Lecture 4: Numerical differentiation

Transcription

1 Handout 5 05/08/02 1 Lecture 4: Numercal dfferentaton Fnte dfference formulas Suppose you are gven a data set of N non-equspaced ponts at x = x wth values f(x ) as shown n Fgure 1. Because the data are not equspaced n general, then x x +1. Fgure 1: Data set of N ponts at x = x wth values f(x ) Let s say we wanted to compute the dervatve f (x) at x = x. For smplcty of notaton, we wll refer to the value of f(x) at x = x as f. Because, n general, we do not know the form of f(x) when dealng wth dsrete ponts, then we need to determne the dervatves of f(x) at x = x n terms of the known quanttes f. Formulas for the dervatves of a data set can be derved usng Taylor seres. The value of f(x) at x = x +1 can be wrtten n terms of the Taylor seres expanson of f about x = x as f +1 = f + x +1 f + x x O ( x 4 +1). (1) Ths can be rearranged to gve us the value of the frst dervatve at x = x as f = f +1 f x +1 x +1 2 x2 +1 f + O ( x+1) 3. (2) 6 If we assume that the value of f does not change sgnfcantly wth changes n x +1, then ths s the frst order dervatve of f(x) at x = x, whch s wrtten as f = f +1 f x +1 + O ( x +1 ). (3)

2 Handout 5 05/08/02 2 Ths s known as a forward dfference. The frst order backward dfference can be obtaned by wrtng the Taylor seres expanson about f to obtan f 1 as f 1 = f x f + x2 2 f x3 6 f + O ( ) x 4, (4) whch can be rearranged to yeld the backward dfference of f(x) at x as f = f f 1 x + O ( x ). (5) The frst order forward and backward dfference formulas are frst order accurate approxmatons to the frst dervatve. Ths means that decreasng the grd spacng by a factor of two wll only ncrease the accuracy of the approxmaton by a factor of two. We can ncrease the accuracy of the fnte dfference formula for the frst dervatve by usng both of the Taylor seres expansons about f, f +1 = f + x +1 f + x x f + O ( ) x 4 +1 f 1 = f x f + x2 2 f x3 6 f + O ( ) x 4. (7) Subtractng equaton (7) from (6) yelds f +1 f 1 = ( x +1 + x ) f + x2 +1 x O ( ) ( ) x O x 4 + x x 3 6 f +1 f 1 = f + x2 +1 x 2 x +1 + x 2 ( x +1 + x ) f + x x 3 6 ( x +1 + x ) f ( ) ( ) + O x 4 +1 x +1 + x + O x 4 x +1 + x (6), (8) whch can be rearranged to yeld f = f +1 f 1 x +1 + x x2 +1 x 2 2 ( x +1 + x ) f + O ( x x 3 ) 6 ( x +1 + x ) (9) In most cases f the spacng of the grd ponts s not too eratc, such that x +1 x, equaton (9) can be wrtten as the central dfference formula for the frst dervatve as f = f +1 f 1 2 x + O ( x 2 ). (10) What s meant by the order of accuracy? Suppose we are gven a data set of N = 16 ponts on an equspaced grd as shown n Fgure 2, and we are asked to compute the frst dervatve f at = 2,..., N 1 usng the forward, backward, and central dfference formulas (3), (5), and (10). If we refer to the approxmaton

3 Handout 5 05/08/02 3 Fgure 2: A data set consstng of N = 16 ponts. of the frst dervatve as, then these three formulas for the frst dervatve on an equspaced δx grd wth x = x can be approxmated as Forward dfference Backward dfference Central dfference δx = f +1 f, x (11) δx = f f 1, x (12) δx = f +1 f 1. 2 x (13) These three approxmatons to the frst dervatve of the data shown n Fgure 2 are shown n Fgure 3. Now let s say we are gven fve more data sets, each of whch defnes the same functon f(x ), but each one has twce as many grd ponts as the prevous one to defne the functon, as shown n Fgure 4. The most accurate approxmatons to the frst dervatves wll be those that use the most refned data wth N = 512 data ponts. In order to quantfy how much more accurate the soluton gets as we add more data ponts, we can compare the dervatve computed wth each data set to the most resolved data set. To compare them, we can plot the dfference n the dervatve at x = 0.5 and call t the error, such that ) Error = ) (14) δx δx where n = 1,..., 5 s the data set and n = 6 corresponds to the most refned data set. The result s shown n Fgure 5 on a log-log plot. For all three cases we can see that the error closely follows the form Error = k x n, (15) where k = 1.08 and n = 1 for the forward and backward dfference approxmatons, and k = 8.64 and n = 2 for the central dfference approxmaton. When we plot the error of a numercal method and t follows the form of equaton (15), then we say that the method s n n=6

4 Handout 5 05/08/02 4 Fgure 3: Approxmaton to the frst dervatve of the data shown n Fgure 2 usng three dfferent approxmatons. n th order and that the error can be wrtten as O ( x n ). Because n = 1 for the forward and backward approxmatons, they are sad to be frst order methods, whle snce n = 2 for the central approxmaton, t s a second order method. Taylor tables The frst order fnte dfference formulas n the prevous sectons were wrtten n the form df dx = + Error, (16) δx where df s the approxmate form of the frst dervatve wth some error that determnes δx dx the order of accuracy of the approxmaton. In ths secton we defne a general method of estmatng dervatves of arbtrary order of accuracy. We wll assume equspaced ponts, but the analyss can be extended to arbtrarly spaced ponts. The n th dervatve of a dscrete functon f at ponts x = x can be wrtten n the form d n f dx n = δn f x=x δx + O n ( xm ), (17) where δ n j=+nr f δx = a n j+nl f +j, (18) j= N l and m s the order of accuracy of the approxmaton, a j+nl are the coeffcents of the approxmaton, and N l and N r defne the wdth of the approxmaton stencl. For example, n the central dfference approxmaton to the frst dervatve, f = 1 2 x f 1 + 0f x f +1 + O ( x 2), (19) = a 0 f 1 + a 1 f + a 2 f +1 + O ( x 2). (20)

5 Handout 5 05/08/02 5 Fgure 4: The orgnal data set and 5 more, each wth twce as many grd ponts as the prevous one. In ths case, N l = 1, N r = 1, a 0 = 1/2 x, a 1 = 0, and a 2 = +1/2 x. In equaton (18) the dscrete values f +j can be wrtten n terms of the Taylor seres expanson about x = x as f +j = f + j xf + (j x)2 2 (j x) k = f + k! k= (21) f (k). (22) Usng ths Taylor seres approxmaton wth m + 2 terms for the f +j n equaton (18), where m s the order of accuracy of the fnte dfference formula, we can substtute these values nto equaton (17) and solve for the coeffcents a j+nl to derve the approprate fnte dfference formula. As an example, suppose we would lke to determne a second order accurate approxmaton to the second dervatve of a functon f(x) at x = x usng the data at x 1, x, and

6 Handout 5 05/08/02 6 Fgure 5: Depcton of the error n computng the frst dervatve for the forward, backward, and central dfference formulas x +1. Wrtng ths n the form of equaton (17) yelds d 2 f dx 2 = δx + O ( x 2), (23) where, from equaton (18), δx = a 0f 1 + a 1 f + a 2 f +1. (24) The Taylor seres approxmatons to f 1 and f +1 to O ( x 4 ) are gven by f 1 f xf + x2 2 f x3 6 f + x4 24 f v, (25) f +1 f + xf + x2 2 f + x3 6 f + x4 24 f v, (26) Rather than substtute these nto equaton (24), we create a Taylor table, whch requres much less wrtng, as follows. If we add the columns n the table then we have a 0 f 1 + a 1 f + a 2 f +1 = (a 0 + a 1 + a 2 )f + ( a 0 + a 2 ) xf + (a 0 + a 2 ) x2 2 f + ( a 0 + a 2 ) x3 6 f + (a 0 + a 2 ) x4 24 f v. (27)

7 Handout 5 05/08/02 7 Term n (24) f xf x 2 f x 3 f x 4 f v a 0 f 1 a 0 a 0 a 0 /2 a 0 /6 a 0 /24 a 1 f a a 2 f +1 a 2 a 2 a 2 /2 a 2 /6 a 2 / ?? Because we would lke the terms contanng f and f on the rght hand sde to vansh, then we must have a 0 + a 1 + a 2 = 0 and a 0 + a 2 = 0. Furthermore, snce we want to retan the second dervatve on the rght hand sde, then we must have a 0 + a 2 = 1. Ths yelds three equatons n three unknowns for a 0, a 1, and a 2, namely, a 0 + a 1 + a 2 = 0 a 0 + a 2 = 0 a 0 /2 + a 2 /2 = 1, (28) n whch the soluton s gven by a 0 = a 2 = 1 and a 1 = 2. Substtutng these values nto equaton (27) results n f 1 2f + f +1 = x 2 + x4 12 f v, (29) whch, after rearrangng, yelds the second order accurate fnte dfference formula for the second dervatve as f = f 1 2f + f +1 + O ( x 2), (30) x 2 where the error term s gven by Error = x2 12 f v. (31) As another example, let us compute the second order accurate one-sded dfference formula for the frst dervatve of f(x) at x = x usng x, x 1, and x 2. The Taylor table for ths example s gven below. By requrng that a 0 + a 1 + a 2 = 0, 2a 0 a 1 = 1, and Term f xf x 2 f x 3 f x 4 f v a 0 f 2 a 0 2a 0 2a 0 4a 0 /3 2a 0 /3 a 1 f 1 a 1 a 1 +a 1 /2 a 1 /6 a 1 /24 a 2 f a ?? 2a 0 + a 1 /2 = 0, we have a 0 = 1/2, a 1 = 2, and a 2 = 3/2. Therefore, the second order accurate one-sded fnte dfference formula for the frst dervatve s gven by where the error term s gven by df dx = f 2 4f 1 + 3f + O ( x 2), (32) 2 x Error = x2 3. (33)

8 Handout 5 05/08/02 8 Hgher order fnte dfference formulas can be derved usng the Taylor table method descrbed n ths secton. These are shown n Appled numercal analyss, sxth edton, by C. F. Gerald & P. O. Wheatley, Addson-Welsley, 1999., pp