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1 1 Notes for Numerical Analysis Math 54 by S. Adjerid Virginia Polytechnic Institute and State University (A Rough Draft)

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3 Contents 1 Polynomial Interpolation Review Introduction Lagrange Interpolation Interpolation error and convergence Interpolation error Convergence Interpolation at Chebyshev points Hermite interpolation Lagrange form of Hermite interpolation polynomials Newton form of Hermite interpolation polynomial Hermite interpolation error Spline Interpolation Piecewise Lagrange interpolation Cubic spline interpolation Convergence of cubic splines B-splines Interpolation in multiple dimensions Least-squares Approximations

4 4 CONTENTS

5 Chapter 1 Polynomial Interpolation 1.1 Review Mean-value theorem: Let f 2 C[a; b] and differentiable on (a; b) then there exists c 2 (a; b) such that f 0 (c) = f(b) f(a) : b a Weighted Mean-value theorem: If f 2 C[a; b] and g(x) > 0 on [a; b], then there exists c 2 (a; b) such that Z b a f(x)g(x)dx = f(c) Z b a g(x)dx: Rolle's theorem: If f 2 C[a; b], differentiable on (a; b) and f(a) = f(b), then there exists c 2 (a; b) such that f 0 (c) =0. Generalized Rolle's theorem: If f 2 C[a; b], n + 1 times differentiable on (a; b) and admits n +2 zeros in [a; b], then there exists c 2 (a; b) such that f (n+1) (c) =0. Intermediate value theorem: If f 2 C[a; b] such that f(a) = f(b), then for each y between f(a) and f(b) there exists c 2 (a; b) such that f(c) =y. 5

6 CHAPTER 1. POLYNOMIAL INTERPOLATION 1.2 Introduction From the Webster dictionary the definition of interpolation reads as follows: Interpolation is the act of introducing something, especially, spurious and foreign, the act of calculating values of functions between values already known" Our goal is approximate a set of data points or a function by a simpler polynomial function. Given a set of data points x i ; i = 0; 1; n; x i = x j ; if i = j we would like toconstruct a polynomial p m (x) such that p (k) m (x i )=f (k) (x i ); i =0; 1; m; k =0; 1; n i ; with n = mx i=0 n i 1: Lagrange Interpolation: n i =0,m 1,p n (x i )=f(x i ); i =0; 1; ;n Taylor Interpolation: n 0 > 1, m =0,p (k) n (x 0 )=f (k) (x 0 ); k =0; 1; ;n 0. Hermite Interpolation: n i 1, m 1, p (k) n (x i )=f (k) (x i ); i =0; 1; ;m; k =0; 1; n i. Why Interpolation? For instance interpolation is used to approximate integrals derivatives Z b a f(x)dx ß Z b a p m (x)dx f (k) (x) ß p (k) m (x) and plays a major role in approximating differential equations. Taylor interpolation: We first study Taylor polynomials defined as p n (x) =f(x 0 )+(x x 0 )f 0 (x 0 )+ (x x 0) 2 f (2) (x 0 )+ + (x x 0) n f (n) (x 0 ): 2 n!

7 1.3. LAGRANGE INTERPOLATION 7 The interpolation error or remainder formula in Taylor expansions is written as f(x) p n (x) = (x x 0) n+1 f (n+1) (ο) (n + 1)! Example: f(x) =sin(x); x 0 =0 p 2n+1 (x) = nx k=0 The interpolation error can be written as jsin(x) p 2n+1 (x)j = ( 1) k x 2k+1 (2k + 1)! jxj2n+3 jxj2n+3 jcos(ο)j < (2n + 3)! (2n + 3)! On the interval 0 <x<1=2 the interpolation error is bounded as jsin(x) p 2n+1 (x)j < 1 2 2n+3 (2n + 3)! : 1.3 Lagrange Interpolation Lagrange form: Given a set of points (x i ;f(x i )); i = 0; 1; 2; n; x j = x i we define the Lagrange coefficient polynomials l i (x); i =0; 1; n: such as ( 1; if i = j l i (x j )= 0; otherwise and is defined as l i (x) = (x x 0)(x x 1 ) (x x i 1 )(x x i+1 ) (x x n ) (x i x 0 )(x i x 1 ) (x i x i 1 )(x i x i+1 ) (x i x n ) : The Lagrange form of the interpolation polynomial is p n (x) = nx i=0 f(x i )l i (x)

8 8 CHAPTER 1. POLYNOMIAL INTERPOLATION Example: Let us consider the data set x = [0; 1; 2]; f =[ 2; 1; 2] l 0 (x) = (x 1)(x 2) ( 1)( 2) =(x 2 3x +2)=2 l 1 (x) = l 2 (x) = x(x 2) (1)( 1) = x2 +2x x(x 1) (2)(1) =(x 2 x)=2 p 2 (x) = 2l 0 (x) l 1 (x)+2l 2 (x) =x 2 2 Example: f(x) = cos(x) 5 using 8 points x = [0; 1; 2; 3; 4; 5; ; 7] and 14 points x i = i Λ 0:5;i=0; 2; 14 A Matlab example x=[ ]; y=cos(x).^5; c = polyfit(x,y,length(x)-1); xi = 0:0.1:7; zi = cos(xi).^5; yi =polyval(c,xi); subplot(2,1,1) title('interpolation') plot(xi,yi,'-.',xi,zi,x,y,'*'); subplot(2,1,2) title('interpolation Error') plot(xi,zi-yi,x,zeros(1,length(x)),'-*'); Newton form and divided differences: We develop a procedure to compute a 0 ; a 1 ; ; a n such that the interpolation polynomial has the form p n (x) =a 0 + a 1 (x x 0 )+a 2 (x x 0 )(x x 1 )+ + a n (x x 0 ) (x x n 1 )

9 1.3. LAGRANGE INTERPOLATION 9 where a 0 = f(x 0 ), a 1 = f (x 1) f (x 0 ) x 1 x 0 and such that a k = f[x 0 ;x 1 ; ;x k ] is called the k th divided difference. All divided differences are generated by the following recurrence formula f[x i ]=f(x i ) f[x i ;x k ]= f[x k] f[x i ] x k x i f[x 0 ;x 1 ; ;x k 1 ;x k ]= f[x 1; ;x k ] f[x 0 ; ;x k 1 ] x k x 0 x i f(x i ) 1 st DD 2 nd DD 3 rd DD 4 th DD x 0 f(x 0 ) f[x 0 ;x 1 ] x 1 f(x 1 ) f[x 0 ;x 1 ;x 2 ] f[x 1 ;x 2 ] f[x 0 ;x 1 ;x 2 ;x 3 ] x 2 f(x 2 ) f[x 1 ;x 2 ;x 3 ] f[x 0 ;x 1 ;x 2 ;x 3 ;x 4 ] f[x 2 ;x 3 ] f[x 1 ;x 2 ;x 3 ;x 4 ] x 3 f(x 3 ) f[x 2 ;x 3 ;x 4 ] f[x 3 ;x 4 ] x 4 f(x 4 ) The forward Newton polynomial can be written as p 4 (x) =f(x 0 )+f[x 0 ;x 1 ](x x 0 )+f[x 0 ;x 1 ;x 2 ](x x 0 )(x x 1 )+ f[x 0 ;x 1 ;x 2 ;x 3 ](x x 0 )(x x 1 )(x x 2 ) +f[x 0 ;x 1 ;x 2 ;x 3 ;x 4 ](x x 0 )(x x 1 )(x x 2 )(x x 3 ) The backward Newton polynomial can be written as p 4 (x) =f(x 4 )+f[x 3 ;x 4 ](x x 4 )+f[x 2 ;x 3 ;x 4 ](x x 4 )(x x 3 )+ f[x 1 ;x 2 ;x 3 ;x 4 ](x x 4 )(x x 3 )(x x 2 ) +f[x 0 ;x 1 ;x 2 ;x 3 ;x 4 ](x x 4 )(x x 3 )(x x 2 )(x x 1 )

10 10 CHAPTER 1. POLYNOMIAL INTERPOLATION Example: x i f(x i ) 1 st DD 2 nd DD 3 rd DD 4 th DD 5 th DD = = / = 24 40= The forward Newton polynomial p 5 (x) =1 8(x +2)+2(x + 2)(x +1) 10 3 (x + 2)(x +1)x (x + 2)(x +1)x(x 1) (x + 2)(x +1)x(x 1)(x 2): The Backward Newton polynomial is given by p 5 (x) = 4 12(x 3)18(x 3)(x 2) 40 (x 3)(x 2)(x 1) Remarks: 11 (x 3)(x 2)(x 1)x (x 3)(x 2)(x 1)x(x 1): (i) The upper diagonal contains the coefficients for the forward Newton polynomials. (ii) The lower diagonal contains the coefficients for the backward Newton polynomials. (iii) p k (x) interpolates f at x 0 ;x 1 ; ;x k and is obtained as

11 1.4. INTERPOLATION ERROR AND CONVERGENCE 11 p k (x) =f(x 0 )+ kx j=1 (iv) p 2 (x) that interpolates f at x 2 ;x 3 and x 4 is Y j 1 f[x 0 ;x 1 ; ;x j ] (x x i ): p 2 (x) =f(x 2 )+f[x 2 ;x 3 ](x x 2 )+f[x 2 ;x 3 ;x 4 ](x x 2 )(x x 3 ): i=0 (v) If we decide to add an additional point, it should be added at the bottom for forward Newton polynomials and at the top for backward Newton polynomials. (vi) f[x 0 ;x 1 ; ;x k ]= f (k) (ο) k! ; ο 2 [a; b]: Nested mutliplication An efficient algorithm to evaluate Newton polynomial can be obtained by writing p n (x) =a 1 +(a 2 + (a n 2 +(a n 1 +(a n + a n+1 (x x n )) Matlab program (x x n 1 ))(x x n 2 ) )(x x 1 )): %input a(i), i=1,2,...,n+1, x(i),i=1,...,n+1, and x % p = a(n+1); for i=n:-1:1 p = a(i) + p*(x-x(i)); end; %p = p_n(x) 1.4 Interpolation error and convergence In this section we study the interpolation error and convergence of interpolation polynomials to the interpolated function.

12 12 CHAPTER 1. POLYNOMIAL INTERPOLATION Interpolation error Theorem Let f 2 C[a; b] and x 0 ;x 1 ;x 2 ; x n ; be n+1 distinct points in [a; b]. Then there exists a unique polynomial p n of degree at most n such that p n (x i )=f(x i ); i =0; 1; n. Proof. Existence: we define and One can verify that Uniqueness: nq nq j=0;j=i (x x j ) L i (x) = (x i x j ) j=0;j=i ( 1; i=j L i (x j )=ffi ij = 0; otherwise p n (x) = nx i=0 f(x i )L i (x) p n (x j )=f(x j ): Assume there are two polynomials q n (x) and p n (x) such that and consider the difference q n (x j )=p n (x j )=f(x j ); j =0; 1; 2; ;n d n (x) =p n (x) q n (x): d n (x j ) = 0; i = 0; 1; ;n so d n (x) has n +1 roots. By the fundamental theorem of Algebra d n (x) =0.

13 1.4. INTERPOLATION ERROR AND CONVERGENCE 13 Theorem Let f 2 C[a; b] (n + 1) differentiable on (a; b) and let x 0 ; x 1 ; ; x n ; be (n +1) distinct points in [a; b]. If p n (x) is such that p n (x i ) = f(x i ); i = 0; 1; ;n, then for each x 2 [a; b] there exists ο(x) 2 [a; b] such that Q where W (x) = n (x x i ). i=0 f(x) p n (x) = f n+1 (ο(x)) W (x); (n + 1)! Proof. Let x 2 [a; b] and x = x i ; i =0; 1; ;n and define the function g(t) =f(t) p n (t) f(x) p n(x) W (t): W (x) We note that g has (n + 2) roots, i.e., g(x i )=0; i =0; 1; n and g(x) =0. Using the generalized Rolle's Theorem there exits ο(x) 2 (a; b) such that which leads to g (n+1) (ο(x)) = 0 g (n+1) (ο(x)) = f (n+1) (ο(x)) 0 f(x) p n(x) (n + 1)! = 0; (1.1.1) W (x) We solve (1.1.1) to find f(x) p n (x) = f (n+1) (ο(x)) W (x) which completes the (n+1)! proof. Corollary 1. Assume that max jf (n+1) (x)j = M n+1 then x2[a;b] and jf(x) p n (x)j» M n+1 jw (x)j; 8 x 2 [a; b]: (n + 1)! max jf(x) p n (x)j» M n+1 x2[a;b] (n + 1)! max jw (x)j: x2[a;b] Proof. The proof is straight forward.

14 14 CHAPTER 1. POLYNOMIAL INTERPOLATION Examples: jjf P 1 jj 1» M 2h 2 ; [x 0 ; x 0 + h]: 8 jjf p 2 jj 1» M 3h 3 jjf p 3 jj 1» M 4h 4 9 p 3 ; [x 0;x 0 +2h]: 24 ; [x 0;x 0 +3h]: Example: Let us interpolate f(x) =e x 3 on [0; 1] at x 0 ;x 1 ; ;x n. The n +1 derivative isf (n+1) (x) = e x 3 3 n+1 where M n+1 = max jf (n+1) (x)j = e 1 3 x2[0;1] 3 : n+1 The interpolation error can be bounded as jf(x) p n (x)j» e1=3 jw (x)j ; x 2 [0; 1]: 3 n+1 (n + 1)! For instance, for n =4and x =[0; 1=4; 1=2; 3=4; 1], W (x) =x(x 1=4)(x 1=2)(x 3=4)(x 1) The error at x =0:3 can be bounded as jf(0:3) p 4 (0:3)j» e1=3 jw (0:3)j 3 5 5! ß 0: : Example: Let us consider f(x) =cos(x)+x on [0; 2] which satisfies jf (k) (x)j»1. max x2[0;2] The interpolation error can be bounded as Case 1: Two interpolation points with h =2, max jf(x) p 1 (x)j» h2 M 2 x2[0;2] 8» 4=8 =0:5:

15 1.4. INTERPOLATION ERROR AND CONVERGENCE 15 Case 2: Three interpolation points with h =1, max jf(x) p 2 (x)j» h2 M 3 x2[0;2] 9 p 3» 1=(9p 3 ß 0:041: Case 3: Four interpolation points with h = 2=3, max jf(x) p 3 (x)j» h4 M 4 x2[0;2] Convergence» (2=3) 4 =24 ß 0:0082: We start by reviewing the convergence of functions and defining simple and uniform convergence of sequences of functions. Let f n (x); n =0; 1; be a sequence of continuous functions on [a; b]. Definition 1. (Simple convergence): f n (x) converges simply to f(x) if and only if at every x 2 [a; b] lim n!1 jf n (x) f(x)j =0: Definition 2. (Uniform convergence): f n converges uniformly to f if and only if lim max jf n (x) f(x)j =0. n!1 x2[a;b] Example: Let us consider the sequence f n (x) = 1 ;n=0; 1; ; x 2 [0; 1]: 1+nx (i) The sequence f n converges simply to 0 for all 0 <x» 1 while f n (0) = 1. However, f n does not converge uniformly since jjf n jj 1 =1. (ii) The sequence f n converges uniformly to 0 on [2; 3] since jjf n jj 1 =1=(1 + 2n). Next, we will study the uniform convergence of interpolation polynomials on a fixed interval [a; b] as the number of interpolation points approaches

16 1 CHAPTER 1. POLYNOMIAL INTERPOLATION infinity. Let h =(b a)=n and x i = a + ih; i =0; 1; 2; ;n; equidistant interpolation points. Let p n (x) denote the Lagrange interpolation polynomial, i.e., p n (x i )=f(x i ); i =0; ;n and let us study the limit lim jjf p njj 1 = lim max jf(x) p n (x)j: n!1 n!1 x2[a;b] For x 2 [a; b], jx x i j»(b a) which leads to jw (x)j»(b a) n+1. Thus, the interpolation error is bounded as jjf p n jj 1» M n+1 (n + 1)! (b a)n+1 : We have uniform convergence when M n+1 (n+1)! (b a)n+1! 0 as n!1. Theorem Let f be an analytic function on a disk centered at(a+b)=2 with a radius r > 3(b a)=2. Then, the interpolation polynomial p n (x) satisfying p n (x i )=f(x i ); i =0; 1; 2; n; converges to f as n!1, i.e., lim jjf p njj 1 =0: n!1 Proof. A function is analytic at (b+a)=2 if it admits a power series expansion that converges on a disk of radius r and centered at (a + b)=2. Applying Cauchy's formula I f (k) (x) = k! 2ßi C r f(z) dz; x 2 [a; b]: k+1 (z x) I jf (k) (x)j» k! 2ß C r jf(z)j dz; k+1 j(z x)j x 2 [a; b]:

17 y 1.4. INTERPOLATION ERROR AND CONVERGENCE z pole 0.5 * r r 0 d a (a+b)/2 x b 0.5 C r x Let z be a point on the circle C r and x 2 [a; b]. From the triangle with vertices z, (a + b)=2 and x the following triangle inequality holds jz xj + d r: Noting that d» (b a)=2 the triangle inequality yields jf (k) (x)j» k! 2ß jz xj r (b a)=2: Assume r> b a 2 ([a; b] ρ C r ) to obtain M k» max jf(z)j z2c r jr (b a)=2j (k+1) 2ßr: r r (b a)=2 max k! jf(z)j z2c r (r (b a)=2) : k Using k = n + 1 the interpolation error may bebounded as jf(x) p n (x)j» M n+1 (n + 1)! (b a)(n+1)» max jf(z)j z2c r ψ r(b a) r (b a) 2 Finally, wehave uniform convergence if establishes the theorem.! b a r (b a)=2 n b a < 1, i.e., r> 3 (b a) which r (b a)=2 2 :

18 18 CHAPTER 1. POLYNOMIAL INTERPOLATION Examples of analytic functions are sin(z), e z, cos(z 2 ). Runge phenomenon: Let f(x) = 1 4+x 2 is C 1 [ 10; 10] but we do not have uniform convergence on [ 10; 10] when using the n +1equally spaced interpolation points x i = 10 + ih; i =0; 1; ;n, h =20=n. The function f(z) has two poles z = ±2i, thus, can't be analytic on disk C r with radius r > 3(b a)=2 = 30 and center at (a + b)=2. We cannot apply the previous theorem and we should expect convergence problems for x far away from the origin. The largest interval [ a; a] satisfying r > 3(b a)=2 = 3a corresponds to a < 2=3. Actually, it may converge on a larger interval because this is a sufficient condition. Interpolation errors and divided differences: The Newton form of p n (x) that interpolates f at x i ; i =0; 1; ;n is p n (x) =f[x 0 ]+f[x 0 ;x 1 ](x x 0 )+ nx f[x 0 ; ;x i ] i 1 Y i=2 j=0 (x x j ): We proof a theorem relating the interpolation errors and divided differences. Theorem If f 2 C[a; b] and n +1 times differentiable, then for every x 2 [a; b] and f(x) p n (x) =f[x 0 ;x 1 ; ;x n ;x] ny i=0 (x x i ) f[x 0 ;x 1 ;x 2 ; ;x k ]= f (k) (ο) ; ο 2 [ min k! x i; max x i]: i=0; ;k i=0; ;n Proof. Let us introduce another point x distinct from x i ;i = 0; 1; n and let p n+1 interpolate f at x 0 ;x 1 ; x n and x, thus

19 1.5. INTERPOLATION AT CHEBYSHEV POINTS 19 p n+1 (x) =p n (x)+f[x 0 ;x 1 ; ;x n ;x] ny i=0 (x x i ): Combining the equation p n+1 (x) =f(x) and the interpolation error formula we write f(x) p n (x) =f[x 0 ;x 1 ; ;x n ;x] ny i=0 (x x i )= f (n+1) (ο(x)) (n + 1)! ny i=0 (x x i ): This leads to f[x 0 ;x 1 ; ;x k ]= f (k) (ο) ; ο 2 [ min k! x i; max x i]; i=0; ;k i=0; ;k which completes the proof. Remark: lim f[x 0;x 1 ; ;x k ]= f (k) (x 0 ) x i!x 0 ;i=1; k k! 1.5 Interpolation at Chebyshev points In the previous section we have shown that uniform convergence does not occur using uniform interpolation points for some functions. Now, we study the interpolation error on [ 1; 1] where the (n + 1) interpolation points, x Λ i ; i =0; 1; ;n; in [ 1; 1] are selected such that jjw Λ (:)jj 1 = min Q2 ~ P n+1 jjq(:)jj 1 where ~P n is the set of the monic polynomials ~P n =[Q 2P n jq = x n + n 1 X i=1 c i x i ];

20 20 CHAPTER 1. POLYNOMIAL INTERPOLATION Q and W Λ (x) = n (x x Λ i ). i Question: Are there interpolation points x Λ i ; i =0; 1; 2; ;n in [ 1; 1] such that jjw Λ jj 1 = min x i 2[a;b];i=0;1; ;n jjw jj 1 If the above statement is true, the interpolation error can be bounded by jje n jj 1» M n+1 (n+1)! jjw Λ jj 1 The Answer : The best interpolation points x Λ i ; i = 0; 1; 2; ;n are the roots of the Chebyshev polynomial T n+1 (x) defined as T k (x) =cos(karcos(x)); k =0; 1; 2; : In the following theorem we will prove some properties of Chebyshev polynomials. Theorem The Chebyshev polynomials T k (x); k =0; 1; 2;, satisfy the following properties: (i) jt k (x)j»1; for all 1» x» 1 (ii) T k+1 (x) =2xT k (x) T k 1 (iii) T k (x) has k roots x Λ j = cos( 2j+1 ß); j =0; 1; ;k 1; 2 [ 1; 1] 2k Q k 1 (iv) T k (x) =2 k 1 (x x Λ j) j=0 (v) If ~T k (x) = T k(x) then max j ~T 2 k 1 k (x)j = 1. 2 x2[ 1;1] k 1 Proof. We obtain (i) by noting that the range of the cosine function is [ 1; 1]. To obtain (ii) we write T k+1 (x) =cos(karcos(x)+arcos(x)) = cos(k + )

21 1.5. INTERPOLATION AT CHEBYSHEV POINTS 21 where = arcos(x) and write T k+1 = cos(k + ) T k 1 = cos(k ) Use the trigonometric identity cos(a ± b) = cos(a)cos(b) sin(a)sin(b) to obtain and cos(k + ) =cos(k )cos( ) sin(k )sin( ) cos(k ) =cos(k )cos( )+sin(k )sin( ) Adding the previous equations to obtain T k+1 (x)+t k 1 (x) =2cos(k )cos( ) =2xT k (x) This proves (ii). To obtain the roots we set cos(karcos(x)) = 0 which leads to karcos(x) = If we solve for x, we obtain leads to the roots x = cos( x Λ j (2j +1) ß; j =0; ±1; ±2; : 2 (2j +1) ß);j =0; ±1; ±2; ; 2k +1) = cos((2j ß);j =0; 1; ;k 1: 2k Use induction to prove (iv) step 1: T 1 (x) =2 0 x, T 2 (x) =2 1 x 2 1 (iv) is true for k =1; 2. P Step 2: Assume T k = 2 k 1 x k + k 1 c i x i, for k = 1; 2; ;n and use (ii) we write i=0 T n+1 (x) =2xT n T n 1 (x) =2 n x n+1 + nx i=0 a i x i

22 22 CHAPTER 1. POLYNOMIAL INTERPOLATION k 1 This establishes (iv), i.e., T k =2 k 1 (x x Λ j). Q j=0 Applying (iv) we show that (v) is true. Corollary 2. If ~T n (x) is the monic Chebyshev polynomial of degree n, then max j Q ~ n (x)j max j T ~ n (x)j = 1 1»x»1 1»x»1 2 ; 8 Q ~ n 2 P ~ n : n 1 Proof. Assume there is another monic polynomial ~ R n (x) such that We also note that max j ~R n (x)j < 1 1»x»1 2 n 1 ~T n (z k )= ( 1)k 2 n 1 ; z k = cos(kß=n); k =0; 1; 2; ;n: The (n 1)-degree polynomial d n 1 (x) = ~T n (x) ~R n (x) satisfies d n 1 (z 0 ) > 0, d n 1 (z 1 ) < 0, d n 1 (z 2 ) > 0, d n 1 (z 3 ) < 0 So d n 1 (x) changes sign between each pair z k and z k+1 ; k =0; 1; 2; ;n and thus has n roots. Thus d n 1 (x) =0,identically, i.e., ~T (x) and ~R n (x) are identical. This leads to a contradiction with the assumption above. Below are the first five Chebyshev polynomials. T 0 (x) = 1 T 1 (x) = x T 2 (x) = 2x 2 1 T 3 (x) = 4x 3 3x T 4 (x) = 8x 4 8x 2 +1 Example of Chebyshev points:

23 1.5. INTERPOLATION AT CHEBYSHEV POINTS 23 k x Λ 0 x Λ 1 x Λ 2 x Λ 3 1 p 0 p 2 p 2=2 2=2 p 3 3=2 0 3=2 4 cos(ß=8) cos(3ß=8) cos(5ß=8) cos(7ß=8) Application to interpolation: Let p n (x) 2 P n interpolate f(x) 2 C n+1 [ 1; 1] at the roots of T n+1 (x), x Λ j; j = 0; 1; 2; ;n. Thus, we can write the interpolation error formula as f(x) p n (x) = f n+1 (ο(x)) (n + 1)! ~T n+1 (x) Using (v) from the previous theorem and assuming jjf n+1 jj 1;[ 1;1]» M n+1 we obtain Remarks: max jf(x) p n (x)j» M n+1 x2[ 1;1] 2 n (n + 1)! 1. We note that this choice of interpolation points reduces the error significantly. 2. With Chebyshev points, p n converges uniformly to f when f 2 C 1 [ 1; 1] only. The function f does not have to be analytic (see Gautschi). Example 1: Consider f(x) =e x ; x 2 [ 1; 1] Case 1: with three points; n=2: Case 2: with points; n=5: jje 2 jj 1» M 3 = e=24 = 0:113: 2 3!2

24 24 CHAPTER 1. POLYNOMIAL INTERPOLATION jje 5 jj 1» M!2 5 = e=(720 32) = 0: : How many Chebyshev points are needed to have jje n jj 1 < 10 8 jje n jj 1» M n+1 2 n (n + 1)! = e (n + 1)!2 n =0: ; for n=9: Thus, 10 Chebyshev points are needed. Chebyshev points on [a; b]: Chebyshev points can be used on an arbitrary interval [a; b] using the linear transformation x = a + b 2 We also need the inverse mapping t =2 x a b a First, we order the Chebyshev nodes in [ 1; 1] as + b a t; 1» t» 1: (1.1.2a) 2 1; a» x» b: (1.1.2b) t Λ 2k k = cos( ß ß) = cos(2k ß); k =0; 1; 2; ;n: 2n +2 2n +2 we define the interpolation nodes on an arbitrary interval [a; b] as Remarks: x Λ k = a + b 2 + b a 2 tλ k; k =0; 1; 2; ;n 1. x Λ 0 <x Λ 1 < <x Λ n 2. x Λ k are symmetric with respect to the center (a + b)=2 3. x Λ k are independent of the interpolated function f

25 1.5. INTERPOLATION AT CHEBYSHEV POINTS 25 Theorem Let f 2 C n+1 [a; b] and p n interpolate f at the Chebyshev nodes x Λ k ;k =0; 1; 2; ;n, in [a,b]. Then (b a) n+1 max jf(x) p n (x)j»2m n+1 x2[a;b] 4 n+1 (n + 1)! : Where M n+1 = max jf (n+1) (x)j. x2[a;b] Q Proof. It suffices to rewrite W (x) = n (x x i Λ) using the mapping (1.1.2) to find that i=0 (x x Λ i )= b a 2 (t tλ i ); and W (x) = ny i=0 (x x i Λ)=( b a 2 )n+1 ny i=0 (t t Λ i )=( b a 2 )n+1 ~ T n+1 (t): Finally, using jj ~T n+1 jj 1;[ 1;1] = 1 2 n we complete proof. Example 2: Consider f(x) = 3 x = e ln(3)x ; x 2 [0; 1] whose derivative is f (n+1) (x) = ln(3) n+1 e ln(3)x. Noting that f (n+1) is a monotonically increasing function, M n+1 = f (n+1) (1)=3ln(3) n+1. Therefore, jje n jj 1» 2 3 ln(3)n+1 4 n+1 (n + 1)! = ln(3)n+1 4 n+1 (n + 1)! :

26 2 CHAPTER 1. POLYNOMIAL INTERPOLATION # of Chebyshev points Error bound Hermite interpolation We restate the general Hermite interpolation by Letting x 0 <x 1 <x 2 x m be m + 1 distinct points such that f (k) (x i )=p (k) n (x i ); k =0; 1; n i 1; i =0; 1; m; (1.1.3) P where m n i = n + 1 and n i 1. We note that n i =1; i =0; ;m; leads to i=0 Lagrange interpolation Lagrange form of Hermite interpolation polynomials Theorem There exists a unique polynomial p n (x) that satisfies (1.1.3) with n i =2and n =2m +1. Proof. Existence: Next, we study the special case n i =2; i =0; 1; 2; ; where l i;1 (x) =(x x i )l i (x) 2 : (1.1.4)

27 1.. HERMITE INTERPOLATION 27 and l i;0 = l i (x) 2 2l 0 i(x i )(x x i )l i (x) 2 : (1.1.5) Now, we can verify that l i;1 (x j )=0; j =0; 1; 2; ;m Thus, l 0 i;1(x j )=ffi ij. l 0 i;1(x) =2(x x i )l 0 i(x)l i (x)+l i (x) 2 : One can easily check that l i;0 (x j )=ffi ij. For l 0 i;0(x) we have l 0 i;0(x) =(1 2l 0 i(x i )(x x i ))2l 0 i(x)l i (x) 2l 0 i(x i )l i (x) 2 : Thus, l 0 i;0(x j )=0; j =0; 1; ;m: Existence of Hermite interpolation polynomial is established by writing the Lagrange form of Hermite polynomial as p n (x) = mx i=0 f(x i )l i;0 (x)+ mx i=0 f 0 (x i )l i;1 (x): (1.1.) Uniqueness: Assume there are two polynomials p n (x) and q n (x) that satisfy (1.1.3) and consider the difference d n (x) =p n (x) q n (x) which satisfies d (s) n (x j )=0; s =0; 1; j =0; 1; ;m: Thus, d n (x) is a polynomial of degree at most n and has (n+1) roots counting the multiplicity of each root. The fundamental theorem of Algebra shows that d n (x) is identically zero. With this we establish the uniqueness of p n and finish the proof of the theorem.

28 28 CHAPTER 1. POLYNOMIAL INTERPOLATION 1..2 Newton form of Hermite interpolation polynomial Using the following relation f[x 0 ;x 0 + h; x 0 +2h; ;x 0 + kh] = f (k) (ο) ; x 0 <ο<x 0 + kh (1.1.7) k! and taking the limit when h! 0we obtain that f[x 0 ;x 0 ;x 0 ; ;x 0 ]= f k (x 0 ) : (1.1.8) k! The divided difference table for the data (x k + ih; f(x 0 + kh)); i =0; 1; 2; 3; 4 will converge to the table shown below where we recover the Taylor polynomial about x k and with n k =5. Using this observation we initialize the table for x k and n k =5asfollows: (i) every point x i is repeated n i times (ii) we set z i = x k ; z i+1 = x k ; ; z i+4 = x k (iii) we initialize f[z i ;z i+1 ; ; z i+s ]= f (s) (x k ) s! as shown in the following table z i x k f(x k ) z i+1 x k f(x k ) f 0 (x k ) z i+2 x k f(x k ) f 0 (x k ) f 00 (x k )=2! z i+3 x k f(x k ) f 0 (x k ) f 00 (x k )=2! f 000 (x k )=3! z i+4 x k f(x k ) f 0 (x k ) f 00 (x k )=2! f 000 (x k )=3! f (4) (x k )=4! The general formula for divided differences with repeated arguments for x 0» x 1» :::<x n is given by f[x i ;x i+1 ;::: ;x i+k ]= ( f [xi+1 ;x i+2 ;::: ;x i+k ] f [x i ;::: ;x k 1 ] f (k) (ο) k! x i+k x i ; if x i = x i+k Example: Let f(x) = x 4 +1, f 0 (x) = 4x 3. We will construct a polynomial p 5 (x) such that p(x i ) = f(x i ) and p 0 (x i ) = f 0 (x i ) with x i = 1; 0; 1. The Hermite divided difference table is given as

29 1.. HERMITE INTERPOLATION 29 z i f(z i ) 1DD 2 DD 3 DD 4 DD 5DD z z f 0 ( 1) = 4 z z f 0 (0) = z z f 0 (1) = The forward Hermite polynomial is given as p 5 (x) =2 4(x +1)+3(x +1) 2 2(x +1) 2 x +(x +1) 2 x 2 =1+x 4 (1.1.9) The Hermite polynomial that interpolates f and f 0 at x = 1; 0isgiven as p 3 (x) =2 4(x +1)+3(x +1) 2 2(x +1) 2 x: (1.1.10) The backward Hermite polynomial is given as p 5 (x) =2 4(x 1)+3(x 1) 2 +2(x 1) 2 x +(x 1) 2 x 2 =1+x 4 (1.1.11) The Hermite polynomial that interpolates f and f 0 at x =0; 1 is given by p 3 (x)=2+4(x 1) + 3(x 1) 2 +2(x 1) 2 x: (1.1.12) Example: Consider the data with m =1; n 0 =1; n 1 = 2 given in the following table x i 0 1 f(x i ) 1 2 f 0 (x i ) 0 1 f 00 (x i ) NA 2 We write the divided differences table as

30 30 CHAPTER 1. POLYNOMIAL INTERPOLATION z I f(z i ) 1DD 2DD 3DD 4DD z z f 0 (x 0 )=0 z z f 0 (x 1 )=1 0-1 z f 0 (x 1 )=1 f 00 (x 1 )=2!=1 1 2 The Hermite polynomial is given as H 4 (x)=1+0(x 0)+(x 0) 2 (x 0) 2 (x 1) + 2(x 0) 2 (x 1) 2 =1+2x 2 x 3 +2x 2 (x 1) 2 : (1.1.13) 1..3 Hermite interpolation error Theorem Let f(x) 2 C[a; b] be 2m +2 differentiable on (a; b) and consider x 0 <x 1 <x 2 ; ;x m in [a; b] with n i =2; i =0; 1; ;m. If p 2m+1 is the Hermite polynomial such that p (k) 2m+1(x i ) = f (k) (x i ), i = 0; 1; ; m, k =0; 1, then there exists ο(x) 2 [a; b] such that f(x) p 2m+1 (x) = f (2m+2) (ο(x)) W (x) (2m + 2)! (1.1.14a) where W (x) = my i=0 (x x i ) 2 : (1.1.14b) Proof. We consider the special case n i = 2; i.e., n = 2m +1, select an arbitrary point x 2 [a; b]; x = x i ; i =0; ;m and define the function g(t) =f(t) p 2m+1 (t) f(x) p 2m+1(x) W (t): (1.1.15) W (x)

31 1.. HERMITE INTERPOLATION 31 We note that g has (m+2) roots, i.e., g(x i )=0; i =0; 1; m and g(x) =0. Applying the generalized Rolle's Theorem we show that g 0 (ο i )=0;i=0; 1; ;m where ο i 2 [a; b], ο i = x j, ο i = x. Using (1.1.3) with n i = 2 we have g 0 (x i ) = 0; i = 0; 1; ;m. Thus, g 0 (t) has 2m +2roots in [a; b]. Applying the generalized Rolle's theorem we show that there exists ο 2 (a; b) such that g (2m+2) (ο) =0: Combining this with (1.1.15) yields 0=f (2m+2) (ο) f(x) p 2m+1(x) (2m + 2)! W (x) Solving for f(x) p 2m+1 (x) leads to (1.1.14). Corollary 3. If f(x) and p 2m+1 (x) are as in the previous theorem, then and jf(x) p 2m+1 (x)j < M 2m+2 jw (x)j; x 2 [a; b] (1.1.1) (2m + 2)! jjf(x) p 2m+1 (x)jj 1;[a;b]» M 2m+2 (2m + 2)! (b a)2m+2 : (1.1.17) Proof. The proof is straight forward. At this point wewould like to note that we can prove a uniform convergence result under the same conditions as for Lagrange interpolation. Example: Let f(x) =sin(x); x 2 [0;ß=2] and p 5 (x) interpolate f and f 0 at x i =0; 0:2; ß 2, with n i = 2 and 2m +2=

32 32 CHAPTER 1. POLYNOMIAL INTERPOLATION Using the error bound with M 2m+2 =1,n i = 2 and W (x) =[(x 0)(x 0:2)(x ß 2 )]2 ; we obtain je 5 (1:1)j» jw (1:1)j! ß 0:4082! ß 3: : (1.1.18) 1.7 Spline Interpolation In this section we will study piecewise polynomial interpolation and write the interpolation errors in terms of the subdivision size and the degree of polynomials Piecewise Lagrange interpolation We construct the piecewise linear interpolation for the data (x i ;f(x i )); i = 0; 1; ;n such that x 0 <x 1 < <x n as P 1 (x) = 8 >< >: p 1;0 (x) =f(x 0 ) (x x 1) x 0 x 1 + f(x 1 ) (x x 0) x 1 x 0 ; x 2 [x 0 ;x 1 ]; p 1;i (x) =f(x i ) (x x i+1) x i x i+1 + f(x i+1 ) (x x i) x i+1 x i ; x 2 [x i ;x i+1 ]: i =0; 1; ;n 1: (1.1.19) The interpolation error on (x i ;x i+1 ) is bounded as where M 2;i = jje 1;i (x)jj» M 2;i 2 j(x x i)(x x i+1 )j; x 2 (x i ;x i+1 ); (1.1.20) max jf (2) (x)j. This can be written as x i»x»x i+1 The global error is jje 1;i jj 1» M 2;ih 2 i 8 ; h i = x i+1 x i : (1.1.21)

33 1.7. SPLINE INTERPOLATION 33 jje 1 jj 1» M 2H 2 8 ; H = max h i : (1.1.22) i=0; ;n 1 Theorem Let f 2 C 0 [a; b] be twice differentiable on (a; b). If P 1 (x) is the piecewise linear interpolant of f at x i = a + i Λ h; i = 0; 1; ;n, h =(b a)=n, then P 1 converges uniformly to f as n!1. Proof. We prove this theorem using the error estimate (1.1.22). For piecewise quadratic interpolation we select h =(b a)=n and construct p 2;i that interpolates f at x i, (x i + x i+1 )=2 and x i+1. In this case the interpolation error is bounded as A global bound is jje 2;i jj 1» M 3;i(h i =2) 3 9 p ; h i = x i+1 x i : (1.1.23) 3 jje 2 jj 1» M 3(H=2) 3 9 p 3 ; H = max h i : (1.1.24) i=0; ;2n 1 Theorem Let f 2 C 0 [a; b] and be m+1 times differentiable on (a; b) and x 0 < x 1 < < x n with h i = x i+1 x i and H = max h i. If P m (x) is i the piecewise polynomial of degree m on each subinterval [x i ;x i+1 ] and P m (x) interpolates f on [x i ;x i+1 ] at x i;k = x i + k Λ h ~ i ; k =0; 1; ;m, h ~ i = h i =m, then P m converges uniformly to f as H! 0. Proof. Again we prove this theorem using the error bound jje m jj 1» M m+1h m+1 ; H = max h i : (1.1.25) (m + 1)! i=0;1; ;nm 1 Similarly,we may construct piecewise Hermite interpolation polynomials following the same line of reasoning as for Lagrange interpolation.

34 34 CHAPTER 1. POLYNOMIAL INTERPOLATION Cubic spline interpolation We use piecewise polynomials of degree three that are C 2 and interpolate the data such as S(x k )=f(x k )=y k ; k =0; 1; n. Algorithm (i) Order the points x k ; x =0; 1; n such that a = x 0 <x 1 <x 2 < x n 1 <x n = b: (ii) Let S(x) be a piecewise spline defined by n cubic polynomials such that S(x) =S k (x) =a k + b k (x x k )+c k (x x k ) 2 + d k (x x k ) 3 ; x k» x» x k+1 (iii) find a k ; b k ; c k ; d k ; k =0; 1; n 1 such that (1) S(x k )=y k ; k =0; 1; ;n (2) S k (x k+1 )=S k+1 (x k+1 ); k =0; ;n 2 (3) S 0 k (x k+1) =S 0 k+1 (x k+1); k =0; ;n 2 (4) S 00 k (x k+1) =S 00 k+1 (x k+1); k =0; ;n 2 Theorem If A is an n n strictly diagonally dominant matrix, i.e., ja kk j > n P i=1;i=k ja k;i j; k =1; 2; n, then A is nonsingular. Proof. By contradiction, we assume that A is singular, i.e., there exists a nonzero vector x such that Ax = 0 and let x k such that jx k j = maxjx i j. This leads to a kk x k = nx i=1;i=k a ki x i : (1.1.2) Taking the absolute value and using the triangle inequality we obtain ja kk jjx k j» nx i=1;i=k ja ki jjx i j: (1.1.27)

35 1.7. SPLINE INTERPOLATION 35 Dividing both terms by jx k j we get ja kk j» nx i=1;i=k ja ki j jx ij jx k j» nx i=1;i=k ja ki j (1.1.28) This leads to a contradiction since A is strictly diagonally dominant. Theorem Let us consider the set of data points (x i ;f(x i )); i = 0; 1; ;n; such that x 0 < x 1 < ;x n. If S 00 (x 0 ) = S 00 (x n ) = 0, then there exists a unique piecewise cubic polynomial that satisfies the conditions (iii). Proof. Existence: we assume S " (x k ) = m k where h k = x k+1 x k and use piecewise linear interpolation of S " to write x x S 00 k+1 x x k k (x) =m k + m k+1 x k x k+1 x k+1 x k = m k h k (x x k+1 )+ m k+1 h k (x x k ); x k» x» x k+1 With this definition of S ", condition (4) is automatically satisfied. Integrating S 00 k (x) we obtain S k (x) = m k h k (x x k+1 ) 3 + m k+1 h k (x x k ) 3 + p k (x k+1 x)+q k (x x k ) Need to find m k ; q k and p k ; k =0; 1; 2; ;n 1: In order to enforce the conditions (1) and (2) we write S k (x k )=y k = m k h2 k + p k h k S k (x k+1 )=y k+1 = m k+1 h2 k + q k h k Solve for p k and q k to solve p k = y k m kh k h k (1.1.29a)

36 3 CHAPTER 1. POLYNOMIAL INTERPOLATION q k = y k+1 h k m k+1h k : (1.1.29b) We note that if m k ; k =0; 1;::: ;n are known, The previous equations may be used to compute p k and q k. Now, substitute p k and q k in the equation for S k to have S k (x) = m k (x x k+1 ) 3 + m k+1 (x x k ) 3 +( y k m kh k )(x k+1 x)+ h k h k h k ( y k+1 h k m k+1h k )(x x k ) (1.1.30) Applying condition (3) to enforce the continuity of S 0 (x) where S 0 k(x k+1 )=S 0 k+1(x k+1 ); k =0; 1; ;n 1: (1.1.31) and S 0 k(x) = m k 2h k (x x k+1 ) 2 + m k+1 2h k (x x k ) 2 ( y k m kh k )+( y k+1 h k h k m k+1h k ); x k» x» x k+1 ; (1.1.32) S 0 k+1(x) = m k+1 2h k+1 (x x k+2 ) 2 + m k+2 2h k+1 (x x k+1 ) 2 ( y k+1 m k+1h k+1 )+( y k+2 m k+2h k+1 ); x k+1» x» x k+2 : (1.1.33) h k+1 h k+1 Taking the limit from the left at x k+1 leads to S 0 k(x k+1 )= m k+1h k 3 + m kh k + d k : (1.1.34) Taking the limit from the right atx k+1 yields

37 1.7. SPLINE INTERPOLATION 37 where S 0 k+1(x k+1 )= m k+1h k+1 3 m k+2h k+1 d k = y k+1 y k h k ; k =0; 1; ;n 1 + d k+1 Using (1.1.31) we obtain the following system having n + 1 unknowns and n 1 equations. (I) ( m k h k +2m k+1 (h k + h k+1 )+m k+2 h k+1 =(d k+1 d k ); k =0; 1; 2; ;n 2: (1.1.35) Nowwe need to close the system by adding two more equations from S 00 (x 0 )= 0 and S 00 (x n ) = 0 which leads to This is called the natural spline. The system (1.1.35) and (1.1.3) lead to (I:NAT) 8 >< >: In matrix form we write 2 4 m 0 =0; m n =0: (1.1.3) (2h 0 +2h 1 )m 1 + h 1 m 2 = u 0 m k h k +2m k+1 (h k + h k+1 )+m k+1 h k+1 = u k ; 1» k» n 3 h n 2 m n 2 +2(h n 2 + h n 1 )m n 1 = u n 2 2(h 0 + h 1 ) h h 1 2(h 1 + h 2 ) h h 2 2(h 2 + h 3 ) h n h n 2 2(h n 2 + h n 1 ) m 1 m 2 m 3. m n = The resulting matrix is strictly symmetric positive definite and diagonally dominant and yields a unique solution. 2 4 u 0 u 1 u 2. u n

38 38 CHAPTER 1. POLYNOMIAL INTERPOLATION Other splines include: Not-a-Knot Spline: We add the following two conditions: S (x 1 )=S (x 1 ) which leads to and the condition S 000 n 2(x n 1 )=S 000 n 1(x n 1 ) which leads to m 1 m 0 h 0 = m 2 m 1 h 1 (1.1.37) m n m n 1 h n 1 = m n 1 m n 2 h n 2 (1.1.38) Solve (1.1.37) and (1.1.38) for m 0 and m n to obtain m 0 =(1+h 0 =h 1 )m 1 (h 0 =h 1 )m 2 (1.1.39) m n = h n =h n1 m n 2 +(1+h n =h n 1 )m n 1 (1.1.40) Substitute into the system (1.1.35) to obtain (I:NK) 8 >< >: (3h 0 +2h 1 + h 2 0=h 1 )m 1 +(h 1 h 2 0=h 1 )m 2 = u 0 m k h k +2m k+1 (h k + h k+1 )+m k+1 h k+1 = u k ; k =1; ;n 3 (h n 2 h 2 n 1=h n 2 )m n 2 +(2h n 2 +3h n 1 + h 2 n 1=h n 2 )m n 1 = u n 2 where u k = (d k+1 d k ); k = 0; 1; ;n 2. We solve the system for m 1 ;m 2 ; m n 1 and use (1.1.39) and (1.1.40) to find m 0 and m n. Use (1.1.29) to find p k and q k ;k =0; 1; n 1. Finally, we use the formula (1.1.30) that defines S k (x).

39 1.7. SPLINE INTERPOLATION 39 Clamped Spline: We close the system (I) using the following conditions S0(x 0 0 )=f 0 (x 0 ) Sn 1(x 0 n )=f 0 (x n ) S 0 0(x) = m 0 2h 0 (x x 1 ) 2 + m 1 2h 0 (x x 0 ) 2 ( y 0 h 0 m 0h 0 )+(y 1 h 0 m 1h 0 ) S 0 0(x 0 )= m 0h 0 3 m 1h 0 The boundary equation S 0 0(x 0 )=f 0 (x 0 ) leads to + y 1 y 0 h 0 : 2m 0 h 0 + m 1 h 0 =(d 0 f 0 (x 0 )) (1.1.41) The boundary condition S 0 n 1(x n )=f 0 (x n ) yields the equation S 0 n 1(x n )= m nh n m n 1h n 1 + d n 1 = f 0 (x n ); which leads Now, the system (I) is reduced to (I:CL) 8 >< >: ( 3h 0 2m n h n 1 + m n 1 h n 1 =(f 0 (x n ) d n 1 ) (1.1.42) +2h 2 1 )m 1 + h 1 m 2 = u 0 3(d 0 f 0 (x 0 )) m k h k +2m k+1 (h k + h k+1 )+m k+1 h k+1 = u k ; k =1; 2; ;n 3 h n 2 m n 2 +(2h n 2 + 3h n 1 )m 2 n 1 = u n 2 3(f 0 (x n ) d n 1 ) Matrix formulation for the not-a-knot spline AM = U

40 40 CHAPTER 1. POLYNOMIAL INTERPOLATION where A = 2 4 3h 0 +2h 1 + h2 0 h 1 h 1 h2 0 h h 1 2(h 1 + h 2 ) h h 2 2(h 2 + h 3 ) h n h n 2 h2 n 1 h n 2 2h n 2 +3h n 1 + h2 n 1 h n 2 M = 2 4 m 1 m 2 m ; B = 2 4 u 0 u 1 u m n 1 u n 2 The system admits a unique solution since the matrix is strictly diagonally dominant. Example 1: consider the function f(x) =x=(2 + x) at 1; 1; 2; 3 x i f(x i ) 1 st DD *(2nd Diff) 1 1 2=3 1 1=3 3 1= 2 1=2 2=5 1=10 3 3=5 h 0 =2,h 1 =1,h 2 =1.» 3h0 +2h 1 + h 2 0=h 1 h 1 h 2 0=h 1» m1 h 1 h 2 2=h 1 2h 1 +3h 2 + h 2 2=h 1 )»» 12 3 m1 = 0 m 2» 3 2=5 The solution is m 1 = 4=15, m 2 = 1=15 = m 2» u0 u 1

41 1.7. SPLINE INTERPOLATION 41 Matrix formulation for the clamped spline 2 4 ( 3h 0 +2h 2 1 ) h h 1 2(h 1 + h 2 ) h h 2 2(h 2 + h 3 ) h n h n 2 (2h n 2 + 3h n 1 = 2 4 u 0 3(d 0 f 0 (x 0 )) u 1 u 2. u n 2 3(f 0 (x n ) d n 1 ) ) m 1 m 2 m 3. m n 1 The system admits a unique solution since the matrix is symmetric diagonally dominant and positive definite (SPD). Example 1: Let us interpolate the function f(x) = x=(2 + x) where f 0 (x) =2=(2 + x) 2. Thus S 0 0(x 0 )=2=f 0 ( 1) and S 0 n 1(x 3 )=2=25 = f 0 (3). To compute the u i ; i =0; 1we use the following table x i f(x i ) 1 st DD *(2nd Diff) 1 1 2=3 1 1=3 3 1= 2 1=2 2=5 1=10 3 3=5 h 0 =2,h 1 =1,h 2 =1.»» 3h0 =2+2h 1 h 1 m1 = h 1 2h 1 +3h 2 =2) m 2» u0 3(d 0 f 0 (x 0 )) u 1 3(f 0 (x 3 ) d 2 ) 3 7 5

42 42 CHAPTER 1. POLYNOMIAL INTERPOLATION»»» 5 1 m1 1 = 1 7=2 m 2 17=50 The solution is m 1 =4=275, m 2 = 9=55 m 0 =3 f [x 0;x 1 ] f 0 (x 0 ) h 0 m 1 =2= 582=275 m 3 =3 f 0 (x 3 ) f [x 2 ;x 3 ] h 2 m 2 =2==275 On [-1,1] : p 0 = y 0 =h 0 (m 0 h 0 )= = 113=550, q 0 = y 1 =h 0 (m 1 h 0 )= =49=550 On [1; 2] S 0 (x) = (x 4 1)3 + (x +1) p 1 = y 1 =h 1 (m 1 h 1 )= =81=275, q 1 = y 2 =h 1 (m 2 h 1 )= =29= (1 x)+ (x +1) S 1 (x) = (x 2)3 9 (x 1) (2 x)= (x 1)=55 On [2,3] p 2 = y 2 =h 2 (m 2 h 2 )= =29=55, q 2 = y 3 =h 2 (m 3 h 2 )= = 14=275 S 2 (x) = 9 55 (x 3)3 1 (x 2) (3 x)= (x 2)=275 Examples of natural spline approximations

43 1.7. SPLINE INTERPOLATION 43 Example 1: Let f(x) = jxj and construct the cubic spline interpolation at the points x i = 2+i; i =0; 1; 2; 3; 4 x i f(x i ) 1 st DD *(2nd Diff) We note that h i = h; i =0; 1; 2; 3, m 0 = m 4 = 0 which leads to the following system m 1 m 2 m = The solution is m 1 = =7, m 2 =24=7, m 3 = =7. On [ 2; 1] p 0 =(y 0 m 0h 2 0 )=h 0 =(2 0)=1 =2q 0 =(y 1 m 1h 2 0 )=h 0 =(1+1=7)=8=7 S 0 (x) = 1 7 (x +2)3 +2( 1 x)+ 8 (x +2) 7 On [ 1; 0] p 1 =(y 1 m 1h 2 1 )=h 1 =(2 0)=1 =8=7 q 1 =(y 2 m 2h 2 1 )=h 1 =(1+1=7) = 4=7 S 1 (x) = 1 7 x (x +1) (0 x) 4 (x +1) 7

44 44 CHAPTER 1. POLYNOMIAL INTERPOLATION On [0; 1] p 2 =(y 2 m 2h 2 2 )=h 2 =( )= 4=7 q 2 =(y 3 m 3h 2 2 )=h 2 =( )=8=7 S 2 (x) = 4 7 (x 1)3 x 3 =7 4 7 (1 x)+8 7 x On [1; 2] p 3 =(y 3 m 3h 2 3 )=h 3 =8=7, q 3 =(y 4 m 4h 2 3 )=h 3 =2 S 3 (x) = (2 x) 3 =7+2(x 1) + 8(2 x)=7 p =(2; 8=7; 4=7; 8=7), q =(8=7; 4=7; 8=7; 2) Example 2: x i f(x i ) 1 st DD *(2nd Diff) 1 1 2=3 1 1=3 3 1= 2 1=2 2=5 1=10 3 3=5 h 0 =2,h 1 =1,h 2 =1. Natural cubic spline leads to m 0 = m 3 =0. The other coefficients satisfy the system»»» 2(h0 + h 1 ) h 1 m1 3 = h 1 2(h 1 + h 2 ) m 2 2=5»» 1 m1 = 1 4 m 2» 3 2=5

45 1.7. SPLINE INTERPOLATION 45 which admits the following solution m 1 = , m 2 = On [ 1; 1] p 0 =(y 0 m 0h 2 0 )=h 0 = 1=2, q 0 =(y 1 m 1h 2 0 )=h 0 =77=230 On [1; 2] p 1 =(y 1 m 1h 2 1 )=h 1 =48=115, q 1 =(y 2 m 2h 2 1 )=h 1 =57=115 On [2; 3] p 2 =(y 2 m 2h 2 2 )=h 2 =57=115, q 2 =(y 3 m 3h 2 2 )=h 2 =3=5 p =[ 1=2; 48=115; 57=115], q = [77=230; 57=115; 3=5] Matlab commands for splines x=0:1:10; y=sin(x); xi=0:0.2:10; yi = sin(xi); %piecewise linear interpolation y1 = interp1(x,y,xi) plot(x,y,'0',xi,yi) %plot the exact function hold on; plot(xi,y1); % y2 = interp1(x,y,xi,'spline') % spline interpolation y3 = interp1(x,y,'cubic') % piecewise cubic interpolation y4 = spline(x,y,xi) %not-a-knot spline plot(xi,y4); plot(xi,y2); plot(xi,y3);

46 4 CHAPTER 1. POLYNOMIAL INTERPOLATION Convergence of cubic splines We will study the uniform convergence of the clamped cubic spline for f 2 C 4 [a; b]. We first write the matrix formulation for the clamped cubic spline as AM = B where M =[m 0 ;m 1 ; ;m n ] t, B =[b 0 ;b 1 ; ;b n ] t. Let us recall that and S 0 0(x 0 )= h 0m 0 3 S 0 n 1(x n )= h n 1m n 1 + h 0m 1 + h n 1m n 3 = (y 1 y 0 ) h 0 f 0 (x 0 ) = f 0 (x n ) (y n y n 1 ) h n 1 : We combine (1.1.41), (1.1.42) and (1.1.35) to write the (n +1) (n +1) matrix A as a i;j = 8 >< >: 2; if i = j 1; if (i; j) =(1; 2) or (i; j) =(n; n 1) h i h i +h i 1 ; if j = i +1; 1 <i<n h i 1 h i +h i 1 ; if j = i 1; 1» i<n 1 0; otherwise; : (1.1.43) the n +1by n +1matrix A can be written as A = ::: h... i 1 h h i +h i 1 2 i h i +h i 1 ::: ::: (1.1.44) The right-hand side B is defined by b 0 = h 0 ( y 1 y 0 h 0 f 0 (x 0 )); (1.1.45)

47 1.7. SPLINE INTERPOLATION 47 b i = h i + h i 1 ( y i+1 y i h i y i y i 1 h i 1 ); i =1; 2; ;n 1; (1.1.4) b n = h n 1 (f 0 (x n ) y n y n 1 h n 1 ): (1.1.47) Lemma Let A be defined in (1.1.43) such that Az = w. Then, jjzjj 1»jjwjj 1 Proof. Let z k be such that jz k j = jjzjj 1. leads to The k th equation from Az = w Applying the triangle inequality we have a k;k 1 z k 1 +2z k + a k;k+1 z k+1 = w k (1.1.48) jjwjj 1 jw k j = ja k;k 1 z k 1 +2z k + a k;k+1 z k+1 j (1.1.49) 2jz k j a k;k 1 jz k 1 j a k;k+1 jz k+1 j (1.1.50) (2 (a k;k 1 + a k;k+1 )jz k j: (1.1.51) Using the fact that a k;k 1 + a k;k+1 =1we complete the proof. In order to state the following lemma we let F =[f 00 (x 0 );f 00 (x 1 ); ;f 00 (x n )] t, R = B AF = A(M F) and H = h i. max i=0; n 1 Lemma If f 2 C 4 [a; b] and jjf (4) jj 1» M 4, then jjm Fjj 1»jjRjj 1» 3 4 M 4H 2 : (1.1.52)

48 48 CHAPTER 1. POLYNOMIAL INTERPOLATION Proof. r 0 = b 0 2f 00 (x 0 ) f 00 (x 1 )= h 0 ( y 1 y 0 h 0 f 0 (x 0 )) 2f 00 (x 0 ) f 00 (x 1 ): Using Taylor expansion we write = 1 h 0 (h 0 f 0 (x 0 )+ h2 0f 00 (x 0 ) 2 (1.1.53) y 1 y 0 h 0 = f(x 0 + h 0 ) f(x 0 ) h 0 (1.1.54) + h3 0f 000 (x 0 ) + h4 0f (4) (fi 1 ) ) (1.1.55) 24 which leads to f 00 (x 0 + h 0 )=f 00 (x 0 )+h 0 f 000 (x 0 )+ h2 0f (4) (fi 2 ) 2 (1.1.5) Thus, r 0 = h2 0f (4) (fi 1 ) 4 h2 0f (4) (fi 2 ) 2 (1.1.57) jr 0 j < 3H 2 M 4 =4: (1.1.58) Similarly for Using Taylor series r n = b n f 00 (x n 1 ) 2f 00 (x n ): (1.1.59) b n = h n 1 (f 0 (x n ) y n y n 1 h n 1 ) (1.1.0) f(x n ) f(x n 1 ) h n 1 = f(x n 1) f(x n ) h n 1 = 1 h n 1 ( h n 1 f 0 (x n )+ h2 n 1 2 f 00 (x n ) h3 n 1 f 000 (x n )+ h4 n 1 24 f (4) (fi 1 )): (1.1.1)

49 1.7. SPLINE INTERPOLATION 49 f 00 (x n 1 )=f 00 (x n h n 1 )=f 00 (x n ) h n 1 2 f 000 (x n )+ h2 n 1 2 f (4) (fi 2 ) (1.1.2) jr n j < 3H 2 M 4 =4: (1.1.3) r j = b j μ j f 00 (x j 1 ) 2f 00 (x j ) j f 00 (x j+1 ); (1.1.4a) where and μ j = h j 1 h j + h j 1 ; j = b j = Using Taylor expansion we write h j h j + h j 1 (1.1.4b) h j 1 + h j ( y j+1 y j h j y j y j 1 h j 1 ): (1.1.4c) y j+1 y j =[f 0 (x j )+ h jf 00 (x j ) + h2 jf 000 (x j ) h j 2 y j y j 1 =[+f 0 (x j ) h j 1f 00 (x j ) + h2 j 1f 000 (x j ) h j h3 jf (4) (fi 1 ) ]; (1.1.4d) 24 h3 j 1f (4) (fi 2 ) ]; 24 (1.1.4e) f 00 (x j 1 )=f 00 (x j ) h j 1 f 000 (x j )+h 2 j 1f (4) (fi 3 )=2; (1.1.4f) f 00 (x j+1 )=f 00 (x j )+h j f 000 (x j )+h 2 jf (4) (fi 4 )=2: (1.1.4g) Note that fi i 2 (x j 1 ;x j+1 ). Combining (1.1.4) we obtain r j = 1 [ h3 jf (4) (fi 1 ) + h3 j 1f (4) (fi 2 ) h3 j 1f (4) (fi 3 ) h j + h j h3 jf (4) (fi 4 ) ]: 2 (1.1.5)

50 50 CHAPTER 1. POLYNOMIAL INTERPOLATION This can be bounded as jr j j» 3 4 M h 3 j + h 3 j 1 4 (1.1.) h j + h j 1 Without loss of generality we assume h j h j 1 and write h 3 j + h 3 j 1 h j + h j 1 = h 2 j 1+( h j 1 h j ) 3 1+ h j 1 h j» h 2 j» H 2 : (1.1.7) Thus, jr j j < 3 4 M 4H 2 : (1.1.8) Finally, using Lemma we have which completes the proof. jjm Fjj 1»jjRjj 1» 3 4 M 4H 2 ; (1.1.9) Theorem Le f(x) 2 C 4 [a; b], a» x 0 < x 1 < < x n» b, h j = x j+1 x j and H = h j. Assume there exits K > 0 independent of H such that max i=0; ;n 1 H h j» K; j =0; 1; ;n 1: (1.1.70) If S(x) is the clamped cubic spline approximation of f at x i ; i =0; 1; ;x n, then there exists C k > 0 independent of H such that jjf (k) S (k) jj 1;[a;b]» C k M 4 KH 4 k ; k =0; 1; 2; 3; (1.1.71) where M 4 = jjf (4) jj 1. Proof. For k = 3 and x 2 [x j 1 x j ]by adding and subtracting few auxiliary terms the error can be written e 000 (x) =f 000 (x) S 000 (x) =f 000 (x) m j m j 1 h j 1

51 1.7. SPLINE INTERPOLATION 51 = f 000 (x) m j f 00 (x j ) h j 1 + m j 1 f 00 (x j 1 ) h j 1 f 00 (x j ) f 00 (x) h j 1 Using Lemma we bound the following terms + f 00 (x j 1 ) f 00 (x) h j 1 : (1.1.72) j m j f 00 (x j ) h j 1 j» 3M 4H 2 4h j 1 ; j m j 1 f 00 (x j 1 ) h j 1 j» 3M 4H 2 4h j 1 : (1.1.73) We use Taylor series to obtain and f 00 (x j ) f 00 (x) =(x j x)f 000 (x)+ (x j x) 2 f (4) (fi 1 ) 2 f 00 (x j 1 ) f 00 (x) =(x j 1 x)f 000 (x)+ (x j 1 x) 2 f (4) (fi 2 ) 2 we bound the error je 000 (x)j» 3M 4H 2 4h j h j 1 j(x j x)f 000 (x)+(x j x) 2 f (4) (fi 1 ) (1.1.74) (x j 1 x)f 000 (x) (x j 1 x) 2 f (4) (fi 2 ) h j 1 f 000 (x)j (1.1.75) 2 The f 000 terms cancel out to give Using we write je 000 (x)j» 3M 4H 2 2h j 1 + M 4 2h j [(x j x) 2 +(x j 1 x) 2 ]: jj(x j x) 2 +(x j 1 x) 2 jj 1;[xj 1 ;x j ] = h 2 j 1» H 2 ; Since H=h j» K we have je 000 (x)j» 3M 4H 2 2h j 1 + M 4H 2 2h j 1 : (1.1.7)

52 52 CHAPTER 1. POLYNOMIAL INTERPOLATION jf 000 (x) S 000 (x)j»2m 4 KH; 8x: (1.1.77) For k =2,we note that for each x 2 (a; b), there is x j such that jx j xj» H=2. Now we rewrite e 00 (x) as e 00 (x) =f 00 (x) S 00 (x) =f 00 (x j ) S 00 (x j )+ Z x x j (f 000 (t) S 000 (t))dt (1.1.78) Using j R gj < R jgjdx and Lemma we obtain je 00 (x)j» 3M 4H j(x x j )j max jjf 000 S 000 jj (1.1.79) x2[a;b] Thus,» 3M 4H M 4 KH 2» 7KM 4H 2 ; K > 1: (1.1.80) 4 jjf 00 (x) S 00 (x)jj 1» 7KM 4H 2 : (1.1.81) 4 For k =1,we consider e(t) =f(t) S(t), since e(x j ) = 0, by Rolle's theorem there exist ο j ;j =0; 1; ;n 1 such that e 0 (ο i )=0ande 0 (x 0 )=e 0 (x n )=0. For every x 2 [a; b] there exists ο i such that jx ο i j» H and e 0 (x) can be written as f 0 (x) S 0 (x) = Thus, Z x ο i (f 00 (t) S 00 (t))dt (1.1.82) je 0 (x)j»jx ο i jjje 00 jj 1» 7 4 M 4KH 3 : (1.1.83)

53 1.7. SPLINE INTERPOLATION 53 For k = 0, for every x 2 [a; b] there is x j such that jx x j j» H=2 we also write f(x) S(x) = Z x x j (f 0 (t) S 0 (t))dt (1.1.84) je(x)j»jx x j jjje 0 jj 1» 7 8 M 4KH 4 : (1.1.85) We conclude that S (k) converges uniformly to f (k) for k =0; 1; 2; 3 and H! 0. Optimal bounds are proved by Birkhoff and De Boor (Burden and Faires) as We recall the Hermite interpolation error jf H 3 j < jjf Sjj 1 < M 4H 4 : (1.1.8) M 4H 4 = M 4H ; (1.1.87) Comparing the cubic spline and Hermite interpolation errors, we see that the ratio between the spline and the Hermite errors is only 5. We also note that Hermite interpolation requires the derivative at all the interpolation points while the clamped spline needs the derivatives at the end points only. Optimality of Splines: The optimality is in the sense that cubic spline has the smallest curvature. For a curve defined by y = f(x) the curvature is defined as fi = jf 00 (x)jj [1+(f 0 (x)) 2 ] 3=2 : Here the curvature is approximated by jf 00 (x)j and R b a S00 (x) 2 dx is minimized. More precisely we state the following theorem.

54 54 CHAPTER 1. POLYNOMIAL INTERPOLATION Theorem If f 2 C 2 [a; b] and S(x) is the natural cubic spline that interpolates f at n +1 points x i ; i =0; 1; ;n; then Z b a S 00 (x) 2 dx» Z b a f 00 (x) 2 dx: (1.1.88) Proof. We consider the function e(x) = f(x) S(x) with e(x i ) = 0; i = 0; 1; ;n; and write the approximate curvature of f = S + e as Z b a Z b a S 00 (x) 2 dx + f 00 (x) 2 dx = Z b a Z b a (S 00 (x)+e 00 (x)) 2 dx = Z b e 00 (x) 2 dx +2 S 00 (x)e 00 (x)dx: (1.1.89) a We complete the proof by showing that the last term in the right-hand side of (1.1.89) is 0. Integrating by parts we obtain Z xi+1 x i e 00 (x)s 00 (x)dx = S 00 (x)e 0 (x)j x=x i+1 x=x i Z xi+1 x i S 000 (x)e 0 (x)dx: Summing over all intervals, using the fact that e 2 C 2 and S 00 (a) =S 00 (b) =0. Noting that S 000 (x) =C i is a constant on (x i ;x i+1 ) we obtain n 1 Z xi+1 X i=0 x i e 00 (x)s 00 (x)dx = = n 1 X i=0 n 1 X i=0 Z xi+1 C i e 0 (x)dx x i C i [e(x i+1 ) e(x i )]=0 We used the fact that e(x i )=0we establish R b a S00 (x)e 00 (x)dx =0. Combining this with (1.1.89) leads to (1.1.88). The same result holds for the clamped cubic spline with S 0 (a) = f 0 (a) and S 0 (b) =f 0 (b). We follow the same line of reasoning to prove it. Thus, among all C 2 functions interpolating f at x 0 ;::: ;x n, the natural cubic spline has the smallest curvature. This includes the clamped spline and nota=knot splines.

55 1.7. SPLINE INTERPOLATION B-splines We describe a system of B-splines (B stands for basis) from which other splines can be obtained. We first start with B-splines of degree 0, i.e., piecewise constant splines defined as ( 1 x Bi 0 i» x<x i+1 (x) = ;i2 Z (1.1.90) 0 otherwise Properties of B 0 i (x): 1. B 0 i (x) 0; for all x and i 2. 1P i= 1 B 0 i (x) = 1, for all x 3. The support of B 0 i (x) is [x i ;x i+1 ) 4. B 0 i (x) 2 C 1. We show property (2) by noting that for arbitrary x there exists m such that x 2 [x m ;x m+1 ) then write 1X i= 1 B 0 i (x) =B 0 m(x) =1: Use the recurrence formula to generate the next basis functions B k i (x) = x x i x i+k x i B k 1 i (x)+ x i+k+1 x x i+k+1 x i+1 B k 1 i+1 (x); for k>0: (1.1.91) Properties of B 1 i : 1. B 1 i (x) are the classical piecewise linear hat functions equal to 1 at x i+1 and zero all other nodes. 2. B 1 i (x) 2 C P Bi 1 i= 1 (x) = 1 for all x

56 5 CHAPTER 1. POLYNOMIAL INTERPOLATION 4. The support of B 1 i (x) is (x i ;x i+2 ) 5. B 1 i (x) 0 for all x and i In general for arbitrary k one can show that: 1. B k i (x) are piecewise polynomials of degree k 2. B k i (x) 2 C k P i= 1 B k i (x) = 1 for all x 4. The support of B k i (x) is (x i ;x i+k+1 ) 5. B k i (x) 0 for all x and i. B k i (x), 1 <i<1 are linearly independent, i.e., they form a basis. See Figure for plots of the first four b-splines. Interpolation using B-splines: (i) For k = 0, we construct a piecewise constant spline interpolation by writing f(x) ß P 0 (x) = 1X i= 1 Using the properties of B 0 i (x) we show that c i = f(x i ). c i B 0 i (x) (1.1.92) (ii) For k = 1, we construct a piecewise linear spline interpolation by writing f(x) ß P 1 (x) = 1X i= 1 Again using B 1 i (x j+1 )=ffi ij we show that c i = f(x i+1 ). c i B 1 i (x) (1.1.93) (ii) For k = 3, we construct a piecewise cubic spline interpolation by writing f(x) ß P 3 (x) = 1X i= 1 c i B 3 i (x) (1.1.94)