1. Approximating Derivatives Using Taylor Series and Vandermonde Matrices

2. The Problem  Approximate f (k) (a) using the values of the function f (x) at Any n points to approximate f (a), f ’ (a), …, f (n-1) (a)

3. The Problem  Approximate f (k) (a) using the values of the function f (x) at Any n points 2n+1 points f (–n), …, f (0), …, f (n) to approximate f (a), f ’ (a), …, f (n-1) (a) f (a), f ’ (a), …, f (2n) (a)

4. The Problem  Approximate f (k) (a) using the values of the function f (x) at Any n points 2n+1 points f (–n), …, f (0), …, f (n) 3 points f (–1), f (0) and f (1) to approximate f (a), f ’ (a), …, f (n-1) (a) f (a), f ’ (a), …, f (2n) (a) f (a), f ’ (a), f ’’ (a)

5. Taylor Series  We will use Taylor Series expanded around a. For example,

6. Since we want an approximation for f (a), we need all of the coefficients of f (a) to add up to 1. To make the approximation as good as possible (i.e. to reduce error), we would like all of the coefficients of f ’ (a) and f ’’ (a) to add up to 0. We can expand each of these points around a :

8.  This leads to

10. Taylor coefficients matrix A

11.  This leads to  Then = the first column of A - 1. Taylor coefficients matrix A

12.  This leads to  Then = the first column of A - 1. Taylor coefficients matrix A

13.  This leads to  Then = the second column of A - 1. Taylor coefficients matrix A

14.  This leads to  Then = the third column of A - 1. Taylor coefficients matrix A

15. The Problem  Approximate f (k) (a) using the values of the function f (x) at Any n points 2n+1 points f (–n), …, f (0), …, f (n) 3 points f (–1), f (0) and f (1) to approximate f (a), f ’ (a), …, f (n-1) (a) f (a), f ’ (a), …, f (2n) (a) f (a), f ’ (a), f ’’ (a)

16. The Problem  Approximate f (k) (a) using the values of the function f (x) at Any n points 2n+1 points f (–n), …, f (0), …, f (n) 3 points f (–1), f (0) and f (1) to approximate f (a), f ’ (a), …, f (n-1) (a) f (a), f ’ (a), …, f (2n) (a) f (a), f ’ (a), f ’’ (a)  Weightings c 1, c 2, … are simply the columns of the inverse of the Taylor coefficients matrix A.

17. Problem with 2n+1 points, n=2  Goal: approximate f (a), f ’ (a), f ‘’ (a), f (3) (a), f (4) (a) using values f (–2), f (–1), f (0), f (1), f (2).  This leads to the Taylor coefficients matrix

19.  For example, with a = 0:

20. With a =0  The columns of A - 1 give desired weightings.

21. With a =0  The columns of A - 1 give desired weightings.  For example,

22. With a =0  The columns of A - 1 give desired weightings.  For example,

23. Vandermonde matrices: 5 x 5 case

24. Compare to

25. Vandermonde matrices: 5 x 5 case

28. L - 1 with our points: (example)  Where  Each element of L - 1 will be a constant.  For example, take row 3, column 2 of L - 1 :

29. L - 1 with our points:  L - 1 will simplify to:

30. U - 1 with our points:  U - 1 will simplify to:

31. Taylor coefficient matrix A = DV T DVTVT

32. ( U -1 ) T =

34. Recall, this means…  If we want to approximate the value of f (3) (a), we use the corresponding column of A - 1 as weights for our points.

35. With a =0  The columns of A - 1 give desired weightings.

36. With a =0  The columns of A - 1 give desired weightings.  For example,

37. With a =0  The columns of A - 1 give desired weightings.  For example,

38. Problem with 2n+1 points, n=4  Let’s look at the case where we have 9 points -4, …, 0, … 4 and we want to find the best approximation for f (6) (0).

39. Problem with 2n+1 points, n=4  Let’s look at the case where we have 9 points -4, …, 0, … 4 and we want to find the best approximation for f (6) (0).

40. In summary  We can approximate f(a) for any a using information at 2n+1 points centered around 0.  The weights will be the columns of A - 1, the Taylor coefficients matrix

41. Can compute one element of A -1  The optimal weights for f (n) (a) are simply the values in the k+1 st column of A -1.  The k+1 st column of A -1 is (L -1 ) T (k+1 st column of (U -1 ) T ) k! that is, (L -1 ) T (k+1 st row of U -1 ) k!  In fact, we can compute individual entries (and thus any single column) of A -1 using the nice formulas for individual entries of L -1 and U -1 that we saw earlier.

42. Comparison with Matlab

43. In summary  We can approximate f(a) for any a using information at 2n+1 points centered around 0.  The weights will be the columns of A - 1, the Taylor coefficients matrix  We can quickly and easily find A - 1 using transposed Vandermonde matrices  We use LU factorization  L - 1 and U - 1 have “nice” formulas  We can compute a single column (or even value) of U - 1.

44. Current work Generalizations:  Approximations using:  Points that are not centered at 0.  Points that are not space one unit apart.  Further work:  Approximation using arbritary points.  General formula for approximation error.  Comparisons to existing techniques.