// Linear polynomial approximating y[n] y_n = 9.0/15.0*y[n] + 6.0/15.0*y[n-1] + 3.0/15.0*y[n-2] - 3.0/15.0*y[n-4]; // Linear polynomial approximating y[n + 1] y_n_1 = 8.0/10.0*y[n] + 5.0/10.0*y[n-1] + 2.0/10.0*y[n-2] - 1.0/10.0*y[n-3] - 4.0/10.0*y[n-4]; // Linear polynomial approximating the change per period at y[n] dy_n = 4.0/20.0*y[n] + 2.0/20.0*y[n-1] - 2.0/20.0*y[n-3] - 4.0/20.0*y[n-4]; // Linear polynomial approximating the average value over the last period average_y_n = 60.0/120.0*y[n] + 42.0/120.0*y[n-1] + 24.0/120.0*y[n-2] + 6.0/120.0*y[n-3] - 12.0/120.0*y[n-4]; // Linear polynomial approximating y[n] y_n = 9.0/15.0*y[n] + 6.0/15.0*y[n-1] + 3.0/15.0*y[n-2] - 3.0/15.0*y[n-4]; // Linear polynomial approximating the change per period at y[n] dy_n = 4.0/20.0*y[n] + 2.0/20.0*y[n-1] - 2.0/20.0*y[n-3] - 4.0/20.0*y[n-4]; // Linear correction for jitter y[n] += epsilon*(-36.0/540.0*y[n] - 126.0/540.0*y[n-1] - 36.0/540.0*y[n-2] + 54.0/540.0*y[n-3] + 144.0/540.0*y[n-4]); // Quadratic polynomial approximating y[n] y[n] = 31.0/35.0*y[n] + 9.0/35.0*y[n-1] - 3.0/35.0*y[n-2] - 5.0/35.0*y[n-3] + 3.0/35.0*y[n-4]; // Quadratic polynomial approximating y[n+1] y_n_1 = 18.0/10.0*y[n] - 8.0/10.0*y[n-2] - 6.0/10.0*y[n-3] + 6.0/10.0*y[n-4]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 162.0/210.0*y[n] - 39.0/210.0*y[n-1] - 120.0/210.0*y[n-2] - 81.0/210.0*y[n-3] + 78.0/210.0*y[n-4]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 6.0/21.0*y[n] - 3.0/21.0*y[n-1] - 6.0/21.0*y[n-2] - 3.0/21.0*y[n-3] + 6.0/21.0*y[n-4]; // Quadratic polynomial approximating the average value over the last period average_y_n = 690.0/1260.0*y[n] + 411.0/1260.0*y[n-1] + 192.0/1260.0*y[n-2] + 33.0/1260.0*y[n-3] - 66.0/1260.0*y[n-4]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 162.0/210.0*y[n] - 39.0/210.0*y[n-1] - 120.0/210.0*y[n-2] - 81.0/210.0*y[n-3] + 78.0/210.0*y[n-4]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 6.0/21.0*y[n] - 3.0/21.0*y[n-1] - 6.0/21.0*y[n-2] - 3.0/21.0*y[n-3] + 6.0/21.0*y[n-4]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 6.0/21.0*y[n] - 3.0/21.0*y[n-1] - 6.0/21.0*y[n-2] - 3.0/21.0*y[n-3] + 6.0/21.0*y[n-4]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 162.0/210.0*y[n] - 39.0/210.0*y[n-1] - 120.0/210.0*y[n-2] - 81.0/210.0*y[n-3] + 78.0/210.0*y[n-4]; // Quadratic polynomial approximating y[n] y[n] = 31.0/35.0*y[n] + 9.0/35.0*y[n-1] - 3.0/35.0*y[n-2] - 5.0/35.0*y[n-3] + 3.0/35.0*y[n-4]; // Quadratic correction for jitter y[n] += epsilon*(-4374.0/6510.0*y[n] - 249.0/6510.0*y[n-1] + 4206.0/6510.0*y[n-2] + 3321.0/6510.0*y[n-3] - 2904.0/6510.0*y[n-4]); With 5 points: 8.773099999999999, 9.691599999999999, 10.5857, 12.0946, 12.5924 = y[n] Linear Approximating y[ 9]: 12.7558 (11.9 or -1.2) Approximating y[10]: 13.75996 (12.6 or -2.5) The slope at 9: 1.004159999999999 (0.7 or -1.2) The integral from 8 to 9: 12.25372 (11.55 or -0.63333) Quadratic Approximating y[ 9]: 12.72342857142857 (11.9 or -1.2) Approximating y[10]: 13.64666 (12.6 or -2.5) The slope at 9: 0.9394171428571428 (0.7 or -1.2) The concavity at 9: -0.03237142857142824 (0.0 or -0.2) The integral from 8 to 9: 12.24832476190476 (11.55 or -0.63333) A maximum or minimum: 29.01994704324831 (undefined or -6) A root (linear): -12.70295570427024 (-17 or undefined) Root (quadratic): -11.33160005065282, 69.37149413714944 (-17, undefined or -10.89898, -1.10102) // Linear polynomial approximating y[n] y_n = 19.0/55.0*y[n] + 16.0/55.0*y[n-1] + 13.0/55.0*y[n-2] + 10.0/55.0*y[n-3] + 7.0/55.0*y[n-4] + 4.0/55.0*y[n-5] + 1.0/55.0*y[n-6] - 2.0/55.0*y[n-7] - 5.0/55.0*y[n-8] - 8.0/55.0*y[n-9]; // Linear polynomial approximating y[n + 1] y_n_1 = 18.0/45.0*y[n] + 15.0/45.0*y[n-1] + 12.0/45.0*y[n-2] + 9.0/45.0*y[n-3] + 6.0/45.0*y[n-4] + 3.0/45.0*y[n-5] - 3.0/45.0*y[n-7] - 6.0/45.0*y[n-8] - 9.0/45.0*y[n-9]; // Linear polynomial approximating the change per period at y[n] dy_n = 9.0/165.0*y[n] + 7.0/165.0*y[n-1] + 5.0/165.0*y[n-2] + 3.0/165.0*y[n-3] + 1.0/165.0*y[n-4] - 1.0/165.0*y[n-5] - 3.0/165.0*y[n-6] - 5.0/165.0*y[n-7] - 7.0/165.0*y[n-8] - 9.0/165.0*y[n-9]; // Linear polynomial approximating the average value over the last period average_y_n = 315.0/990.0*y[n] + 267.0/990.0*y[n-1] + 219.0/990.0*y[n-2] + 171.0/990.0*y[n-3] + 123.0/990.0*y[n-4] + 75.0/990.0*y[n-5] + 27.0/990.0*y[n-6] - 21.0/990.0*y[n-7] - 69.0/990.0*y[n-8] - 117.0/990.0*y[n-9]; // Linear polynomial approximating y[n] y_n = 19.0/55.0*y[n] + 16.0/55.0*y[n-1] + 13.0/55.0*y[n-2] + 10.0/55.0*y[n-3] + 7.0/55.0*y[n-4] + 4.0/55.0*y[n-5] + 1.0/55.0*y[n-6] - 2.0/55.0*y[n-7] - 5.0/55.0*y[n-8] - 8.0/55.0*y[n-9]; // Linear polynomial approximating the change per period at y[n] dy_n = 9.0/165.0*y[n] + 7.0/165.0*y[n-1] + 5.0/165.0*y[n-2] + 3.0/165.0*y[n-3] + 1.0/165.0*y[n-4] - 1.0/165.0*y[n-5] - 3.0/165.0*y[n-6] - 5.0/165.0*y[n-7] - 7.0/165.0*y[n-8] - 9.0/165.0*y[n-9]; // Linear correction for jitter y[n] += epsilon*(459.0/9405.0*y[n] - 831.0/9405.0*y[n-1] - 636.0/9405.0*y[n-2] - 441.0/9405.0*y[n-3] - 246.0/9405.0*y[n-4] - 51.0/9405.0*y[n-5] + 144.0/9405.0*y[n-6] + 339.0/9405.0*y[n-7] + 534.0/9405.0*y[n-8] + 729.0/9405.0*y[n-9]); // Quadratic polynomial approximating y[n] y[n] = 136.0/220.0*y[n] + 84.0/220.0*y[n-1] + 42.0/220.0*y[n-2] + 10.0/220.0*y[n-3] - 12.0/220.0*y[n-4] - 24.0/220.0*y[n-5] - 26.0/220.0*y[n-6] - 18.0/220.0*y[n-7] + 28.0/220.0*y[n-9]; // Quadratic polynomial approximating y[n+1] y_n_1 = 108.0/120.0*y[n] + 60.0/120.0*y[n-1] + 22.0/120.0*y[n-2] - 6.0/120.0*y[n-3] - 24.0/120.0*y[n-4] - 32.0/120.0*y[n-5] - 30.0/120.0*y[n-6] - 18.0/120.0*y[n-7] + 4.0/120.0*y[n-8] + 36.0/120.0*y[n-9]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 2052.0/7920.0*y[n] + 876.0/7920.0*y[n-1] - 30.0/7920.0*y[n-2] - 666.0/7920.0*y[n-3] - 1032.0/7920.0*y[n-4] - 1128.0/7920.0*y[n-5] - 954.0/7920.0*y[n-6] - 510.0/7920.0*y[n-7] + 204.0/7920.0*y[n-8] + 1188.0/7920.0*y[n-9]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 36.0/792.0*y[n] + 12.0/792.0*y[n-1] - 6.0/792.0*y[n-2] - 18.0/792.0*y[n-3] - 24.0/792.0*y[n-4] - 24.0/792.0*y[n-5] - 18.0/792.0*y[n-6] - 6.0/792.0*y[n-7] + 12.0/792.0*y[n-8] + 36.0/792.0*y[n-9]; // Quadratic polynomial approximating the average value over the last period average_y_n = 23580.0/47520.0*y[n] + 15636.0/47520.0*y[n-1] + 9102.0/47520.0*y[n-2] + 3978.0/47520.0*y[n-3] + 264.0/47520.0*y[n-4] - 2040.0/47520.0*y[n-5] - 2934.0/47520.0*y[n-6] - 2418.0/47520.0*y[n-7] - 492.0/47520.0*y[n-8] + 2844.0/47520.0*y[n-9]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 2052.0/7920.0*y[n] + 876.0/7920.0*y[n-1] - 30.0/7920.0*y[n-2] - 666.0/7920.0*y[n-3] - 1032.0/7920.0*y[n-4] - 1128.0/7920.0*y[n-5] - 954.0/7920.0*y[n-6] - 510.0/7920.0*y[n-7] + 204.0/7920.0*y[n-8] + 1188.0/7920.0*y[n-9]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 36.0/792.0*y[n] + 12.0/792.0*y[n-1] - 6.0/792.0*y[n-2] - 18.0/792.0*y[n-3] - 24.0/792.0*y[n-4] - 24.0/792.0*y[n-5] - 18.0/792.0*y[n-6] - 6.0/792.0*y[n-7] + 12.0/792.0*y[n-8] + 36.0/792.0*y[n-9]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 36.0/792.0*y[n] + 12.0/792.0*y[n-1] - 6.0/792.0*y[n-2] - 18.0/792.0*y[n-3] - 24.0/792.0*y[n-4] - 24.0/792.0*y[n-5] - 18.0/792.0*y[n-6] - 6.0/792.0*y[n-7] + 12.0/792.0*y[n-8] + 36.0/792.0*y[n-9]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 2052.0/7920.0*y[n] + 876.0/7920.0*y[n-1] - 30.0/7920.0*y[n-2] - 666.0/7920.0*y[n-3] - 1032.0/7920.0*y[n-4] - 1128.0/7920.0*y[n-5] - 954.0/7920.0*y[n-6] - 510.0/7920.0*y[n-7] + 204.0/7920.0*y[n-8] + 1188.0/7920.0*y[n-9]; // Quadratic polynomial approximating y[n] y[n] = 136.0/220.0*y[n] + 84.0/220.0*y[n-1] + 42.0/220.0*y[n-2] + 10.0/220.0*y[n-3] - 12.0/220.0*y[n-4] - 24.0/220.0*y[n-5] - 26.0/220.0*y[n-6] - 18.0/220.0*y[n-7] + 28.0/220.0*y[n-9]; // Quadratic correction for jitter y[n] += epsilon*(-106704.0/1077120.0*y[n] - 291504.0/1077120.0*y[n-1] - 82104.0/1077120.0*y[n-2] + 70056.0/1077120.0*y[n-3] + 164976.0/1077120.0*y[n-4] + 202656.0/1077120.0*y[n-5] + 183096.0/1077120.0*y[n-6] + 106296.0/1077120.0*y[n-7] - 27744.0/1077120.0*y[n-8] - 219024.0/1077120.0*y[n-9]); With 10 points: 5.7344, 6.7585, 6.4353, 7.9155, 8.479699999999999, 8.773099999999999, 9.691599999999999, 10.5857, 12.0946, 12.5924 = y[n] Linear Approximating y[ 9]: 12.3274 (11.9 or -1.2) Approximating y[10]: 13.08769333333333 (12.6 or -2.5) The slope at 9: 0.7602933333333333 (0.7 or -1.2) The integral from 8 to 9: 11.94725333333333 (11.55 or -0.63333) Quadratic Approximating y[ 9]: 12.72801363636364 (11.9 or -1.2) Approximating y[10]: 13.82215166666667 (12.6 or -2.5) The slope at 9: 1.06075356060606 (0.7 or -1.2) The concavity at 9: 0.06676893939393946 (0.0 or -0.2) The integral from 8 to 9: 12.20876501262626 (11.55 or -0.63333) A maximum or minimum: -15.88693141203833 (undefined or -6) A root (linear): -16.21400512083056 (-17 or undefined) Root (quadratic): nan, nan (-17, undefined or -10.89898, -1.10102) // Linear polynomial approximating y[n] y_n = 9.0/15.0*y[n] + 6.0/15.0*y[n-1] + 3.0/15.0*y[n-2] - 3.0/15.0*y[n-4]; // Linear polynomial approximating y[n + 1] y_n_1 = 8.0/10.0*y[n] + 5.0/10.0*y[n-1] + 2.0/10.0*y[n-2] - 1.0/10.0*y[n-3] - 4.0/10.0*y[n-4]; // Linear polynomial approximating the change per period at y[n] dy_n = 4.0/20.0*y[n] + 2.0/20.0*y[n-1] - 2.0/20.0*y[n-3] - 4.0/20.0*y[n-4]; // Linear polynomial approximating the average value over the last period average_y_n = 60.0/120.0*y[n] + 42.0/120.0*y[n-1] + 24.0/120.0*y[n-2] + 6.0/120.0*y[n-3] - 12.0/120.0*y[n-4]; // Linear polynomial approximating y[n] y_n = 9.0/15.0*y[n] + 6.0/15.0*y[n-1] + 3.0/15.0*y[n-2] - 3.0/15.0*y[n-4]; // Linear polynomial approximating the change per period at y[n] dy_n = 4.0/20.0*y[n] + 2.0/20.0*y[n-1] - 2.0/20.0*y[n-3] - 4.0/20.0*y[n-4]; // Linear correction for jitter y[n] += epsilon*(-36.0/540.0*y[n] - 126.0/540.0*y[n-1] - 36.0/540.0*y[n-2] + 54.0/540.0*y[n-3] + 144.0/540.0*y[n-4]); // Quadratic polynomial approximating y[n] y[n] = 31.0/35.0*y[n] + 9.0/35.0*y[n-1] - 3.0/35.0*y[n-2] - 5.0/35.0*y[n-3] + 3.0/35.0*y[n-4]; // Quadratic polynomial approximating y[n+1] y_n_1 = 18.0/10.0*y[n] - 8.0/10.0*y[n-2] - 6.0/10.0*y[n-3] + 6.0/10.0*y[n-4]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 162.0/210.0*y[n] - 39.0/210.0*y[n-1] - 120.0/210.0*y[n-2] - 81.0/210.0*y[n-3] + 78.0/210.0*y[n-4]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 6.0/21.0*y[n] - 3.0/21.0*y[n-1] - 6.0/21.0*y[n-2] - 3.0/21.0*y[n-3] + 6.0/21.0*y[n-4]; // Quadratic polynomial approximating the average value over the last period average_y_n = 690.0/1260.0*y[n] + 411.0/1260.0*y[n-1] + 192.0/1260.0*y[n-2] + 33.0/1260.0*y[n-3] - 66.0/1260.0*y[n-4]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 162.0/210.0*y[n] - 39.0/210.0*y[n-1] - 120.0/210.0*y[n-2] - 81.0/210.0*y[n-3] + 78.0/210.0*y[n-4]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 6.0/21.0*y[n] - 3.0/21.0*y[n-1] - 6.0/21.0*y[n-2] - 3.0/21.0*y[n-3] + 6.0/21.0*y[n-4]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 6.0/21.0*y[n] - 3.0/21.0*y[n-1] - 6.0/21.0*y[n-2] - 3.0/21.0*y[n-3] + 6.0/21.0*y[n-4]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 162.0/210.0*y[n] - 39.0/210.0*y[n-1] - 120.0/210.0*y[n-2] - 81.0/210.0*y[n-3] + 78.0/210.0*y[n-4]; // Quadratic polynomial approximating y[n] y[n] = 31.0/35.0*y[n] + 9.0/35.0*y[n-1] - 3.0/35.0*y[n-2] - 5.0/35.0*y[n-3] + 3.0/35.0*y[n-4]; // Quadratic correction for jitter y[n] += epsilon*(-4374.0/6510.0*y[n] - 249.0/6510.0*y[n-1] + 4206.0/6510.0*y[n-2] + 3321.0/6510.0*y[n-3] - 2904.0/6510.0*y[n-4]); With 5 points: 3.0347, 2.2269, 0.4966, 0.1939, -1.9873 = y[n] Linear Approximating y[ 9]: -1.62244 (11.9 or -1.2) Approximating y[10]: -2.83014 (12.6 or -2.5) The slope at 9: -1.2077 (0.7 or -1.2) The integral from 8 to 9: -1.01859 (11.55 or -0.63333) Quadratic Approximating y[ 9]: -1.810897142857143 (11.9 or -1.2) Approximating y[10]: -3.48974 (12.6 or -2.5) The slope at 9: -1.584614285714286 (0.7 or -1.2) The concavity at 9: -0.1884571428571428 (0.0 or -0.2) The integral from 8 to 9: -1.049999523809524 (11.55 or -0.63333) A maximum or minimum: -8.408353547604611 (undefined or -6) A root (linear): -1.343413099279622 (-17 or undefined) Root (quadratic): -15.58346868118825, -1.233238414020976 (-17, undefined or -10.89898, -1.10102) // Linear polynomial approximating y[n] y_n = 19.0/55.0*y[n] + 16.0/55.0*y[n-1] + 13.0/55.0*y[n-2] + 10.0/55.0*y[n-3] + 7.0/55.0*y[n-4] + 4.0/55.0*y[n-5] + 1.0/55.0*y[n-6] - 2.0/55.0*y[n-7] - 5.0/55.0*y[n-8] - 8.0/55.0*y[n-9]; // Linear polynomial approximating y[n + 1] y_n_1 = 18.0/45.0*y[n] + 15.0/45.0*y[n-1] + 12.0/45.0*y[n-2] + 9.0/45.0*y[n-3] + 6.0/45.0*y[n-4] + 3.0/45.0*y[n-5] - 3.0/45.0*y[n-7] - 6.0/45.0*y[n-8] - 9.0/45.0*y[n-9]; // Linear polynomial approximating the change per period at y[n] dy_n = 9.0/165.0*y[n] + 7.0/165.0*y[n-1] + 5.0/165.0*y[n-2] + 3.0/165.0*y[n-3] + 1.0/165.0*y[n-4] - 1.0/165.0*y[n-5] - 3.0/165.0*y[n-6] - 5.0/165.0*y[n-7] - 7.0/165.0*y[n-8] - 9.0/165.0*y[n-9]; // Linear polynomial approximating the average value over the last period average_y_n = 315.0/990.0*y[n] + 267.0/990.0*y[n-1] + 219.0/990.0*y[n-2] + 171.0/990.0*y[n-3] + 123.0/990.0*y[n-4] + 75.0/990.0*y[n-5] + 27.0/990.0*y[n-6] - 21.0/990.0*y[n-7] - 69.0/990.0*y[n-8] - 117.0/990.0*y[n-9]; // Linear polynomial approximating y[n] y_n = 19.0/55.0*y[n] + 16.0/55.0*y[n-1] + 13.0/55.0*y[n-2] + 10.0/55.0*y[n-3] + 7.0/55.0*y[n-4] + 4.0/55.0*y[n-5] + 1.0/55.0*y[n-6] - 2.0/55.0*y[n-7] - 5.0/55.0*y[n-8] - 8.0/55.0*y[n-9]; // Linear polynomial approximating the change per period at y[n] dy_n = 9.0/165.0*y[n] + 7.0/165.0*y[n-1] + 5.0/165.0*y[n-2] + 3.0/165.0*y[n-3] + 1.0/165.0*y[n-4] - 1.0/165.0*y[n-5] - 3.0/165.0*y[n-6] - 5.0/165.0*y[n-7] - 7.0/165.0*y[n-8] - 9.0/165.0*y[n-9]; // Linear correction for jitter y[n] += epsilon*(459.0/9405.0*y[n] - 831.0/9405.0*y[n-1] - 636.0/9405.0*y[n-2] - 441.0/9405.0*y[n-3] - 246.0/9405.0*y[n-4] - 51.0/9405.0*y[n-5] + 144.0/9405.0*y[n-6] + 339.0/9405.0*y[n-7] + 534.0/9405.0*y[n-8] + 729.0/9405.0*y[n-9]); // Quadratic polynomial approximating y[n] y[n] = 136.0/220.0*y[n] + 84.0/220.0*y[n-1] + 42.0/220.0*y[n-2] + 10.0/220.0*y[n-3] - 12.0/220.0*y[n-4] - 24.0/220.0*y[n-5] - 26.0/220.0*y[n-6] - 18.0/220.0*y[n-7] + 28.0/220.0*y[n-9]; // Quadratic polynomial approximating y[n+1] y_n_1 = 108.0/120.0*y[n] + 60.0/120.0*y[n-1] + 22.0/120.0*y[n-2] - 6.0/120.0*y[n-3] - 24.0/120.0*y[n-4] - 32.0/120.0*y[n-5] - 30.0/120.0*y[n-6] - 18.0/120.0*y[n-7] + 4.0/120.0*y[n-8] + 36.0/120.0*y[n-9]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 2052.0/7920.0*y[n] + 876.0/7920.0*y[n-1] - 30.0/7920.0*y[n-2] - 666.0/7920.0*y[n-3] - 1032.0/7920.0*y[n-4] - 1128.0/7920.0*y[n-5] - 954.0/7920.0*y[n-6] - 510.0/7920.0*y[n-7] + 204.0/7920.0*y[n-8] + 1188.0/7920.0*y[n-9]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 36.0/792.0*y[n] + 12.0/792.0*y[n-1] - 6.0/792.0*y[n-2] - 18.0/792.0*y[n-3] - 24.0/792.0*y[n-4] - 24.0/792.0*y[n-5] - 18.0/792.0*y[n-6] - 6.0/792.0*y[n-7] + 12.0/792.0*y[n-8] + 36.0/792.0*y[n-9]; // Quadratic polynomial approximating the average value over the last period average_y_n = 23580.0/47520.0*y[n] + 15636.0/47520.0*y[n-1] + 9102.0/47520.0*y[n-2] + 3978.0/47520.0*y[n-3] + 264.0/47520.0*y[n-4] - 2040.0/47520.0*y[n-5] - 2934.0/47520.0*y[n-6] - 2418.0/47520.0*y[n-7] - 492.0/47520.0*y[n-8] + 2844.0/47520.0*y[n-9]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 2052.0/7920.0*y[n] + 876.0/7920.0*y[n-1] - 30.0/7920.0*y[n-2] - 666.0/7920.0*y[n-3] - 1032.0/7920.0*y[n-4] - 1128.0/7920.0*y[n-5] - 954.0/7920.0*y[n-6] - 510.0/7920.0*y[n-7] + 204.0/7920.0*y[n-8] + 1188.0/7920.0*y[n-9]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 36.0/792.0*y[n] + 12.0/792.0*y[n-1] - 6.0/792.0*y[n-2] - 18.0/792.0*y[n-3] - 24.0/792.0*y[n-4] - 24.0/792.0*y[n-5] - 18.0/792.0*y[n-6] - 6.0/792.0*y[n-7] + 12.0/792.0*y[n-8] + 36.0/792.0*y[n-9]; // Quadratic polynomial approximating the concavity at y[n] ddy_n = 36.0/792.0*y[n] + 12.0/792.0*y[n-1] - 6.0/792.0*y[n-2] - 18.0/792.0*y[n-3] - 24.0/792.0*y[n-4] - 24.0/792.0*y[n-5] - 18.0/792.0*y[n-6] - 6.0/792.0*y[n-7] + 12.0/792.0*y[n-8] + 36.0/792.0*y[n-9]; // Quadratic polynomial approximating the change per period at y[n] dy_n = 2052.0/7920.0*y[n] + 876.0/7920.0*y[n-1] - 30.0/7920.0*y[n-2] - 666.0/7920.0*y[n-3] - 1032.0/7920.0*y[n-4] - 1128.0/7920.0*y[n-5] - 954.0/7920.0*y[n-6] - 510.0/7920.0*y[n-7] + 204.0/7920.0*y[n-8] + 1188.0/7920.0*y[n-9]; // Quadratic polynomial approximating y[n] y[n] = 136.0/220.0*y[n] + 84.0/220.0*y[n-1] + 42.0/220.0*y[n-2] + 10.0/220.0*y[n-3] - 12.0/220.0*y[n-4] - 24.0/220.0*y[n-5] - 26.0/220.0*y[n-6] - 18.0/220.0*y[n-7] + 28.0/220.0*y[n-9]; // Quadratic correction for jitter y[n] += epsilon*(-106704.0/1077120.0*y[n] - 291504.0/1077120.0*y[n-1] - 82104.0/1077120.0*y[n-2] + 70056.0/1077120.0*y[n-3] + 164976.0/1077120.0*y[n-4] + 202656.0/1077120.0*y[n-5] + 183096.0/1077120.0*y[n-6] + 106296.0/1077120.0*y[n-7] - 27744.0/1077120.0*y[n-8] - 219024.0/1077120.0*y[n-9]); With 10 points: 2.1715, 0.7925, 3.0172, 4.0302, 2.7889, 3.0347, 2.2269, 0.4966, 0.1939, -1.9873 = y[n] Linear Approximating y[ 9]: 0.05687818181818187 (11.9 or -1.2) Approximating y[10]: -0.3030400000000001 (12.6 or -2.5) The slope at 9: -0.3599181818181818 (0.7 or -1.2) The integral from 8 to 9: 0.2368372727272726 (11.55 or -0.63333) Quadratic Approximating y[ 9]: -1.875008181818182 (11.9 or -1.2) Approximating y[10]: -3.844831666666668 (12.6 or -2.5) The slope at 9: -1.808832954545455 (0.7 or -1.2) The concavity at 9: -0.3219810606060606 (0.0 or -0.2) The integral from 8 to 9: -1.024255214646465 (11.55 or -0.63333) A maximum or minimum: -5.61782407679729 (undefined or -6) A root (linear): 0.158030865600768 (-17 or undefined) Root (quadratic): -10.08025040904465, -1.155397744549929 (-17, undefined or -10.89898, -1.10102)