. - Piecewise Linear-Quadratic Interpolation Piecewise-polynomial Approximation: Problem: Givenn pairs of data points x i, y i, i,,...,n, find a piecewise-polynomial Sx S x if x x x Sx S x if x x x 2 : : S n x if x n x x n where S i x are polynomials and Sx i y i, i,,...,n. Observe that each S i can be in a different degree of polynomial or all S i are in a same degree.. Linear Splines: 2. Quadratic Splines: L i x a i b i x x i, i,,...,n, and Lx L x if x x x : : L n x if x n x x n Q i x a i b i x x i c i x x i 2, i,,...,n, and Qx Q x if x x x : : Q n x if x n x x n. Cubic Splines: (Section.) S i x a i b i x x i c i x x i 2 d i x x i, i,,...,n. Question: How can we solve coefficients a i,andb i for a linear spline; a i, b i and c i for a quadratic spline; and a i, b i, c i and d i for a cubic spline?. Linear Splines: L i x a i b i x x i, i,,...,n. We need to find 2n unknowns: a, a,..., a n, b, b,..., b n so that L i x satisfy the following conditions: () L i x i y i, i,,2,..,n - condition for interpolation, a total of n equations (2) L i x i L i x i, i,,...,n 2 - condition for continuity at interior points, a total of n equations Sincewehave2nequations for 2n unknowns, we can solve a i and b i uniquely. Observe the following. From the condition in (), L i x i y i,i,,2,..,n, we can obtain a i for i,,...,n : L i x i a i y i, i,,2,..,n and b n L n x n b n x n x n y n y n, b n y n y n x n x n. From the the condition in (2), L i x i L i x i, i,,...,n 2, we can solve b i for i,,...,n 2: L i x i b i x i x i y i L i x i y i, b i y i y i x i x i, i,,...,n 2.
L x y y y x x x x if x x x Hence, Lx L i x y i y i y i x i x i x x i if x i x x i. L n x y n y n y n x n x n x x n if x n x x n Approximation error: Theorem Let a x x x n b, h k n x k x k and f x be continuous on a,b. Then x a,b fx Lx 8 h2 c a,b f c. Proof Let x x i,x i. Then fx L i x y i y i y i x i x i x x i. Note that fx Newton Form fx i f x i fx i x i x i Note that y i fx i, i,...,n. So, fx L i x f c i x 2! x x i f ci x 2! x x i x x i. x x i x x i. Approximation error fx Lx fx L i x f c i x 2! x x i x x i 2 c i x i,x i f c i x x i,x i x x ix x i. What is x xi,x i x x i x x i? Let q i x x x i x x i. The graph of q i x is a concave up parabola so the vertex x i,y i of the parabola is the lowest point on the graph. Hence, x xi,x i x x i x x i y i. Because of the symmetry of a parabola, x i 2 x i x i and then y i x 2 i x i x i x i 2 x i x i x i x i 2 h2. Hence, fx Lx 2 c i x i,x i f c i x x i,x i x x ix x i 2 c i x i,x i 8 h2 c a,b f c i h 2 f c Note: If we know x x i,x i, then fx Lx fx L i x 8 x i x i 2 f x. x x i,x i 2
Example Let fx x. Given,,,2,8,, (i) construct a linear spline Lx; (ii) approximate f2 by L2 and f5 by L5; and (iii) estimate the approximate errors for f2 L2, f5 L5 and fx Lx. y (i) Lx L x a b x if x L x a b x if x 8 L x x if x L x 2 x 5 if x 8, 2 2 5 6 7 8 x --y fx, y Lx (ii) f2 L 2 2 5 f5 L 5 2 5 5 2 5 (iii) f x 2 x /2, f x x /2, f x x /2 For x 2, x. f2 L2 8 2 c, x c 8 2.2825, and for x 5, f5 L5 8 8 2 c,8 x c 8 52 To see how accurate these approximations are, we check the true errors: f2 L2 2 5.658 8.97656. f5 L5 5 2 5 Now for any x in,8, fx Lx 8 h2 f c c,8.99. 8 h2 c,8 x c 8 52.7825.
2. Quadratic Splines: Q i x a i b i x x i c i x x i 2, i,,...,n. We need to find n unknowns: a, a,..., a n, b, b,..., b n, c, c,...,c n so that Q i x satisfy the following conditions: () Q i x i y i, i,,2,..,n - condition for interpolation, a total of n equations (2) Q i x i Q i x i, i,,...,n 2 - condition for continuity at interior points, a total of n equations () Q i x i Q i x i, i,,...,n 2 - condition for continuous slope at interior points - n equations () one additional condition from the following three conditions: (.) Q x, (.2) Q x n, or (.) Q x Q x n depending on information about fx, equation Sincewehavenequations for n unknowns, we can again solve a i, b i and c i uniquely. From the condition in (), Q i x i y i,i,,2,..,n, we can obtain a i for i,,...,n : Q i x i a i y i, i,,2,..,n, andanequationinb n and c n : Q n x n y n b n x n x n c n x n x n 2 y n. (.) From the condition in (2), Q i x i Q i x i, i,,...,n 2, we can obtain n equations in b i and c i for i,,...,n 2: Q i x i y i b i x i x i c i x i x i 2 Q i x i y i, i,,...,n 2. (2.) Similarly, from the condition in (), Q i x i Q i x i, i,,...,n 2, we can obtain n 2 equations in b i and c i for i,,...,n 2: Q i x i b i 2c i x i x i Q i x i b i, i,,..., n 2. (.) with one more equation from the condition given in (), (.) Q x Q x b 2c x x (.2) Q x n Q n x b n 2c n x n x n or (.) Q x b 2c x x Q x n b n 2c n x n x n, we have 2n equations to solve b i s and c i s. Example Let fx x. Given,,,2,8, (i) construct a quadratic spline Qx suppose that we also know f 2 ; (ii) approximate f2 by Q2 and f5 by Q5; (iii) approximate f by Q and f 5 by Q 5; and (iv) approximate fxdx by Qxdx. (i)
Qx Q x a b x c x 2 if x Q x a b x c x 2 if x 8 Q x b x c x 2 if x Q x 2 b x c x 2 if x 8 We solve b, b, c and c as follows. From the condition (.): Q 8 2 b 5 c 5 2 2 5b 25c y 2. (.2) From the condition in (2.), we have the equation: Q b c 2 b 9c Q y 2. (2.2) From the condition (.), we have the equation: Q b 2c b 6c Q b. (.2) From the condition (.), we have the equation: Q b 2. Solve c in (2.2): c 9 2 b 9 2 8. Solve b in (.2): and solve c in (.2): Hence, Qx b 2 6 8 c 25 2 5 6 6 5. y Q x x 2 8 x2 if x Q x 2 x 6 5 2 if x 8 2 2 5 6 7 8 --y fx, y Qx (ii) f2 Q2 Q 2 2 2 8 22 6 9. 77777778 f5 Q5 Q 5 2 2 6 5 22 59 25 2. 6 True errors: 2 6 9.57, 5 59 25.89897. 5 x
(iii) f Q Q Q 6 f 5 Q 5 Q 5 2 29 5.9. 6 5 5 (iv) fxdx Qxdx Q xdx 2 x 8 x2 dx. 75 Exercises:. The following table gives the viscosity of sulfuric acid, in millipascal-seconds (centipoises), as a function of concentration, in mass percent: Concentration (C) 2 6 8 Viscosity (V).8. 2.5 5.7 7. 2.2 a. Using given date, construct a linear spline LC to approximate the viscosity VC. b. Estimate the viscosity by LC when the concentration is 5% and 6% and 92%. 2. Let fx e x.given,e,,,,e, construct a linear spline Lx to approximate fx. a. Approximate f by L and f by L ;and 2 2 2 2 b. Estimate the approximate errors for f L, f L and 2 2 2 2 fx Lx.. Givenx i,y i,,,2, 2, where y i fx i for some fx, construct a quadratic spline Qx to approximate fx if we also know f. a. Approximate f by Q and f by Q. 2 2 2 2 b. Approximate f by Q and f by 2 Q. 2 c. Approximate fxdx by Qxdx. 6