Mathematics by T. Arens, F. Hettlich, Ch. Karpfinger, U. Kockelkorn, K. Lichtenegger, H. Stachel (chapter 14: Linear systems of equations)restart; Solving linear system of equationsIn order to solve a linear system of equations by a computer algebra sysetm, we first have to enter the coefficient matrix. Therefore Maple provides the datatyp matrix . A:=matrix(4,3,[[1, 0, 1],[1, 2, 0],[1, 1, 1],[2,1,0]]);The first argument sets the number of rows, thus the number of equations in our system. The second arguments defines the number of columns equivilant to the number of unknowns. The third argument collects all coefficients of the system as a list of lists (the rows).We may access single elements of the matrix. For exampe the third entry in the second row byA[2,3];Now we like to work on the linear system of equations from the appllication of page 479 in the Book. We begin by entering the coefficient matrix. The columns are defined by arranging the six unknows in the following order 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 coefficient matrix looks likeA:=matrix(6,6,[[-2040,-2040,2020,0,1140,0],[2040,0,-2020,-2020,0,1140],[-2450,-2450,2570,0,1380,0],[2450,0,-2570,-2570,0,1380],[-4280,-4280,2960,0,1560,0],[4280,0,-2960,-2960,0,1560]]);The rhs vector of the LSE is still required and is also entered in matrix form.b:=matrix(6,1,[-300,-120,-340,-180,-810,-50]);We are almost ready to apply the Gauss elimination method. The program and lots of other tools for linear algebra are part of the linalg package. So we load the package:with(linalg):(Remark: If we terminate the command with a semicolon instead of the colon a list of all included commands will be returned. Give them a look, may be you already know some of them.)Now a lot of commands to work with LSEs are available. A solution of the LSE is determined by the command linsolve .l:=linsolve(A,b);We would like to get the result as floating point numbers, so we tryevalf(l);... that doesn't work! If we like to regard the matrix l , we have to evaluate with evalm first, thus commands of the formevalm(l);
evalf(evalm(l));are necessary. Now we can compare our result with the book.With the command rank the rank of a matrix is determined, so that we can verify whether the system is solvable by comparingrank(A);with the rank of the extended coefficient matrix. Therefore we assemble the coefficient matrix and the rhs with the command concat (or augment) horizontally to one matrix and obtainconcat(A,b);
rank(concat(A,b));(Remark: If the entries of the matrix are floating point numbers the rank evaluation is done by a singular value decomposition. The procedur will later be explained in the book.)Since we are working with a CAS, we may even solve systems depending on parameters similar to the example on page 474. Let us consider the 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,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 .We define the coefficient matrix and the rhs:A:=matrix(3,3,[[1,1,0],[1,c,3],[0,1,1]]);
b:=matrix(3,1,[1, 1, d]);and get the resultlinsolve(A,b);This is a formal solution, the critical case for 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 is not examined seperately. We substitute c by 4A:=subs(c=4,evalm(A));and regard the new resultlinsolve(A,b);Since there is no solution if d is unequal to 0, we do not get a result. But with 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b:=subs(d=0,evalm(b));
linsolve(A,b);Maple introduced a help variable for the infinite solutions of the system of equations. All these solutions are part of the parametric solution (elements of the straight line), while the help variable could be any arbitrary number.If necessary the help variables are included in the solution set, if there are less equations than unknows in the system. For example two help variables forA:=matrix(2,4,[[1,2,3,4],[0,1,1,1]]);
b:=matrix(2,1,[1, 1]);
linsolve(A,b); Single Gaussian elimination stepsMistakes easily appear while solving systems of equations, so it is often helpful to have single gaussian steps checked by Maple. Therefore the command addrow(A,r1,r2,p) is used. It substitudes the row r2 by r2+p*r1 . For example we get A:= matrix(3,3,[[1, 0, 1],[1, 2, 0],[1, 1, 1]]);if we add 3 times the second row to the first rowaddrow(A,2,1,3);To easily check your own computations on the way two the reduced modification of the Gauss table, it makes sense to have an appropriate program. Maple's proc commands offers this option. Try to understand the steps of the program. The program lets you interactively choose the pivot elements.Copy the program into a new worksheet and use it to check your homework exercises.gjcheck:=proc(a,b)
local al, bl, piv, seperat, tmp, j:
#Eingabe testen
al:= a: bl:=b:
if type(al,array) then
seperat:=array([seq([`|`],j=1..rowdim(al))]):
if (type(bl,array) and rowdim(al)=rowdim(bl)) then
tmp := concat(al,bl);
print(evalm(concat(al,seperat,bl)));
piv:=readstat("Pivotelement ?"):
#Gauss Schritte
while (piv <> 0) do
tmp:=pivot(tmp,piv[1],piv[2]):
#Ausgabe und Eingabe des neuen Pivotelements
al:=submatrix(tmp,1..rowdim(al),1..coldim(al)):
bl:=submatrix(tmp,1..rowdim(bl),(coldim(al)+1)..coldim(tmp)):
print(evalm(concat(al,seperat,bl)));
piv:=readstat("Pivotelement ?"):
od:
"Tschuess";
else
"falsche Eingabe";
fi:
else
"falsche Eingabe";
fi:
end:(In particular the command pivot is used. It computes the according step by elementary linear row transformations)We define the rhsb:=matrix(3,1,[ 1,1,0]);Now we are ready to use the program. Therefore we enter the desired pivot element of the form " (3,1); " right after the question mark. If you enter " 0; " instead of the indices pair, the program will be terminated.gjcheck(A,b);(2,3);(1,1);0;(Note: You can also write procedurs to any external editor file and implement them by read(filename) in the current Maple session. Mind that after the end: a return has to be entered moving the cursor to the next line. Otherwise the code will not be executed.
To read data from external formated files, for example a matrix, the fscanf command is used, you probably already know from c-libaries)In our program we used two ways two enter matrices by the aid of operators.matrix(3,5,(i,j)->1/(i^2+j^2));Instead of a list, in this version a function defines how the (i,j)-th entry of the matrix is allocated. Exercises1. Solve the linear system of 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 ,the 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 ,as well as the complex LSE (see. Ex. 14.9)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 obtainA:=matrix(4,4,[[2, 1, 4, 3],[-1, 2, 1, -1],[3, 4, -1, -2],[4, 3, 2, 1]]);
b:=matrix(4,1,[0,4,0,0]);
linsolve(A,b);andA:=matrix(2,3,[[1, 2, -3],[3, -1, 2]]);
b:=matrix(2,1,[-1,7]);
linsolve(A,b);andA:=matrix(3,3,[[2, 0, I],[1, -3, -I],[I, 1, 1]]);
b:=matrix(3,1,[I,2*I,1+I]);
linsolve(A,b);2. By determining the rank of the coefficient matrix, verify whether the following LSE is solvable,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:=matrix(4,3,[[5, 3, -4],[1, 2, -3],[2, -1, 4],[4, 3, -2]]);
b:=matrix(4,1,[2,6,2,14]);
evalb(rank(A) = rank(concat(A,b)));
And the solutionlinsolve(A,b);3. Solve exercise 14.8 of the book, finding the solution of the 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 dependency of parameter a.SolutionA:=matrix(3,3,[[a+1, -a^2+6*a-9, a-2],[a^2-2*a-3, a^2-6*a+9, 3],[a+1, -a^2+6*a-9, a+1]]);
b:=matrix(3,1,[1,a-3,1]);
linsolve(A,b);Obviously the LSE is solvable for 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 . The command does not yield a complete analysis of the solution set. Thus it is still possible that infinite solutions exist. We obtain the different cases by doing row modifications of the extended coefficient matrix. For the case 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 and the two other special cases 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 and 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 we get the resultC:=subs(a=-1,evalm(A));
d:=subs(a=-1,evalm(b));
linsolve(C,d);C:=subs(a=2,evalm(A));
d:=subs(a=2,evalm(b));
linsolve(C,d);andC:=subs(a=3,evalm(A));
d:=subs(a=3,evalm(b));
linsolve(C,d);
For a=-1 the LSE has no solution. In the other two cases infinite solutions that are identified by the according help variables exist.