Gaussian Elimination Method : Without Pivoting, with Partial Pivoting & Complete Pivoting

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Lucy Jefferson
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1 بدون محورگیري روش حذفی گاوس با محورگیري جزي ی و کامل Gaussian Elimination Method : Without Pivoting, with Partial Pivoting & Complete Pivoting 1392

2 جهت حل دستگاه معادلات خطی به روش حذفی گاوس ااراي ه شده است. در این نوشتار سه برنامه مطلب 1- برنامه فاقد محورگیري % this program presents Gaussian Elimination method for solving linear % system Ax=b, where A is a square matrix of order n without pivoting % and with backward substitution clc disp(' '); disp('input the coefficients of matrix [A,b]'); disp('sample : [a11 a12 a13 b1;a21 a22 a23 b2;a31 a32 a33 b3]'); A = input('input matrix [A,b] = '); tic % start Timer [n,m]=size(a); if det(a(:,1:m-1))==0 disp(' A is singular and there is no unique answer') else % Display (A Backslash b) for Compare With Gaussian Eliminations Method disp('a Backslash b [A\b] ='); disp(a(:,1:m-1)\(a(:,m)) )); % Gaussian Elimination Method Statement for k=1:n-1 for i=k+1:n b=a(i,k)/a(k,k) ); for j=1:m A(i,j)=A(i,,j)-b*A(k,j); % Backward Substitution x=zeros(n,1); x(n)=a(n,m)/a(n,n); for i=n-1:-1:1 s=0; for j=n:-1:i+1 s=s+a(i,j)*x(j) ); x(i)=(a(i,m)-s)/a(i,i); disp('upper Triangular A : '); disp(a); disp('final X result is [Gaussian Elimination Method] : '); disp(x); toc % Stop Timer and Display Run-time input the coefficients of matrix [A,b] Sample : [a11 a12 a13 b1;a21 a22 a23 b2;a31 a32 a33 b3] Input matrix [A,b] = [ ; ; ] A Backslash b [A\b] = Upper Triangular A :

3 Final X result is [Gaussian Elimination Method] : Elapsed time is seconds. 2- برنامه با محورگیري جزي ی % this program presents Gaussian Elimination method for solving linear % system Ax=b, where A is a square matrix of order n with partial pivoting % and backward substitution clc disp(' '); disp('input the coefficients of matrix [A,b]'); disp('sample : [a11 a12 a13 b1;a21 a22 a23 b2;a31 a32 a33 b3]'); A = input('input matrix [A,b] = '); tic % start Timer [m,n]=size(a); %m=n-1 if det(a(:,1:n-1))==0 disp(' A is singular and there is no unique answer') else % Display (A Backslash b) for Compare With Gaussian Eliminations Method disp('a Backslash b [A\b] ='); disp(a(:,1:n-1)\(a(:,n)) )); % Partial Pivoting Statement % Swap the Rows for j=1:m-1 A2 = A(j:m,j); r = find (A2>=max(A2),1,'first')+j-1; if r~=j k=a(r,:); A(r,:)=A(j,:); A(j,:)=k; % Gaussian Elimination Method Statement for i=j+1:m A(i,:)=A(i,:)-( (A(i,j)/A(j,j))*A(j,:); % Backward Substitution Statement x=zeros(m,1); x(m)=a(m,n)/a(m,m); for i=m-1:-1:1 s=0; for j=m:-1:i+1 s=s+a(i,j)*x(j) ); x(i)=(a(i,n)-s)/a(i,i); disp('upper Triangular A : '); disp(a); 3

4 disp('final X result is [Gaussian Elimination Method] : '); disp(x); toc % Stop Timer and Display Run-time input the coefficients of matrix [A,b] Sample : [a11 a12 a13 b1;a21 a22 a23 b2;a31 a32 a33 b3] Input matrix [A,b] = [ ; ; ] A Backslash b [A\b] = Upper Triangular A : Final X result is [Gaussian Elimination Method] : Elapsed time is seconds. 3- برنامه با محورگیري کامل % this program presents Gaussian Elimination method for solving linear % system Ax=b, where A is a square matrix of order n with complete pivoting % and backward substitution function G_E_2 clc disp(' '); A = input('input the coefficients of square matrix A = '); b = input('input the vector b Like [b1 b2 b3] = '); tic % start Timer [n,m]=size(a); if det(a)==0 n~=m disp(' A is singular Or A is not Square') else % Display (A Backslash b) for Compare With Gaussian Method disp('a Backslash b [A\b] ='); disp((a\(b)')'); for u = 1:n % Generate Array For Keep Permutation of X new_indis_x(u) = u; for i=1:n-1 % Complete Pivoting Statement % User-Defined Function that Return Index of largest absolute entry in (i:n,i:n) sub matrix % MaxMatrix([Input Matrix],[Start Row & Columns to Search]); % MaxMatrix is More Efficient & Flexible than [find(a>=max(a),1,'first')] ; [R_Max,C_Max] = MaxMatrix(A,i); % Swap Rows Statement if i~=r_max 4 Eliminations

5 k1=a(i,:); A(i,:)=A(R_Max,,:); A(R_Max,:)=k1; b1=b(i); b(i)=b(r_max); b(r_max)=b1; % Swap Columns Statement if i~=c_max k2=a(:,i); A(:,i)=A(:,C_Max); A(:,C_Max)=k2; % This Statement Apply Permutation of X on [new_indis_x] Array new_indis_x_temp=new_indis_x(i); new_indis_x(i)= =new_indis_x(c_max); new_indis_x(c_max)=new_indis_x_temp; % Gaussian Elimination Method Statement for j=i+1:n b(j)=b(j)-(a(j,,i)/a(i,i))*b(i); A(j,:)=A(j,:)-( (A(j,i)/A(i,i))*A(i,:); % Backward Substitution Statement x=zeros(n,1); % Calculate X(n) x(n)=b(n)/a(n,n); % Calculate X(n-1:1) for i=n-1:-1:1 s=0; for j=n-1:-1:i s=s+a(i,j+1)*x( (j+1); x(i)=(b(i)-s)/a(i,i); % This Statement Use Permutation of X in [new_indis_x] Array for Apply % True Permutation on [New_X] for o=1:n new_x(new_indis_x(o))=x(o) ; disp('upper Triangular A : '); disp(a); disp('final X result is [Gaussian Elimination Method] : '); disp(new_x); toc % Stop Timer and Display Run-time % User Defined Functions % Return Index of largest absolute entry in (i:n,i:n) sub matrix % MaxMatrix([Input Matrix],,[Start Row & Columns to Search]) ); function [indis_r,indis_c] = MaxMatrix(A,First_Indis) [m,n]=size(a); Best_Max = 0; for i=first_indis:m for j=first_indis:nn if abs(a(i,j)) > Best_Max 5

6 Best_Max = indis_r=i; indis_c=j; abs(a(i,j)); input the coefficients of square matrix A = [2 4 8;3 3 8;2 7 6] input the vector b Like [b1 b2 b3] = [6 3 1] A Backslash b [A\b] = Upper Triangular A : Final X result is [Gaussian Elimination Method] : Elapsed time is seconds. گاوس سه اشکال عمده وجود دارد : البته لازم به ذکر است در روش حذفی صفر شدن عضو محوي 1. رشد خطاي گرد کردن 2. بزرگ شدن درایه هاي ماتریس میانی به طوري که قابل ذخیره نباشند. 3. در پایدارسازي الگوریتم حذفی گاوس و حل مشکل صفر شدن عضو محوري و جلوگیري از رشد خطاي گرد کردن اهمیت محورگیري هاي جزي ی و کامل نمودار می گردد. براي جلوگیري از بزرگ شدن درایه هاي ماتریس میانی معمولا از روش Scaling استفاده می شود. با تشکر قاسمی فرد 6

Final Exam Sample Questions Signals and Systems Course Fall 91

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