MATLAB Workshop 13 - Linear Systems of Equations

Transcription

1 MATLAB: Workshop - Liner Systems of Equtions pge MATLAB Workshop - Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB Fetures: colon (:) opertor - specil vector mtrix opertor Nottion Action Result :c cretes vector with elements prt [ c] :b:c cretes vector with elements b prt [ +b +2b... ( c)] A(m,:) selects m th row of A A(:,j) selects j th column of A A(m:n,:) selects m th through n th rows of A A(:,j:k) selects j th through k th columns of A A(m:n,j:k) selects m th through n th rows of A nd j th through k th columns of A mtrix functions number of rows nd columns workspce commnd Commnd vrnme = lod(filenme) [rows, cols] = size(x) Action retrieves the contents of file s ASCII text nd ssigns the contents to vrnme. filenme is string tht specifies the nme nd loction of the file. if the file is not in the current directory, the entire pth must be specified.

2 MATLAB: Workshop - Liner Systems of Equtions pge 2 Liner systems of equtions nd mtrix lgebr Mtrix lgebr provides convenient shorthnd nottion for liner systems of equtions. Consider the system of equtions = x + 2 x2 + x b 2x + 22 x2 + 2x b2 x + 2 x2 + x b = = where the ij represent the coefficients of the unknown vribles x i. This system of equtions cn be written in mtrix lgebr nottion s [ A ] N x N ( X ) N x = ( B) N x where [A] NxN is squre (sme number of columns s rows) mtrix of coefficients, (X) Nx is column vector of unknowns, nd (B) Nx is the forcing vector (lso column vector). Note tht the multipliction is defined within the rules of mtrix lgebr., i.e., n (NxN) mtrix times n (Nx) vector yields n (Nx) vector. In expnded nottion, the mtrix lgebr eqution cn be written s x x x 2 b = b b 2 You cn verify tht pplying the rules of mtrix multipliction to the bove mtrix lgebr eqution reproduces the originl set of equtions. Note tht the coefficient mtrix is obtined by tking the coefficients of ech eqution nd plcing them in the corresponding row. The following exmple illustrtes the use of mtrix lgebr for liner systems of equtions. The set of liner equtions 2 x + 47 y 2z = 25 5 x + 2z = 46 6 x 6y + 5z = 225 cn be written in mtrix lgebr form s x 25 2 y = 46 5 z 225 Note tht one of the coefficients in the second eqution is zero. This needs to be noted explicitly in the coefficient mtrix.

3 MATLAB: Workshop - Liner Systems of Equtions pge A Civil Engineering problem Liner systems of equtions rise frequently in every engineering discipline. For exmple, consider the following problem tht might rise in rod construction project. A civil engineer needs to purchse 85 tons of rod mteril consisting of 25 tons of grvel, 40 tons of pebbles, nd 20 tons of stones. She contcts three locl qurries nd obtins the following specifictions on their product mix. Qurry, wt% Product Mix 2 Grvel Pebbles Stones None of the qurries individully hs product vilble in the percentges tht re required for the project. So the next question she sks is whether she cn meet her requirements through buying some product from ech qurry. To do this, she sets up mteril blnces for the mounts of grvel, pebbles nd stones she requires. If she purchses Q tons from Qurry, Q 2 tons from Qurry 2, nd Q tons from Qurry, then mteril blnce for grvel, obtined by multiplying the frction of grvel from qurry by the totl tons to be purchsed from the qurry, becomes ( 0.40) Q + (0.20) Q2 + (0.00) Q = 25 where the totl mount of grvel purchsed from the three qurries must be equl to the mount needed for the project. Likewise, mteril blnces for pebbles nd stones yield the equtions ( 0.50) Q + (0.25) Q2 + (0.50) Q 0.0) Q + (0.55) Q + (0.50) Q = ( 2 = These equtions cn be put into mtrix lgebr form s Q 0.50 Q 0.50 Q 2 = MATLAB cn be used to solve this system of equtions for Q, Q 2, nd Q. Solution of liner equtions in MATLAB Solution of set of liner equtions in MATLAB is rther strightforwrd. In mtrix lgebr terminology the solution to the set of liner equtions [ A ]( x) = ( b) where A is the coefficient mtrix, x is the vector of unknowns, nd b is the right hnd side vector is given by (in MATLAB terminology)» x = A\b Tht is, the bckslsh opertor, \, is used to find the solution. () Solving system of liner equtions in MATLAB.

4 MATLAB: Workshop - Liner Systems of Equtions pge 4 To solve system of liner equtions, one needs the coefficient mtrix, the right hnd side vector, nd the bckslsh opertor, \. To find the solution to the qurry problem, enter the following t the commnd prompt:» Coeff = [0.40, 0.20, 0.00; 0.50, 0.25, 0.50; 0.0, 0.55, 0.50];» mount = [25; 40; 20];» Q = Coeff\mount Q = The first line entered the coefficient mtrix, Coeff, by rows. The right hnd side vector, mount, ws entered s column vector. This is very importnt. Wht hppens if row vector is used insted? Try it! The solution vector, Q, ws obtined by using the bckslsh opertor. Note tht Q is column vector, s would be expected (why?). Creting n ugmented coefficient mtrix file While solution of liner systems of equtions is rther strightforwrd in MATLAB, doing so in the Commnd Window hs couple of drwbcks. The first drwbck is in entering the coefficient mtrix. This is rther esy for system of three equtions. However, systems of liner equtions frequently consist of hundreds of equtions. Entering the coefficient mtrix in the Commnd Window for such lrge system is not only tedious, but is frught with the potentil for error. Wht would hppen if one coefficient in the 0,000 required coefficients for system of 00 equtions is entered incorrectly? Would you even ctch such n error? Similr issues re encountered with entering the right hnd side vector in the Commnd Window. To void these problems, engineers (nd others) typiclly crete text files with the coefficients. Rther thn hve two text files for problem (one contining the coefficients nd the other contining the right hnd side vector), the two sets of numbers re generlly combined into single file clled the ugmented coefficient mtrix file. The ugmented coefficient mtrix file contins the coefficient mtrix in rows nd columns with the right hnd side vector ppended s the (n+) th column, where n is the number of equtions in the system. For exmple, the ugmented coefficient mtrix for the qurry problem would look like where the first three rows nd columns re the coefficient mtrix nd the fourth (lst) column is the right hnd side vector. A file contining these numbers hs ll of the informtion necessry to get find solution to the set of equtions. (2) Creting text file with the ugmented coefficient mtrix. There re severl methods to crete n ugmented coefficient mtrix file. You could use your fvorite text editor such s Notepd or Wordpd (these will utomticlly sve s n ASCII text file). If you use word processor, such s Word, you will need to specify tht the file is to be sved s ASCII text when you sve it. You could lso use spredsheet, such s Excel, to crete the file. Using spredsheet hs the dvntge tht ech coefficient cn be loded into seprte cell. You could then export the file s ASCII text.

5 MATLAB: Workshop - Liner Systems of Equtions pge 5 Pick method nd crete n ASCII file with the qurry problem coefficients. Sve it s qurry.dt in your ctive MATLAB directory. The.dt extension for file is frequently used to denote n ASCII file tht contins dt. Loding n ugmented file into the Workspce An ugmented coefficient mtrix is esily loded into the MATLAB Workspce by using vrition of the lod commnd described in Workshop 4. The prticulr form of the commnd to lod dt file is vrnme = lod(filenme); where vrnme is the nme ssigned to hold the ugmented coefficient mtrix vlues nd filenme is string tht identifies the nme nd loction of the file. () Loding text file contining n ugmented coefficient mtrix. Lod the ugmented coefficient mtrix vlues into the Workspce by entering» ugmt = lod('qurry.dt') ugmt = Why did the vlues disply? How would you suppress disply? Notes on filenme filenme cn be hrdwired s shown bove (not very useful). filenme cn be vrible nme tht contins string s shown here: fnme = 'qurry.dt'; ugmt = lod(fnme); or, better yet, fnme = input('plese enter ugmented mtrix file nme ==> ','s'); ugmt = lod(fnme); If the required file is locted in the current directory, only the file nme (with extension) is required. If the required file is not in the current directory, the entire pth must be specified, i.e., c:\temp\qurry.dt. Extrcting the coefficient nd right hnd side vector from the ugmented mtrix Once the ugmented coefficient mtrix is in the Workspce, we need to extrct the coefficient mtrix nd the right hnd side vector. Although we hve been working with system of three equtions nd three unknowns, in generl, system of liner equtions my be of ny size. Fortuntely, MATLAB provides the tools to hndle this. (4) Extrcting the coefficient mtrix nd right hnd side vector. Determine the size of the system of equtions by entering» [nrows, ncols] = size(ugmt) nrows =

6 MATLAB: Workshop - Liner Systems of Equtions pge 6 ncols = 4 The number of rows, nrows, tells the number of equtions in the ugmented coefficient mtrix. ncols should be one greter thn nrows. Complete the following MATLAB commnds to extrct the coefficient mtrix nd right hnd side vector from the ugmented mtrix» Coeff = ugmt(????) Coeff = » rhs = ugmt(????) rhs = Refer to Workshop 2 if necessry. Crete script for solving systems of equtions Systems of liner equtions occur so frequently in engineering tht hving user friendly script for their solution would be useful. Remember the generl prdigm for script development: () get informtion (2) do computtions () report results. Becuse MATLAB cn solve system of liner equtions with single ssignment sttement, designing function to get the necessry informtion (i.e., problem nme, coefficient mtrix nd right hnd side vector) is the most complex prt of designing the script. The output function design is lso rther strightforwrd. (5) Design n informtion input function for liner eqution solver script. Following the design methodology of the previous workshops, design function tht will ) sk the user for the nme of the problem to be solved 2) sk the user for the nme/loction of the ugmented coefficient mtrix file ) lod the ugmented coefficient mtrix file 4) extrct the coefficient mtrix 5) extrct the rhs vector The function should return the problem nme, coefficient mtrix, nd rhs vector (6) Solve liner system of equtions. Becuse MATLAB solves system of liner equtions with single ssignment sttement, there is no need to design seprte function for this tsk. (7) Disply the results.

7 MATLAB: Workshop - Liner Systems of Equtions pge 7 Design function tht will disply the results. To be user friendly, the results from solving the system of liner equtions should hve lbel prior to disply. From where might n pproprite lbel come? How might it be pssed to this function for use? (8) Crete the script lineqsolver. You now hve the requisite functions to crete script tht puts it ll together. Crete script, lineqsolver.m, tht does the following displys script heder informtion (cll to lineqsolver_heder) gets the necessry input informtion (cll to lineqsolver_info) solves the system of equtions (simple ssignment sttement) displys the results (cll to lineqsolver_results) Be sure your script hs n pproprite heder section nd vrible dictionry. Bonus: Add while loop to your script so tht the script sks the user whether to repet for nother system of liner equtions nd, if the nswer is yes, repets the input, clcultion, disply portion of the script. Exercises: Use your script, lineqsolver, to solve the following problems.. Six people, Alice, Brb, Crol, Den, Eric, nd Fred, ech hve bg of money. Find the mount in ech bg if: The totl of ll six bgs is $.56; Crol hs twice s much money s Alice; Brb nd Crol together hve s much s Den; Alice nd Brb together hve s much s Eric; Eric s mount subtrcted from Fred s mount is twice Alice s mount; nd $.5 is left fter subtrcting Brb s mount nd Eric s mount from Fred s mount. 2. Find the five-digit number tht stisfies the following properties: The sum of ll the digits is 8; The third digit is the sum of the first nd second digits; Subtrcting the third digit from the fourth digit yields 4; The first nd third digits dded together give the fifth digit; nd The first digit is twice the second digit.. Find the six-digit number tht stisfies the following properties: The first nd third digit sum to 0; The sum of the second, third, nd fifth digits equls the fourth digit; The third nd lst digits re the sme; The sum of the second nd third digit equls the fifth digit; The fourth digit is the sme s the lst digit minus the first digit; nd The sum of ll the digits is Five people, Alice, Ben, Cindy, Den, nd Evn, decide to invest in the stock mrket. Wht is ech person s profit fter one yer if:

8 MATLAB: Workshop - Liner Systems of Equtions pge 8 The sum of ll their profits is $299.25; Evn s profit is three times s lrge s Den s profit; Ben s nd Cindy s profits together totl to Den s profit; Alice nd Cindy together mde $89.20 profit; nd Alice s profit ws $0.55 more thn Den s profit. Recp: You should hve lerned Wht n ugmented coefficient mtrix is nd its reltion to system of liner equtions. How to use the lod commnd to retrieve n ugmented coefficient file. How to use the size commnd to determine the number of equtions. How to extrct the coefficient mtrix nd rhs vector from the ugmented coefficient mtrix. How to use the bckslsh opertor, \, to solve system of equtions.