add nodes A, . . , D, E or delete nodes (max 9)[br]Input in AlgebraView or move nodes A, B, C, D in Graphics[br][br]XY={A, B, C, D[color=#ff0000], E[/color]} [br][br]polynom degree = count nodes - 1 (n=4)[br](1)[i] f_a:Take({a_0,a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9},1,[color=#ff0000]n[/color])[/i] [color=#0000ff]→polynom coefficents[/color][br](2) [i]f_X: Sequence(x^j , j, 0, n-1) [color=#0000ff]→ {1, x, x[sup]2[/sup], x[sup]3[/sup]} polynom basis[/color][/i][br](3)[i] fo(x):=Sum(f_a f_X) [color=#0000ff]→ a[sub]3[/sub] x[sup]3[/sup]+a[sub]2[/sub] x[sup]2[/sup]+a[sub]1[/sub] x +a_0[/color][/i][color=#0000ff] general polynom function[/color][br](5) [i]Sequence(fo(X(j)) = Y(j),j,1,n)[/i] [color=#0000ff]→[i]{fo(X(1)) = Y(1), fo(X(2)) = Y(2), fo(X(3)) = Y(3), fo(X(4)) = Y(4)}[/i][/color] [br](6) [i]V:=Substitute(LeftSide($5), f_a=Identity([color=#ff0000]n[/color])) [/i][color=#0000ff]→ Vandermonde Matrix [/color][br][br] [math]V(x)\,:=\,\left(\begin{array}{rrrr}1&x_1&x_1^{2}&x_1^{3}\\1&x_2&x_2^{2}&x_2^{3}\\1&x_3&x_3^{2}&x_3^{3}\\1&x_4&x_4^{2}&x_4^{3}\\\end{array}\right)[/math][br][url=https://www.geogebra.org/m/qvbtpky9][icon]/images/ggb/toolbar/mode_conic5.png[/icon]App Kubisches Polynom (Vandermonde-Matrix) with annotations[math]\nwarrow[/math][br][/url][br][br]Newton Polynom Coefficient Matrix (27) n=4[br][math]\large \omega_{Ci} = [br]\left(\begin{array}{r}c_{0} + \sum_{i=1}^{1}{\prod_{k=1}^{i-1} x_{1}-x_{k} } \\c_{0} + \sum_{i=1}^{2}{\prod_{k=1}^{i-1} x_{2}-x_{k} }\\c_{0} + \sum_{i=1}^{3}{\prod_{k=1}^{i-1} x_{3}-x_{k} }\\c_{0} + \sum_{i=1}^{4}{\prod_{k=1}^{i-1} x_{4}-x_{k} }\\\end{array}\right) -\LARGE \vec{Y}[/math][br][br][url=https://www.lernhelfer.de/schuelerlexikon/mathematik-abitur/artikel/newtonsches-und-lagrangesches-interpolationsverfahren]Newtonsches und Lagrangesches Interpolationsverfahren[math]\nearrow[/math][/url]