InterpolationTabulated Data:≔f((x))sin⎛⎜⎝⋅―4x⎞⎟⎠≔vx00.511.52⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦≔i ‥04≔vyif⎛⎝vxi⎞⎠=vy00.3830.7070.9241⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦0.10.20.30.40.50.60.70.80.91-0.101.10.25 0.5 0.75 1 1.25 1.5 1.75 2-0.25 0 2.25vyvxwe wish to estimate the y value for an untabulated x value of 0.7 (between 0.5 and 1)Linear Interpolation:≔x .7≔x0vx1≔y0vy1≔x1vx2≔y1vy2≔y_linear +y0⋅―――−y1y0−x1x0⎛⎝−xx0⎞⎠=y_linear 0.512≔ytf((x))=yt0.522499≔εt|||―――――−yty_linearyt|||=εt%1.923MathCAD method:changed Formatting-Display-Precision to 5 decimal places=linterp((,,vx vy .7))0.51245QdiI liNon-Commercial Use OnlyQuadratic Interpolation:≔x2vx3≔y2vy3adding another tabulated point≔y_quadratic +y_linear ⋅⋅――――――−―――−y2y1−x2x1―――−y1y0−x1x0−x2x0⎛⎝−xx0⎞⎠⎛⎝−xx1⎞⎠=y_quadratic 0.525≔εt|||――――――−yty_quadraticyt|||=εt%0.55Cubic Interpolation:≔x3vx4≔y3vy4adding another tabulated point≔fdd1⎛⎝,xixj⎞⎠――――−f⎛⎝xi⎞⎠f⎛⎝xj⎞⎠−xixj≔fdd2⎛⎝,,xixjxk⎞⎠―――――――――−fdd1⎛⎝,xixj⎞⎠fdd1⎛⎝,xjxk⎞⎠−xixk≔fdd3⎛⎝,,,xixjxkxl⎞⎠―――――――――――−fdd2⎛⎝,,xixjxk⎞⎠fdd2⎛⎝,,xjxkxl⎞⎠−xixl≔b0f⎛⎝x0⎞⎠=b00.383≔b1fdd1⎛⎝,x1x0⎞⎠=b10.649≔b2fdd2⎛⎝,,x2x1x0⎞⎠=b2−0.215≔b3fdd3⎛⎝,,,x3x2x1x0⎞⎠=b3−0.044≔y_linear +b0⋅b1⎛⎝−xx0⎞⎠=y_linear 0.512≔y_quadratic +y_linear ⋅⋅b2⎛⎝−xx0⎞⎠⎛⎝−xx1⎞⎠=y_quadratic 0.525Non-Commercial Use Only≔y_cubic +y_quadratic ⋅⋅⋅b3⎛⎝−xx0⎞⎠⎛⎝−xx1⎞⎠⎛⎝−xx2⎞⎠=y_cubic 0.523≔εt|||――――−yty_cubicyt|||=εt%0.145Alternative (full) equation:≔y_cubic +++b0⋅b1⎛⎝−xx0⎞⎠⋅⋅b2⎛⎝−xx0⎞⎠⎛⎝−xx1⎞⎠⋅⋅⋅b3⎛⎝−xx0⎞⎠⎛⎝−xx1⎞⎠⎛⎝−xx2⎞⎠=y_cubic 0.52326Lagrange Interpolating Polynomials:≔vxT0.511.52[[]]≔L((,,,vx i x n))‖‖‖‖‖‖‖‖‖‖|||||||||←L 1for ∊|||||||j ‥0 n‖‖‖‖‖‖‖|||||if≠ji‖‖‖‖‖←L ⋅L――――⎛⎝−xvxj⎞⎠⎛⎝−vxivxj⎞⎠checking some of the Lagrange factors [Li(x)=1 for x=xi; Li(x)=0 at x = xj where j<>i]:=L⎛⎝,,,vx 0 vx01⎞⎠1=L⎛⎝,,,vx 0 vx12⎞⎠0=L⎛⎝,,,vx 2 vx23⎞⎠1=L⎛⎝,,,vx 2 vx33⎞⎠0≔Lagrange_poly((,,nvxx))∑=i 0n⎛⎝⋅L((,,,vx i x n))f⎛⎝vxi⎞⎠⎞⎠=Lagrange_poly((,,1 vx .7))0.51245=Lagrange_poly((,,2 vx .7))0.52537=Lagrange_poly((,,3 vx .7))0.52326Fi di C ffi i f I l i P l i lNon-Commercial Use OnlyFinding Coefficients of an Interpolation Polynomial:＝f((x))+++a0⋅a1x ⋅a2x2⋅a3x3＝⋅1 x0⎛⎝x0⎞⎠2⎛⎝x0⎞⎠31 x1⎛⎝x1⎞⎠2⎛⎝x1⎞⎠31 x2⎛⎝x2⎞⎠2⎛⎝x2⎞⎠31 x3⎛⎝x3⎞⎠2⎛⎝x3⎞⎠3⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦a0a1a2a3⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦y0y1y2y3⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦≔x0.511.52⎡⎢⎢⎢⎣⎤⎥⎥⎥⎦≔i ‥03 ≔yif⎛⎝xi⎞⎠=y0.3830.7070.9241⎡⎢⎢⎢⎣⎤⎥⎥⎥⎦≔j ‥03≔A,ij⎛⎝xi⎞⎠j=A1 0.5 0.25 0.12511 1 11 1.5 2.25 3.37512 4 8⎡⎢⎢⎢⎣⎤⎥⎥⎥⎦≔a0a1a2a3⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦⋅A−1y0y1y2y3⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦=a0a1a2a3⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦−0.0160.851−0.083−0.044⎡⎢⎢⎢⎣⎤⎥⎥⎥⎦≔f((x))∑i⎛⎜⎝⋅aixi⎞⎟⎠=f((.7))0.52326same result as Newton and Lagrangecubic interpolations (because the polynomial is unique)Non-Commercial Use OnlyComparing the cubic polynomial fit to the tabulated data and the theoretical function:≔xr ,‥.5 .51 2 ≔ft((x))sin⎛⎜⎝⋅―4x⎞⎟⎠0.50.560.620.680.740.80.860.920.980.380.441.040.8 0.95 1.1 1.25 1.4 1.55 1.7 1.850.5 0.65 2yft((xr))f((xr))x xr xrNon-Commercial Use