document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192827>Question 1192827</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate the equations below using the Gauss-seidel Method, at-least 5 iterations.\r<BR>\n" );
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document.write( "3x + 2y + 4z = 12<BR>\n" );
document.write( "x + 5y + 2z = -9<BR>\n" );
document.write( "7x + 2y + z = 5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824793 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824793\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "3x + 2y + 4z = 12<BR>\n" );
document.write( "x + 5y + 2z = -9<BR>\n" );
document.write( "7x + 2y + z = 5<br><BR>\n" );
document.write( "I have to admit that this method is new to me; I have never run across it before.  Thanks for submitting a question that lets me learn something new.<br><BR>\n" );
document.write( "After viewing a couple of online videos, it appears the method is well defined.<br><BR>\n" );
document.write( "(1) Rearrange the equations so that the system is in \"diagonally dominant\" form:<br><BR>\n" );
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document.write( "NOTE: See the response from tutor @Edwin for a description of \"diagonally dominant form\"....<br><BR>\n" );
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document.write( "7x + 2y + z = 5<BR>\n" );
document.write( "x + 5y + 2z = -9<BR>\n" );
document.write( "3x + 2y + 4z = 12<br><BR>\n" );
document.write( "(2) Solve the first equation for x; the second for y; and the third for z:<br><BR>\n" );
document.write( "[1] x=(5-2y-z)/7<BR>\n" );
document.write( "[2] y=(-9-x-2z)/5<BR>\n" );
document.write( "[3] z=(12-3x-2y)/4<br><BR>\n" );
document.write( "(3) Choose random starting values x(0), y(0), and z(0) for the three variables.<br><BR>\n" );
document.write( "(4) Use the most recent values of y and z in [1] to compute the next iteration of x;<BR>\n" );
document.write( "(5) Use the most recent values of x and z in [2] to compute the next iteration of y;<BR>\n" );
document.write( "(6) Use the most recent values of x and y in [3] to compute the next iteration of z;<BR>\n" );
document.write( "(7) Repeat (4) to (6) until the desired number of iterations have been made.<br><BR>\n" );
document.write( "The calculations are impossible by hand; and they are extremely awkward using a calculator.  However, they are quite easy to perform using a spreadsheet program.<br><BR>\n" );
document.write( "Here is a description of how to do that.<br><BR>\n" );
document.write( "Make column A the iteration number: A1=0, A2=1, ..., A6=5 (since you are doing 5 iterations).<BR>\n" );
document.write( "Columns B, C, and D will be the iterated values of x, y, and z.  Put your initial random guesses (iteration 0) in row 1: B1=0; C1=0; D1=0. (you can use any numbers there....)<BR>\n" );
document.write( "Cell B2 will contain the first iteration of x, using equation [1] and the most recent values of variables y and z:  B2=(5-2*C1-D1)/7.<BR>\n" );
document.write( "Cell C2 will contain the first iteration of y, using equation [2] and the most recent values of variables x and z:  C2=(-9-B2-2*C1)/5.<BR>\n" );
document.write( "Cell D2 will contain the first iteration of z, using equation [3] and the most recent values of variables x and y:  D2=(12-3*B2-2*C2)/4<br><BR>\n" );
document.write( "Now for the beauty of excel....<br><BR>\n" );
document.write( "To compute the remaining iterations 2 through 5, simply copy the formulas in cells B2 to D2 down into rows 3 through 6 by highlighting cells B2 to D2 and using ctrl-D to drag those formulas into the other rows.<br><BR>\n" );
document.write( "I don't know how to copy the results of my excel file here, so I will recreate it....<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
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document.write( " iteration   x          y          z\r\n" );
document.write( "     0       0          0          0\r\n" );
document.write( "     1    0.714286  -1.94286   3.435714\r\n" );
document.write( "     2    0.778571  -3.33      4.081071\r\n" );
document.write( "     3    1.082704  -3.64897   4.012457\r\n" );
document.write( "     4    1.18364   -3.64171   3.933125\r\n" );
document.write( "     5    1.192899  -3.61183   3.91124</PRE><BR>\n" );
document.write( "So the solution after 5 iterations is<br><BR>\n" );
document.write( "(1.192899, -3.61183, 3.91124)<br><BR>\n" );
document.write( "The exact solution is (122/103, -371/103, 403/103)<br><BR>\n" );
document.write( "which to several decimal places is<br><BR>\n" );
document.write( "(1.184466, -3.60194, 3.912621)<br><BR>\n" );
document.write( "Note if you do this exercise in excel on your own and continue for about 14 iterations, the solutions will be exact to 5 or 6 decimal places.<br>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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