MATLAB: M-files; Numerical Integration Last revised : March, 2003

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1 MATLAB: M-files; Numericl Integrtion Lst revised : Mrch, 00 Introduction to M-files In this tutoril we lern the bsics of working with M-files in MATLAB, so clled becuse they must use.m for their filenme extension. There re two types of M-files: A script is collection of MATLAB commnds which re executed just s if they hd been typed in t the commnd line. When the script is finished running, ny vribles defined in the script re still vilble in the current MATLAB commnd-line environment. Executing MATLAB commnds from within script, even for reltively simple tsks, mkes it much esier to reproduce results lter, or to rerun the sme clcultions with different prmeter settings. At the sme time the script serves s record of your ctivity nd cn serve s useful strting point for future projects. A function ccepts input rguments nd returns output rguments, similr to functions in structured progrmming lnguges such s JAVA, C, nd FORTRAN. Vribles defined in function re locl to the function. The vribles creted in the function re not directly ccessible from the commnd line once the function completes execution. The Windows version of MATLAB provides text editor for creting nd modifying M-files. Throughout this document it is ssumed tht the reder is running the student version of MATLAB6 on the Windows pltform. In this cse the defult working directory is typiclly C:\mtlb_sv\work nd the M-file editor will sve the M-files to this directory unless the user specifies otherwise. The MATLAB principles will be introduced in the context of lerning severl methods of numericl integrtion. At the of the tutoril we will hve produced MATLAB code tht cn be used to pproximte definite integrls. Numericl integrtion To pproximte the definite integrl f (x dx we first divide the intervl [, b] into n subintervls, ech of length h = (b /n. The points of the subintervls re denoted by the points x, x,..., +, where x = nd + = b. We write y i to denote the evlution of f (x t node x i, y i = f (x i. The i th subintervl is given by [x i, x i+ ], with x i+ x i = h, for i =,..., n. [The points re indexed s,,..., n + to coincide with the indexing of rry elements in Mtlb. In the textbook the grid points re indexed s 0,,..., n.] The left-point pproximtion uses n rectngles of width h to pproximte the integrl. The height of the rectngle for intervl [x i, x i+ ] is given by f (x i, the vlue of the function t the left point. The re of the i th rectngle is A i = h f (x i = hy i. Summing over the n rectngles yields the pproximtion: Left Endpoint Approx (n=8 hy i = h ( y + y + + y n. = Ln x x + Use the pwd commnd to confirm the current working directory. To chnge the current directory, enter cd dir_nme from the commnd line, or choose MATLAB Current Directory from the MATLAB Lunch Pd.

2 For the right-point pproximtion the height of the rectngle for the intervl [x i, x i+ ] is given by f (x i+, the vlue of the function t the right point. hy i+ = h ( y + y + + y n+. = Rn x Right Endpoint Approx (n=8 x + The trpezoidl rule uses n trpezoids of width h to pproximte the definite integrl. The top edge of the trpezoid for the subintervl [x i, x i+ ] is the line segment connecting the two points (x i, y i nd (x i+, y i+. The re of the i th trpezoid is given by A i = h(y i + y i+ /. Summing over the n trpezoids, h ( h (. yi + y i+ = y + y + y + + y n + y n+ = Tn ( The trpezoidl rule is equivlent to verging the left-point nd right-point pproximtions, Creting MATLAB script T n = ( L n + R n /. ( We first write MATLAB script tht clcultes the left-point, right-point, nd trpezoidl pproximtions for prticulr definite integrl. After the script is verified to be working, we will convert it into MATLAB function nd lter incorporte two dditionl pproximtions, the midpoint rule nd Simpson s rule. As test cse for our numericl integrtion, we will pproximte the definite integrl of f (x = 4/(+x on the intervl [0, ]. In section.6 of the Stewrt text we used implicit differentition to show tht the derivtive of rctn x is the function /( + x. In the lnguge of section 4.9 we sy rctn x is n ntiderivtive of the function /( + x. Applying the evlution theorem in section 5., x dx = 4 rctn x 0 = 4 ( π 4 0 = π ( After strting up MATLAB enter the commnd pwd to confirm the nme of the current working directory. This current directory is where your M-files will be stored. >> pwd ns = C:\mtlb_sv\work % print nme of working directory % M-files will be sved here Note: When copying commnds from this document into your own M-files or to the commnd line, the percent sign (% mrks the rest of the line s comment. It is not necessry to copy the comments. To crete new M-file, select File New M-file from the menu br. The M-file editor strts up with n empty window. Copy the following MATLAB commnds into the new M-file. = 0.0; b =.0; n = 8; h = (b - / n; cler x y; x( = ; y( = 4 / ( + x(ˆ; % integrte f(x from =0 to b= % number of subintervls; n EVEN % width of subintervls % remove current vlues of d y % first node is the left point

3 for i = :n x(i+ = x(i + h; y(i+ = 4 / ( + x(i+ˆ; Ln = 0; Rn = 0; for i = :n Ln = Ln + y(i; Rn = Rn + y(i+; Ln = h*ln; Rn = h*rn; Tn = (Ln + Rn / ; % i =,,..., n-, n % x(=x(+h; x(=x(+h;... % y(i = 4 / ( + x(i*x(i % initilize the sums to 0 % left-point % right-point % left-point pproximtion % right-point pproximtion % trpezoidl pprox; eqution ( fprintf(, n = %d\n, n; % disply some results fprintf(, Ln = %.f error= %.e\n, Ln, Ln - pi; fprintf(, Rn = %.f error= %.e\n, Rn, Rn - pi; fprintf(, Tn = %.f error= %.e\n, Tn, Tn - pi; The function fprintf is for writing formtted dt to file. To sve the new M-file to the disk, choose File Sve from the editor s pull-down menu nd follow the dilog box. To test the new script, enter the nme of the script from the Commnd Window. This writeup ssumes the script hs been nmed ms.m. If MATLAB reports n error, use the editor to mke the corrections, sve the revised file, nd rerun ms. Once the progrm is working correctly you should see the following output: >> ms n = 8 Ln = Rn = Tn = error=.e-0 error= -.8e-0 error= -.60e-0 Converting the script to function Incresing the vlue of n in the script ms.m will improve the ccurcy of the numericl integrtion. However, ech new vlue of n requires editing the M-file, chnging the vlue of n, nd sving the revised file. Here it will be helpful to first convert the script into function so tht the number of subintervls is controlled through the vlue of n input rgument. The function will require single input rgument, the number of subintervls n, nd return single output rgument, the trpezoidl pproximtion for the definite integrl. Note tht n must be positive even integer. Rther thn overwrite the working script, first crete new file mf.m tht is just copy of the file ms.m. In this new file, delete the line n = 8 nd, t the very top of the file, dd the following line of Mtlb code: function Tn = mf(n The first line of the file identifies mf s function tht cn be clled with single input rgument. The input vlue will be copied to the vrible n inside the function. The contents of the vrible Tn re returned to the With the first rgument of fprintf set to the output goes to the Commnd Window. The second rgument is string for specifying the output formt. For exmple, %d specifies deciml integer output, %.f specifies rel numbers with twelve digits to the right of the deciml point, %.e specifies exponentil nottion with two digits to the right of the deciml point (three significnt digits, nd \n produces linefeed.

4 progrm tht clled mf (just the Commnd Window in this cse. After sving these chnges, run the new function for different vlues of n. For exmple, pss the number 8 to duplicte the erlier results. >> mf(8 n = 8 Ln = Rn = Tn = ns =.90 error=.e-0 error= -.8e-0 error= -.60e-0 Clling the function with the commnd S = mf(8 will sve the output in the vrible S. Exercise : Run your function mf.m using incresing vlues of the input rgument p. Wht is the smllest vlue of n tht mkes the bsolute vlue of the error in Tn less thn 0 6? Less thn 0 8? The midpoint rule For the sme collection of n subintervls with points x, x,...,, +, the midpoint pproximtion uses n/ rectngles of width h to pproximte the integrl. For k =,..., n/, the k th subintervl is [x k, x k+ ] with midpoint x k. The height of the rectngle for this intervl is tken to be f (x k, the vlue of the function t the midpoint of the intervl, nd the re of the k th rectngle is A k = hy k. x Midpoint Approx (n=8 x 4 Summing the res from the n/ rectngles yields the midpoint pproximtion: n/ hy k = h (. y + y y n + y n = Mn (4 k= To include the midpoint pproximtion in your MATLAB function, dd the following code to the function mf.m. This new for loop cn be dded just fter the clcultion of Tn. Mn = 0; for k = :n/ Mn = Mn + y(*k; Mn = *h*mn; % initilize the sum to 0 % k =,,..., n/ % see eqution (4 % midpoint pproximtion Include nother printf sttement t the of the M-file. fprintf(, Mn = %.f error= %.e\n, Mn, Mn - pi; Exercise : Run mf.m nd compre the ccurcy of the midpoint nd trpezoidl pproximtions for severl vlues of the input rgument n. Comment on your observtions. Simpson s rule In ech of these integrtion methods the pproximtion formul is just the definite integrl for simple function tht pproximtes f (x. For exmple, the trpezoidl rule uses piecewise liner function. On It is ALWAYS good ide to first mke copy of the existing progrm before dding new fetures. If you re unble to get the revisions to work properly, you cn still return to working version of the progrm. 4

5 ech subintervl [x i, x i+ ] we integrte liner function in plce of f (x. The pproximting function is piecewise becuse the eqution of the liner function chnges from one subintervl to the next. Ech of the other three methods use piecewise constnt functions to pproximte f (x. With this point of view, we look to improve the ccurcy of the numericl integrtion by pproximting f (x with higher order polynomil. On ech subintervl [x k, x k+ ] of length h, there is exctly one second-order polynomil tht psses through the three points (x k, y k, (x k, y k, nd (x k+, y k+. The re under this polynomil is given by A k = h(y k + 4y k + y k+ /6, for k =,,..., n/. Summing these res (there re n/ subintervls of length h results in Simpson s Rule: n/ k= h ( h (. yk + 4y k + y k+ = y + 4y + y + 4y y n + y n+ = Sn (5 Simpson s rule cn lso be expressed s weighted verge of the trpezoidl nd midpoint rules. S n = ( T n + M n / (6 Exercise : Add the clcultion for Simpson s rule from eqution (6 to the function mf.m, including n dditionl fprintf sttement nd error clcultion. Chnge the function so it returns the vlue of the Simpson pproximtion rther thn the trpezoidl pproximtion. Compre the ccurcy of the different integrtion methods. Wht is the smllest vlue of n tht mkes the bsolute vlue of the error in S n less thn 0 6? Less thn 0 8? Exercise 4 : Modify the function mf.m to estimte the definite integrl 4 0 e x dx using Simpson s pproximtion. Clculte the Simpson pproximtion for incresing vlues of n = {, 4, 8, 6,...} until the difference S n S n/ < 0 0. In this exercise you must chnge in the M-file both the vlue of b nd the expressions for clculting y = f (x. The error clcultions (Tn - pi,... re no longer vlid for this exercise. The following code shows how one might proceed to cll mf once the function hs been revised. >> formt long; >> s8 = mf(8 ns =... >> s6 = mf(6 ns =... >> bs(s6 - s8 ns =... >>... % disply more significnt digits % disply Simpson pprox with n = 8 % disply Simpson pprox with n = 6 % disply the difference S_6 - S_8 % continue until difference <.0e-0 Record the finl vlue of n, the estimte S n with t lest significnt digits, nd the difference S n S n/. Exercise 5 : If f (x is ny third-order polynomil, show tht Simpson s rule with n = is exct for the definite integrl f (x dx. In other words, verify the following equlity: (Ax + Bx + Cx + D dx = h ( y + 4y + y First clculte the integrl on the left-hnd side in terms of the prmeters A, B, C, D, nd. Then clculte the right-hnd side where y i = f (x i = Ax i + Bx i + Cx i + D. In evluting the right-hnd side, you must first identify the vlue of h, nd the nodes x, x, x, in terms of the prmeter. You should be ble to show tht the two sides re equl for ny choice of the constnts A, B, C, D, nd. [ This is one reson for the populrity of Simpson s rule. We strted by pproximting the integrnd with piecewise second-order polynomil nd the method turns out to be exct for ll third-order polynomils. ] 5