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# 1 to 11 Point - Polynomial Wave Projection Equations

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The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3675 Posts
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 Posted: August 4, 2007, 4:53 pm - IP Logged

1 to 11 Point - Polynomial Wave Projection Equations

The different point projections are an expansion process of finding the Variable Coefficient Equations and the Solved Projection Equations. The Cofactors in the Coefficient Equations are solved through an algorithmic process. You can see the process for this example of 1 to 11 points as an Excel sheet, link: Excel Sheet Example of 1 to 11 Point Algorithmic Cofactors Process

Each Projection is based on a Known Point Set, n, and is Related to a given (n - 1)th Degree Polynomial Equation. The Variable Coefficient Equations are solved for that given (n - 1)th Degree Polynomial Equation and are then plugged into the Solved (n + 1)th Point Projection Equation. Point Projections 3 through 11 are curve fittings for the most probable wave projections. As the number of Point Projections increases, the Projection process can handle higher frequency data. As the number of Point Projections decreases, the process is more tuned to low frequency data. Use the links above to get a feel for where each Point Projection can best do the work.

If you're having a problem seeing the on LP, go here, link: 1 to 11 Point - Polynomial Wave Projection Equations

 1 Point Projection Known Point Set {y0} Related 0th Degree Polynomial Equation y = a0 x0 Variable Coefficient Equation a0 = 1y0 Solved 2nd Point Projection y1 = (1a0) / 1

 2 Point Projection Known Point Set {y0, y1} Related 1st Degree Polynomial Equation y = a0 x0 + a1 x1 Variable Coefficient Equations a0 = 2y0 - 1y1 a1 = -1y0 + 1y1 Solved 3rd Point Projection y2 = (1a0 + 3a1) / 1

 3 Point Projection Known Point Set {y0, y1, y2} Related 2nd Degree Polynomial Equation y = a0 x0 + a1 x1 + a2 x2 Variable Coefficient Equations a0 = 6y0 - 6y1 + 2y2 a1 = -5y0 + 8y1 - 3y2 a2 = 1y0 - 2y1 + 1y2 Solved 4th Point Projection y3 = (1a0 + 4a1 + 16a2) / 2

 4 Point Projection Known Point Set {y0, y1, y2, y3} Related 3rd Degree Polynomial Equation y = a0 x0 + a1 x1 + a2 x2 + a3 x3 Variable Coefficient Equations a0 = 24y0 - 36y1 + 24y2 - 6y3 a1 = -26y0 + 57y1 - 42y2 + 11y3 a2 = 9y0 - 24y1 + 21y2 - 6y3 a3 = -1y0 + 3y1 - 3y2 + 1y3 Solved 5th Point Projection y4 = (1a0 + 5a1 + 25a2 + 125a3) / 6

 5 Point Projection Known Point Set {y0, y1, y2, y3, y4} Related 4th Degree Polynomial Equation y = a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 Variable Coefficient Equations a0 = 120y0 - 240y1 + 240y2 - 120y3 + 24y4 a1 = -154y0 + 428y1 - 468y2 + 244y3 - 50y4 a2 = 71y0 - 236y1 + 294y2 - 164y3 + 35y4 a3 = -14y0 + 52y1 - 72y2 + 44y3 - 10y4 a4 = 1y0 - 4y1 + 6y2 - 4y3 + 1y4 Solved 6th Point Projection y5 = (1a0 + 6a1 + 36a2 + 216a3 + 1296a4) / 24

 6 Point Projection Known Point Set {y0, y1, y2, y3, y4, y5} Related 5th Degree Polynomial Equation y = a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 Variable Coefficient Equations a0 = 720y0 - 1800y1 + 2400y2 - 1800y3 + 720y4 - 120y5 a1 = -1044y0 + 3510y1 - 5080y2 + 3960y3 - 1620y4 + 274y5 a2 = 580y0 - 2305y1 + 3720y2 - 3070y3 + 1300y4 - 225y5 a3 = -155y0 + 685y1 - 1210y2 + 1070y3 - 475y4 + 85y5 a4 = 20y0 - 95y1 + 180y2 - 170y3 + 80y4 - 15y5 a5 = -1y0 + 5y1 - 10y2 + 10y3 - 5y4 + 1y5 Solved 7th Point Projection y6 = (1a0 + 7a1 + 49a2 + 343a3 + 2401a4 + 16807a5) / 120

 7 Point Projection Known Point Set {y0, y1, y2, y3, y4, y5, y6} Related 6th Degree Polynomial Equation y = a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 Variable Coefficient Equations a0 = 5040y0 - 15120y1 + 25200y2 - 25200y3 + 15120y4 - 5040y5 + 720y6 a1 = -8028y0 + 31644y1 - 56940y2 + 59040y3 - 36180y4 + 12228y5 - 1764y6 a2 = 5104y0 - 23574y1 + 46680y2 - 50900y3 + 32160y4 - 11094y5 + 1624y6 a3 = -1665y0 + 8520y1 - 18285y2 + 21120y3 - 13875y4 + 4920y5 - 735y6 a4 = 295y0 - 1620y1 + 3705y2 - 4520y3 + 3105y4 - 1140y5 + 175y6 a5 = -27y0 + 156y1 - 375y2 + 480y3 - 345y4 + 132y5 - 21y6 a6 = 1y0 - 6y1 + 15y2 - 20y3 + 15y4 - 6y5 + 1y6 Solved 8th Point Projection y7 = (1a0 + 8a1 + 64a2 + 512a3 + 4096a4 + 32768a5 + 262144a6) / 720

 8 Point Projection Known Point Set {y0, y1, y2, y3, y4, y5, y6, y7} Related 7th Degree Polynomial Equation y = a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + a7 x7 Variable Coefficient Equations a0 = 40320y0 - 141120y1 + 282240y2 - 352800y3 + 282240y4 - 141120y5 + 40320y6 - 5040y7 a1 = -69264y0 + 312984y1 - 673008y2 + 870660y3 - 710640y4 + 360024y5 - 103824y6 + 13068y7 a2 = 48860y0 - 256942y1 + 602532y2 - 815920y3 + 684740y4 - 353430y5 + 103292y6 - 13132y7 a3 = -18424y0 + 107023y1 - 270144y2 + 384755y3 - 334040y4 + 176589y5 - 52528y6 + 6769y7 a4 = 4025y0 - 25060y1 + 67095y2 - 100240y3 + 90335y4 - 49140y5 + 14945y6 - 1960y7 a5 = -511y0 + 3346y1 - 9387y2 + 14630y3 - 13685y4 + 7686y5 - 2401y6 + 322y7 a6 = 35y0 - 238y1 + 693y2 - 1120y3 + 1085y4 - 630y5 + 203y6 - 28y7 a7 = -1y0 + 7y1 - 21y2 + 35y3 - 35y4 + 21y5 - 7y6 + 1y7 Solved 9th Point Projection y8 = (1a0 + 9a1 + 81a2 + 729a3 + 6561a4 + 59049a5 + 531441a6 + 4782969a7) / 5040

 9 Point Projection Known Point Set {y0, y1, y2, y3, y4, y5, y6, y7, y8} Related 8th Degree Polynomial Equation y = a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + a7 x7 + a8 x8 Variable Coefficient Equations a0 = 362880y0 - 1451520y1 + 3386880y2 - 5080320y3 + 5080320y4 - 3386880y5 + 1451520y6 - 362880y7 + 40320y8 a1 = -663696y0 + 3380544y1 - 8452416y2 + 13101984y3 - 13356000y4 + 9016896y5 - 3898944y6 + 981216y7 - 109584y8 a2 = 509004y0 - 3000528y1 + 8127728y2 - 13142304y3 + 13746600y4 - 9442384y5 + 4133808y6 - 1050048y7 + 118124y8 a3 = -214676y0 + 1394456y1 - 4045104y2 + 6845944y3 - 7382200y4 + 5180616y5 - 2304176y6 + 592424y7 - 67284y8 a4 = 54649y0 - 380072y1 + 1165332y2 - 2059064y3 + 2294110y4 - 1650264y5 + 748132y6 - 195272y7 + 22449y8 a5 = -8624y0 + 63056y1 - 202104y2 + 371056y3 - 427000y4 + 315504y5 - 146216y6 + 38864y7 - 4536y8 a6 = 826y0 - 6272y1 + 20832y2 - 39536y3 + 46900y4 - 35616y5 + 16912y6 - 4592y7 + 546y8 a7 = -44y0 + 344y1 - 1176y2 + 2296y3 - 2800y4 + 2184y5 - 1064y6 + 296y7 - 36y8 a8 = 1y0 - 8y1 + 28y2 - 56y3 + 70y4 - 56y5 + 28y6 - 8y7 + 1y8 Solved 10th Point Projection y9 = (1a0 + 10a1 + 100a2 + 1000a3 + 10000a4 + 100000a5 + 1000000a6 + 10000000a7 + 100000000a8) / 40320

 10 point Projection Known Point Set {y0, y1, y2, y3, y4, y5, y6, y7, y8, y9} Related 9th Degree Polynomial Equation y = a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + a7 x7 + a8 x8 + a9 x9 Variable Coefficient Equation a0 = 3628800y0 - 16329600y1 + 43545600y2 - 76204800y3 + 91445760y4 - 76204800y5 + 43545600y6 - 16329600y7 + 3628800y8 - 362880y9 a1 = -6999840y0 + 39664080y1 - 113028480y2 + 204150240y3 - 249552576y4 + 210500640y5 - 121322880y6 + 45787680y7 - 10225440y8 + 1026576y9 a2 = 5753736y0 - 37559052y1 + 115366752y2 - 216787536y3 + 271479600y4 - 232741656y5 + 135711072y6 - 51667632y7 + 11617416y8 - 1172700y9 a3 = -2655764y0 + 19063224y1 - 62458416y2 + 122402616y3 - 157623480y4 + 137809224y5 - 81526704y6 + 31384296y7 - 7118676y8 + 723680y9 a4 = 761166y0 - 5844573y1 + 20183688y2 - 41154876y3 + 54581940y4 - 48787326y5 + 29356488y6 - 11453148y7 + 2625966y8 - 269325y9 a5 = -140889y0 + 1136961y1 - 4096764y2 + 8654436y3 - 11815398y4 + 10811934y5 - 6630876y6 + 2627604y7 - 610281y8 + 63273y9 a6 = 16884y0 - 141498y1 + 527688y2 - 1149624y3 + 1612800y4 - 1511244y5 + 946008y6 - 381528y7 + 89964y8 - 9450y9 a7 = -1266y0 + 10926y1 - 41904y2 + 93744y3 - 134820y4 + 129276y5 - 82656y6 + 33984y7 - 8154y8 + 870y9 a8 = 54y0 - 477y1 + 1872y2 - 4284y3 + 6300y4 - 6174y5 + 4032y6 - 1692y7 + 414y8 - 45y9 a9 = -1y0 + 9y1 - 36y2 + 84y3 - 126y4 + 126y5 - 84y6 + 36y7 - 9y8 + 1y9 Solved 11th Point Projection y10 = (1a0 + 11a1 + 121a2 + 1331a3 + 14641a4 + 161051a5 + 1771561a6 + 19487171a7 + 214358881a8 + 2357947691a9) / 362880

 11 point projection Known Point Set {y0, y1, y2, y3, y4, y5, y6, y7, y8, y9, y10} Related 10th Degree Polynomial Equation y = a0 x0 + a1 x1 + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + a7 x7 + a8 x8 + a9 x9 + a10 x10 Variable Coefficient Equations a0 = 47174400y0 - 214099200y1 + 606009600y2 - 1197504000y3 + 1676505600y4 - 1676505600y5 + 1197504000y6 - 598752000y7 + 199584000y8 - 39916800y9 + 3628800y10 a1 = -95142240y0 + 531957600y1 - 1623088800y2 + 3316939200y3 - 4727540160y4 + 4783423680y5 - 3445243200y6 + 1733313600y7 - 580543200y8 + 116552160y9 - 10628640y10 a2 = 79362936y0 - 521270280y1 + 1736650440y2 - 3698304480y3 + 5393046960y4 - 5541317712y5 + 4035361680y6 - 2047105440y7 + 690085080y8 - 139262760y9 + 12753576y10 a3 = -36781540y0 + 278356040y1 - 1004826060y2 + 2233166160y3 - 3342229800y4 + 3497286240y5 - 2581262040y6 + 1322982960y7 - 449614260y8 + 91331800y9 - 8409500y10 a4 = 11028590y0 - 92615030y1 + 355598730y2 - 821580360y3 + 1263374700y4 - 1348939620y5 + 1011120180y6 - 524563080y7 + 180021510y8 - 36862550y9 + 3416930y10 a5 = -2310945y0 + 20390160y1 - 81560115y2 + 194790960y3 - 307585530y4 + 335437200y5 - 255740310y6 + 134522640y7 - 46695285y8 + 9653280y9 - 902055y10 a6 = 326613y0 - 2992710y1 + 12376665y2 - 30429000y3 + 49260330y4 - 54871236y5 + 42592410y6 - 22748040y7 + 7999425y8 - 1672230y9 + 157773y10 a7 = -30810y0 + 290760y1 - 1235790y2 + 3115440y3 - 5159700y4 + 5866560y5 - 4638060y6 + 2517840y7 - 898290y8 + 190200y9 - 18150y10 a8 = 1860y0 - 17970y1 + 78120y2 - 201240y3 + 340200y4 - 394380y5 + 317520y6 - 175320y7 + 63540y8 - 13650y9 + 1320y10 a9 = -65y0 + 640y1 - 2835y2 + 7440y3 - 12810y4 + 15120y5 - 12390y6 + 6960y7 - 2565y8 + 560y9 - 55y10 a10 = 1y0 - 10y1 + 45y2 - 120y3 + 210y4 - 252y5 + 210y6 - 120y7 + 45y8 - 10y9 + 1y10 Solved 12th Point Projection y11 = (1a0 + 12a1 + 144a2 + 1728a3 + 20736a4 + 248832a5 + 2985984a6 + 35831808a7 + 429981696a8 + 5159780352a9 + 61917364224a10) / 3628800

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