Interpolation ChEn 2450 Given (xi,yi), find a function f(x) to interpolate these points. 1 Interpolation.key - September 8, 2014 Motivation & Concept Properties of air at atmospheric pressure Motivation: T rho lambda viscosity! K kg/m3 W/(m K) N s/m2! • Often we have discrete data (tabulated, ----------------------------------------! from experiments, etc) that we need to 100 3.5562 0.0093 7.110e-06! interpolate. 150 2.3364 0.0138 1.034e-05! 200 1.7458 0.0181 1.325e-05! • Interpolating functions form the basis for 250 1.3947 0.0223 1.596e-05! numerical integration and differentiation 300 1.1614 0.0263 1.846e-05! techniques 350 0.9950 0.0300 2.082e-05! 400 0.8711 0.0338 2.301e-05! Used for solving ODEs & PDEs ‣ 450 0.7750 0.0373 2.507e-05! ‣ we will cover this later 500 0.6864 0.0407 2.701e-05! 550 0.6329 0.0439 2.884e-05! Concept: 600 0.5804 0.0469 3.058e-05! • Choose a polynomial function to fit to 650 0.5356 0.0497 3.225e-05! the data (connect the dots) 700 0.4975 0.0524 3.388e-05! 750 0.4643 0.0549 3.546e-05! • Solve for the coefficients of the 800 0.4354 0.0573 3.698e-05! polynomial 850 0.4097 0.0596 3.843e-05! 900 0.3868 0.0620 3.981e-05! • Evaluate the polynomial wherever you 950 0.3666 0.0643 4.113e-05! want (interpolation) 1000 0.3482 0.0667 4.244e-05 Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, 4th ed. 2 Interpolation.key - September 8, 2014 Linear Interpolation y1 = mx1 + b x 1 m y y 1 = 1 y = mx + b x 1 b y y2 = mx2 + b 2 2 ⇤ ⌅ ⇥ ⇥ y2 solve for m, b and substitute into the original equation... Program this into y2 y1 your calculator. y = − (x x1) + y1 y1 x2 x1 − − ⇥ x1 x2 x Advantages: • Easy to use (homework, exams) In MATLAB: Disadvantages: yi=interp1(x,y,xi,’linear’) • Not very accurate for • x - independent variable entries (vector) nonlinear functions 4 • y - dependent variable entries (vector) 3.5 linear interp y=x2 3 2.5 • xi - value(s) where you want to interpolate y 2 1.5 • yi - interpolated value(s) at xi. 1 0.5 0 0 0.5 1 1.5 2 x 3 Interpolation.key - September 8, 2014 Hoffman §4.3 Polynomial Interpolation np 2 n k 1 x1 x1 x1 a0 y1 p(x) = akx 2 · · · n 1 x2 x2 x2 a1 y2 k=0 ⇤ . · · · . ⌅ . ⇥ = . ⇥ . Given n+1 data points, we can ⌥ 2 · · · n ↵ ↵ ⌥ 1 xn+1 x x ↵ an ↵ yn+1 th ⌥ n+1 · · · n+1 ↵ ↵ fit an n -degree polynomial. ⇧ ⌃ ⌦ ⌦ Given: (xi,yi), solve for ai In MATLAB:! Two steps: • p=polyfit(x,y,n) 1. Obtain polynomial ‣ Forms & solves the above system. coefficients by solving the set of linear equations. ‣ requires at least n+1 points 2. Evaluate the value of the ‣ NOTE: if you supply more than n+1 points, then polynomial at the desired regression will be performed (more later). location (xi) • yi=polyval(p,xi) ‣ evaluates polynomial at point(s) given by xi. 4 Interpolation.key - September 8, 2014 Can Apply Polynomial Interpolation “Globally” or “Locally” Properties of air at atmospheric pressure What is the value T rho lambda viscosity! K kg/m3 W/(m K) N s/m2! of the viscosity at ----------------------------------------! ? 100 3.5562 0.0093 7.110e-06! T=412 K 150 2.3364 0.0138 1.034e-05! 200 1.7458 0.0181 1.325e-05! 250 1.3947 0.0223 1.596e-05! 300 1.1614 0.0263 1.846e-05! 350 0.9950 0.0300 2.082e-05! Polynomial interpolation, n=2 400 0.8711 0.0338 2.301e-05! Polynomial interpolation, n=3 450 0.7750 0.0373 2.507e-05! 500 0.6864 0.0407 2.701e-05! Linear interpolation (n=1) 550 0.6329 0.0439 2.884e-05! 600 0.5804 0.0469 3.058e-05! 650 0.5356 0.0497 3.225e-05! 700 0.4975 0.0524 3.388e-05! 750 0.4643 0.0549 3.546e-05! 800 0.4354 0.0573 3.698e-05! 850 0.4097 0.0596 3.843e-05! np 900 0.3868 0.0620 3.981e-05! k 950 0.3666 0.0643 4.113e-05! p(x) = akx 1000 0.3482 0.0667 4.244e-05 k=0 Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, 4th ed. 5 Interpolation.key - September 8, 2014 Hoffman §4.9 Cubic Spline Interpolation Concept: use cubic polynomial and “hook” For n+1 points, we form n splines. them together over a wide range of data... We must specify 4 variables per spline ⇒ we need 4n equations. 3 2 f3(x) = a3x + b3x + c3x + d3 3 2 f2(x) = a2x + b2x + c2x + d2 3 2 f1(x) = a1x + b1x + c1x + d1 1. At each data point, the values of adjacent splines must be the same. This applies to all interior points (where two functions meet) ⇒ 2(n-1) constraints. 2. At each point, the first derivatives of adjacent splines must (x3,y3) be equal (applies to all interior points) ⇒ (n-1) constraints. (x2,y2) 3. At each point, the second derivative of adjacent splines must be equal (applies to all interior points) ⇒ (n-1) constraints. (x1,y1) 4. The first and last splines must pass through the first and last points, respectively ⇒ 2 constraints. 5. The curvature (d2f / dx2) must be specified at the end points ⇒ 2 constraints. • d2f / dx2 = 0 ⇒ “natural spline” 4n constraints 6 Interpolation.key - September 8, 2014 Cubic Spline Interpolation Advantages: • Provides a “smooth” interpolant. Disadvantages: • Usually more accurate than linear • Requires a bit more work than interpolation. linear interpolation to implement. • Doesn’t usually get “wiggly” like higher-order polynomial interpolation can. yi=interp1(x,y,xi,’spline’) • x - independent variable entries (vector) MATLAB Implementation: • y - dependent variable entries (vector) • xi - value(s) where you want to interpolate • yi - interpolated value(s) at xi. 7 Interpolation.key - September 8, 2014 Comparison Between Linear, Spline, & Polynomial Interpolation (x + b)2 f(x) = exp − c ⇥ f(x) Interpolating Points f(x) Polynomial 0.8 Interpolating Points 0.8 Cubic Spline Polynomial Linear Cubic Spline 0.6 Linear 0.6 Increase f(x) number f(x) 0.4 of data 0.4 points. 0.2 0.2 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x Spline & polynomial are indistinguishable on this plot. 8 Interpolation.key - September 8, 2014 x (x + b)2 Data points follow f(x) = tanh + exp a − c ⇥ ⇤ ⌅ 1 0.5 f(x) f(x) 0 Interpolating Points Polynomial, L =2.6e+03 2 Cubic Spline, L =9.3e−01 2 Linear, L =7.9e−01 −0.5 2 −1 −3 −2 −1 0 1 2 3 x Polynomial interpolation can be bad if we use high-order polynomials 9 Interpolation.key - September 8, 2014 Data points follow f(x)=x5 x4 15x − − 15 f(x) 10 f(x) Interpolating Points 10 Interpolating Points Polynomial 5 Polynomial Cubic Spline 5 Cubic Spline Linear Linear 0 0 −5 f(x) f(x) −5 −10 −10 −15 −20 −15 −25 −20 −2 −1 0 1 2 −2 −1 0 1 2 x x • Cubic spline interpolation is usually quite accurate and relatively cost effective. • Linear interpolation is quick and easy, and may be adequate for well-resolved data. • Polynomial interpolation can be problematic, unless the underlying data is truly a polynomial! 10 Interpolation.key - September 8, 2014 Hoffman §4.8.1 2-D Linear Interpolation Interpolate Interpolate If you have “structured” (tabular) data: y φ1,2 φ2,2 1. Interpolate in one direction (two 1-D y2 interpolations) φ 2. Interpolate in second direction. y first φ1,1 φ Use this for simple homework assignments, in-class y1 2,1 Bilinear interpolation exams, etc. (x2 x)(y2 y) x1 x2 x φ(x, y) φ1,1 − − ⇡ (x x )(y y ) Interpolate 2 − 1 2 − 1 (x x1)(y2 y) y + φ2,1 − − φ1,2 φ2,2 (x x )(y y ) y2 2 − 1 2 − 1 (x x)(y y ) + φ 2 − − 1 φ 1,2 (x x )(y y ) 2 − 1 2 − 1 (x x1)(y y1) x + φ2,2 − − φ1,1 φ first (x x )(y y ) y1 2,1 2 − 1 2 − 1 ϕ=interp2(x,y,ϕ,xi,yi,’method’) x1 x2 x • ‘linear’ - 2D linear interpolation (default) • ‘spline’ - 2D spline interpolation x,y may be vectors (matlab assumes tabular form) ϕ must be a matrix (unique ϕ for each x-y pair) 11 Interpolation.key - September 8, 2014 Hoffman §4.8.2 General 2-D Linear Interpolation Equation of a plane: φ = ax + by + c φ φ3 y φ1 = ax1 + by1 + c Data is NOT “structured” φ2 = ax2 + by2 + c φ1 φ3 = ax3 + by3 + c φ2 φ (y y ) + φ (y y ) + φ (y y ) a = 3 1 − 2 2 3 − 1 1 2 − 3 , x Solve 3 x (y y ) + x (y y ) + x (y y ) 3 1 − 2 2 3 − 1 1 2 − 3 equations for φ (x x ) + φ (x x ) + φ (x x ) b = 3 2 − 1 2 1 − 3 1 3 − 2 , 3 unknowns: y (x x ) + y (x x ) + y (x x ) 3 2 − 1 2 1 − 3 1 3 − 2 φ (x y x y ) + φ (x y x y ) + φ (x y x y ) c = 3 1 2 − 2 1 2 3 1 − 1 3 1 2 3 − 3 2 x (y y ) + x (y y ) + x (y y ) 1 2 − 3 2 3 − 1 3 1 − 2 ϕ=interp2(x,y,ϕ,xi,yi,’method’) • ‘linear’ - linear interpolation • ‘spline’ - spline interpolation Note: multidimensional higher- x, y, ϕ are matrices (unique x,y for each ϕ).