03 0.316085 0.119332E-02 0.329824 0.223231E-02 0.345982 0.357916E-02 0.363662 0.528719E-02 0.381967 0.740969E-02 0.400000 0.100000E-01 TEST19 Bivariate interpolation using B-splines. The data to be interpolated: 1.0 2.0 3.0 4.0 5.0 6.0 1.0 0.00000E+00 0.00000E+00 0.00000E+00 0.10000E+01 0.80000E+01 0.27000E+02 2.0 0.00000E+00 0.00000E+00 0.00000E+00 0.10000E+01 0.80000E+01 0.27000E+02 3.0 0.00000E+00 0.00000E+00 0.00000E+00 0.10000E+01 0.80000E+01 0.27000E+02 4.0 0.25000E+00 0.25000E+00 0.25000E+00 0.12500E+01 0.82500E+01 0.27250E+02 5.0 0.22500E+01 0.22500E+01 0.22500E+01 0.32500E+01 0.10250E+02 0.29250E+02 6.0 0.62500E+01 0.62500E+01 0.62500E+01 0.72500E+01 0.14250E+02 0.33250E+02 7.0 0.12250E+02 0.12250E+02 0.12250E+02 0.13250E+02 0.20250E+02 0.39250E+02 Interpolating function: 1.0 2.0 3.0 4.0 5.0 6.0 1.0 0.00000E+00 0.00000E+00 0.00000E+00 0.10000E+01 0.80000E+01 0.27000E+02 2.0 0.00000E+00-0.77037E-33-0.41087E-32 0.10000E+01 0.80000E+01 0.27000E+02 3.0 0.38519E-33 0.15407E-32 0.82173E-32 0.10000E+01 0.80000E+01 0.27000E+02 4.0 0.25000E+00 0.25000E+00 0.25000E+00 0.12500E+01 0.82500E+01 0.27250E+02 5.0 0.22500E+01 0.22500E+01 0.22500E+01 0.32500E+01 0.10250E+02 0.29250E+02 6.0 0.62500E+01 0.62500E+01 0.62500E+01 0.72500E+01 0.14250E+02 0.33250E+02 7.0 0.12250E+02 0.12250E+02 0.12250E+02 0.13250E+02 0.20250E+02 0.39250E+02 Interpolation error: 1.0 2.0 3.0 4.0 5.0 6.0 1.0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 2.0 0.00000E+00 0.77037E-33 0.41087E-32 0.00000E+00 0.00000E+00 0.00000E+00 3.0-0.38519E-33-0.15407E-32-0.82173E-32 0.00000E+00 0.00000E+00 0.00000E+00 4.0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00-0.35527E-14 5.0 0.00000E+00 0.00000E+00 0.00000E+00 0.44409E-15 0.00000E+00 0.00000E+00 6.0 0.00000E+00-0.17764E-14 0.00000E+00 0.00000E+00-0.35527E-14 0.00000E+00 7.0 0.00000E+00-0.17764E-14 0.00000E+00-0.17764E-14 0.00000E+00 0.00000E+00 TEST20 Bivariate interpolation using piecewise polynomials. The data to be interpolated: 1.0 2.0 3.0 4.0 5.0 6.0 1.0 0.00000E+00 0.00000E+00 0.00000E+00 0.10000E+01 0.80000E+01 0.27000E+02 2.0 0.00000E+00 0.00000E+00 0.00000E+00 0.10000E+01 0.80000E+01 0.27000E+02 3.0 0.00000E+00 0.00000E+00 0.00000E+00 0.10000E+01 0.80000E+01 0.27000E+02 4.0 0.25000E+00 0.25000E+00 0.25000E+00 0.12500E+01 0.82500E+01 0.27250E+02 5.0 0.22500E+01 0.22500E+01 0.22500E+01 0.32500E+01 0.10250E+02 0.29250E+02 6.0 0.62500E+01 0.62500E+01 0.62500E+01 0.72500E+01 0.14250E+02 0.33250E+02 7.0 0.12250E+02 0.12250E+02 0.12250E+02 0.13250E+02 0.20250E+02 0.39250E+02 Interpolating function: 1.0 2.0 3.0 4.0 5.0 6.0 1.0 0.00000E+00 0.00000E+00 0.00000E+00 0.10000E+01 0.80000E+01 0.27000E+02 2.0 0.00000E+00 0.00000E+00-0.38519E-32 0.10000E+01 0.80000E+01 0.27000E+02 3.0 0.00000E+00 0.00000E+00 0.61630E-32 0.10000E+01 0.80000E+01 0.27000E+02 4.0 0.25000E+00 0.25000E+00 0.25000E+00 0.12500E+01 0.82500E+01 0.27250E+02 5.0 0.22500E+01 0.22500E+01 0.22500E+01 0.32500E+01 0.10250E+02 0.29250E+02 6.0 0.62500E+01 0.62500E+01 0.62500E+01 0.72500E+01 0.14250E+02 0.33250E+02 7.0 0.12250E+02 0.12250E+02 0.12250E+02 0.13250E+02 0.20250E+02 0.39250E+02 Interpolation error: 1.0 2.0 3.0 4.0 5.0 6.0 1.0 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 2.0 0.00000E+00 0.00000E+00 0.38519E-32-0.22204E-15-0.17764E-14-0.10658E-13 3.0 0.00000E+00 0.00000E+00-0.61630E-32 0.11102E-15-0.17764E-14-0.10658E-13 4.0 0.00000E+00 0.27756E-16 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 5.0 0.00000E+00 0.00000E+00 0.44409E-15 0.88818E-15-0.53291E-14-0.28422E-13 6.0 0.00000E+00 0.00000E+00 0.00000E+00-0.88818E-15-0.35527E-14-0.71054E-14 7.0 0.00000E+00 0.00000E+00 0.35527E-14 0.00000E+00 0.35527E-14 0.28422E-13 TEST21 Least squares approximation by splines. Run number 1 Approximation by splines of order 2 using 6 intervals. Breakpoints: 0.166667 0.333333 0.500000 0.666667 0.833333 Least square error = 0.116995E-02 Average error = 0.856121E-03 Maximum error = 0.243739E-02 Approximation by splines of order 2 using 6 intervals. Breakpoints: 0.121920 0.239427 0.354096 0.467520 0.616353 Least square error = 0.485002E-02 Average error = 0.372762E-02 Maximum error = 0.999725E-02 Run number 2 Approximation by splines of order 3 using 1 intervals. Breakpoints: Least square error = 0.460310 Average error = 0.391244 Maximum error = 1.34714 Approximation by splines of order 3 using 1 intervals. Breakpoints: Least square error = 0.460310 Average error = 0.391244 Maximum error = 1.34714 Approximation by splines of order 3 using 2 intervals. Breakpoints: 1.50000 Least square error = 0.121451 Average error = 0.913469E-01 Maximum error = 0.428157 Approximation by splines of order 3 using 3 intervals. Breakpoints: 1.00000 2.00000 Least square error = 0.385040E-01 Average error = 0.276762E-01 Maximum error = 0.138238 Approximation by splines of order 3 using 4 intervals. Breakpoints: 0.750000 1.50000 2.25000 Least square error = 0.171150E-01 Average error = 0.123528E-01 Maximum error = 0.589345E-01 Approximation by splines of order 3 using 5 intervals. Breakpoints: 0.600000 1.20000 1.80000 2.40000 Least square error = 0.938277E-02 Average error = 0.696340E-02 Maximum error = 0.291671E-01 Approximation by splines of order 3 using 6 intervals. Breakpoints: 0.500000 1.00000 1.50000 2.00000 2.50000 Least square error = 0.526016E-02 Average error = 0.396308E-02 Maximum error = 0.160495E-01 Approximation by splines of order 3 using 7 intervals. Breakpoints: 0.428571 0.857143 1.28571 1.71429 2.14286 2.57143 Least square error = 0.419508E-02 Average error = 0.330550E-02 Maximum error = 0.104223E-01 Approximation by splines of order 3 using 8 intervals. Breakpoints: 0.375000 0.750000 1.12500 1.50000 1.87500 2.25000 2.62500 Least square error = 0.369242E-02 Average error = 0.289470E-02 Maximum error = 0.102887E-01 Approximation by splines of order 3 using 9 intervals. Breakpoints: 0.333333 0.666667 1.00000 1.33333 1.66667 2.00000 2.33333 2.66667 Least square error = 0.319237E-02 Average error = 0.256675E-02 Maximum error = 0.737150E-02 Approximation by splines of order 3 using 10 intervals. Breakpoints: 0.300000 0.600000 0.900000 1.20000 1.50000 1.80000 2.10000 2.40000 2.70000 Least square error = 0.287403E-02 Average error = 0.247918E-02 Maximum error = 0.613093E-02 Approximation by splines of order 3 using 11 intervals. Breakpoints: 0.272727 0.545455 0.818182 1.09091 1.36364 1.63636 1.90909 2.18182 2.45455 2.72727 Least square error = 0.270043E-02 Average error = 0.230457E-02 Maximum error = 0.578737E-02 Approximation by splines of order 3 using 12 intervals. Breakpoints: 0.250000 0.500000 0.750000 1.00000 1.25000 1.50000 1.75000 2.00000 2.25000 2.50000 2.75000 Least square error = 0.260370E-02 Average error = 0.224555E-02 Maximum error = 0.574599E-02 Approximation by splines of order 3 using 13 intervals. Breakpoints: 0.230769 0.461538 0.692308 0.923077 1.15385 1.38462 1.61538 1.84615 2.07692 2.30769 2.53846 2.76923 Least square error = 0.271116E-02 Average error = 0.234604E-02 Maximum error = 0.611600E-02 Approximation by splines of order 3 using 14 intervals. Breakpoints: 0.214286 0.428571 0.642857 0.857143 1.07143 1.28571 1.50000 1.71429 1.92857 2.14286 2.35714 2.57143 2.78571 Least square error = 0.268365E-02 Average error = 0.224667E-02 Maximum error = 0.628692E-02 Approximation by splines of order 3 using 15 intervals. Breakpoints: 0.200000 0.400000 0.600000 0.800000 1.00000 1.20000 1.40000 1.60000 1.80000 2.00000 2.20000 2.40000 2.60000 2.80000 Least square error = 0.252216E-02 Average error = 0.211183E-02 Maximum error = 0.571827E-02 Approximation by splines of order 3 using 16 intervals. Breakpoints: 0.187500 0.375000 0.562500 0.750000 0.937500 1.12500 1.31250 1.50000 1.68750 1.87500 2.06250 2.25000 2.43750 2.62500 2.81250 Least square error = 0.249301E-02 Average error = 0.209679E-02 Maximum error = 0.566659E-02 Approximation by splines of order 3 using 17 intervals. Breakpoints: 0.176471 0.352941 0.529412 0.705882 0.882353 1.05882 1.23529 1.41176 1.58824 1.76471 1.94118 2.11765 2.29412 2.47059 2.64706 2.82353 Least square error = 0.245884E-02 Average error = 0.206502E-02 Maximum error = 0.587878E-02 Approximation by splines of order 3 using 18 intervals. Breakpoints: 0.166667 0.333333 0.500000 0.666667 0.833333 1.00000 1.16667 1.33333 1.50000 1.66667 1.83333 2.00000 2.16667 2.33333 2.50000 2.66667 2.83333 Least square error = 0.247271E-02 Average error = 0.207767E-02 Maximum error = 0.582376E-02 Approximation by splines of order 3 using 19 intervals. Breakpoints: 0.157895 0.315789 0.473684 0.631579 0.789474 0.947368 1.10526 1.26316 1.42105 1.57895 1.73684 1.89474 2.05263 2.21053 2.36842 2.52632 2.68421 2.84211 Least square error = 0.246767E-02 Average error = 0.205841E-02 Maximum error = 0.554117E-02 Approximation by splines of order 3 using 20 intervals. Breakpoints: 0.150000 0.300000 0.450000 0.600000 0.750000 0.900000 1.05000 1.20000 1.35000 1.50000 1.65000 1.80000 1.95000 2.10000 2.25000 2.40000 2.55000 2.70000 2.85000 Least square error = 0.240785E-02 Average error = 0.202511E-02 Maximum error = 0.555014E-02 Run number 3 Approximation by splines of order 5 using 8 intervals. Breakpoints: 730.985 794.414 844.476 880.060 907.814 938.001 976.752 Least square error = 0.555261E-01 Average error = 0.361613E-01 Maximum error = 0.215728 Approximation by splines of order 5 using 8 intervals. Breakpoints: 687.118 765.727 820.087 860.590 894.212 927.676 970.563 Least square error = 0.987152E-01 Average error = 0.686795E-01 Maximum error = 0.316400 Approximation by splines of order 5 using 10 intervals. Breakpoints: 667.284 732.749 788.903 836.593 879.401 915.739 951.377 990.468 1032.73 Least square error = 0.585322E-01 Average error = 0.367868E-01 Maximum error = 0.214116 Approximation by splines of order 5 using 12 intervals. Breakpoints: 654.435 712.100 762.722 805.042 840.704 870.930 899.340 927.423 956.855 990.363 1031.45 Least square error = 0.568376E-01 Average error = 0.371164E-01 Maximum error = 0.188978 Approximation by splines of order 5 using 14 intervals. Breakpoints: 646.169 697.610 745.369 785.334 818.071 846.056 871.687 899.956 928.076 953.183 980.134 1010.06 1042.04 Least square error = 0.449888E-01 Average error = 0.274609E-01 Maximum error = 0.165889 TEST22 The Runge example Use cubic spline interpolation of order N. Boundary conditions are Not-a-knot. N Max error Decay exponent 2 0.8968E+00 0.00 4 0.6995E+00 -0.36 6 0.4290E+00 -1.21 8 0.2459E+00 -1.93 10 0.1423E+00 -2.45 12 0.8380E-01 -2.91 14 0.5027E-01 -3.31 16 0.3077E-01 -3.68 18 0.1924E-01 -3.99 20 0.1229E-01 -4.25 Boundary conditions are Derivatives at endpoints. N Max error Decay exponent 2 0.8599E+00 0.00 4 0.6754E+00 -0.35 6 0.4192E+00 -1.18 8 0.2448E+00 -1.87 10 0.1423E+00 -2.43 12 0.8380E-01 -2.90 14 0.5027E-01 -3.31 16 0.3077E-01 -3.68 18 0.1924E-01 -3.99 20 0.1229E-01 -4.25 Boundary conditions are Second derivatives at endpoints. N Max error Decay exponent 2 0.1002E+01 0.00 4 0.6915E+00 -0.53 6 0.4213E+00 -1.22 8 0.2448E+00 -1.89 10 0.1423E+00 -2.43 12 0.8380E-01 -2.90 14 0.5027E-01 -3.32 16 0.3077E-01 -3.68 18 0.1924E-01 -3.99 20 0.1229E-01 -4.25 PPPACK_PRB Normal end of execution. 25 March 2017 10:04:27.615 AM