diff options
Diffstat (limited to '191')
160 files changed, 4227 insertions, 0 deletions
diff --git a/191/CH1/EX1.1/Example1_1.sce b/191/CH1/EX1.1/Example1_1.sce new file mode 100755 index 000000000..0c77d737c --- /dev/null +++ b/191/CH1/EX1.1/Example1_1.sce @@ -0,0 +1,30 @@ +//Illustrating that a small error in data provided can result in big errors.
+//with original equations
+//X+Y=2 & X+1.01Y=2.01
+clear;
+clc;
+close();
+A=[1 1;1 1.01];
+B=[2 2.01]';
+x=A\B;
+disp(x,'Solutions are :')
+x=linspace(-0.5,1.5);
+y1=2-x;
+y2=(2.01-x)/1.01;
+subplot(2,1,1);
+plot(x,y1)
+plot(x,y2,'r')
+xtitle('plot of correct equations','x axis','y axis')
+//with the equations having some error in data
+//X+Y=2 & X+1.01Y=2.02
+A=[1 1;1 1.01];
+B=[2 2.02]';
+x=A\B;
+disp(x,'Solutions are :')
+subplot(2,1,2);
+x=linspace(-1,1);
+y1=2-x;
+y2=(2.02-x)/1.01;
+plot(x,y1)
+plot(x,y2,'r')
+xtitle('plot of error having equations','x axis','y axis')
\ No newline at end of file diff --git a/191/CH1/EX1.1/Figure1_1.png b/191/CH1/EX1.1/Figure1_1.png Binary files differnew file mode 100755 index 000000000..15344ae76 --- /dev/null +++ b/191/CH1/EX1.1/Figure1_1.png diff --git a/191/CH1/EX1.1/Result1_1.txt b/191/CH1/EX1.1/Result1_1.txt new file mode 100755 index 000000000..8f46f665a --- /dev/null +++ b/191/CH1/EX1.1/Result1_1.txt @@ -0,0 +1,9 @@ + Solutions are :
+
+ 1.
+ 1.
+
+ Solutions are :
+
+ 0.
+ 2.
\ No newline at end of file diff --git a/191/CH1/EX1.4/Example1_4.sce b/191/CH1/EX1.4/Example1_4.sce new file mode 100755 index 000000000..7be2ed732 --- /dev/null +++ b/191/CH1/EX1.4/Example1_4.sce @@ -0,0 +1,11 @@ +//illustrating the induced instability through the deflation method of polynomial factorisation.
+clear;
+clc;
+close();
+x=poly(0,'x');
+p3=x^3-13*x^2+32*x-20;//Given Polynomial
+roots(p3)
+//suppose that an estimate of its largest zero is taken as 10.1.Now devide p3 by (x-10.1)
+p2=x^2-2.9*x+2.71;//the quotient
+roots(p2)
+disp('induced a large error in roots')
\ No newline at end of file diff --git a/191/CH1/EX1.4/Result1_4.txt b/191/CH1/EX1.4/Result1_4.txt new file mode 100755 index 000000000..3a052bc34 --- /dev/null +++ b/191/CH1/EX1.4/Result1_4.txt @@ -0,0 +1 @@ + induced a large error in roots
\ No newline at end of file diff --git a/191/CH2/EX2.1/Example2_1.sce b/191/CH2/EX2.1/Example2_1.sce new file mode 100755 index 000000000..ef643b23b --- /dev/null +++ b/191/CH2/EX2.1/Example2_1.sce @@ -0,0 +1,48 @@ +//Illutrates the effect of the partial pivoting using 3 significant //figure arithmetic with rounding
+//first done without pivoting and then with partial pivoting
+clear;
+close();
+clc;
+A=[0.610,1.23,1.72;1.02,2.15,-5.51;-4.34,11.2,-4.25];
+B=[0.792;12.0;16.3];
+C=[A,B];
+format('v',10);
+n=3;
+for k=1:n-1
+ for i=k+1:n
+ c=C(i,k)/C(k,k);
+ for j=k:n+1
+ C(i,j)=C(i,j)-c*C(k,j);
+ end
+ end
+end
+x3=C(3,4)/C(3,3);
+x2=(C(2,4)-C(2,3)*x3)/C(2,2);
+x1=(C(1,4)-C(1,3)*x3-C(1,2)*x2)/C(1,1);
+disp([x1,x2,x3],'Answers without partial pivoting : ')
+
+
+C=[A,B];
+format('v',5);
+n=3;
+for k=1:n-1
+ m = max(abs(A(:,k)));
+ for l=k:n
+ if C(l,k)==m
+ temp = C(l,:);
+ C(l,:)= C(k,:);
+ C(k,:)=temp;
+ break;
+ end
+ end
+ for i=k+1:n
+ c=C(i,k)/C(k,k);
+ for j=k:n+1
+ C(i,j)=C(i,j)-c*C(k,j);
+ end
+ end
+end
+x3=C(3,4)/C(3,3);
+x2=(C(2,4)-C(2,3)*x3)/C(2,2);
+x1=(C(1,4)-C(1,3)*x3-C(1,2)*x2)/C(1,1);
+disp([x1,x2,x3],'Answers using partial pivoting : ')
\ No newline at end of file diff --git a/191/CH2/EX2.1/Result2_1.txt b/191/CH2/EX2.1/Result2_1.txt new file mode 100755 index 000000000..ab29826d1 --- /dev/null +++ b/191/CH2/EX2.1/Result2_1.txt @@ -0,0 +1,3 @@ + Answers using partial pivoting :
+
+ 1.61 1.6 - 1.26
\ No newline at end of file diff --git a/191/CH2/EX2.2/Example2_2.sce b/191/CH2/EX2.2/Example2_2.sce new file mode 100755 index 000000000..99edbd5cc --- /dev/null +++ b/191/CH2/EX2.2/Example2_2.sce @@ -0,0 +1,25 @@ +//Illustrates the decomposition of every matrix into product of lower //and upper triangular matrix if diagonal elements of any one is '1' //then L and U could uniquely be determined.
+clear;
+close();
+clc;
+format('v',5);
+A = {4,-2,2;4,-3,-2;2,3,-1];
+L(1,1)=1;L(2,2)=1;L(3,3)=1;
+for i=1:3
+ for j=1:3
+ s=0;
+ if j>=i
+ for k=1:i-1
+ s=s+L(i,k)*U(k,j);
+ end
+ U(i,j)=A(i,j)-s;
+ else
+ for k=1:j-1
+ s=s+L(i,k)*U(k,j);
+ end
+ L(i,j)=(A(i,j)-s)/U(j,j);
+ end
+ end
+end
+disp(L,'L =')
+disp(U,'U =')
\ No newline at end of file diff --git a/191/CH2/EX2.2/Result2_2.txt b/191/CH2/EX2.2/Result2_2.txt new file mode 100755 index 000000000..674b9252f --- /dev/null +++ b/191/CH2/EX2.2/Result2_2.txt @@ -0,0 +1,12 @@ + L =
+
+ 1. 0. 0.
+ 1. 1. 0.
+ 0.5 - 4. 1.
+
+
+ U =
+
+ 4. - 2. 2.
+ 0. - 1. - 4.
+ 0. 0. - 18.
\ No newline at end of file diff --git a/191/CH2/EX2.3/Example2_3.sce b/191/CH2/EX2.3/Example2_3.sce new file mode 100755 index 000000000..8a05465fe --- /dev/null +++ b/191/CH2/EX2.3/Example2_3.sce @@ -0,0 +1,32 @@ +//Applying LU factorization method for solving the system of equation
+
+clear;
+close();
+clc;
+format('v',5);
+A = {4,-2,2;4,-3,-2;2,3,-1];
+for l=1:3
+ L(l,l)=1;
+end
+for i=1:3
+ for j=1:3
+ s=0;
+ if j>=i
+ for k=1:i-1
+ s=s+L(i,k)*U(k,j);
+ end
+ //disp(s,'sum :');
+ U(i,j)=A(i,j)-s;
+ else
+ //s=0;
+ for k=1:j-1
+ s=s+L(i,k)*U(k,j);
+ end
+ L(i,j)=(A(i,j)-s)/U(j,j);
+ end
+ end
+end
+b=[6;-8;5];
+c=L\b;
+x=U\c;
+disp(x,'Solution of equations :')
\ No newline at end of file diff --git a/191/CH2/EX2.3/Result2_3.txt b/191/CH2/EX2.3/Result2_3.txt new file mode 100755 index 000000000..06999f147 --- /dev/null +++ b/191/CH2/EX2.3/Result2_3.txt @@ -0,0 +1,5 @@ + Solution of equations :
+
+ 1.
+ 2.
+ 3.
\ No newline at end of file diff --git a/191/CH2/EX2.4/Example2_4.sce b/191/CH2/EX2.4/Example2_4.sce new file mode 100755 index 000000000..b80a44757 --- /dev/null +++ b/191/CH2/EX2.4/Example2_4.sce @@ -0,0 +1,33 @@ +//Application of LU factorisation method for solving the system of equation
+//In this case A(1,1)=0 so to avoid the division by 0 we will have to interchange the rows.
+
+clear;
+close();
+clc;
+format('v',5);
+A = {2,2,-2,4;0,1,5,3;1,5,7,-10;-1,1,6,-5];
+for l=1:4
+ L(l,l)=1;
+end
+for i=1:4
+ for j=1:4
+ s=0;
+ if j>=i
+ for k=1:i-1
+ s=s+L(i,k)*U(k,j);
+ end
+ //disp(s,'sum :');
+ U(i,j)=A(i,j)-s;
+ else
+ //s=0;
+ for k=1:j-1
+ s=s+L(i,k)*U(k,j);
+ end
+ L(i,j)=(A(i,j)-s)/U(j,j);
+ end
+ end
+end
+b=[4;-6;14;0];
+c=L\b;
+x=U\c;
+disp(x,'Solution of equations :')
\ No newline at end of file diff --git a/191/CH2/EX2.4/Result2_4.txt b/191/CH2/EX2.4/Result2_4.txt new file mode 100755 index 000000000..a904f8d84 --- /dev/null +++ b/191/CH2/EX2.4/Result2_4.txt @@ -0,0 +1,6 @@ + Solution of equations :
+
+ 1.
+ 2.
+ - 1.
+ - 1.
\ No newline at end of file diff --git a/191/CH2/EX2.5/Example2_5.sce b/191/CH2/EX2.5/Example2_5.sce new file mode 100755 index 000000000..9be1cf9ae --- /dev/null +++ b/191/CH2/EX2.5/Example2_5.sce @@ -0,0 +1,28 @@ +//Solving the problem using Choleski decomposition
+//Decomposition of a matrix "A" to L and L'
+
+clear;
+close();
+clc;
+format('v',7)
+A = [4,2,-2;2,10,2;-2,2,3];
+n = 3;
+for i = 1:n
+ for j = 1:i
+ s=0;
+ if i==j
+ for k = 1:j-1
+ s=s+L(j,k)*L(j,k);
+ end
+ L(j,j)= sqrt(A(j,j)-s);
+ else
+ for k = 1:j-1
+ s=s+L(i,k)*L(j,k);
+ end
+ L(i,j)= (A(i,j)-s)/L(j,j);
+ end
+ end
+end
+U = L';
+disp(L,'Lower triangular matrix is :')
+disp(U,'Upper triangular matrix is :')
diff --git a/191/CH2/EX2.5/Result2_5.txt b/191/CH2/EX2.5/Result2_5.txt new file mode 100755 index 000000000..fb02d620c --- /dev/null +++ b/191/CH2/EX2.5/Result2_5.txt @@ -0,0 +1,13 @@ + Lower triangular matrix is :
+
+ 2. 0. 0.
+ 1. 3. 0.
+ - 1. 1. 1.
+
+
+ Upper triangular matrix is :
+
+ 2. 1. - 1.
+ 0. 3. 1.
+ 0. 0. 1.
+
\ No newline at end of file diff --git a/191/CH2/EX2.6/Example2_6.sce b/191/CH2/EX2.6/Example2_6.sce new file mode 100755 index 000000000..cf16c43e0 --- /dev/null +++ b/191/CH2/EX2.6/Example2_6.sce @@ -0,0 +1,47 @@ +//Solving the problem using Jacobi method
+//the first case in converging and the 2nd is diverging ..... drawback
+//of jacobi method
+//the ans is correct to 2D place
+
+clear;
+close();
+clc;
+format('v',7);
+x1=[0,0];
+x2=[0,0];
+x3=[0,0];
+x1(1,2)=0.2*(6-2*x2(1,1)+x3(1,1));
+x2(1,2)=0.16667*(4-x1(1,1)+3*x3(1,1));
+x3(1,2)=0.25*(7-2*x1(1,1)-x2(1,1));
+i=1;
+while (abs(x1(1,1)-x1(1,2))>0.5*10^-2 | abs(x2(1,1)-x2(1,2))>0.5*10^-2 | abs(x3(1,1)-x3(1,2))>0.5*10^-2 )
+ x1(1,1)=x1(1,2);
+ x2(1,1)=x2(1,2);
+ x3(1,1)=x3(1,2);
+ x1(1,2)=0.2*(6-2*x2(1,1)+x3(1,1));
+ x2(1,2)=0.16667*(4-x1(1,1)+3*x3(1,1));
+ x3(1,2)=0.25*(7-2*x1(1,1)-x2(1,1));
+ i=i+1;
+end
+disp([x1; x2; x3],'Answers are :')
+disp(i,'Number of Iterations :')
+
+
+x1=[0,0];
+x2=[0,0];
+x3=[0,0];
+x1(1,2)=4-6*x2(1,1)+3*x3(1,1);
+x2(1,2)=0.5*(6-5*x1(1,1)+x3(1,1));
+x3(1,2)=0.25*(7-2*x1(1,1)-x2(1,1));
+i=1;
+while (abs(x1(1,1)-x1(1,2))>0.5*10^-2 | abs(x2(1,1)-x2(1,2))>0.5*10^-2 | abs(x3(1,1)-x3(1,2))>0.5*10^-2 )
+ x1(1,1)=x1(1,2);
+ x2(1,1)=x2(1,2);
+ x3(1,1)=x3(1,2);
+ x1(1,2)=(4-6*x2(1,1)+3*x3(1,1));
+ x2(1,2)=0.5*(6-5*x1(1,1)+x3(1,1));
+ x3(1,2)=0.25*(7-2*x1(1,1)-x2(1,1));
+ i=i+1;
+end
+disp([x1; x2; x3],'Answers are :')
+disp(i,'Number of Iterations :')
diff --git a/191/CH2/EX2.6/Result2_6.txt b/191/CH2/EX2.6/Result2_6.txt new file mode 100755 index 000000000..7fd9f1901 --- /dev/null +++ b/191/CH2/EX2.6/Result2_6.txt @@ -0,0 +1,17 @@ + Answers are :
+
+ 0.9976 1.0007
+ 0.9976 0.9998
+ 0.9989 1.0018
+Number of Iterations :
+
+ 11.
+
+Answers are :
+
+ Inf Inf
+ -Inf -Inf
+ - 2.+307 Nan
+Number of Iterations :
+
+ 538.
\ No newline at end of file diff --git a/191/CH2/EX2.7/Example2_7.sce b/191/CH2/EX2.7/Example2_7.sce new file mode 100755 index 000000000..31fec40df --- /dev/null +++ b/191/CH2/EX2.7/Example2_7.sce @@ -0,0 +1,25 @@ +//the problem is solved using Gauss-Seidel method
+//it is fast convergent as compared to jacobi method
+
+clear;
+close();
+clc;
+format('v',7);
+x1=[0,0];
+x2=[0,0];
+x3=[0,0];
+x1(1,2)=0.2*(6-2*x2(1,1)+x3(1,1));
+x2(1,2)=0.16667*(4-x1(1,2)+3*x3(1,1));
+x3(1,2)=0.25*(7-2*x1(1,2)-x2(1,2));
+i=1;
+while (abs(x1(1,1)-x1(1,2))>0.5*10^-2 | abs(x2(1,1)-x2(1,2))>0.5*10^-2 | abs(x3(1,1)-x3(1,2))>0.5*10^-2 )
+ x1(1,1)=x1(1,2);
+ x2(1,1)=x2(1,2);
+ x3(1,1)=x3(1,2);
+ x1(1,2)=0.2*(6-2*x2(1,1)+x3(1,1));
+ x2(1,2)=0.16667*(4-x1(1,2)+3*x3(1,1));
+ x3(1,2)=0.25*(7-2*x1(1,2)-x2(1,2));
+ i=i+1;
+end
+disp([x1; x2; x3],'Answers are :')
+disp(i,'Number of Iterations :')
diff --git a/191/CH2/EX2.7/Result2_7.txt b/191/CH2/EX2.7/Result2_7.txt new file mode 100755 index 000000000..7e850efe7 --- /dev/null +++ b/191/CH2/EX2.7/Result2_7.txt @@ -0,0 +1,9 @@ + Answers are :
+
+ 1.0031 0.9990
+ 1.0015 0.9992
+ 0.9981 1.0007
+
+Number of Iterations :
+
+ 7.
\ No newline at end of file diff --git a/191/CH2/EX2.8/Example2_8.sce b/191/CH2/EX2.8/Example2_8.sce new file mode 100755 index 000000000..c393eb799 --- /dev/null +++ b/191/CH2/EX2.8/Example2_8.sce @@ -0,0 +1,27 @@ +//The method used to solve is SOR(Successive over-relaxation) method
+//only marginal improvement is possible for this pasticular system since
+//Gauss-Seidel iteration itself converges quite rapidly
+
+clear;
+close();
+clc;
+format('v',7);
+x1=[0,0];
+x2=[0,0];
+x3=[0,0];
+w =0.9;
+x1(1,2)=x1(1,1)+0.2*w*(6-5*x1(1,1)-2*x2(1,1)+x3(1,1));
+x2(1,2)=x2(1,1)+0.16667*w*(4-x1(1,2)-6*x2(1,1)+3*x3(1,1));
+x3(1,2)=x3(1,1)+0.25*w*(7-2*x1(1,2)-x2(1,2)-4*x3(1,1));
+i=1;
+while (abs(x1(1,1)-x1(1,2))>0.5*10^-2 | abs(x2(1,1)-x2(1,2))>0.5*10^-2 | abs(x3(1,1)-x3(1,2))>0.5*10^-2 )
+ x1(1,1)=x1(1,2);
+ x2(1,1)=x2(1,2);
+ x3(1,1)=x3(1,2);
+ x1(1,2)=x1(1,1)+0.2*w*(6-5*x1(1,1)-2*x2(1,1)+x3(1,1));
+ x2(1,2)=x2(1,1)+0.16667*w*(4-x1(1,2)-6*x2(1,1)+3*x3(1,1));
+ x3(1,2)=x3(1,1)+0.25*w*(7-2*x1(1,2)-x2(1,2)-4*x3(1,1));
+ i=i+1;
+end
+disp([x1; x2; x3],'Answers are:')
+disp(i,'Number of Iterations :')
diff --git a/191/CH2/EX2.8/Result2_8.txt b/191/CH2/EX2.8/Result2_8.txt new file mode 100755 index 000000000..6c1b94343 --- /dev/null +++ b/191/CH2/EX2.8/Result2_8.txt @@ -0,0 +1,9 @@ + Answers are:
+
+ 1.0058 1.0021
+ 0.9947 0.9982
+ 0.9979 0.9992
+
+Number of Iterations :
+
+ 6.
\ No newline at end of file diff --git a/191/CH2/EX2.9/Example2_9.sce b/191/CH2/EX2.9/Example2_9.sce new file mode 100755 index 000000000..a6a83da54 --- /dev/null +++ b/191/CH2/EX2.9/Example2_9.sce @@ -0,0 +1,54 @@ +//Solving four linear system of equations with Gauss-Seidel and SOR method
+//the convergence is much faster in SOR method
+
+clear;
+close();
+clc;
+format('v',7);
+x1=[0,0];
+x2=[0,0];
+x3=[0,0];
+x4=[0,0];
+x1(1,2)=-0.33333*(1-x2(1,1)-3*x4(1,1));
+x2(1,2)=0.16667*(1-x1(1,2)-x3(1,1));
+x3(1,2)=0.16667*(1-x2(1,2)-x4(1,1));
+x4(1,2)=-0.33333*(1-3*x1(1,2)-x3(1,2));
+i=1;
+while (abs(x1(1,1)-x1(1,2))>0.5*10^-2 | abs(x2(1,1)-x2(1,2))>0.5*10^-2 | abs(x3(1,1)-x3(1,2))>0.5*10^-2 | abs(x4(1,1)-x4(1,2))>0.5*10^-2)
+ x1(1,1)=x1(1,2);
+ x2(1,1)=x2(1,2);
+ x3(1,1)=x3(1,2);
+ x4(1,1)=x4(1,2);
+ x1(1,2)=-0.33333*(1-x2(1,1)-3*x4(1,1));
+ x2(1,2)=0.16667*(1-x1(1,2)-x3(1,1));
+ x3(1,2)=0.16667*(1-x2(1,2)-x4(1,1));
+ x4(1,2)=-0.33333*(1-3*x1(1,2)-x3(1,2));
+ i=i+1;
+end
+disp([x1; x2; x3; x4],'Answers are:')
+disp(i,'Number of Iterations :')
+
+
+w=1.6;
+x1=[0,0];
+x2=[0,0];
+x3=[0,0];
+x4=[0,0];
+x1(1,2)=x1(1,1)-0.33333*w*(1+3*x1(1,1)-x2(1,1)-3*x4(1,1));
+x2(1,2)=x2(1,1)+0.16667*w*(1-x1(1,2)-6*x2(1,2)-x3(1,1));
+x3(1,2)=x3(1,1)+0.16667*w*(1-x2(1,2)-6*x3(1,2)-x4(1,1));
+x4(1,2)=x4(1,1)-0.33333*w*(1-3*x1(1,2)-x3(1,2)+3*x4(1,1));
+i=1;
+while (abs(x1(1,1)-x1(1,2))>0.5*10^-2 | abs(x2(1,1)-x2(1,2))>0.5*10^-2 | abs(x3(1,1)-x3(1,2))>0.5*10^-2 | abs(x4(1,1)-x4(1,2))>0.5*10^-2)
+ x1(1,1)=x1(1,2);
+ x2(1,1)=x2(1,2);
+ x3(1,1)=x3(1,2);
+ x4(1,1)=x4(1,2);
+ x1(1,2)=x1(1,1)-0.33333*w*(1+3*x1(1,1)-x2(1,1)-3*x4(1,1));
+ x2(1,2)=x2(1,1)+0.16667*w*(1-x1(1,2)-6*x2(1,2)-x3(1,1));
+ x3(1,2)=x3(1,1)+0.16667*w*(1-x2(1,2)-6*x3(1,2)-x4(1,1));
+ x4(1,2)=x4(1,1)-0.33333*w*(1-3*x1(1,2)-x3(1,2)+3*x4(1,1));
+ i=i+1;
+end
+disp([x1; x2; x3; x4],'Answers are :')
+disp(i,'Number of Iterations :')
\ No newline at end of file diff --git a/191/CH2/EX2.9/Result2_9.txt b/191/CH2/EX2.9/Result2_9.txt new file mode 100755 index 000000000..d7ac2d2ed --- /dev/null +++ b/191/CH2/EX2.9/Result2_9.txt @@ -0,0 +1,21 @@ + Answers are:
+
+ - 5.9516 - 5.9561
+ 0.9933 0.9939
+ 0.9927 0.9934
+ - 5.954 - 5.9583
+
+ Number of Iterations :
+
+ 49.
+
+Answers are :
+
+ - 6.0029 - 6.0003
+ 0.9999 1.0000
+ 1.0005 1.
+ - 6.0012 - 5.9997
+
+ Number of Iterations :
+
+ 15.
\ No newline at end of file diff --git a/191/CH3/EX3.1/Example3_1.sce b/191/CH3/EX3.1/Example3_1.sce new file mode 100755 index 000000000..5a09ca883 --- /dev/null +++ b/191/CH3/EX3.1/Example3_1.sce @@ -0,0 +1,35 @@ +//Bisection Method
+clc;
+clear;
+close();
+format('v',9);
+b(1)=1;a(1)=0;k=5;
+deff('[fx]=bisec(x)','fx =(x+1).^2.*exp(x.^2-2)-1');
+x = linspace(0,1);
+plot(x,((x+1).^2).*(exp(x.^2-2))-1);
+//in interval [0,1]changes its sign thus has a root
+//k = no of decimal place of accuracy
+//a = lower limit of interval
+//b = upper limit of interval
+//n = no of iterations required
+n = log2((10^k)*(b-a));
+n = ceil(n);
+disp(n,'Number of iterations : ')
+for i = 1:n-1
+ N(i) = i;
+ c(i) = (a(i)+b(i))/2;
+ bs(i) = bisec(c(i));
+ if (bisec(b(i))*bisec(c(i))<0)
+ a(i+1)=c(i);
+ b(i+1)=b(i);
+ else
+ b(i+1)=c(i);
+ a(i+1)=a(i);
+ end
+end
+N(i+1)=i+1;
+c(i+1) = (a(i+1)+b(i+1))/2;
+bs(i+1) = bisec(c(i));
+ann = [N a b c bs];
+disp(ann , 'The Table : ');
+disp(c(i),'The root of the function is : ')
\ No newline at end of file diff --git a/191/CH3/EX3.1/Figure3_1.png b/191/CH3/EX3.1/Figure3_1.png Binary files differnew file mode 100755 index 000000000..55872eef9 --- /dev/null +++ b/191/CH3/EX3.1/Figure3_1.png diff --git a/191/CH3/EX3.1/Result3_1.txt b/191/CH3/EX3.1/Result3_1.txt new file mode 100755 index 000000000..c78d978b7 --- /dev/null +++ b/191/CH3/EX3.1/Result3_1.txt @@ -0,0 +1,28 @@ +
+ Number of iterations :
+
+ 17.
+
+ The Table :
+
+ 1. 0. 1. 0.5 - 0.609009
+ 2. 0.5 1. 0.75 - 0.272592
+ 3. 0.75 1. 0.875 0.023105
+ 4. 0.75 0.875 0.8125 - 0.139662
+ 5. 0.8125 0.875 0.84375 - 0.062448
+ 6. 0.84375 0.875 0.859375 - 0.020775
+ 7. 0.859375 0.875 0.867188 0.000881
+ 8. 0.859375 0.867188 0.863281 - 0.010017
+ 9. 0.863281 0.867188 0.865234 - 0.004585
+ 10. 0.865234 0.867188 0.866211 - 0.001857
+ 11. 0.866211 0.867188 0.866699 - 0.000489
+ 12. 0.866699 0.867188 0.866943 0.000196
+ 13. 0.866699 0.866943 0.866821 - 0.000147
+ 14. 0.866821 0.866943 0.866882 0.000025
+ 15. 0.866821 0.866882 0.866852 - 0.000061
+ 16. 0.866852 0.866882 0.866867 - 0.000018
+ 17. 0.866867 0.866882 0.866875 - 0.000018
+
+ The root of the function is :
+
+ 0.866867
\ No newline at end of file diff --git a/191/CH3/EX3.2/Example3_2.sce b/191/CH3/EX3.2/Example3_2.sce new file mode 100755 index 000000000..706f6bb6f --- /dev/null +++ b/191/CH3/EX3.2/Example3_2.sce @@ -0,0 +1,35 @@ +//The solution using false position method
+clc;
+clear;
+close();
+b(1)=1;a(1)=0;k=5;i=1;
+format('v',9);
+deff('[fx]=bisec(x)','fx =(x+1)^2*exp(x^2-2)-1');
+x = linspace(0,1);
+plot(x,((x+1).^2).*(exp(x.^2-2))-1);
+//in interval [0,1]changes its sign thus has a root
+//k = no of decimal place of accuracy
+//a = lower limit of interval
+//b = upper limit of interval
+c(i) = (a(i)*bisec(b(i))-b(i)*bisec(a(i)))/(bisec(b(i))-bisec(a(i)));
+bs(1)=bisec(c(1));
+N(1) = 1;
+a(i+1)=c(i);
+b(i+1)=b(i);
+while abs(bisec(c(i)))>(0.5*(10^-k))
+ i = i+1;
+ N(i)=i;
+ c(i) = (a(i)*bisec(b(i))-b(i)*bisec(a(i)))/(bisec(b(i))-bisec(a(i)));
+ bs(i) = bisec(c(i));
+ if (bisec(b(i))*bisec(c(i))<0)
+ a(i+1)=c(i);
+ b(i+1)=b(i);
+ else
+ b(i+1)=c(i);
+ a(i+1)=a(i);
+ end
+end
+a(10)=[];b(10)=[];
+ann = [N a b c bs];
+disp(ann , 'The Table : ');
+disp('The root of the function is bracketed by [0.647116 1] ');
\ No newline at end of file diff --git a/191/CH3/EX3.2/Figure3_2.png b/191/CH3/EX3.2/Figure3_2.png Binary files differnew file mode 100755 index 000000000..9303d1a44 --- /dev/null +++ b/191/CH3/EX3.2/Figure3_2.png diff --git a/191/CH3/EX3.2/Result3_2.txt b/191/CH3/EX3.2/Result3_2.txt new file mode 100755 index 000000000..847340152 --- /dev/null +++ b/191/CH3/EX3.2/Result3_2.txt @@ -0,0 +1,14 @@ +
+ The Table :
+
+ 1. 0. 1. 0.647116 - 0.441885
+ 2. 0.647116 1. 0.817834 - 0.127033
+ 3. 0.817834 1. 0.856496 - 0.028617
+ 4. 0.856496 1. 0.864707 - 0.006057
+ 5. 0.864707 1. 0.866422 - 0.001265
+ 6. 0.866422 1. 0.866780 - 0.000263
+ 7. 0.866780 1. 0.866854 - 0.000055
+ 8. 0.866854 1. 0.866869 - 0.000011
+ 9. 0.866869 1. 0.866873 - 0.000002
+
+ The root of the function is bracketed by [0.647116 1]
\ No newline at end of file diff --git a/191/CH3/EX3.3/Example3_3.sce b/191/CH3/EX3.3/Example3_3.sce new file mode 100755 index 000000000..0cd02ef26 --- /dev/null +++ b/191/CH3/EX3.3/Example3_3.sce @@ -0,0 +1,40 @@ +//We have quadratic equation x^2-2*x-8=0 with roots -2 and 4
+//for solving it we use fixed point iteration method for that we rearrange it in 3 ways.
+//first way x=(2*x+8)^(1/2)
+//here x0 is chosen arbitrarily
+
+clear;
+clc;
+close();
+format('v',5)
+funcprot(0);
+deff('[fixed_point]=fx(x)','fixed_point=(2*x+8)^0.5')
+x0=5;
+while abs(x0-fx(x0))>0.5*10^(-2)
+ x0=fx(x0);
+end
+disp(x0,'root is :')
+
+//second way x=(2*x+8)/x
+
+format('v',5)
+funcprot(0);
+deff('[fixed_point]=fx(x)','fixed_point=(2*x+8)/x')
+x0=5;
+while abs(x0-fx(x0))>0.5*10^(-2)
+ x0=fx(x0);
+end
+disp(x0,'root is :')
+
+//third way x=(x^2-8)/2
+
+format('v',10)
+funcprot(0);
+deff('[fixed_point]=fx(x)','fixed_point=(x^2-8)/2')
+x0=5;
+for i=1:5
+ x0=fx(x0);
+ disp(x0,'value is :')
+end
+disp(x0,'As you can see that the root is not converging.So this method is not applicable.')
+
diff --git a/191/CH3/EX3.3/Result3_3.txt b/191/CH3/EX3.3/Result3_3.txt new file mode 100755 index 000000000..09b6b4261 --- /dev/null +++ b/191/CH3/EX3.3/Result3_3.txt @@ -0,0 +1,10 @@ +(i)
+ root is :
+
+ 4.
+(ii)
+ root is :
+
+ 4.
+(iii)
+root is not converging.So this method is not applicable.
\ No newline at end of file diff --git a/191/CH3/EX3.4/Example3_4.sce b/191/CH3/EX3.4/Example3_4.sce new file mode 100755 index 000000000..0dc785a70 --- /dev/null +++ b/191/CH3/EX3.4/Example3_4.sce @@ -0,0 +1,40 @@ +//checking for the convergence and divergence of different functions we are getting after rearrangement of the given quadratic equation x^2-2*x-8=0.
+//after first type of arrangement we get a function gx=(2*x+8)^(1/2).for this we have..
+
+clear;
+clc;
+close();
+alpha=4;
+I=alpha-1:alpha+1;//required interval
+deff('[f1]=gx(x)','f1=(2*x+8)^(1/2)');
+deff('[f2]=diffgx(x)','f2=(2*x+8)^(-0.5)');
+x=linspace(3,5);
+subplot(2,1,1);
+plot(x,(2*x+8)^(1/2))
+plot(x,x)
+x0=5;
+if diffgx(I)>0
+ disp('Errors in two consecutive iterates are of same sign so convergence is monotonic')
+end
+if abs(diffgx(x0))<1
+ disp('So this method converges')
+end
+
+//after second type of arrangement we get a function gx=(2*x+8)/x.for this we have..
+
+deff('[f1]=gx(x)','f1=(2*x+8)/x');
+deff('[f2]=diffgx(x)','f2=(-8)/(x^2)');
+x=linspace(1,5);
+for i=1:100
+ y(1,i)=2+8/x(1,i);
+end
+subplot(2,1,2);
+plot(x,y)
+plot(x,x)
+x0=5;
+if diffgx(I)<0
+ disp('Errors in two consecutive iterates are of opposite sign so convergence is oscillatory')
+end
+if abs(diffgx(x0))<1
+ disp('So this method converges')
+end
diff --git a/191/CH3/EX3.4/Figure3_4.png b/191/CH3/EX3.4/Figure3_4.png Binary files differnew file mode 100755 index 000000000..9d261d93e --- /dev/null +++ b/191/CH3/EX3.4/Figure3_4.png diff --git a/191/CH3/EX3.4/Result3_4.txt b/191/CH3/EX3.4/Result3_4.txt new file mode 100755 index 000000000..7964935a7 --- /dev/null +++ b/191/CH3/EX3.4/Result3_4.txt @@ -0,0 +1,2 @@ +(i) Errors in two consecutive iterates are of same sign so convergence is monotonic
+(ii) Errors in two consecutive iterates are of opposite sign so convergence is oscillatory.
\ No newline at end of file diff --git a/191/CH3/EX3.5/Example3_5.sce b/191/CH3/EX3.5/Example3_5.sce new file mode 100755 index 000000000..782e72a77 --- /dev/null +++ b/191/CH3/EX3.5/Example3_5.sce @@ -0,0 +1,37 @@ +//Newton's Method
+//the first few iteration converges quikcly in negative root as compared to positive root
+clc;
+clear;
+close();
+funcprot(0);
+format('v',9);
+deff('[Newton]=fx(x)','Newton=exp(x)-x-2');
+deff('[diff]=gx(x)','diff=exp(x)-1');
+x = linspace(-2.5,1.5);
+plot(x,exp(x)-x-2)
+//from the graph the function has 2 roots
+//considering the initial negative root -10
+x1 = -10;
+x2 = x1-fx(x1)/gx(x1);
+i=0;
+while abs(x1-x2)>(0.5*10^-7)
+ x1=x2;
+ x2 = x1-fx(x1)/gx(x1);
+ i=i+1;
+end
+disp(i,'Number of iterations : ')
+disp(x2,'The negative root of the function is : ')
+
+
+//considering the initial positive root 10
+x1 = 10;
+x2 = x1-fx(x1)/gx(x1);
+i=0;
+while abs(x1-x2)>(0.5*10^-7)
+ x1=x2;
+ x2 = x1-fx(x1)/gx(x1);
+ i=i+1;
+end
+disp(i,'Number of iteration : ')
+disp(x2,'The positive root of the function is : ')
+//number of iterations showing fast and slow convergent
\ No newline at end of file diff --git a/191/CH3/EX3.5/Figure3_5.png b/191/CH3/EX3.5/Figure3_5.png Binary files differnew file mode 100755 index 000000000..a14280044 --- /dev/null +++ b/191/CH3/EX3.5/Figure3_5.png diff --git a/191/CH3/EX3.5/Result3_5.txt b/191/CH3/EX3.5/Result3_5.txt new file mode 100755 index 000000000..189069cc1 --- /dev/null +++ b/191/CH3/EX3.5/Result3_5.txt @@ -0,0 +1,17 @@ +
+ Number of iterations :
+
+ 4.
+
+ The negative root of the function is :
+
+ - 1.841406
+
+ Number of iteration :
+
+ 13.
+
+ The positive root of the function is :
+
+ 1.146193
+
\ No newline at end of file diff --git a/191/CH3/EX3.6/Example3_6.sce b/191/CH3/EX3.6/Example3_6.sce new file mode 100755 index 000000000..3ac4c3f2d --- /dev/null +++ b/191/CH3/EX3.6/Example3_6.sce @@ -0,0 +1,51 @@ +//Secant Method
+//the first few iteration converges quikcly in negative root as compared to positive root
+clc;
+clear;
+close();
+funcprot(0);
+format('v',9);
+deff('[Secant]=f(x)','Secant=exp(x)-x-2');
+x = linspace(0,1.5);
+subplot(2,1,1);
+plot(x,exp(x)-x-2);
+plot(x,0);
+//from the graph the function has 2 roots
+//considering the initial negative root -10
+x0 = -10
+x1 = -9;
+x2 = (x0*f(x1)-x1*f(x0))/(f(x1)-f(x0));
+i=0;
+while abs(x1-x2)>(0.5*10^-7)
+ x0=x1;
+ x1=x2;
+ x2 = (x0*f(x1)-x1*f(x0))/(f(x1)-f(x0));
+ i=i+1;
+end
+disp(i,'Number of iterations : ')
+disp(x2,'The negative root of the function is : ')
+
+
+//considering the initial positive root 10
+subplot(2,1,2);
+x = linspace(-2.5,0);
+plot(x,exp(x)-x-2);
+plot(x,0);
+x0 = 10
+x1 = 9;
+x2 = (x0*f(x1)-x1*f(x0))/(f(x1)-f(x0));
+i=0;
+while abs(x1-x2)>(0.5*10^-7)
+ x0=x1;
+ x1=x2;
+ x2 = (x0*f(x1)-x1*f(x0))/(f(x1)-f(x0));
+ i=i+1;
+end
+disp(i,'Number of iteration : ')
+disp(x2,'The positive root of the function is : ')
+//number of iterations showing fast and slow convergent
+
+format('v',6)
+//Order of secant method (p)
+ p = log(31.52439)/log(8.54952);
+ disp(p,'Order of Secant Method : ')
\ No newline at end of file diff --git a/191/CH3/EX3.6/Figure3_6.png b/191/CH3/EX3.6/Figure3_6.png Binary files differnew file mode 100755 index 000000000..d9b1f5e69 --- /dev/null +++ b/191/CH3/EX3.6/Figure3_6.png diff --git a/191/CH3/EX3.6/Result3_6.txt b/191/CH3/EX3.6/Result3_6.txt new file mode 100755 index 000000000..b017e4d29 --- /dev/null +++ b/191/CH3/EX3.6/Result3_6.txt @@ -0,0 +1,20 @@ +
+ Number of iterations :
+
+ 5.
+
+ The negative root of the function is :
+
+ - 1.841406
+
+ Number of iteration :
+
+ 18.
+
+ The positive root of the function is :
+
+ 1.146193
+
+ Order of Secant Method :
+
+ 1.608
\ No newline at end of file diff --git a/191/CH3/EX3.7/Example3_7.sce b/191/CH3/EX3.7/Example3_7.sce new file mode 100755 index 000000000..153500b6c --- /dev/null +++ b/191/CH3/EX3.7/Example3_7.sce @@ -0,0 +1,32 @@ +//Non-Linear Equation
+clc;
+clear;
+close();
+funcprot(0);
+format('v',9);
+i = 1;
+deff('[func1]=f(x,y)','func1=x^2+y^2-4');
+deff('[func2]=g(x,y)','func2=2*x-y^2');
+deff('[difffx]=fx(x)','difffx=2*x');
+deff('[difffy]=fy(y)','difffy=2*y');
+deff('[diffgx]=gx(x)','diffgx=2');
+deff('[diffgy]=gy(y)','diffgy=-2*y');
+x1(i)=1;y1(i)=1;
+J = [fx(x1(i)),fy(y1(i));gx(x1(i)),gy(y1(i))];
+n=[x1(i);y1(i)]-inv(J)*[f(x1(i),y1(i));g(x1(i),y1(i))];
+x2(i)=n(1,1);y2(i)=n(2,1);
+N(1)=i-1;
+while (abs(x2(i)-x1(i))>0.5*10^-7) | (abs(y2(i)-y1(i))>0.5*10^-7)
+ i=i+1;
+ N(i)=i-1;
+ x1(i)=x2(i-1);
+ y1(i)=y2(i-1);
+ J = [fx(x1(i)),fy(y1(i));gx(x1(i)),gy(y1(i))];
+ n=[x1(i);y1(i)]-inv(J)*[f(x1(i),y1(i));g(x1(i),y1(i))];
+ x2(i)=n(1,1);y2(i)=n(2,1);
+end
+N(i+1)=i;
+x1(i+1) = x2(i);
+y1(i+1) = y2(i);
+n = [N x1 y1];
+disp(n,'The value of n x and y :')
\ No newline at end of file diff --git a/191/CH3/EX3.7/Result3_7.txt b/191/CH3/EX3.7/Result3_7.txt new file mode 100755 index 000000000..9d256e2bd --- /dev/null +++ b/191/CH3/EX3.7/Result3_7.txt @@ -0,0 +1,9 @@ +
+ The value of n x and y :
+
+ 0. 1. 1.
+ 1. 1.25 1.75
+ 2. 1.236111 1.581349
+ 3. 1.236068 1.572329
+ 4. 1.236068 1.572303
+ 5. 1.236068 1.572303
\ No newline at end of file diff --git a/191/CH3/EX3.8/Example3_8.sce b/191/CH3/EX3.8/Example3_8.sce new file mode 100755 index 000000000..d1325132a --- /dev/null +++ b/191/CH3/EX3.8/Example3_8.sce @@ -0,0 +1,34 @@ +//Non-Linear Equation
+clc;
+clear;
+close();
+funcprot(0);
+format('v',9);
+deff('[func1]=f(x1,x2)','func1=-2.0625*x1+2*x2-0.0625*x1^3+0.5');
+deff('[func2]=g(x1,x2,x3)','func2=x3-2*x2+x1-0.0625*x2^3+0.125*x2*(x3-x1)');
+deff('[func3]=h(x2,x3,x4)','func3=x4-2*x3+x2-0.0625*x3^3+0.125*x3*(x4-x2)');
+deff('[func4]=k(x3,x4)','func4=-1.9375*x4+x3-0.0625*x4^3-0.125*x3*x4+0.5');
+//define the corresponding partial differenciation of the function = 16
+deff('[difffx1]=fx1(x1)','difffx1=-2.0625-3*0.0625*x1^2');
+deff('[difffx2]=fx2(x2)','difffx2=2');
+
+deff('[diffgx1]=gx1(x2)','diffgx1=1-0.125*x2');
+deff('[diffgx2]=gx2(x1,x2,x3)','diffgx2=-2-3*0.0625*x2^2+0.125*(x3-x1)');
+deff('[diffgx3]=gx3(x2)','diffgx3=1+0.125*x2');
+
+deff('[diffhx2]=hx2(x3)','diffhx2=1-0.125*x3');
+deff('[diffhx3]=hx3(x3,x4)','diffhx3=-2-0.0625*3*x3^2+0.125*x4');
+deff('[diffhx4]=hx4(x3)','diffhx4 = 1+0.125*x3');
+
+deff('[diffkx3]=kx3(x4)','diffkx3=1-0.125*x4');
+deff('[diffkx4]=kx4(x3,x4)','diffkx4=-1.9375-3*0.0625*x4^2-0.125*x3');
+
+x = [1.5 1.25 1.0 0.75]';
+for i=1:6
+ N(i)=i-1;
+ x1(i) = x(1);x2(i)=x(2);x3(i) = x(3);x4(i)=x(4);
+ J = [fx1(x(1)),fx2(x(2)),0,0;gx1(x(2)),gx2(x(1),x(2),x(3)),gx3(x(2)),0;0,hx2(x(3)),hx3(x(3),x(4)),hx4(x(3));0,0,kx3(x(4)),kx4(x(3),x(4))];
+ x = x - inv(J)*[f(x(1),x(2));g(x(1),x(2),x(3));h(x(2),x(3),x(4));k(x(3),x(4))];
+end
+n = [N x1 x2 x3 x4];
+disp(n,'The values of N x1 x2 x3 x4 respectively : ');
\ No newline at end of file diff --git a/191/CH3/EX3.8/Result3_8.txt b/191/CH3/EX3.8/Result3_8.txt new file mode 100755 index 000000000..e2ec3b9f7 --- /dev/null +++ b/191/CH3/EX3.8/Result3_8.txt @@ -0,0 +1,10 @@ +
+ The values of N x1 x2 x3 x4 respectively :
+
+ 0. 1.5 1.25 1. 0.75
+ 1. 1.056454 0.851376 0.684750 0.584436
+ 2. 0.979212 0.788578 0.659124 0.568218
+ 3. 0.979061 0.788984 0.660975 0.569006
+ 4. 0.978892 0.788795 0.660741 0.568901
+ 5. 0.978913 0.788819 0.660771 0.568914
+
\ No newline at end of file diff --git a/191/CH4/EX4.1/Example4_1.sce b/191/CH4/EX4.1/Example4_1.sce new file mode 100755 index 000000000..79593dae9 --- /dev/null +++ b/191/CH4/EX4.1/Example4_1.sce @@ -0,0 +1,17 @@ +//The Power Method of finding largest Eigen value of given matrix
+clear;
+clc;
+close();
+A=[3 0 1;2 2 2;4 2 5]; //Given Matrix
+u0=[1 1 1]'; //Intial vector
+v=A*u0;
+a=max(u0);
+while abs(max(v)-a)>0.05 //for accuracy
+ a=max(v);
+ u0=v/max(v);
+ v=A*u0;
+end
+format('v',4);
+disp(max(v),'Eigen value :')
+format('v',5);
+disp(u0,'Eigen vector :')
\ No newline at end of file diff --git a/191/CH4/EX4.1/Result4_1.txt b/191/CH4/EX4.1/Result4_1.txt new file mode 100755 index 000000000..6b7b1abed --- /dev/null +++ b/191/CH4/EX4.1/Result4_1.txt @@ -0,0 +1,9 @@ + Eigen value :
+
+ 7.
+
+ Eigen vector :
+
+ 0.25
+ 0.5
+ 1.
\ No newline at end of file diff --git a/191/CH4/EX4.10/Example4_10.sce b/191/CH4/EX4.10/Example4_10.sce new file mode 100755 index 000000000..e6771b545 --- /dev/null +++ b/191/CH4/EX4.10/Example4_10.sce @@ -0,0 +1,16 @@ +//Householder Matrix +clc; +clear; +close(); +format('v',7); +e = [1;0;0]; +x = [-1;1;4]; +disp(e , 'e = '); +disp(x , 'x = '); +//considering the positive k according to sign convention +k = sqrt(x'*x); +disp(k,'k = '); +u = x - k*e; +disp(u,'u = '); +Q = eye(3,3) - 2*u*u'/(u'*u); +disp(Q,'Householder Matrix : ')
\ No newline at end of file diff --git a/191/CH4/EX4.10/Result4_10.txt b/191/CH4/EX4.10/Result4_10.txt new file mode 100755 index 000000000..1e49606ac --- /dev/null +++ b/191/CH4/EX4.10/Result4_10.txt @@ -0,0 +1,28 @@ +
+ e =
+
+ 1.
+ 0.
+ 0.
+
+ x =
+
+ - 1.
+ 1.
+ 4.
+
+ k =
+
+ 4.2426
+
+ u =
+
+ - 5.2426
+ 1.
+ 4.
+
+ Householder Matrix :
+
+ - 0.2357 0.2357 0.9428
+ 0.2357 0.9550 - 0.1798
+ 0.9428 - 0.1798 0.2807
\ No newline at end of file diff --git a/191/CH4/EX4.11/Example4_11.sce b/191/CH4/EX4.11/Example4_11.sce new file mode 100755 index 000000000..da9bd7dd0 --- /dev/null +++ b/191/CH4/EX4.11/Example4_11.sce @@ -0,0 +1,28 @@ +//Householder methods
+clc;
+clear;
+close();
+format('v',7);
+A = [2 -1 1 4;-1 3 1 2;1 1 5 -3;4 2 -3 6];
+disp(A, 'A = ');
+n=4;
+for r=1:n-2
+ x = A(r+1:n,r);
+ f = eye(n-r,n-r);
+ e = f(:,1)
+ I = eye(r,r);
+ O(1:n-r,r) = 0;
+ //calculating Q
+ k = sqrt(x'*x);
+ u = x - k*e;
+ Q = eye(n-r,n-r) - 2*u*u'/(u'*u);
+ //substituting in P
+ P(1:r,1:r)= I;
+ P(r+1:n,1:r)=0;
+ P(1:r,r+1:n)=0;
+ P(r+1:n,r+1:n)=Q;
+ A = P*A*P;
+ disp(A,Q,P,'The P Q and A matrix are ; ')
+end
+C = A;
+disp(C,'The tridiagonal matrix by householder method is : ')
\ No newline at end of file diff --git a/191/CH4/EX4.11/Result4_11.txt b/191/CH4/EX4.11/Result4_11.txt new file mode 100755 index 000000000..ccb902b53 --- /dev/null +++ b/191/CH4/EX4.11/Result4_11.txt @@ -0,0 +1,46 @@ +
+ A =
+
+ 2. - 1. 1. 4.
+ - 1. 3. 1. 2.
+ 1. 1. 5. - 3.
+ 4. 2. - 3. 6.
+
+ The P Q and A matrix are ;
+
+ 1. 0. 0. 0.
+ 0. - 0.2357 0.2357 0.9428
+ 0. 0.2357 0.9550 - 0.1798
+ 0. 0.9428 - 0.1798 0.2807
+
+ - 0.2357 0.2357 0.9428
+ 0.2357 0.9550 - 0.1798
+ 0.9428 - 0.1798 0.2807
+
+ 2. 4.2426 - 1.D-16 0.
+ 4.2426 3.4444 - 2.2729 2.9293
+ - 1.D-16 - 2.2729 6.2324 - 0.7448
+ 0. 2.9293 - 0.7448 4.3232
+
+ The P Q and A matrix are ;
+
+ 1. 0. 0. 0.
+ 0. 1. 0. 0.
+ 0. 0. - 0.6130 0.7901
+ 0. 0. 0.7901 0.6130
+
+ - 0.6130 0.7901
+ 0.7901 0.6130
+
+ 2. 4.2426 7.D-17 - 9.D-17
+ 4.2426 3.4444 3.7077 - 4.D-16
+ 7.D-17 3.7077 5.7621 - 1.1097
+ - 9.D-17 - 4.D-16 - 1.1097 4.7934
+
+ The tridiagonal matrix by householder method is :
+
+ 2. 4.2426 7.D-17 - 9.D-17
+ 4.2426 3.4444 3.7077 - 4.D-16
+ 7.D-17 3.7077 5.7621 - 1.1097
+ - 9.D-17 - 4.D-16 - 1.1097 4.7934
+
\ No newline at end of file diff --git a/191/CH4/EX4.12/Example4_12.sce b/191/CH4/EX4.12/Example4_12.sce new file mode 100755 index 000000000..c205c7fae --- /dev/null +++ b/191/CH4/EX4.12/Example4_12.sce @@ -0,0 +1,16 @@ +//stable LR method
+clc;
+clear;
+close();
+format('v',7);
+A = [2 1 3 1;-1 2 2 1;1 0 1 0;-1 -1 -1 1];
+disp(A, 'A = ');
+for i = 1:6
+ [L,R,P]= lu(A);
+ A = R*P*L;
+ disp(A,R,L,'The L R and A matrix are : ');
+end
+disp(A,'The (1,1) and (4,4) elements have converged to real eigenvalues')
+X = [A(2,2) A(2,3);A(3,2) A(3,3)];
+E = spec(X);
+disp(E,'Although submatrix themselves are not converging their eigen values converges.')
\ No newline at end of file diff --git a/191/CH4/EX4.12/Result4_12.txt b/191/CH4/EX4.12/Result4_12.txt new file mode 100755 index 000000000..4718926fd --- /dev/null +++ b/191/CH4/EX4.12/Result4_12.txt @@ -0,0 +1,124 @@ +
+ A =
+
+ 2. 1. 3. 1.
+ - 1. 2. 2. 1.
+ 1. 0. 1. 0.
+ - 1. - 1. - 1. 1.
+
+ The L R and A matrix are :
+
+ 1. 0. 0. 0.
+ - 0.5 1. 0. 0.
+ - 0.5 - 0.2 1. 0.
+ 0.5 - 0.2 0.1667 1.
+
+ 2. 1. 3. 1.
+ 0. 2.5 3.5 1.5
+ 0. 0. 1.2 1.8
+ 0. 0. 0. - 0.5
+
+ 2.5 0.2 1.5 3.
+ - 0.25 1.5 2.0833 3.5
+ - 0.3 - 0.6 2. 1.2
+ 0.25 0.1 - 0.5 0.
+
+ The L R and A matrix are :
+
+ 1. 0. 0. 0.
+ - 0.1 1. 0. 0.
+ - 0.12 - 0.3789 1. 0.
+ 0.1 0.0526 - 0.2536 1.
+
+ 2.5 0.2 1.5 3.
+ 0. 1.52 2.2333 3.8
+ 0. 0. 3.0263 3.
+ 0. 0. 0. 0.2609
+
+ 2.6 - 0.2105 0.7391 3.
+ - 0.04 0.8737 1.2696 3.8
+ - 0.0632 - 0.9889 2.2654 3.
+ 0.0261 0.0137 - 0.0662 0.2609
+
+ The L R and A matrix are :
+
+ 1. 0. 0. 0.
+ - 0.0243 1. 0. 0.
+ - 0.0154 - 0.8757 1. 0.
+ 0.0100 - 0.0159 - 0.0113 1.
+
+ 2.6 - 0.2105 0.7391 3.
+ 0. - 0.9940 2.2834 3.0729
+ 0. 0. 3.2804 6.537
+ 0. 0. 0. 0.3538
+
+ 2.6154 0.8757 - 0.2445 3.
+ - 0.0093 3.1049 - 1.0289 3.0729
+ - 0.0141 3.1763 - 0.0741 6.537
+ 0.0036 - 0.0056 - 0.0040 0.3538
+
+ The L R and A matrix are :
+
+ 1. 0. 0. 0.
+ - 0.0054 1. 0. 0.
+ - 0.0036 0.9771 1. 0.
+ 0.0014 - 0.0021 0.0040 1.
+
+ 2.6154 0.8757 - 0.2445 3.
+ 0. 3.181 - 0.0754 6.5531
+ 0. 0. - 0.9561 - 3.3192
+ 0. 0. 0. 0.3772
+
+ 2.6176 0.6046 0.8877 3.
+ - 0.0021 3.0185 3.2073 6.5531
+ 0.0006 - 0.9489 - 0.0133 - 3.3192
+ 0.0005 - 0.0008 0.0015 0.3772
+
+ The L R and A matrix are :
+
+ 1. 0. 0. 0.
+ - 0.0008 1. 0. 0.
+ 0.0002 - 0.3144 1. 0.
+ 0.0002 - 0.0003 0.0023 1.
+
+ 2.6176 0.6046 0.8877 3.
+ 0. 3.019 3.208 6.5555
+ 0. 0. 0.9949 - 1.2591
+ 0. 0. 0. 0.3815
+
+ 2.618 0.3246 0.8947 3.
+ - 0.0003 2.0085 3.2233 6.5555
+ - 1.D-17 - 0.3124 0.992 - 1.2591
+ 7.D-05 - 0.0001 0.0009 0.3815
+
+ The L R and A matrix are :
+
+ 1. 0. 0. 0.
+ - 0.0001 1. 0. 0.
+ - 4.D-18 - 0.1555 1. 0.
+ 3.D-05 - 6.D-05 0.0007 1.
+
+ 2.618 0.3246 0.8947 3.
+ 0. 2.0085 3.2234 6.5559
+ 0. 0. 1.4933 - 0.2395
+ 0. 0. 0. 0.3820
+
+ 2.618 0.1853 0.8969 3.
+ - 4.D-05 1.5067 3.2281 6.5559
+ - 7.D-06 - 0.2323 1.4932 - 0.2395
+ 1.D-05 - 2.D-05 0.0003 0.3820
+
+ The (1,1) and (4,4) elements have converged to real eigenva
+ lues
+
+ 2.618 0.1853 0.8969 3.
+ - 4.D-05 1.5067 3.2281 6.5559
+ - 7.D-06 - 0.2323 1.4932 - 0.2395
+ 1.D-05 - 2.D-05 0.0003 0.3820
+
+ Although submatrix themselves are not converging their eige
+ n values converges.
+
+ 1.5 + 0.8658i
+ 1.5 - 0.8658i
+
\ No newline at end of file diff --git a/191/CH4/EX4.13/Example4_13.sce b/191/CH4/EX4.13/Example4_13.sce new file mode 100755 index 000000000..13af20e4e --- /dev/null +++ b/191/CH4/EX4.13/Example4_13.sce @@ -0,0 +1,31 @@ +//Orthogonal decomposition - QR method
+//reduce A to tridiagonal form
+clc;
+clear;
+close();
+format('v',7);
+A1 = [1 4 2;-1 2 0;1 3 -1];
+disp(A1, 'A = ');
+// zero is created in lower triangle
+//by taking the rotation matrix X1=[c s 0;-s c 0;0 0 1]; where c=cos and s=sin
+//O is theta
+
+Q = eye(3,3);
+for i=2:3
+ for j=1:i-1
+ p=i;q=j;
+ O = -atan(A1(p,q)/(A1(q,q)));
+ c = cos(O);
+ s = sin(O);
+ X = eye(3,3);
+ X(p,p)=c;
+ X(q,q)=c;
+ X(p,q)=-s;
+ X(q,p)=s;
+ A1 = X'*A1;
+ Q = Q*X;
+ disp(A1,X,'The X and A matrix : ');
+ end
+end
+R = A1;
+disp(R,Q,'Hence the original matrix can be decomposed as : ')
\ No newline at end of file diff --git a/191/CH4/EX4.13/Result4_13.txt b/191/CH4/EX4.13/Result4_13.txt new file mode 100755 index 000000000..961550a90 --- /dev/null +++ b/191/CH4/EX4.13/Result4_13.txt @@ -0,0 +1,47 @@ +
+ A =
+
+ 1. 4. 2.
+ - 1. 2. 0.
+ 1. 3. - 1.
+
+ The X and A matrix :
+
+ 0.7071 0.7071 0.
+ - 0.7071 0.7071 0.
+ 0. 0. 1.
+
+ 1.4142 1.4142 1.4142
+ - 1.D-16 4.2426 1.4142
+ 1. 3. - 1.
+
+ The X and A matrix :
+
+ 0.8165 0. - 0.5774
+ 0. 1. 0.
+ 0.5774 0. 0.8165
+
+ 1.7321 2.8868 0.5774
+ - 1.D-16 4.2426 1.4142
+ 0. 1.633 - 1.633
+
+ The X and A matrix :
+
+ 1. 0. 0.
+ 0. 0.9333 - 0.3592
+ 0. 0.3592 0.9333
+
+ 1.7321 2.8868 0.5774
+ - 1.D-16 4.5461 0.7332
+ 4.D-17 0. - 2.032
+
+ Hence the original matrix can be decomposed as :
+
+ 0.5774 0.5133 - 0.635
+ - 0.5774 0.8066 0.127
+ 0.5774 0.2933 0.762
+
+ 1.7321 2.8868 0.5774
+ - 1.D-16 4.5461 0.7332
+ 4.D-17 0. - 2.032
+
\ No newline at end of file diff --git a/191/CH4/EX4.14/Example4_14.sce b/191/CH4/EX4.14/Example4_14.sce new file mode 100755 index 000000000..1925ccf13 --- /dev/null +++ b/191/CH4/EX4.14/Example4_14.sce @@ -0,0 +1,34 @@ +//Redduction to upper Hessenberg form
+clc;
+clear;
+close();
+format('v',7);
+A1 = [4 2 1 -3;2 4 1 -3;3 2 2 -3;1 2 1 0];
+disp(A1, 'A = ' );
+//the element with largest modulus below diagonal in first column need to be at the top and then similarly for column 2
+A1=gsort(A1,'lr');
+temp = A1(:,3);
+A1(:,3) = A1(:,2);
+A1(:,2) = temp;
+M1 = eye(4,4);
+M1(3,2) = A1(3,1)/A1(2,1);
+M1(4,2) = A1(4,1)/A1(2,1);
+A2 = inv(M1)*A1*M1;
+disp(A2,M1, 'M1 and A2 : ')
+A2=gsort(A2,'lr');
+temp = A2(:,3);
+A2(:,3) = A2(:,4);
+A2(:,4) = temp;
+M2 = eye(4,4);
+M2(4,3) = A2(4,2)/A2(3,2);
+A3 = inv(M2)*A2*M2;
+disp(M2,'M2 = ');
+disp(A3,'Upper Hessenberg Matrix :')
+
+
+//for i=2:n
+// M =eye(4,4);
+// for j=i+1:n
+// M(j,i) = A(j,)
+// end
+//end
\ No newline at end of file diff --git a/191/CH4/EX4.14/Result4_14.txt b/191/CH4/EX4.14/Result4_14.txt new file mode 100755 index 000000000..a2ad145c2 --- /dev/null +++ b/191/CH4/EX4.14/Result4_14.txt @@ -0,0 +1,34 @@ +
+ A =
+
+ 4. 2. 1. - 3.
+ 2. 4. 1. - 3.
+ 3. 2. 2. - 3.
+ 1. 2. 1. 0.
+
+ M1 and A2 :
+
+ 1. 0. 0. 0.
+ 0. 1. 0. 0.
+ 0. 0.6667 1. 0.
+ 0. 0.3333 0. 1.
+
+ 4. 1.3333 2. - 3.
+ 3. 2.3333 2. - 3.
+ 0. 1.1111 2.6667 - 1.
+ 0. 1.5556 1.3333 1.
+
+ M2 =
+
+ 1. 0. 0. 0.
+ 0. 1. 0. 0.
+ 0. 0. 1. 0.
+ 0. 0. 0.7143 1.
+
+ Upper Hessenberg Matrix :
+
+ 4. 1.3333 - 1.5714 2.
+ 3. 2.3333 - 1.5714 2.
+ 0. 1.5556 1.9524 1.3333
+ 0. 0. - 0.4898 1.7143
+
\ No newline at end of file diff --git a/191/CH4/EX4.15/Example4_15.sce b/191/CH4/EX4.15/Example4_15.sce new file mode 100755 index 000000000..7de6c501d --- /dev/null +++ b/191/CH4/EX4.15/Example4_15.sce @@ -0,0 +1,37 @@ +//Redduction to upper Hessenberg form and calculating eigen values
+clc;
+clear;
+close();
+format('v',7);
+A1 = [4 2 1 -3;2 4 1 -3;3 2 2 -3;1 2 1 0];
+//the element with largest modulus below diagonal in first column need to be at the top and then similarly for column 2
+A1=gsort(A1,'lr');
+temp = A1(:,3);
+A1(:,3) = A1(:,2);
+A1(:,2) = temp;
+M1 = eye(4,4);
+M1(3,2) = A1(3,1)/A1(2,1);
+M1(4,2) = A1(4,1)/A1(2,1);
+A2 = inv(M1)*A1*M1;
+
+A2=gsort(A2,'lr');
+temp = A2(:,3);
+A2(:,3) = A2(:,4);
+A2(:,4) = temp;
+M2 = eye(4,4);
+M2(4,3) = A2(4,2)/A2(3,2);
+A3 = inv(M2)*A2*M2;
+H = A3;
+disp(H,'Upper Hessenberg Matrix :')
+l =0;
+for i=4:-1:1
+ K =H(1:i,1:i);
+ while abs(K(i,i)-l)>0.005
+ l=K(i,i);
+ [Q,R]=qr(K-K(i,i)*eye(i,i));
+ K = R*Q + K(i,i)*eye(i,i);
+ end
+ l = 0;
+ EV(i) = K(i,i);
+end
+disp(EV,'Eigen Values : ')
\ No newline at end of file diff --git a/191/CH4/EX4.15/Result4_15.txt b/191/CH4/EX4.15/Result4_15.txt new file mode 100755 index 000000000..22335d36b --- /dev/null +++ b/191/CH4/EX4.15/Result4_15.txt @@ -0,0 +1,14 @@ +
+ Upper Hessenberg Matrix :
+
+ 4. 1.3333 - 1.5714 2.
+ 3. 2.3333 - 1.5714 2.
+ 0. 1.5556 1.9524 1.3333
+ 0. 0. - 0.4898 1.7143
+
+ Eigen Values :
+
+ 4.
+ 1.
+ 3.
+ 1.9999
\ No newline at end of file diff --git a/191/CH4/EX4.2/Example4_2.sce b/191/CH4/EX4.2/Example4_2.sce new file mode 100755 index 000000000..12be3d482 --- /dev/null +++ b/191/CH4/EX4.2/Example4_2.sce @@ -0,0 +1,18 @@ +//The Power Method of finding largest Eigen value of given matrix
+clear;
+clc;
+close();
+A=[3 0 1;2 2 2;4 2 5];
+new_A=A-7*eye(3,3); //Given Matrix
+u0=[1 1 1]'; //Intial vector
+v=new_A*u0;
+a=max(abs(u0));
+while abs(max(abs(v))-a)>0.005 //for accuracy
+ a=max(abs(v));
+ u0=v/max(abs(v));
+ v=new_A*u0;
+end
+format('v',5);
+disp(max(v),'Eigen value :')
+format('v',5);
+disp(u0,'Eigen vector :')
\ No newline at end of file diff --git a/191/CH4/EX4.2/Result4_2.txt b/191/CH4/EX4.2/Result4_2.txt new file mode 100755 index 000000000..90494f57e --- /dev/null +++ b/191/CH4/EX4.2/Result4_2.txt @@ -0,0 +1,9 @@ + Eigen value :
+
+ 5.78
+
+ Eigen vector :
+
+ - 0.51
+ - 0.96
+ 1.
\ No newline at end of file diff --git a/191/CH4/EX4.3/Example4_3.sce b/191/CH4/EX4.3/Example4_3.sce new file mode 100755 index 000000000..4747e2ac4 --- /dev/null +++ b/191/CH4/EX4.3/Example4_3.sce @@ -0,0 +1,25 @@ +//Convergence of Inverse Iteration
+clc;
+clear;
+close();
+format('v',4);
+A = [3 0 1;2 2 2; 4 2 5];
+e1 = 7.00;
+e2 = 1.02;
+p = sum(diag(A))-e1-e2;
+disp(A, 'A = ');
+A = A - p*eye(3,3);
+disp(A,'A-1.98I = ');
+L = [1 0 0; 0.50 1 0; 0.26 0.52 1];
+U = [4 2 3.02; 0 -.98 0.49; 0 0 -.03];
+disp(L,U,'The decomposition of A - 1.98I (L,U): ');
+u = [1,1,1]';
+I = inv(U)*inv(L);
+for i = 1:3
+ v = inv(U)*inv(L)*u;
+ disp(max(v),v,u,i-1,'The values of s u(s) v(s+1) and max(v(s+1)) : ');
+ u = v./max(v);
+end
+disp(u,'The Eigen Vector : ');
+ev = p+1/max(v);
+disp(ev,'The approx eigen value :');
\ No newline at end of file diff --git a/191/CH4/EX4.3/Result4_3.txt b/191/CH4/EX4.3/Result4_3.txt new file mode 100755 index 000000000..ba83a8d3f --- /dev/null +++ b/191/CH4/EX4.3/Result4_3.txt @@ -0,0 +1,74 @@ +
+ A =
+
+ 3. 0. 1.
+ 2. 2. 2.
+ 4. 2. 5.
+
+ A-1.98I =
+
+ 1. 0. 1.
+ 2. 0.0 2.
+ 4. 2. 3.
+
+ The decomposition of A - 1.98I (L,U):
+
+ 4. 2. 3.
+ 0. - 1.0 0.5
+ 0. 0. - 0.0
+
+ 1. 0. 0.
+ 0.5 1. 0.
+ 0.3 0.5 1.
+
+ The values of s u(s) v(s+1) and max(v(s+1)) :
+
+ 0.
+
+ 1.
+ 1.
+ 1.
+
+ 17.
+ - 8.5
+ - 16.
+
+ 17.
+
+ The values of s u(s) v(s+1) and max(v(s+1)) :
+
+ 1.
+
+ 1.
+ - 0.5
+ - 1.0
+
+ - 24.
+ 13.
+ 23.
+
+ 23.
+
+ The values of s u(s) v(s+1) and max(v(s+1)) :
+
+ 2.
+
+ - 1.
+ 0.5
+ 1.
+
+ 24.
+ - 13.
+ - 24.
+
+ 24.
+
+ The Eigen Vector :
+
+ 1.
+ - 0.5
+ - 1.0
+
+ The approx eigen value :
+
+ 2.
\ No newline at end of file diff --git a/191/CH4/EX4.4/Example4_4.sce b/191/CH4/EX4.4/Example4_4.sce new file mode 100755 index 000000000..dce228d08 --- /dev/null +++ b/191/CH4/EX4.4/Example4_4.sce @@ -0,0 +1,21 @@ +//Deflation
+clc;
+clear;
+close();
+A = [10 -6 -4; -6 11 2; -4 2 6];
+P = [1 0 0;-1 1 0;-0.5 0 1];
+disp(P,A,'The A and the P(transformation matrix) are : ');
+B = inv(P)*A*P;
+disp(B,'Hence B = ')
+C = B;
+C(1,:) = [];
+C(:,1) = [];
+disp(C,'The deflated matrix : ');
+Y = spec(C);
+disp(Y,'The matrix A therefore has eigen values : ');
+e1 = [1/3,1,-1/2]';
+e2 = [2/3,1,1]';
+disp(e1,e2,'The eigen values of B are : ');
+x1 = P*e1;
+x2 = P*e2;
+disp(3/2.*x1,3/2.*x2,'The eigen vextors of the orginal matrix A : ')
\ No newline at end of file diff --git a/191/CH4/EX4.4/Result4_4.txt b/191/CH4/EX4.4/Result4_4.txt new file mode 100755 index 000000000..aa90b7457 --- /dev/null +++ b/191/CH4/EX4.4/Result4_4.txt @@ -0,0 +1,46 @@ +
+ The A and the P(transformation matrix) are :
+
+ 10. - 6. - 4.
+ - 6. 11. 2.
+ - 4. 2. 6.
+
+ 1. 0. 0.
+ - 1. 1. 0.
+ - 0.5 0. 1.
+
+ Hence B =
+
+ 18. - 6. - 4.
+ 0. 5. - 2.
+ 0. - 1. 4.
+
+ The deflated matrix :
+
+ 5. - 2.
+ - 1. 4.
+
+ The matrix A therefore has eigen values :
+
+ 6.
+ 3.
+
+ The eigen values of B are :
+
+ 0.7
+ 1.
+ 1.
+
+ 0.3
+ 1.
+ - 0.5
+
+ The eigen vextors of the orginal matrix A :
+
+ 1.
+ 0.5
+ 1.
+
+ 0.5
+ 1.
+ - 1.
\ No newline at end of file diff --git a/191/CH4/EX4.5/Example4_5.sce b/191/CH4/EX4.5/Example4_5.sce new file mode 100755 index 000000000..35ce2d32c --- /dev/null +++ b/191/CH4/EX4.5/Example4_5.sce @@ -0,0 +1,44 @@ +//Threshold serial Jacobi Method
+//taking threshold values 0.5 and 0.05
+clc;
+clear;
+close();
+format('v',9);
+A = [3 0.4 5;0.4 4 0.1;5 0.1 -2];
+//for first cycle |0.4|<0.5 trasnformation is omitted
+//|5|>0.5 a zero is created at (1,3)
+//by taking the rotation matrix P1=[c 0 s; 0 1 0;-s 0 c]; where c=cos and s=sin
+//O is theta
+p=1;q=3;
+O = 0.5*atan(2*A(p,q)/(A(q,q)-A(p,p)));
+P1 = [cos(O) 0 sin(O);0 1 0;-sin(O) 0 cos(O)];
+A1 = A;
+A2 = inv(P1)*A*P1;
+//as all the off-diagonals < 0.5 the first cycle is complete
+disp(diag(A2),'The eigen values for case 1 : ')
+
+//second cycle for 0.05
+count =0;
+EV = P1;
+for i=1:3
+ for j=i+1:3
+ if A2(i,j)>0.05 then
+ p=i;q=j;
+ O = 0.5*atan(2*A2(p,q)/(A2(q,q)-A2(p,p)));
+ c = cos(O);
+ s = sin(O);
+ P = eye(3,3);
+ P(p,p)=c;
+ P(q,q)=c;
+ P(p,q)=s;
+ P(q,p)=-s;
+ A = inv(P)*A2*P;
+ disp(EV,'value of P*')
+ EV = EV * P;
+ count = count+1;
+ end
+ end
+end
+//eigen values are the diagonal elements of A and the column of P gives eigen vectors
+disp(diag(A),'Eigen values : ')
+disp(EV,'Correspoding eigen vectors : ')
\ No newline at end of file diff --git a/191/CH4/EX4.5/Result4_5.txt b/191/CH4/EX4.5/Result4_5.txt new file mode 100755 index 000000000..e56dd560e --- /dev/null +++ b/191/CH4/EX4.5/Result4_5.txt @@ -0,0 +1,25 @@ +
+ The eigen values for case 1 :
+
+ 6.09017
+ 4.
+ - 5.09017
+
+ value of P*
+
+ 0.850651 0. - 0.525731
+ 0. 1. 0.
+ 0.525731 0. 0.850651
+
+ Eigen values :
+
+ 6.161562
+ 3.928608
+ - 5.09017
+
+ Correspoding eigen vectors :
+
+ 0.836942 - 0.152102 - 0.525731
+ 0.178807 0.983884 0.
+ 0.517259 - 0.094004 0.850651
+
\ No newline at end of file diff --git a/191/CH4/EX4.6/Example4_6.sce b/191/CH4/EX4.6/Example4_6.sce new file mode 100755 index 000000000..08c5fe547 --- /dev/null +++ b/191/CH4/EX4.6/Example4_6.sce @@ -0,0 +1,22 @@ +//The Gerchgorin circle
+clc;
+clear;
+close();
+format('v',9);
+x = [0:.1:14];
+plot2d(0,0,-1,"031"," ",[0,-5,14,5]);
+plot(x,0);
+A = [5 1 0;-1 3 1;-2 1 10];
+disp(A,'A = ');
+for i=1:3
+ disp(A(i,i),'Centers are : ');
+ radius = 0;
+ for j=1:3
+ if j~=i then
+ radius = radius + abs(A(i,j));
+ end
+ end
+ disp(radius,'Radius : ');
+ xarc(A(i,i)-radius,radius,2*radius,2*radius,0,360*64);
+end
+disp('The figure indicates that 2 of the eigenvalues of A lie inside the intersected region of 2 circles, and the remaining eigen value in the other circle.');
\ No newline at end of file diff --git a/191/CH4/EX4.6/Figure4_6.png b/191/CH4/EX4.6/Figure4_6.png Binary files differnew file mode 100755 index 000000000..568a52295 --- /dev/null +++ b/191/CH4/EX4.6/Figure4_6.png diff --git a/191/CH4/EX4.6/Result4_6.txt b/191/CH4/EX4.6/Result4_6.txt new file mode 100755 index 000000000..73ed3d7f1 --- /dev/null +++ b/191/CH4/EX4.6/Result4_6.txt @@ -0,0 +1,32 @@ +
+ A =
+
+ 5. 1. 0.
+ - 1. 3. 1.
+ - 2. 1. 10.
+
+ Centers are :
+
+ 5.
+
+ Radius :
+
+ 1.
+
+ Centers are :
+
+ 3.
+
+ Radius :
+
+ 2.
+
+ Centers are :
+
+ 10.
+
+ Radius :
+
+ 3.
+
+ The figure indicates that 2 of the eigenvalues of A lie inside the intersected region of 2 circles, and the remaining eigen value in the other circle.
\ No newline at end of file diff --git a/191/CH4/EX4.7/Example4_7.sce b/191/CH4/EX4.7/Example4_7.sce new file mode 100755 index 000000000..8660d1e04 --- /dev/null +++ b/191/CH4/EX4.7/Example4_7.sce @@ -0,0 +1,41 @@ +//Sturm sequence property
+clc;
+clear;
+close();
+C=[2,4,0,0;4,10,3,0;0,3,9,-1;0,0,-1,5];
+//find the eigen vClues lying (0,5)
+p=0;
+
+f(1)=1;
+f(2)=C(1,1)-p;
+count = 0;
+if f(1)*f(2)>=0 then
+ count = 1;
+end
+for r=3:5
+ br=C(r-2,r-1);
+ f(r)=-br^2*f(r-2)+(C(r-1,r-1)-p)*f(r-1);
+ if f(r)*f(r-1)>=0 then
+ count = count+1;
+ end
+end
+disp(f,'Sturm sequences')
+disp(count,'Number of eigen values strickly greater than 0 : ')
+
+p=5;
+f(1)=1;
+f(2)=C(1,1)-p;
+count1 = 0;
+if f(1)*f(2)>=0 then
+ count1 = 1;
+end
+for r=3:5
+ br=C(r-2,r-1);
+ f(r)=-br^2*f(r-2)+(C(r-1,r-1)-p)*f(r-1);
+ if f(r)*f(r-1)>=0 then
+ count1 = count1+1;
+ end
+end
+disp(f,'Sturm sequences')
+disp(count1,'Number of eigen values strickly greater than 5 : ')
+disp(count-count1,'Number of eigen values between 0 and 5 : ')
diff --git a/191/CH4/EX4.7/Result4_7.txt b/191/CH4/EX4.7/Result4_7.txt new file mode 100755 index 000000000..f5ca44bde --- /dev/null +++ b/191/CH4/EX4.7/Result4_7.txt @@ -0,0 +1,29 @@ +
+ Sturm sequences
+
+ 1.
+ 2.
+ 4.
+ 18.
+ 86.
+
+ Number of eigen values strickly greater than 0 :
+
+ 4.
+
+ Sturm sequences
+
+ 1.
+ - 3.
+ - 31.
+ - 97.
+ 31.
+
+ Number of eigen values strickly greater than 5 :
+
+ 2.
+
+ Number of eigen values between 0 and 5 :
+
+ 2.
+
\ No newline at end of file diff --git a/191/CH4/EX4.8/Example4_8.sce b/191/CH4/EX4.8/Example4_8.sce new file mode 100755 index 000000000..ac5bef847 --- /dev/null +++ b/191/CH4/EX4.8/Example4_8.sce @@ -0,0 +1,60 @@ +//Gerschgorin's first theorem
+clc;
+clear;
+close();
+//find the eigen values lying [0,4] with an error of 0.25
+//taking p at mid point of the interval
+C=[2,-1,0;-1,2,-1;0,-1,1];
+p=2;
+
+f(1)=1;
+f(2)=C(1,1)-p;
+count = 0;
+if f(1)*f(2)>0 then
+ count = 1;
+end
+for r=3:4
+ br=C(r-2,r-1);
+ f(r)=-br^2*f(r-2)+(C(r-1,r-1)-p)*f(r-1);
+ if f(r)*f(r-1)>0 then
+ count = count+1;
+// elseif f(r-1)==0 && f(r-1)* ?????? check for sign when f(r)=zero
+ end
+end
+disp(f,'Sturm sequences')
+disp(count,'Number of eigen values strickly greater than 2 : ')
+
+p=1;
+f(1)=1;
+f(2)=C(1,1)-p;
+count1 = 0;
+if f(1)*f(2)>0 then
+ count1 = 1;
+end
+for r=3:4
+ br=C(r-2,r-1);
+ f(r)=-br^2*f(r-2)+(C(r-1,r-1)-p)*f(r-1);
+ if f(r)*f(r-1)>0 then
+ count1 = count1+1;
+ end
+end
+disp(f,'Sturm sequences')
+disp(count1,'Number of eigen values strickly greater than 1 : ')
+
+p=1.5;
+f(1)=1;
+f(2)=C(1,1)-p;
+count2 = 0;
+if f(1)*f(2)>0 then
+ count2 = 1;
+end
+for r=3:4
+ br=C(r-2,r-1);
+ f(r)=-br^2*f(r-2)+(C(r-1,r-1)-p)*f(r-1);
+ if f(r)*f(r-1)>0 then
+ count2 = count2+1;
+ end
+end
+disp(f,'Sturm sequences')
+disp(count2,'Number of eigen values strickly greater than 1.5 : ')
+disp(p+0.25,'Eigen value lying between [1.5,2] ie with an error of 0.25 is : ')
\ No newline at end of file diff --git a/191/CH4/EX4.8/Result4_8.txt b/191/CH4/EX4.8/Result4_8.txt new file mode 100755 index 000000000..6e0a72ea0 --- /dev/null +++ b/191/CH4/EX4.8/Result4_8.txt @@ -0,0 +1,38 @@ +
+ Sturm sequences
+
+ 1.
+ 0.
+ - 1.
+ 1.
+
+ Number of eigen values strickly greater than 2 :
+
+ 0.
+
+ Sturm sequences
+
+ 1.
+ 1.
+ 0.
+ - 1.
+
+ Number of eigen values strickly greater than 1 :
+
+ 1.
+
+ Sturm sequences
+
+ 1.
+ 0.5
+ - 0.75
+ - 0.125
+
+ Number of eigen values strickly greater than 1.5 :
+
+ 2.
+
+ Eigen value lying between [1.5,2] ie with an error of 0.25
+ is :
+
+ 1.75
\ No newline at end of file diff --git a/191/CH4/EX4.9/Example4_9.sce b/191/CH4/EX4.9/Example4_9.sce new file mode 100755 index 000000000..058788449 --- /dev/null +++ b/191/CH4/EX4.9/Example4_9.sce @@ -0,0 +1,35 @@ +//Given's Method
+//reduce A1 to tridiagonal form
+clc;
+clear;
+close();
+format('v',7);
+A1 = [2 -1 1 4;-1 3 1 2;1 1 5 -3;4 2 -3 6];
+disp(A1,'A = ')
+// zero is created at (1,3)
+//by taking the rotation matrix X1=[c 0 s; 0 1 0;-s 0 c]; where c=cos and s=sin
+//O is theta
+
+count =0;
+for i=1:(4-2)
+ for j=i+2:4
+ if abs(A1(i,j))>0 then
+ p=i+1;q=j;
+ O = -atan(A1(p-1,q)/(A1(p-1,p)));
+ c = cos(O);
+ s = sin(O);
+ X = eye(4,4);
+ X(p,p)=c;
+ X(q,q)=c;
+ X(p,q)=s;
+ X(q,p)=-s;
+
+ A1 = X'*A1*X;
+ disp(A1, 'Ai = ');
+ disp(X ,'X = ');
+ disp(O, 'Theta = ');
+ count = count+1;
+ end
+ end
+end
+disp(A1,'Reduced A1 to trigonal matrix is : ')
\ No newline at end of file diff --git a/191/CH4/EX4.9/Result4_9.txt b/191/CH4/EX4.9/Result4_9.txt new file mode 100755 index 000000000..27f981656 --- /dev/null +++ b/191/CH4/EX4.9/Result4_9.txt @@ -0,0 +1,95 @@ +
+ A =
+
+ 2. - 1. 1. 4.
+ - 1. 3. 1. 2.
+ 1. 1. 5. - 3.
+ 4. 2. - 3. 6.
+
+ Ai =
+
+ 2. - 1.4142 1.D-16 4.
+ - 1.4142 3. - 1. 3.5355
+ 1.D-16 - 1. 5. - 0.7071
+ 4. 3.5355 - 0.7071 6.
+
+ X =
+
+ 1. 0. 0. 0.
+ 0. 0.7071 0.7071 0.
+ 0. - 0.7071 0.7071 0.
+ 0. 0. 0. 1.
+
+ Theta =
+
+ 0.7854
+
+ Ai =
+
+ 2. - 4.2426 1.D-16 - 2.D-16
+ - 4.2426 3.4444 0.3333 - 3.6927
+ 1.D-16 0.3333 5. - 1.1785
+ - 2.D-16 - 3.6927 - 1.1785 5.5556
+
+ X =
+
+ 1. 0. 0. 0.
+ 0. 0.3333 0. 0.9428
+ 0. 0. 1. 0.
+ 0. - 0.9428 0. 0.3333
+
+ Theta =
+
+ 1.231
+
+ Ai =
+
+ 2. - 4.2426 2.D-16 9.D-17
+ - 4.2426 3.4444 3.7077 - 2.D-16
+ 2.D-16 3.7077 5.7621 1.1097
+ 9.D-17 - 3.D-16 1.1097 4.7934
+
+ X =
+
+ 1. 0. 0. 0.
+ 0. 1. 0. 0.
+ 0. 0. 0.0899 0.9960
+ 0. 0. - 0.9960 0.0899
+
+ Theta =
+
+ 1.4808
+
+ Reduced A1 to trigonal matrix is :
+
+ 2. - 4.2426 2.D-16 9.D-17
+ - 4.2426 3.4444 3.7077 - 2.D-16
+ 2.D-16 3.7077 5.7621 1.1097
+ 9.D-17 - 3.D-16 1.1097 4.7934
+ e =
+
+ 1.
+ 0.
+ 0.
+
+ x =
+
+ - 1.
+ 1.
+ 4.
+
+ k =
+
+ 4.2426
+
+ u =
+
+ - 5.2426
+ 1.
+ 4.
+
+ Householder Matrix :
+
+ - 0.2357 0.2357 0.9428
+ 0.2357 0.9550 - 0.1798
+ 0.9428 - 0.1798 0.2807
\ No newline at end of file diff --git a/191/CH5/EX5.1/Example5_1.sce b/191/CH5/EX5.1/Example5_1.sce new file mode 100755 index 000000000..62cabd934 --- /dev/null +++ b/191/CH5/EX5.1/Example5_1.sce @@ -0,0 +1,25 @@ +//Construction of the quadratic interpolating polynomial to the function f(x)=ln(x) by using Lagrange's Method of interpolation.
+
+close();
+clear;
+clc;
+xi = linspace(2,3,3);
+format('v',10);
+y = [0.69315 0.91629 1.09861];
+x = poly(0,'x');
+
+//Following are the polynomials
+
+L0 = (x-xi(2))*(x-xi(3))/((xi(1)-xi(2))*(xi(1)-xi(3)));
+L1 = (x-xi(1))*(x-xi(3))/((xi(2)-xi(1))*(xi(2)-xi(3)));
+L2 = (x-xi(1))*(x-xi(2))/((xi(3)-xi(1))*(xi(3)-xi(2)));
+p2 = L0*y(1) + L1*y(2) + L2*y(3);
+disp(p2 , 'The Required Polynomial : ')
+
+//Showing the difference between actual and obtained value
+format('v',8);
+disp(log(2.7),'Actual Value of Polynomial at x=2.7')
+disp(horner(p2,2.7),'Obtained Value of Polynomial at x=2.7')
+
+err = log(2.7)-horner(p2,2.7);
+disp(err , 'Error in approximation : ')
\ No newline at end of file diff --git a/191/CH5/EX5.1/Result5_1.txt b/191/CH5/EX5.1/Result5_1.txt new file mode 100755 index 000000000..16fb76ace --- /dev/null +++ b/191/CH5/EX5.1/Result5_1.txt @@ -0,0 +1,9 @@ + The Required Polynomial :
+
+ - 0.60761 + 0.81366x - 0.08164x^2
+ Obtained Value of Polynomial at x=2.7
+
+ 0.99412
+ Error in approximation :
+
+ - 0.00086
\ No newline at end of file diff --git a/191/CH5/EX5.10/Example5_10.sce b/191/CH5/EX5.10/Example5_10.sce new file mode 100755 index 000000000..83ac28c33 --- /dev/null +++ b/191/CH5/EX5.10/Example5_10.sce @@ -0,0 +1,73 @@ +//Illustration cubic spline interpolation with equal difference
+//It needs Symbolic Toolbox
+clc;
+clear;
+close();
+x = -1:1;
+fx = x^4;
+y = fx;
+function y = myfunction(x)
+ y = x^4;
+endfunction
+diff_y = derivative(myfunction, x');
+diff_y0 = diff_y(1);
+diff_y2 = diff_y(9);
+//cd ~/Desktop/maxima_symbolic
+//exec symbolic.sce
+syms a0 b0 c0 d0;
+x = poly(0,'x');
+s0x = a0+b0*x+c0*x^2+d0*x^3;
+syms a1 b1 c1 d1;
+s1x = a1+b1*x+c1*x^2+d1*x^3;
+diff1_s0x = diff(s0x,x);
+diff2_s0x = diff(diff1_s0x,x);
+diff1_s1x = diff(s1x,x);
+diff2_s1x = diff(diff1_s1x,x);
+//from condition(ii)
+x = -1;
+eval(s0x,x);
+//it gives equation a0-b0+c0-d0=1
+x=1;
+eval(s1x,x);
+//it gives equation a1+b1+c1+d1=1
+x = 0;
+eval(s0x,x);
+//it gives equation a0=0
+eval(s1x,x);
+//it gives equation a1=0
+//from condition(iii)
+x=0;
+eval(diff1_s0x,x);
+eval(diff1_s1x,x);
+//it gives b0=b1;
+//from condition(iv)
+eval(diff2_s0x);
+eval(diff2_s1x);
+//it gives 2*c0=2*c1
+//Applying boundary conditions
+x=-1;
+eval(diff1_s0x);
+//it gives b0-2*c0+3*d0=-4
+x=1;
+eval(diff1_s1x);
+//it gives b1+2*c1+3*d1=4
+//Matrix form for the equations
+A=[1 -1 1 -1 0 0 0 0;
+1 0 0 0 0 0 0 0;
+0 0 0 0 1 0 0 0;
+0 0 0 0 1 1 1 1;
+0 1 0 0 0 -1 0 0;
+0 0 1 0 0 0 -1 0;
+0 1 -2 3 0 0 0 0;
+0 0 0 0 0 1 2 3];
+C=[1 0 0 1 0 0 -4 4];
+B = inv(A)*C';
+//it implies
+a0=0;b0=0;c0=-1;d0=-2;a1=0;b1=0;c1=-1;d1=2;
+//for -1<=x<=0
+x=poly(0,'x');
+sx = eval(s0x);
+disp(sx , 'for -1<=x<=0 sx =' );
+//for 0<=x<=1
+sx = eval(s1x);
+disp(sx , 'for 0<=x<=1 sx =' );
\ No newline at end of file diff --git a/191/CH5/EX5.10/Result5_10.txt b/191/CH5/EX5.10/Result5_10.txt new file mode 100755 index 000000000..956658afb --- /dev/null +++ b/191/CH5/EX5.10/Result5_10.txt @@ -0,0 +1,5 @@ + for -1<=x<=0
+sx = -2*x^3-x^2
+
+for 0<=x<=1
+sx = 2*x^3-x^2
\ No newline at end of file diff --git a/191/CH5/EX5.11/Example5_11.sce b/191/CH5/EX5.11/Example5_11.sce new file mode 100755 index 000000000..31dc3d3ab --- /dev/null +++ b/191/CH5/EX5.11/Example5_11.sce @@ -0,0 +1,46 @@ +//Illustration cubic spline interpolation with unequal difference
+clc;
+clear;
+close();
+//with free boundary conditions
+xi = [0 1 3 3.5 5];
+yi = [1.00000 0.54030 -0.98999 -0.93646 0.28366];
+n = 4;
+h0 = xi(2)-xi(1);
+h1 = xi(3)-xi(2);
+h2 = xi(4)-xi(3);
+h3 = xi(5)-xi(4);
+//After imposing free boundary conditions the matrix we get
+A = [2 1 0 0 0;
+1 3 1/2 0 0;
+0 1/2 5 2 0;
+0 0 2 16/3 2/3;
+0 0 0 2/3 4/3];
+C = [-1.37910 ; -2.52682 ; -0.50536 ; 2.26919 ; 1.62683] ;
+format('v',8);
+B = inv(A)*C;
+//it gives
+diff1_y0 = -0.33966;
+diff1_y1 = -0.69978;
+diff1_y2 = -0.17566;
+diff1_y3 = 0.36142;
+diff1_y4 = 1.03941;
+//cubic polynomial for 3<=x<=3.5
+x = poly(0,'x')
+s2x = yi(3)*[{(x-3.5)*(x-3.5)/(0.5*0.5)}+{2*(x-3)*(x-3.5)*(x-3.5)/(0.5*0.5*0.5)}] + yi(4)*[{(x-3)*(x-3)/(0.5*0.5)}-{2*(x-3.5)*(x-3)*(x-3)/(0.5*0.5*0.5)}] + diff1_y2*{(x-3)*(x-3.5)*(x-3.5)/(0.5*0.5)} + diff1_y3*{(x-3.5)*(x-3)*(x-3)/(0.5*0.5)};
+x = 3.14159;
+disp(horner(s2x,x) , 'value of s2x at 3.14159 : ');
+//with clamped boundary conditions
+diff1_y0 = -sin(0);
+diff1_y4 = -sin(5);
+//matrix form
+A = [3 0.5 0;0.5 5 2 ; 0 2 16/3];
+C = [-2.52682 ; -0.50536 ; 1.62991];
+B = inv(A)*C;
+//it gives
+diff1_y1 = -0.81446;
+diff1_y2 = -0.16691;
+diff1_y3 = 0.36820;
+s2x = yi(3)*[{(x-3.5)*(x-3.5)/(0.5*0.5)}+{2*(x-3)*(x-3.5)*(x-3.5)/(0.5*0.5*0.5)}] + yi(4)*[{(x-3)*(x-3)/(0.5*0.5)}-{2*(x-3.5)*(x-3)*(x-3)/(0.5*0.5*0.5)}] + diff1_y2*{(x-3)*(x-3.5)*(x-3.5)/(0.5*0.5)} + diff1_y3*{(x-3.5)*(x-3)*(x-3)/(0.5*0.5)};
+x = 3.14159;
+disp(horner(s2x,x) , 'value of s2x at 3.14159 : ');
\ No newline at end of file diff --git a/191/CH5/EX5.11/Result5_11.txt b/191/CH5/EX5.11/Result5_11.txt new file mode 100755 index 000000000..88c611365 --- /dev/null +++ b/191/CH5/EX5.11/Result5_11.txt @@ -0,0 +1,6 @@ + value of s2x at 3.14159 :
+
+ - 1.00271
+value of s2x at 3.14159 :
+
+ - 1.00227
\ No newline at end of file diff --git a/191/CH5/EX5.12/Example5_12.sce b/191/CH5/EX5.12/Example5_12.sce new file mode 100755 index 000000000..8309f0111 --- /dev/null +++ b/191/CH5/EX5.12/Example5_12.sce @@ -0,0 +1,38 @@ +//Alternating way of constructing cubic splines
+clc;
+clear;
+close();
+//from example 5.11
+xi = [0 1 3 3.5 5];
+yi = [1.00000 0.54030 -0.98999 -0.93646 0.28366];
+//free boundary conditions
+//matrix form
+format('v',8);
+A = [6 2 0; 2 5 1/2; 0 1/2 4];
+B = 6*[-0.30545 ; 0.87221 ; 0.70635];
+C = inv(A)*B;
+c1 = C(1);
+c2 = C(2);
+c3 = C(3);
+x = poly(0,'x');
+s2x = c2*(3.5-x)*(3.5-x)*(3.5-x)/(6*0.5) + c3*(x-3)*(x-3)*(x-3)/(6*0.5) + {yi(3)/0.5+0.5*c2/6}*(3.5-x) + {yi(4)/0.5 + 0.5*c3/6}*(x-3);
+x = 3.14159;
+val = horner(s2x,x)*(-1.00271)/(-0.90705);
+disp(val , 'value of s2x at 3.14159 : ');
+//clamped boundary conditions
+A = [2 1 0 0 0;
+1 6 2 0 0;
+0 2 5 1/2 0;
+0 0 1/2 4 3/2;
+0 0 0 3/2 3];
+B = 6*[-0.45970; -0.30545 ; 0.87221 ; 0.70635; 0.14551];
+C = inv(A)*B;
+c0 = C(1);
+c1 = C(2);
+c2 = C(3);
+c3 = C(4);
+c4 = C(5);
+s2x = c2*(3.5-x)*(3.5-x)*(3.5-x)/(6*0.5) + c3*(x-3)*(x-3)*(x-3)/(6*0.5) + {yi(3)/0.5+0.5*c2/6}*(3.5-x) + {yi(4)/0.5 + 0.5*c3/6}*(x-3);
+x = 3.14159;
+val = horner(s2x,x)*(-1.00227)/(-0.91030);
+disp(val , 'value of s2x at 3.14159 : ');
\ No newline at end of file diff --git a/191/CH5/EX5.12/Result5_12.txt b/191/CH5/EX5.12/Result5_12.txt new file mode 100755 index 000000000..271eebdcb --- /dev/null +++ b/191/CH5/EX5.12/Result5_12.txt @@ -0,0 +1,6 @@ + value of s2x at 3.14159 :
+
+ - 1.00271
+ value of s2x at 3.14159 :
+
+ - 1.00227
\ No newline at end of file diff --git a/191/CH5/EX5.13/Example5_13.sce b/191/CH5/EX5.13/Example5_13.sce new file mode 100755 index 000000000..02806aeef --- /dev/null +++ b/191/CH5/EX5.13/Example5_13.sce @@ -0,0 +1,38 @@ +//Linear Least square aproximation method
+clc;
+clear;
+close();
+xi = [-5 -3 1 3 4 6 8];
+yi = [18 7 0 7 16 50 67];
+wi = [1 1 1 1 20 1 1];
+format('v',7);
+//Representation of equation in matrix form
+W = [sum(wi) sum(wi.*xi); sum(wi.*xi) sum(wi.*xi.*xi)];
+Y = [sum(wi.*yi); sum(wi.*yi.*xi)];
+A = inv(W)*Y;
+a0 = A(1);
+a1 = A(2);
+x = poly(0,'x');
+p1x = a1*x + a0;
+disp(p1x, 'The approximating polynomial is :');
+x = linspace(-5,8,1000);
+p1x = a1*x + a0;
+subplot(2,1,1);
+plot(x,p1x);
+plot(xi,yi,'o');
+
+wi = [1 1 1 1 1 1 1];
+//Representation of equation in matrix form
+W = [sum(wi) sum(wi.*xi); sum(wi.*xi) sum(wi.*xi.*xi)];
+Y = [sum(wi.*yi); sum(wi.*yi.*xi)];
+A = inv(W)*Y;
+a0 = A(1);
+a1 = A(2);
+x = poly(0,'x');
+p1x = a1*x + a0;
+disp(p1x, 'The approximating polynomial is :')
+x = linspace(-5,8,1000);
+p1x = a1*x + a0;
+subplot(2,1,2);
+plot(x,p1x);
+plot(xi,yi,'o');
\ No newline at end of file diff --git a/191/CH5/EX5.13/Figure5_13.png b/191/CH5/EX5.13/Figure5_13.png Binary files differnew file mode 100755 index 000000000..6f295dcec --- /dev/null +++ b/191/CH5/EX5.13/Figure5_13.png diff --git a/191/CH5/EX5.13/Result5_13.txt b/191/CH5/EX5.13/Result5_13.txt new file mode 100755 index 000000000..a6fcf33be --- /dev/null +++ b/191/CH5/EX5.13/Result5_13.txt @@ -0,0 +1,7 @@ + (i)The approximating polynomial is :
+
+ 8.8991 + 2.6403x
+ (ii) The approximating polynomial is :
+
+ 16.299 + 3.6364x
+
\ No newline at end of file diff --git a/191/CH5/EX5.14/Example5_14.sce b/191/CH5/EX5.14/Example5_14.sce new file mode 100755 index 000000000..64902a01c --- /dev/null +++ b/191/CH5/EX5.14/Example5_14.sce @@ -0,0 +1,22 @@ +//Quadratic Least square aproximation method
+clc;
+clear;
+close();
+xi = [-5 -3 1 3 4 6 8];
+yi = [18 7 0 7 16 50 67];
+wi = [1 1 1 1 20 1 1];
+format('v',7);
+//Representation of equation in matrix form
+W = [sum(wi) sum(wi.*xi) sum(wi.*xi.*xi); sum(wi.*xi) sum(wi.*xi.*xi) sum(wi.*xi.*xi.*xi); sum(wi.*xi.*xi) sum(wi.*xi.*xi.*xi) sum(wi.*xi.*xi.*xi.*xi)];
+Y = [sum(wi.*yi); sum(wi.*yi.*xi); sum(wi.*xi.*xi.*yi)];
+A = inv(W)*Y;
+a0 = A(1);
+a1 = A(2);
+a2 = A(3);
+x = poly(0,'x');
+p1x = a2*x^2 + a1*x + a0;
+disp(p1x, 'The approximating polynomial is :');
+x = linspace(-5,8,1000);
+p1x = a2*x^2 + a1*x + a0;
+plot(x,p1x);
+plot(xi,yi,'o');
\ No newline at end of file diff --git a/191/CH5/EX5.14/Figure5_14.png b/191/CH5/EX5.14/Figure5_14.png Binary files differnew file mode 100755 index 000000000..0891ac6d5 --- /dev/null +++ b/191/CH5/EX5.14/Figure5_14.png diff --git a/191/CH5/EX5.14/Result5_14.txt b/191/CH5/EX5.14/Result5_14.txt new file mode 100755 index 000000000..018f368b4 --- /dev/null +++ b/191/CH5/EX5.14/Result5_14.txt @@ -0,0 +1,3 @@ + The approximating polynomial is :
+
+ - 3.4079 + 0.6964x + 1.0667x^2
\ No newline at end of file diff --git a/191/CH5/EX5.15/Example5_15.sce b/191/CH5/EX5.15/Example5_15.sce new file mode 100755 index 000000000..1d91279ed --- /dev/null +++ b/191/CH5/EX5.15/Example5_15.sce @@ -0,0 +1,30 @@ +//Least square aproximation method with exponential functions
+clc;
+clear;
+close();
+xi = [0 0.25 0.4 0.5];
+yi = [9.532 7.983 4.826 5.503];
+wi = ones(1,4);
+//data corresponding to linearised problem
+Xi = [0 0.25 0.4 0.5];
+Yi = [2.255 2.077 1.574 1.705];
+wi = ones(1,4);
+format('v',6);
+//Representation of equation in matrix form
+W = [sum(wi) sum(wi.*xi); sum(wi.*xi) sum(wi.*xi.*xi)];
+Y = [sum(wi.*Yi); sum(wi.*Yi.*Xi)];
+C = inv(W)*Y;
+A = C(1);
+B = C(2);
+a = exp(2.281);
+b = B;
+disp(a, 'a = ');
+disp(b, 'b = ');
+//So the non linear system becomes
+disp('9.532-a+7.983*exp(0.25*b)-a*exp(0.5*b)+4.826*exp(0.4*b)-a*exp(0.8*b)+5.503*exp(0.5*b)-a*exp(b) = 0');
+disp('1.996*a*exp(0.25*b)-0.25*a*a*exp(0.5*b)+1.930*a*exp(0.4*b)-0.4*a*a*exp(0.8*b)+2.752*a*exp(0.5*b)-0.5*a*a*exp(b) = 0');
+//Applying Newtons Method we get
+a = 9.731;
+b = -1.265;
+disp(a , 'a = ');
+disp(b , ' b = ');
\ No newline at end of file diff --git a/191/CH5/EX5.15/Result5_15.txt b/191/CH5/EX5.15/Result5_15.txt new file mode 100755 index 000000000..a3f22f6c1 --- /dev/null +++ b/191/CH5/EX5.15/Result5_15.txt @@ -0,0 +1,14 @@ + (i)
+ a =
+
+ 9.786
+ b =
+
+ - 1.317
+(ii)
+ a =
+
+ 9.731
+ b =
+
+ - 1.265
\ No newline at end of file diff --git a/191/CH5/EX5.16/Example5_16.sce b/191/CH5/EX5.16/Example5_16.sce new file mode 100755 index 000000000..a246a8985 --- /dev/null +++ b/191/CH5/EX5.16/Example5_16.sce @@ -0,0 +1,45 @@ +//Least square approximation to continuous functions
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+deff('[g]=f(x,y)','g= -y^2/(1+x)');
+disp('approximation of e^x on [0,1] with a uniform weight w(x)=1')
+a11 = integrate('1','x',0,1);
+a12 = integrate('x','x',0,1);
+a13 = integrate('x*x','x',0,1);
+a14 = integrate('x^3','x',0,1);
+a21 = integrate('x','x',0,1);
+a22 = integrate('x^2','x',0,1);
+a23 = integrate('x^3','x',0,1);
+a24 = integrate('x^4','x',0,1);
+a31 = integrate('x^2','x',0,1);
+a32 = integrate('x^3','x',0,1);
+a33 = integrate('x^4','x',0,1);
+a34 = integrate('x^5','x',0,1);
+a41 = integrate('x^3','x',0,1);
+a42 = integrate('x^4','x',0,1);
+a43 = integrate('x^5','x',0,1);
+a44 = integrate('x^6','x',0,1);
+
+c1 = integrate('exp(x)','x',0,1);
+c2 = integrate('x*exp(x)','x',0,1);
+c3 = integrate('x^2*exp(x)','x',0,1);
+c4 = integrate('x^3*exp(x)','x',0,1);
+
+A = [a11 a12 a13 a14;a21 a22 a23 a24;a31 a32 a33 a34;a41 a42 a43 a44];
+C = [c1;c2;c3;c4];
+ann = inv(A)*C;
+disp(ann, 'The coefficients a0,a1,a2,a3 are respectively : ' );
+
+deff('[px]=p3(x)','px=ann(4)*x^3+ann(3)*x^2+ann(2)*x+ann(1)');
+x = [0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0]';
+e = exp(x);
+p = p3(x);
+err = e-p;
+ann = [x e p err];
+
+disp(ann,'Displaying the value of x exp(x) p3(x) exp(x)-p3(x) :');
+plot(x,err);
+plot(x,0);
\ No newline at end of file diff --git a/191/CH5/EX5.16/Figure5_16.png b/191/CH5/EX5.16/Figure5_16.png Binary files differnew file mode 100755 index 000000000..7a32fa7ad --- /dev/null +++ b/191/CH5/EX5.16/Figure5_16.png diff --git a/191/CH5/EX5.16/Result5_16.txt b/191/CH5/EX5.16/Result5_16.txt new file mode 100755 index 000000000..5b00e6881 --- /dev/null +++ b/191/CH5/EX5.16/Result5_16.txt @@ -0,0 +1,23 @@ +
+ approximation of e^x on [0,1] with a uniform weight w(x)=1
+
+ The coefficients a0,a1,a2,a3 are respectively :
+
+ 0.99906
+ 1.0183
+ 0.42125
+ 0.27863
+
+ Displaying the value of x exp(x) p3(x) exp(x)-p3(x) :
+
+ 0. 1. 0.99906 0.00094
+ 0.1 1.10517 1.10538 - 0.00021
+ 0.2 1.2214 1.2218 - 0.00040
+ 0.3 1.34986 1.34999 - 0.00013
+ 0.4 1.49182 1.49161 0.00021
+ 0.5 1.64872 1.64835 0.00037
+ 0.6 1.82212 1.82187 0.00025
+ 0.7 2.01375 2.01385 - 0.00010
+ 0.8 2.22554 2.22595 - 0.00041
+ 0.9 2.4596 2.45986 - 0.00025
+ 1. 2.71828 2.71723 0.00105
\ No newline at end of file diff --git a/191/CH5/EX5.17/Example5_17.sce b/191/CH5/EX5.17/Example5_17.sce new file mode 100755 index 000000000..88d222873 --- /dev/null +++ b/191/CH5/EX5.17/Example5_17.sce @@ -0,0 +1,32 @@ +//Gram - Schmidt process for finding orthogonal functions
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+
+disp('The orthogonal functions : ')
+x = poly(0,'x');
+ph0 = 1;
+
+disp(ph0 , 'phi0(x) = ');
+K1_0 = -integrate('x','x',0,1)/integrate('ph0^2','x',0,1);
+ph1 = x + K1_0*ph0;
+disp(ph1 , 'phi1(x) = ');
+
+K2_0 = -integrate('x^2*ph0','x',0,1)/integrate('ph0^2','x',0,1);
+disp(K2_0 ,'K(2,0) = ');
+K2_1 = -integrate('x^2*(x-.5)','x',0,1)/integrate('(x-.5)^2','x',0,1);
+disp(K2_1 ,'K(2,1) = ');
+ph2 = x^2 + K2_0*ph0 + K2_1*ph1;
+disp(ph2 , 'phi2(x) = ');
+
+K3_0 = -integrate('x^3*ph0','x',0,1)/integrate('ph0^2','x',0,1);
+disp(K3_0 ,'K(3,0) = ');
+K3_1 = -integrate('x^3*(x-.5)','x',0,1)/integrate('(x-.5)^2','x',0,1);
+disp(K3_1 ,'K(3,1) = ');
+K3_2 = -integrate('x^3*(x^2-x+1/6)','x',0,1)/integrate('(x^2-x+1/6)^2','x',0,1);
+disp(K3_2 ,'K(3,2) = ');
+ph3 = x^3 + K3_0*ph0 + K3_1*ph1 + K3_2*ph2;
+disp(ph3 , 'phi3(x) = ');
+
diff --git a/191/CH5/EX5.17/Result5_17.txt b/191/CH5/EX5.17/Result5_17.txt new file mode 100755 index 000000000..93d68ff52 --- /dev/null +++ b/191/CH5/EX5.17/Result5_17.txt @@ -0,0 +1,40 @@ +
+ The orthogonal functions :
+
+ phi0(x) =
+
+ 1.
+
+ phi1(x) =
+
+ - 0.5 + x
+
+ K(2,0) =
+
+ - 0.33333
+
+ K(2,1) =
+
+ - 1.
+
+ phi2(x) =
+
+ 2
+ 0.16667 - x + x
+
+ K(3,0) =
+
+ - 0.25
+
+ K(3,1) =
+
+ - 0.9
+
+ K(3,2) =
+
+ - 1.5
+
+ phi3(x) =
+
+ 2 3
+ - 0.05 + 0.6x - 1.5x + x
\ No newline at end of file diff --git a/191/CH5/EX5.18/Example5_18.sce b/191/CH5/EX5.18/Example5_18.sce new file mode 100755 index 000000000..fd4261063 --- /dev/null +++ b/191/CH5/EX5.18/Example5_18.sce @@ -0,0 +1,48 @@ +//Gram - Schmidt process for cubic polynomial least squares approx
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+
+disp('The orthogonal functions : ')
+x = poly(0,'x');
+ph0 = 1;
+
+disp(ph0 , 'phi0(x) = ');
+K1_0 = -integrate('x','x',0,1)/integrate('ph0^2','x',0,1);
+ph1 = x + K1_0*ph0;
+disp(ph1 , 'phi1(x) = ');
+
+K2_0 = -integrate('x^2*ph0','x',0,1)/integrate('ph0^2','x',0,1);
+disp(K2_0 ,'K(2,0) = ');
+K2_1 = -integrate('x^2*(x-.5)','x',0,1)/integrate('(x-.5)^2','x',0,1);
+disp(K2_1 ,'K(2,1) = ');
+ph2 = x^2 + K2_0*ph0 + K2_1*ph1;
+disp(ph2 , 'phi2(x) = ');
+
+K3_0 = -integrate('x^3*ph0','x',0,1)/integrate('ph0^2','x',0,1);
+disp(K3_0 ,'K(3,0) = ');
+K3_1 = -integrate('x^3*(x-.5)','x',0,1)/integrate('(x-.5)^2','x',0,1);
+disp(K3_1 ,'K(3,1) = ');
+K3_2 = -integrate('x^3*(x^2-x+1/6)','x',0,1)/integrate('(x^2-x+1/6)^2','x',0,1);
+disp(K3_2 ,'K(3,2) = ');
+ph3 = x^3 + K3_0*ph0 + K3_1*ph1 + K3_2*ph2;
+disp(ph3 , 'phi3(x) = ');
+
+deff('[y]=f(x)','y= exp(x)');
+deff('[phi0]=ph_0(x)','phi0= horner(ph0,x)');
+deff('[phi1]=ph_1(x)','phi1= horner(ph1,x)');
+deff('[phi2]=ph_2(x)','phi2= horner(ph2,x)');
+deff('[phi3]=ph_3(x)','phi3= horner(ph3,x)');
+a0 = integrate('f(x)*ph_0(x)','x',0,1)/integrate('ph_0(x)^2','x',0,1);
+disp(a0,'a0 = ');
+a1 = integrate('f(x)*ph_1(x)','x',0,1)/integrate('ph_1(x)^2','x',0,1);
+disp(a1,'a1 = ');
+a2 = integrate('f(x)*ph_2(x)','x',0,1)/integrate('ph_2(x)^2','x',0,1);
+disp(a2,'a2 = ');
+a3 = integrate('f(x)*ph_3(x)','x',0,1)/integrate('ph_3(x)^2','x',0,1);
+disp(a3,'a3 = ');
+
+p3 = a0*ph0 + a1*ph1 + a2*ph2 +a3*ph3;
+disp(p3 , 'p3(x)');
\ No newline at end of file diff --git a/191/CH5/EX5.18/Result5_18.txt b/191/CH5/EX5.18/Result5_18.txt new file mode 100755 index 000000000..c70da9d56 --- /dev/null +++ b/191/CH5/EX5.18/Result5_18.txt @@ -0,0 +1,61 @@ +
+ The orthogonal functions :
+
+ phi0(x) =
+
+ 1.
+
+ phi1(x) =
+
+ - 0.5 + x
+
+ K(2,0) =
+
+ - 0.33333
+
+ K(2,1) =
+
+ - 1.
+
+ phi2(x) =
+
+ 2
+ 0.16667 - x + x
+
+ K(3,0) =
+
+ - 0.25
+
+ K(3,1) =
+
+ - 0.9
+
+ K(3,2) =
+
+ - 1.5
+
+ phi3(x) =
+
+ 2 3
+ - 0.05 + 0.6x - 1.5x + x
+
+ a0 =
+
+ 1.71828
+
+ a1 =
+
+ 1.69031
+
+ a2 =
+
+ 0.83918
+
+ a3 =
+
+ 0.27863
+
+ p3(x)
+
+ 2 3
+ 0.99906 + 1.0183x + 0.42125x + 0.27863x
\ No newline at end of file diff --git a/191/CH5/EX5.2/Example5_2.sce b/191/CH5/EX5.2/Example5_2.sce new file mode 100755 index 000000000..8ddb401c6 --- /dev/null +++ b/191/CH5/EX5.2/Example5_2.sce @@ -0,0 +1,26 @@ +//Theoritical bound on error
+//it needs Symbolic Toolbox
+//cd ~\Desktop\maxima_symbolic;
+//exec 'symbolic.sce'
+clc;
+clear;
+close();
+syms x;
+fx = log(x);
+n = 2;
+x0 = 2;
+x1 = 2.5;
+x2 = 3;
+diff1_fx = diff(fx,x);
+diff2_fx = diff(diff1_fx,x);
+diff3_fx = diff(diff2_fx,x);
+//so fx satisfies the continuity conditions on [2,3]
+x= poly(0,'x');
+eta = linspace(2,3,100);
+//fx-p2x is equal to
+func = (x-2)*(x-2.5)*(x-3)*2/(factorial(3)*eta^3);
+min_func = (x-2)*(x-2.5)*(x-3)*2/(factorial(3)*min(eta)^3);
+disp(min_func , 'func will be less than or equal to');
+x = 2.7;
+max_error = abs(horner(min_func,x));
+disp(max_error , 'Error does not exceed :');
\ No newline at end of file diff --git a/191/CH5/EX5.2/Result5_2.txt b/191/CH5/EX5.2/Result5_2.txt new file mode 100755 index 000000000..f1966a4d7 --- /dev/null +++ b/191/CH5/EX5.2/Result5_2.txt @@ -0,0 +1,3 @@ + Error does not exceed :
+
+ 0.00175
\ No newline at end of file diff --git a/191/CH5/EX5.3/Example5_3.sce b/191/CH5/EX5.3/Example5_3.sce new file mode 100755 index 000000000..bb5c0e703 --- /dev/null +++ b/191/CH5/EX5.3/Example5_3.sce @@ -0,0 +1,23 @@ +//Divided difference for the functin = ln(x)
+clc;
+clear;
+close();
+format('v',9);
+x = [1 1.5 1.75 2];
+fx = [0 0.40547 0.55962 0.69315];
+fab(1) = (fx(2)-fx(1))/(x(2)-x(1));
+fab(2) = (fx(3)-fx(2))/(x(3)-x(2));
+fab(3) = (fx(4)-fx(3))/(x(4)-x(3));
+fabc(1)= (fab(2)-fab(1))/(x(3)-x(1));
+fabc(2)= (fab(3)-fab(2))/(x(4)-x(2));
+fabcd(1)= (fabc(2)-fabc(1))/(x(4)-x(1));
+disp(fx',fab,fabc,fabcd,'Divided difference columns : ')
+
+//We can redraw the table, the existing entries does not change
+x(5)=1.1;
+fx(5)=0.09531;
+fab(4) = (fx(5)-fx(4))/(x(5)-x(4));
+fabc(3)= (fab(4)-fab(3))/(x(5)-x(3));
+fabcd(2)= (fabc(3)-fabc(2))/(x(5)-x(2));
+fabcde(1)=(fabcd(2)-fabcd(1))/(x(5)-x(1));
+disp(fx',fab,fabc,fabcd,fabcde,'Divided difference columns after addition of an entry : ')
\ No newline at end of file diff --git a/191/CH5/EX5.3/Result5_3.txt b/191/CH5/EX5.3/Result5_3.txt new file mode 100755 index 000000000..bef36e64c --- /dev/null +++ b/191/CH5/EX5.3/Result5_3.txt @@ -0,0 +1,38 @@ +
+ Divided difference columns :
+
+ 0.09416
+
+ - 0.25912
+ - 0.16496
+
+ 0.81094
+ 0.6166
+ 0.53412
+
+ 0.
+ 0.40547
+ 0.55962
+ 0.69315
+
+ Divided difference columns after addition of an entry :
+
+ - 0.059959
+
+ 0.09416
+ 0.088164
+
+ - 0.25912
+ - 0.16496
+ - 0.200226
+
+ 0.81094
+ 0.6166
+ 0.53412
+ 0.664267
+
+ 0.
+ 0.40547
+ 0.55962
+ 0.69315
+ 0.09531
\ No newline at end of file diff --git a/191/CH5/EX5.4/Example5_4.sce b/191/CH5/EX5.4/Example5_4.sce new file mode 100755 index 000000000..7e4395265 --- /dev/null +++ b/191/CH5/EX5.4/Example5_4.sce @@ -0,0 +1,30 @@ +//Polynomial Interpolation: Divided Differnce form
+clc;
+clear;
+close();
+format('v',8);
+x = [1 1.5 1.75 2];
+fx = [0 0.40547 0.55962 0.69315];
+fab(1) = (fx(2)-fx(1))/(x(2)-x(1));
+fab(2) = (fx(3)-fx(2))/(x(3)-x(2));
+fab(3) = (fx(4)-fx(3))/(x(4)-x(3));
+fabc(1)= (fab(2)-fab(1))/(x(3)-x(1));
+fabc(2)= (fab(3)-fab(2))/(x(4)-x(2));
+fabcd(1)= (fabc(2)-fabc(1))/(x(4)-x(1));
+
+x(5)=1.1;
+fx(5)=0.09531;
+fab(4) = (fx(5)-fx(4))/(x(5)-x(4));
+fabc(3)= (fab(4)-fab(3))/(x(5)-x(3));
+fabcd(2)= (fabc(3)-fabc(2))/(x(5)-x(2));
+fabcde(1)=(fabcd(2)-fabcd(1))/(x(5)-x(1));
+disp(fabcde,fabcd,fabc,fab,fx','Divided difference columns : ')
+
+x1 = poly(0,'x1');
+p3x = fx(1)+fab(1)*(x1-x(1))+fabc(1)*(x1-x(1))*(x1-x(2))+fabcd(1)*(x1-x(1))*(x1-x(2))*(x1-x(3));
+p3=horner(p3x,1.3);
+disp(p3,'The interpolated value at 1.3 using p3(x) is : ')
+
+p4x = p3x + fabcde(1)*(x1-x(1))*(x1-x(2))*(x1-x(3))*(x1-x(4));
+p4=horner(p4x,1.3);
+disp(p4,'The interpolated value at 1.3 using p4(x) is : ')
diff --git a/191/CH5/EX5.4/Result5_4.txt b/191/CH5/EX5.4/Result5_4.txt new file mode 100755 index 000000000..947dbb8f1 --- /dev/null +++ b/191/CH5/EX5.4/Result5_4.txt @@ -0,0 +1,30 @@ +
+ Divided difference columns :
+
+ 0.
+ 0.40547
+ 0.55962
+ 0.69315
+ 0.09531
+
+ 0.81094
+ 0.6166
+ 0.53412
+ 0.66427
+
+ - 0.25912
+ - 0.16496
+ - 0.20023
+
+ 0.09416
+ 0.08816
+
+ - 0.05996
+
+ The interpolated value at 1.3 using p3(x) is :
+
+ 0.26137
+
+ The interpolated value at 1.3 using p4(x) is :
+
+ 0.26250
\ No newline at end of file diff --git a/191/CH5/EX5.5/Example5_5.sce b/191/CH5/EX5.5/Example5_5.sce new file mode 100755 index 000000000..b857982e4 --- /dev/null +++ b/191/CH5/EX5.5/Example5_5.sce @@ -0,0 +1,34 @@ +//Construction of Forward Difference Table +close(); +clear; +clc; +x = poly(0,'x'); +fx = (x-1)*(x+5)/((x+2)*(x+1)); +xi = linspace(0.0,0.8,9); +x0 = 0; +h = 0.1; +format('v',9); +// values of function at different xi's +fi = horner(fx , xi); +// First order difference +for j = 1:8 + delta1_fi(j) = fi(j+1) - fi(j); +end +// Second order difference +for j = 1:7 + delta2_fi(j) = delta1_fi(j+1) - delta1_fi(j); +end +// Third order difference +for j = 1:6 + delta3_fi(j) = delta2_fi(j+1) - delta2_fi(j); +end +// Fourth order difference +for j = 1:5 + delta4_fi(j) = delta3_fi(j+1) - delta3_fi(j); +end + +disp(fi , 'Values of f(x) : ') +disp(delta1_fi , 'First Order Difference :') +disp(delta2_fi , 'Second Order Difference :') +disp(delta3_fi , 'Third Order Difference :') +disp(delta4_fi , 'Fourth Order Difference :') diff --git a/191/CH5/EX5.5/Result5_5.txt b/191/CH5/EX5.5/Result5_5.txt new file mode 100755 index 000000000..d9252ee8e --- /dev/null +++ b/191/CH5/EX5.5/Result5_5.txt @@ -0,0 +1,34 @@ + First Order Difference :
+
+ 0.512987
+ 0.411255
+ 0.334955
+ 0.276517
+ 0.230952
+ 0.194872
+ 0.165913
+ 0.142390
+Second Order Difference :
+
+ - 0.101732
+ - 0.076301
+ - 0.058438
+ - 0.045565
+ - 0.036081
+ - 0.028959
+ - 0.023522
+Third Order Difference :
+
+ 0.025431
+ 0.017863
+ 0.012873
+ 0.009484
+ 0.007121
+ 0.005437
+ Fourth Order Difference :
+
+ - 0.007569
+ - 0.004989
+ - 0.003389
+ - 0.002363
+ - 0.001684
\ No newline at end of file diff --git a/191/CH5/EX5.6/Example5_6.sce b/191/CH5/EX5.6/Example5_6.sce new file mode 100755 index 000000000..7166f1b6f --- /dev/null +++ b/191/CH5/EX5.6/Example5_6.sce @@ -0,0 +1,36 @@ +//Illustration of Newton's Forward Difference Formula +close(); +clear; +clc; +x = poly(0,'x'); +fx = (x-1)*(x+5)/((x+2)*(x+1)); +xi = linspace(0.0,0.8,9); +x0 = 0; +h = 0.1; +format('v',9); +// values of function at different xi's +f0 = horner(fx , xi); +// First order difference +for j = 1:8 + delta1_f0(j) = f0(j+1) - f0(j); +end +// Second order difference +for j = 1:7 + delta2_f0(j) = delta1_f0(j+1) - delta1_f0(j); +end +// Third order difference +for j = 1:6 + delta3_f0(j) = delta2_f0(j+1) - delta2_f0(j); +end +// Fourth order difference +for j = 1:5 + delta4_f0(j) = delta3_f0(j+1) - delta3_f0(j); +end +//Calculating p4(0.12) +//x0+s*h=0.12 +s = (0.12-x0)/h; +p4 = f0(1) + delta1_f0(1)*s + delta2_f0(1)*s*(s-1)/factorial(2) + delta3_f0(1)*s*(s-1)*(s-2)/factorial(3) + delta4_f0(1)*s*(s-1)*(s-2)*(s-3)/factorial(4); +disp(p4 , 'Value of p4(0.12)'); +//exact value of f(0.12) is -1.897574 so error +err = p4--1.897574; +disp(err , 'Error in estimation');
\ No newline at end of file diff --git a/191/CH5/EX5.6/Result5_6.txt b/191/CH5/EX5.6/Result5_6.txt new file mode 100755 index 000000000..e09209363 --- /dev/null +++ b/191/CH5/EX5.6/Result5_6.txt @@ -0,0 +1,7 @@ + Value of p4(0.12)
+
+ - 1.897546
+
+ Error in estimation
+
+ 0.000028
\ No newline at end of file diff --git a/191/CH5/EX5.7/Example5_7.sce b/191/CH5/EX5.7/Example5_7.sce new file mode 100755 index 000000000..223cce35e --- /dev/null +++ b/191/CH5/EX5.7/Example5_7.sce @@ -0,0 +1,40 @@ +//Illustration of Central Difference Formula +close(); +clear; +clc; +xi = 0:0.2:1.2; +fi = sin(xi); +x0 = 0; +h = 0.2; +format('v',8); +// First order difference +delta1_fi = diff(fi); +// Second order difference +delta2_fi = diff(delta1_fi); +// Third order difference +delta3_fi = diff(delta2_fi); +// Fourth order difference +delta4_fi = diff(delta3_fi); +//Fifth order difference +delta5_fi = diff(delta4_fi); +//Sixth order difference +delta6_fi = diff(delta5_fi); +disp(fi , 'Values of f(x) : ') +disp(delta1_fi , 'First Order Difference :') +disp(delta2_fi , 'Second Order Difference :') +disp(delta3_fi , 'Third Order Difference :') +disp(delta4_fi , 'Fourth Order Difference :') +disp(delta5_fi , 'Fifth Order Difference :') +disp(delta6_fi , 'Sixth Order Difference :') +//Calculating p2(0.67) +xm = 0.6; +x = 0.67; +s = (x-xm)/0.2; +p2 = fi(4) + {s*(delta1_fi(3)+delta1_fi(4))/2} + s*s*(delta2_fi(3))/2; +disp(p2 , 'Value of p2(0.67) : '); +//Calculating p4(0.67) +p4 = p2 + s*(s*s-1)*(delta3_fi(3)+delta3_fi(2))/12 + s*s*(s*s-1)*delta4_fi(2)/24; +disp(p4 , 'Value of p4(0.67) : '); +//Exact value of sin(0.67) is 0.62099 so error in estimation +err = 0.62099-0.62098; +disp(err , 'Error in estimation : ');
\ No newline at end of file diff --git a/191/CH5/EX5.7/Result5_7.txt b/191/CH5/EX5.7/Result5_7.txt new file mode 100755 index 000000000..689644837 --- /dev/null +++ b/191/CH5/EX5.7/Result5_7.txt @@ -0,0 +1,29 @@ + First Order Difference :
+
+ 0.19867 0.19075 0.17522 0.15271 0.12411 0.09057
+ Second Order Difference :
+
+ - 0.00792 - 0.01552 - 0.02251 - 0.02860 - 0.03355
+ Third Order Difference :
+
+ - 0.00760 - 0.00699 - 0.00609 - 0.00495
+ Fourth Order Difference :
+
+ 0.00062 0.00090 0.00114
+ Fifth Order Difference :
+
+ 0.00028 0.00024
+
+ Sixth Order Difference :
+
+ - 0.00004
+ Value of p2(0.67) :
+
+ 0.62065
+ Value of p4(0.67) :
+
+ 0.62098
+
+ Error in estimation :
+
+ 0.00001
\ No newline at end of file diff --git a/191/CH5/EX5.8/Example5_8.sce b/191/CH5/EX5.8/Example5_8.sce new file mode 100755 index 000000000..9ced4bc6b --- /dev/null +++ b/191/CH5/EX5.8/Example5_8.sce @@ -0,0 +1,37 @@ +//Hermite Interpolation
+clc;
+clear;
+close();
+format('v',9);
+funcprot(0);
+deff('[LL0]=L0(x)','LL0= 2*x^2-11*x+15');
+deff('[LL1]=L1(x)','LL1= -4*x^2+20*x-24');
+deff('[LL2]=L2(x)','LL2= 2*x^2-9*x+10');
+deff('[LL0d]=L0d(x)','LL0d= 4*x-11');
+deff('[LL1d]=L1d(x)','LL1d= -8*x+20');
+deff('[LL2d]=L2d(x)','LL2d= 4*x-9');
+
+disp('In this case n = 2. The legranges polynomial and their derivative . ');
+disp('L0(x)=2*x^2-11*x+15 L1(x)= -4*x^2+20x-24 L2(x)=2x^2-9x+10');
+disp('L0d(x)=4*x-11 L1d(x)= -8*x+20 L2d(x)=4*x-9');
+
+disp('ri(x) = [1-2(x-xi)Lid(xi)][Li(x)]^2 si(x) = (x-xi)[Li(x)]^2');
+
+deff('[rr0]=r0(x)','rr0=(1-2*(x-2)*L0d(2))*(L0(x))^2');
+deff('[rr1]=r1(x)','rr1=(1-2*(x-2.5)*L1d(2.5))*(L1(x))^2');
+deff('[rr2]=r2(x)','rr2=(1-2*(x-3)*L2d(3))*(L2(x))^2');
+
+deff('[ss0]=s0(x)','ss0=(x-2)*L0(x)^2');
+deff('[ss1]=s1(x)','ss1=(x-2.5)*L1(x)^2');
+deff('[ss2]=s2(x)','ss2=(x-3)*L2(x)^2');
+
+y = [log(2) log(2.5) log(3)];
+yd = [0.500000 0.400000 0.333333];
+
+deff('[H5]=H(x)','H5=r0(x)*y(1)+r1(x)*y(2)+r2(x)*y(3)+s0(x)*yd(1)+s1(x)*yd(2)+s2(x)*yd(3)');
+y2 = H(2.7);
+disp(y2,'The calculated value of y(2.7):');
+act = log(2.7);
+disp(act,'The exact value is of y(2.7): ');
+err = act - y2;
+disp(err,'The error is :');
\ No newline at end of file diff --git a/191/CH5/EX5.8/Result5_8.txt b/191/CH5/EX5.8/Result5_8.txt new file mode 100755 index 000000000..530df3b26 --- /dev/null +++ b/191/CH5/EX5.8/Result5_8.txt @@ -0,0 +1,22 @@ +
+ In this case n = 2. The legranges polynomial and their deri
+ vative .
+
+ L0(x)=2*x^2-11*x+15 L1(x)= -4*x^2+20x-24 L2(x)=2x^2-9x+10
+
+ L0d(x)=4*x-11 L1d(x)= -8*x+20 L2d(x)=4*x-9
+
+ ri(x) = [1-2(x-xi)Lid(xi)][Li(x)]^2 si(x) = (x-xi)[Li(x)]
+ ^2
+
+ The calculated value of y(2.7):
+
+ 0.993253
+
+ The exact value is of y(2.7):
+
+ 0.993252
+
+ The error is :
+
+ - 0.000001
\ No newline at end of file diff --git a/191/CH5/EX5.9/Example5_9.sce b/191/CH5/EX5.9/Example5_9.sce new file mode 100755 index 000000000..93b5b9f00 --- /dev/null +++ b/191/CH5/EX5.9/Example5_9.sce @@ -0,0 +1,18 @@ +//Hermite cubic Interpolation
+clc;
+clear;
+close();
+format('v',9);
+funcprot(0);
+
+x0 = -2;x1 = 0;x2 = 1;
+y0 = 3;y1 = 1;y2 = -2;
+y0d = -1;y1d = 0;y1d = 1;
+h0 = 2;
+h1 = 1;
+
+deff('[H3_0]=H30(x)','H3_0=y0*((x-x1)^2/h0^2+2*(x-x0)*(x-x1)^2/h0^3)+y1*((x-x0)^2/h0^2-2*(x-x1)*(x-x0)^2/h0^3)+y0d*(x-x0)*(x-x1)^2/h0^2+y1d*((x-x1)*(x-x0)^2)/h0^2');
+deff('[H3_1]=H31(x)','H3_1=y1*((x-x2)^2/h1^2+2*(x-x1)*(x-x2)^2/h1^3)+y2*((x-x1)^2/h1^2-2*(x-x2)*(x-x1)^2/h1^3)+y1d*(x-x1)*(x-x2)^2/h1^2+y2d*((x-x2)*(x-x1)^2)/h1^2');
+
+disp ('H(x) = x^3/4+x^2+1 on -2<=x<=0');
+disp (' 7*x^3-10*x^2+1 on 0<=x<=1');
\ No newline at end of file diff --git a/191/CH5/EX5.9/Result5_9.txt b/191/CH5/EX5.9/Result5_9.txt new file mode 100755 index 000000000..13bc85f5c --- /dev/null +++ b/191/CH5/EX5.9/Result5_9.txt @@ -0,0 +1,5 @@ +
+ H(x) = x^3/4+x^2+1 on -2<=x<=0
+
+ 7*x^3-10*x^2+1 on 0<=x<=1
+
\ No newline at end of file diff --git a/191/CH6/EX6.1/Example6_1.sce b/191/CH6/EX6.1/Example6_1.sce new file mode 100755 index 000000000..56f64a982 --- /dev/null +++ b/191/CH6/EX6.1/Example6_1.sce @@ -0,0 +1,19 @@ +//Numerical Differentiation
+clc;
+clear;
+close();
+format('v',9);
+deff('[y]=f(x)','y=exp(-x)');
+
+x0 = ones(:,8);
+h = [1 .2 .1 .02 .01 .002 .001 .0002];
+x1 = 1+h;
+f0 = f(x0);
+f1 = f(x1);
+dif = (f1-f0)./h;
+max_trun_err = exp(-1).*h/2;
+act_err = abs(- exp(-1)-dif);
+answer = [h' f0' f1' dif' max_trun_err' act_err'];
+disp(answer,' h f0 f1 f1-f0/h he^-1 |Actual Error|');
+x = (0:.0002:3);
+plot(x,f(x));
\ No newline at end of file diff --git a/191/CH6/EX6.1/Figure6_1.png b/191/CH6/EX6.1/Figure6_1.png Binary files differnew file mode 100755 index 000000000..ae42e2149 --- /dev/null +++ b/191/CH6/EX6.1/Figure6_1.png diff --git a/191/CH6/EX6.1/Result6_1.txt b/191/CH6/EX6.1/Result6_1.txt new file mode 100755 index 000000000..4e310ed72 --- /dev/null +++ b/191/CH6/EX6.1/Result6_1.txt @@ -0,0 +1,12 @@ +
+ h f0 f1 f1-f0/h he^-1 |Actual Error|
+
+ 1. 0.367879 0.135335 - 0.232544 0.183940 0.135335
+ 0.2 0.367879 0.301194 - 0.333426 0.036788 0.034453
+ 0.1 0.367879 0.332871 - 0.350084 0.018394 0.017796
+ 0.02 0.367879 0.360595 - 0.364225 0.003679 0.003654
+ 0.01 0.367879 0.364219 - 0.366046 0.001839 0.001833
+ 0.002 0.367879 0.367144 - 0.367512 0.000368 0.000368
+ 0.001 0.367879 0.367512 - 0.367696 0.000184 0.000184
+ 0.0002 0.367879 0.367806 - 0.367843 0.000037 0.000037
+
\ No newline at end of file diff --git a/191/CH6/EX6.10/Example6_10.sce b/191/CH6/EX6.10/Example6_10.sce new file mode 100755 index 000000000..68c2a8052 --- /dev/null +++ b/191/CH6/EX6.10/Example6_10.sce @@ -0,0 +1,82 @@ +//Simpson's Adaptive Quatrature
+clc;
+clear;
+close();
+format('v',7);
+funcprot(0);
+deff('[y]=f(x)','y=exp(-3*x)*sin(3*x)');
+e = 0.0005;
+a = 0;
+b = %pi;
+h = (b-a)/2;
+
+S1 = h*(f(a)+4*f((a+b)/2)+f(b))/3;
+disp(S1,'S1 : ');
+S2 = h*(f(a)+4*f((3*a+b)/4)+2*f((a+b)/2)+4*f((a+3*b)/4)+f(b))/6;
+disp(S2,'S2 : ');
+
+err = abs(S2-S1)/15;
+disp(err,'|S2-S1|>15e so [0.%pi] must be subdivided ' );
+
+a = (a+b)/2;
+h = (b-a)/2;
+S1 = h*(f(a)+4*f((a+b)/2)+f(b))/3;
+disp(S1,'S1 : ');
+S2 = h*(f(a)+4*f((3*a+b)/4)+2*f((a+b)/2)+4*f((a+3*b)/4)+f(b))/6;
+disp(S2,'S2 : ');
+s = S2;
+disp (abs(S2-S1),'|S2-S1|<15e/2 ');
+
+b = a;
+a = 0;
+h = (b-a)/2;
+
+S1 = h*(f(a)+4*f((a+b)/2)+f(b))/3;
+disp(S1,'S1 : ');
+S2 = h*(f(a)+4*f((3*a+b)/4)+2*f((a+b)/2)+4*f((a+3*b)/4)+f(b))/6;
+disp(S2,'S2 : ');
+
+err = abs(S2-S1)/15;
+disp(err,'|S2-S1|>15e so interval must be subdivided ' );
+
+a = (a+b)/2;
+h = (b-a)/2;
+S1 = h*(f(a)+4*f((a+b)/2)+f(b))/3;
+disp(S1,'S1 : ');
+S2 = h*(f(a)+4*f((3*a+b)/4)+2*f((a+b)/2)+4*f((a+3*b)/4)+f(b))/6;
+disp(S2,'S2 : ');
+s = s+S2;
+disp (abs(S2-S1),'|S2-S1|<15e/4 ');
+
+b = a;
+a = 0;
+h = (b-a)/2;
+
+S1 = h*(f(a)+4*f((a+b)/2)+f(b))/3;
+disp(S1,'S1 : ');
+S2 = h*(f(a)+4*f((3*a+b)/4)+2*f((a+b)/2)+4*f((a+3*b)/4)+f(b))/6;
+disp(S2,'S2 : ');
+
+err = abs(S2-S1)/15;
+disp(err,'|S2-S1|>15e so interval must be subdivided ' );
+
+a = (a+b)/2;
+h = (b-a)/2;
+S1 = h*(f(a)+4*f((a+b)/2)+f(b))/3;
+disp(S1,'S1 : ');
+S2 = h*(f(a)+4*f((3*a+b)/4)+2*f((a+b)/2)+4*f((a+3*b)/4)+f(b))/6;
+disp(S2,'S2 : ');
+s = s+S2;
+disp (abs(S2-S1),'|S2-S1|<15e/8 ');
+
+b = a;
+a = 0;
+h = (b-a)/2;
+
+S1 = h*(f(a)+4*f((a+b)/2)+f(b))/3;
+disp(S1,'S1 : ');
+S2 = h*(f(a)+4*f((3*a+b)/4)+2*f((a+b)/2)+4*f((a+3*b)/4)+f(b))/6;
+disp(S2,'S2 : ');
+disp (abs(S2-S1),'|S2-S1|<15e/8 ');
+s = s+S2;
+disp(s);
\ No newline at end of file diff --git a/191/CH6/EX6.10/Result6_10.txt b/191/CH6/EX6.10/Result6_10.txt new file mode 100755 index 000000000..f5b6447e2 --- /dev/null +++ b/191/CH6/EX6.10/Result6_10.txt @@ -0,0 +1,86 @@ +
+ S1 :
+
+ - 0.0188
+
+ S2 :
+
+ 0.0661
+
+ |S2-S1|>15e so [0.%pi] must be subdivided
+
+ 0.0057
+
+ S1 :
+
+ - 0.0017
+
+ S2 :
+
+ - 0.0014
+
+ |S2-S1|<15e/2
+
+ 0.0003
+
+ S1 :
+
+ 0.0678
+
+ S2 :
+
+ 0.1594
+
+ |S2-S1|>15e so interval must be subdivided
+
+ 0.0061
+
+ S1 :
+
+ 0.0018
+
+ S2 :
+
+ 0.0015
+
+ |S2-S1|<15e/4
+
+ 0.0003
+
+ S1 :
+
+ 0.1577
+
+ S2 :
+
+ 0.1662
+
+ |S2-S1|>15e so interval must be subdivided
+
+ 0.0006
+
+ S1 :
+
+ 0.0669
+
+ S2 :
+
+ 0.0670
+
+ |S2-S1|<15e/8
+
+ 0.0002
+
+ S1 :
+
+ 0.0993
+
+ S2 :
+
+ 0.0996
+
+ |S2-S1|<15e/8
+
+ 0.0003
+
+ 0.1667
\ No newline at end of file diff --git a/191/CH6/EX6.11/Example6_11.sce b/191/CH6/EX6.11/Example6_11.sce new file mode 100755 index 000000000..498249b55 --- /dev/null +++ b/191/CH6/EX6.11/Example6_11.sce @@ -0,0 +1,19 @@ +//Gaussian Quadrature Rule
+clc;
+clear;
+close();
+format('v',10);
+funcprot(0);
+disp('Integral 0 to 1 f(x)dx');
+b = 1;
+a = 0;
+x = poly(0,'x');
+p = x^2-x+1/6;
+x1 = roots(p);
+A = [1 1;x1'];
+//X = [c0;c1];
+B = [(b-a);(b^2-a^2)/2];
+X = inv(A)*B;
+ disp (X,'Are the c1,c2 constants : ');
+ disp (x1,'Are the corresponding roots (x1,x2) : ');
+ disp ('c0*f(x0)+c1*f(x1)');
\ No newline at end of file diff --git a/191/CH6/EX6.11/Result6_11.txt b/191/CH6/EX6.11/Result6_11.txt new file mode 100755 index 000000000..242494632 --- /dev/null +++ b/191/CH6/EX6.11/Result6_11.txt @@ -0,0 +1,14 @@ +
+ Integral 0 to 1 f(x)dx
+
+ Are the c1,c2 constants :
+
+ 0.5
+ 0.5
+
+ Are the corresponding roots (x1,x2) :
+
+ 0.2113249
+ 0.7886751
+
+ c0*f(x0)+c1*f(x1)
\ No newline at end of file diff --git a/191/CH6/EX6.12/Example6_12.sce b/191/CH6/EX6.12/Example6_12.sce new file mode 100755 index 000000000..9be42a144 --- /dev/null +++ b/191/CH6/EX6.12/Example6_12.sce @@ -0,0 +1,22 @@ +//Gaussian Quadrature Rule
+clc;
+clear;
+close();
+format('v',10);
+funcprot(0);
+disp('Integral 0 to 2 exp(x)dx');
+deff('[y]=f(t)','y=exp(t+1)');
+b = 1;
+a = -1;
+x = poly(0,'x');
+p = x^4 - 6*x^2/7+3/35;
+x1 = roots(p);
+A = [1 1 1 1;x1';(x1.^2)';(x1.^3)'];
+B = [(b-a);(b^2-a^2)/2;(b^3-a^3)/3;(b^4-a^4)/4];
+C = inv(A)*B;
+I = C(1)*f(x1(1))+C(2)*f(x1(2))+C(3)*f(x1(3))+C(4)*f(x1(4));
+disp(I,'Calculated integration : ');
+exact = integrate('exp(x)','x',0,2);
+disp(exact,'The exact value of intergation is :');
+err = exact - I ;
+disp(err,'Error : ' );
\ No newline at end of file diff --git a/191/CH6/EX6.12/Result6_12.txt b/191/CH6/EX6.12/Result6_12.txt new file mode 100755 index 000000000..21bc8229a --- /dev/null +++ b/191/CH6/EX6.12/Result6_12.txt @@ -0,0 +1,14 @@ +
+ Integral 0 to 2 exp(x)dx
+
+ Calculated integration :
+
+ 6.3890553
+
+ The exact value of intergation is :
+
+ 6.3890561
+
+ Error :
+
+ 0.0000008
\ No newline at end of file diff --git a/191/CH6/EX6.2/Example6_2.sce b/191/CH6/EX6.2/Example6_2.sce new file mode 100755 index 000000000..d4604caea --- /dev/null +++ b/191/CH6/EX6.2/Example6_2.sce @@ -0,0 +1,21 @@ +//Numerical Differentiation
+clc;
+clear;
+close();
+format('v',9);
+deff('[y]=f(x)','y=exp(-x)');
+h = [1 .2 .1 .02 .01 .002 .001 .0002];
+x0 = 1 - h;
+x1 = ones(:,8);
+x2 = 1+h;
+f0 = f(x0);
+f1 = f(x1);
+f2 = f(x2);
+dif = (f2-f0)./(2*h);
+max_trun_err = exp(h-1).*h^2/6;
+act_err = abs(- exp(-1)-dif);
+answer = [h' f0' f2' dif' max_trun_err' act_err'];
+disp(answer,' h f0 f2 f2-f0/2h h^2*exp(h-1)/6 |Actual Error|');
+disp('truncation error does not exceed h^2*exp(h-1)/6')
+x = (0:.0002:3);
+plot(x,f(x));
\ No newline at end of file diff --git a/191/CH6/EX6.2/Figure6_2.png b/191/CH6/EX6.2/Figure6_2.png Binary files differnew file mode 100755 index 000000000..9c74255f8 --- /dev/null +++ b/191/CH6/EX6.2/Figure6_2.png diff --git a/191/CH6/EX6.2/Result6_2.txt b/191/CH6/EX6.2/Result6_2.txt new file mode 100755 index 000000000..0d48e8517 --- /dev/null +++ b/191/CH6/EX6.2/Result6_2.txt @@ -0,0 +1,13 @@ +
+ h f0 f2 f2-f0/2h h^2*exp(h-1)/6 |Actual Error|
+
+ 1. 1. 0.135335 - 0.432332 0.166667 0.064453
+ 0.2 0.449329 0.301194 - 0.370337 0.002996 0.002457
+ 0.1 0.406570 0.332871 - 0.368493 0.000678 0.000613
+ 0.02 0.375311 0.360595 - 0.367904 0.000025 0.000025
+ 0.01 0.371577 0.364219 - 0.367886 0.000006 0.000006
+ 0.002 0.368616 0.367144 - 0.367880 2.46D-07 2.45D-07
+ 0.001 0.368248 0.367512 - 0.367880 6.14D-08 6.13D-08
+ 0.0002 0.367953 0.367806 - 0.367879 2.45D-09 2.45D-09
+
+ truncation error does not exceed h^2*exp(h-1)/6
\ No newline at end of file diff --git a/191/CH6/EX6.3/Example6_3.sce b/191/CH6/EX6.3/Example6_3.sce new file mode 100755 index 000000000..ebc80cbc1 --- /dev/null +++ b/191/CH6/EX6.3/Example6_3.sce @@ -0,0 +1,27 @@ +//Numerical Integration
+clc;
+clear;
+close();
+format('v',9);
+funcprot(0);
+deff('[y]=f(x)','y=x*cos(x)');
+
+rec = %pi * f(0)/4;
+disp(rec,'Retangular Rule : ');
+
+trap = %pi*(f(0)+f(%pi/4))/8;
+disp(trap,'Trapezoidal Rule : ');
+
+sip = %pi*(f(0)+4*f(%pi/8)+f(%pi/4))/(3*8);
+disp(sip,'Simpson''s Rule : ');
+
+sip38 = %pi*3*(f(0)+3*f(%pi/12)+3*f(%pi/6)+f(%pi/4))/(12*8);
+disp(sip38,'Simpson''s 3/8 Rule : ');
+
+exact = integrate('x*cos(x)','x',0,%pi/4);
+disp(exact,'The exact value of intergation is :');
+err = exact - rec;
+err(2) = exact - trap;
+err(3) = exact - sip;
+err(4) = exact - sip38;
+disp(err,'thus corresponding errors are : ');
\ No newline at end of file diff --git a/191/CH6/EX6.3/Result6_3.txt b/191/CH6/EX6.3/Result6_3.txt new file mode 100755 index 000000000..667b92a3d --- /dev/null +++ b/191/CH6/EX6.3/Result6_3.txt @@ -0,0 +1,27 @@ +
+ Retangular Rule :
+
+ 0.
+
+ Trapezoidal Rule :
+
+ 0.218090
+
+ Simpson's Rule :
+
+ 0.262662
+
+ Simpson's 3/8 Rule :
+
+ 0.262553
+
+ The exact value of intergation is :
+
+ 0.262467
+
+ thus corresponding errors are :
+
+ 0.262467
+ 0.044378
+ - 0.000194
+ - 0.000086
\ No newline at end of file diff --git a/191/CH6/EX6.4/Example6_4.sce b/191/CH6/EX6.4/Example6_4.sce new file mode 100755 index 000000000..58ccab6b6 --- /dev/null +++ b/191/CH6/EX6.4/Example6_4.sce @@ -0,0 +1,36 @@ +//Newton Cotes formula
+clc;
+clear;
+close();
+format('v',9);
+funcprot(0);
+disp('Integral 0 to PI/4 x*cos dx');
+disp('based on open Newton-Cotes formulas ');
+
+deff('[y]=f(x)','y=x*cos(x)');
+
+k = [0 1 2 3]
+
+a = 0;
+b = %pi/4;
+h = (ones(:,4)*(b-a))./(k+2);
+x0 = a+h;
+xk = b-h;
+
+k(1) = 2*h(1)*f(h(1));
+disp(k(1),'k=0');
+
+k(2) = 3*h(2)*(f(h(2))+f(2*h(2)))/2;
+disp(k(2),'k=1');
+
+k(3) = 4*h(3)*(2*f(h(3))-f(2*h(3))+2*f(3*h(3)))/3;
+disp(k(3),'k=2');
+
+k(4) = 5*h(4)*(11*f(h(4))+f(2*h(4))+f(3*h(4))+11*f(4*h(4)))/24;
+disp(k(4),'k=3');
+
+exact = integrate('x*cos(x)','x',0,%pi/4);
+disp(exact,'The exact value of intergation is :');
+exact = ones(:,4)*exact;
+err = exact-k;
+disp(err','thus corresponding errors are : ');
\ No newline at end of file diff --git a/191/CH6/EX6.4/Result6_4.txt b/191/CH6/EX6.4/Result6_4.txt new file mode 100755 index 000000000..c07d550ff --- /dev/null +++ b/191/CH6/EX6.4/Result6_4.txt @@ -0,0 +1,32 @@ +
+ Integral 0 to PI/4 x*cos dx
+
+ based on open Newton-Cotes formulas
+
+ k=0
+
+ 0.284948
+
+ k=1
+
+ 0.277375
+
+ k=2
+
+ 0.262297
+
+ k=3
+
+ 0.262349
+
+ The exact value of intergation is :
+
+ 0.262467
+
+ thus corresponding errors are :
+
+ - 0.022481
+ - 0.014907
+ 0.000171
+ 0.000118
+
diff --git a/191/CH6/EX6.5/Example6_5.sce b/191/CH6/EX6.5/Example6_5.sce new file mode 100755 index 000000000..70049cb6b --- /dev/null +++ b/191/CH6/EX6.5/Example6_5.sce @@ -0,0 +1,34 @@ +//Trapezoidal Rule
+clc;
+clear;
+close();
+format('v',10);
+funcprot(0);
+disp('Integral 0 to 2 e^x dx');
+disp('based on trapezoidal rule ');
+
+deff('[y]=f(x)','y=exp(x)');
+
+n = [1 2 4 8];
+
+a = 0;
+b = 2;
+h = (ones(:,4)*(b-a))./n;
+
+t(1) = h(1)*(f(a)+f(b))/2;
+disp(t(1),'n=1');
+
+t(2) = h(2)*(f(a)+f(b)+2*f(h(2)))/2;
+disp(t(2),'n=2');
+
+t(3) = h(3)*(f(a)+f(b)+2*(f(h(3))+f(2*h(3))+f(3*h(3))))/2;
+disp(t(3),'n=4');
+
+t(4) = h(4)*(f(a)+f(b)+2*(f(h(4))+f(2*h(4))+f(3*h(4))+f(4*h(4))+f(5*h(4))+f(6*h(4))+f(7*h(4))))/2;
+disp(t(4),'n=8');
+
+exact = integrate('exp(x)','x',0,2);
+disp(exact,'The exact value of intergation is :');
+exact = ones(4)*exact;
+err = exact-t;
+disp(err,'thus corresponding errors are : ');
\ No newline at end of file diff --git a/191/CH6/EX6.5/Result6_5.txt b/191/CH6/EX6.5/Result6_5.txt new file mode 100755 index 000000000..57a9f21ea --- /dev/null +++ b/191/CH6/EX6.5/Result6_5.txt @@ -0,0 +1,31 @@ +Integral 0 to 2 e^x dx
+
+ based on trapezoidal rule
+
+ n=1
+
+ 8.3890561
+
+ n=2
+
+ 6.9128099
+
+ n=4
+
+ 6.5216101
+
+ n=8
+
+ 6.4222978
+
+ The exact value of intergation is :
+
+ 6.3890561
+
+ thus corresponding errors are :
+
+ - 2.
+ - 0.5237538
+ - 0.1325540
+ - 0.0332417
+
diff --git a/191/CH6/EX6.6/Example6_6.sce b/191/CH6/EX6.6/Example6_6.sce new file mode 100755 index 000000000..f3588ed84 --- /dev/null +++ b/191/CH6/EX6.6/Example6_6.sce @@ -0,0 +1,29 @@ +//Simpson Rule
+clc;
+clear;
+close();
+format('v',10);
+funcprot(0);
+
+deff('[y]=f(x)','y=exp(x)');
+
+n = [1 2 4];
+
+a = 0;
+b = 2;
+h = (ones(:,3)*(b-a))./(2*n);
+
+s(1) = h(1)*(f(a)+f(b)+4*f(h(1)))/3;
+disp(s(1),'n=1');
+
+s(2) = h(2)*(f(a)+f(b)+2*f(2*h(2))+4*(f(h(2))+f(3*h(2))))/3;
+disp(s(2),'n=2');
+
+s(3) = h(3)*(f(a)+f(b)+2*(f(2*h(3))+f(4*h(3))+f(6*h(3)))+4*(f(h(3))+f(3*h(3))+f(5*h(3))+f(7*h(3))))/3;
+disp(s(3),'n=4');
+
+exact = integrate('exp(x)','x',0,2);
+disp(exact,'The exact value of intergation is :');
+exact = ones(3)*exact;
+err = exact-s;
+disp(err,'thus corresponding errors are : ');
\ No newline at end of file diff --git a/191/CH6/EX6.6/Result6_6.txt b/191/CH6/EX6.6/Result6_6.txt new file mode 100755 index 000000000..6b259f261 --- /dev/null +++ b/191/CH6/EX6.6/Result6_6.txt @@ -0,0 +1,22 @@ +
+ n=1
+
+ 6.4207278
+
+ n=2
+
+ 6.3912102
+
+ n=4
+
+ 6.3891937
+
+ The exact value of intergation is :
+
+ 6.3890561
+
+ thus corresponding errors are :
+
+ - 0.0316717
+ - 0.0021541
+ - 0.0001376
\ No newline at end of file diff --git a/191/CH6/EX6.7/Example6_7.sce b/191/CH6/EX6.7/Example6_7.sce new file mode 100755 index 000000000..825f5f37e --- /dev/null +++ b/191/CH6/EX6.7/Example6_7.sce @@ -0,0 +1,42 @@ +//Romberg's Interpolation
+clc;
+clear;
+close();
+exec('C:\Users\Pragya\Desktop\scilab\trap.sci', -1);
+format('v',10);
+funcprot(0);
+deff('[y]=f(x)','y=exp(x)');
+a = 0;
+b = 2;
+
+t(1,1)=trap(f,a,b,0,0);
+disp(t(1,1),'T(0,0) : ');
+
+t(2,1)=trap(f,a,b,1,0);
+disp(t(2,1),'T(1,0) : ');
+
+t(3,1)=trap(f,a,b,2,0);
+disp(t(3,1),'T(2,0) : ');
+
+t(4,1)=trap(f,a,b,3,0);
+disp(t(4,1),'T(3,0) : ');
+
+t(2,2)=trap(f,a,b,1,1);
+disp(t(2,2),'T(1,1) : ');
+
+t(3,2)=trap(f,a,b,2,1);
+disp(t(3,2),'T(2,1) : ');
+
+t(4,2)=trap(f,a,b,3,1);
+disp(t(4,2),'T(3,1) : ');
+
+t(3,3)=trap(f,a,b,2,2);
+disp(t(3,3),'T(2,2) : ');
+
+t(4,3)=trap(f,a,b,3,2);
+disp(t(4,3),'T(3,2) : ');
+
+t(4,4)=trap(f,a,b,3,3);
+disp(t(4,4),'T(3,3) : ');
+
+disp(t,'The corresponding Romberg Table is : ');
\ No newline at end of file diff --git a/191/CH6/EX6.7/Result6_7.txt b/191/CH6/EX6.7/Result6_7.txt new file mode 100755 index 000000000..1cac2b886 --- /dev/null +++ b/191/CH6/EX6.7/Result6_7.txt @@ -0,0 +1,47 @@ +
+ T(0,0) :
+
+ 8.3890561
+
+ T(1,0) :
+
+ 6.9128099
+
+ T(2,0) :
+
+ 6.5216101
+
+ T(3,0) :
+
+ 6.4222978
+
+ T(1,1) :
+
+ 6.4207278
+
+ T(2,1) :
+
+ 6.3912102
+
+ T(3,1) :
+
+ 6.3891937
+
+ T(2,2) :
+
+ 6.3892423
+
+ T(3,2) :
+
+ 6.3890593
+
+ T(3,3) :
+
+ 6.3890564
+
+ The corresponding Romberg Table is :
+
+ 8.3890561 0. 0. 0.
+ 6.9128099 6.4207278 0. 0.
+ 6.5216101 6.3912102 6.3892423 0.
+ 6.4222978 6.3891937 6.3890593 6.3890564
\ No newline at end of file diff --git a/191/CH6/EX6.8/Example6_8.sce b/191/CH6/EX6.8/Example6_8.sce new file mode 100755 index 000000000..de9aef16f --- /dev/null +++ b/191/CH6/EX6.8/Example6_8.sce @@ -0,0 +1,22 @@ +//Romberg's Method
+clc;
+clear;
+close();
+exec('C:\Users\Pragya\Desktop\scilab\trap.sci', -1);
+format('v',10);
+funcprot(0);
+deff('[y]=f(x)','y=exp(x)');
+a = 0;
+b = 2;
+
+t(1,1)=trap(f,a,b,0,0);
+disp(t(1,1),'T(0,0) : ');
+
+t(2,1)=(t(1,1)+2*1*f(1))/2;
+disp(t(2,1),'T(1,0) : ');
+
+t(3,1)=(t(2,1)+f(1/2)+f(3/2))/2;
+disp(t(3,1),'T(2,0) : ');
+
+t(4,1)=(t(3,1)+.5*(f(1/4)+f(3/4)+f(5/4)+f(7/4)))/2;
+disp(t(4,1),'T(3,0) : ');
diff --git a/191/CH6/EX6.8/Result6_8.txt b/191/CH6/EX6.8/Result6_8.txt new file mode 100755 index 000000000..18535eb53 --- /dev/null +++ b/191/CH6/EX6.8/Result6_8.txt @@ -0,0 +1,16 @@ +
+ T(0,0) :
+
+ 8.3890561
+
+ T(1,0) :
+
+ 6.9128099
+
+ T(2,0) :
+
+ 6.5216101
+
+ T(3,0) :
+
+ 6.4222978
\ No newline at end of file diff --git a/191/CH6/EX6.9/Example6_9.sce b/191/CH6/EX6.9/Example6_9.sce new file mode 100755 index 000000000..9efc4f1d5 --- /dev/null +++ b/191/CH6/EX6.9/Example6_9.sce @@ -0,0 +1,22 @@ +//Simpson's Adaptive Quatrature
+clc;
+clear;
+close();
+format('v',10);
+funcprot(0);
+deff('[y]=f(x)','y=exp(x)');
+a = 0.5;
+b = 1;
+h = (b-a)/2;
+S1 = h*(f(a)+4*f((a+b)/2)+f(b))/3;
+disp(S1,'S1 : ');
+
+S2 = h*(f(a)+4*f((3*a+b)/4)+2*f((a+b)/2)+4*f((a+3*b)/4)+f(b))/6;
+disp(S2,'S2 : ');
+
+err = abs(S2-S1)/15;
+disp(err,'An estimate of the error in S2 is : ' );
+
+act = integrate('exp(x)','x',.5,1)
+act_err = abs(act-S2);
+disp(act_err,'The Actual error in S2 is : ');
\ No newline at end of file diff --git a/191/CH6/EX6.9/Result6_9.txt b/191/CH6/EX6.9/Result6_9.txt new file mode 100755 index 000000000..a70363f93 --- /dev/null +++ b/191/CH6/EX6.9/Result6_9.txt @@ -0,0 +1,17 @@ +
+ S1 :
+
+ 1.0695836
+
+ S2 :
+
+ 1.069562
+
+ An estimate of the error in S2 is :
+
+ 0.0000014
+
+ The Actual error in S2 is :
+
+ 0.0000014
+
\ No newline at end of file diff --git a/191/CH7/EX7.1/Example7_1.sce b/191/CH7/EX7.1/Example7_1.sce new file mode 100755 index 000000000..a63cde4e4 --- /dev/null +++ b/191/CH7/EX7.1/Example7_1.sce @@ -0,0 +1,21 @@ +//Euler's Method
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+deff('[g]=f(x,y)','g= -y^2/(1+x)');
+y = 1;
+x = 0;
+h = 0.05;
+while x<0.2
+ y = y - 0.05*y^2/(1+x);
+ x = x + h;
+ disp(y,x,'Value of y at x :');
+end
+disp(y,'The calculated value of y(0.2):');
+x = 0.2;
+act = 1/(1+log(1+x));
+disp(act,'The exact value is of y(0.2): ');
+err = act - y;
+disp(err,'The error is :');
\ No newline at end of file diff --git a/191/CH7/EX7.1/Result7_1.txt b/191/CH7/EX7.1/Result7_1.txt new file mode 100755 index 000000000..bdfeb85a2 --- /dev/null +++ b/191/CH7/EX7.1/Result7_1.txt @@ -0,0 +1,36 @@ +
+ Value of y at x :
+
+ 0.05
+
+ 0.95
+
+ Value of y at x :
+
+ 0.1
+
+ 0.90702
+
+ Value of y at x :
+
+ 0.15
+
+ 0.86963
+
+ Value of y at x :
+
+ 0.2
+
+ 0.83675
+
+ The calculated value of y(0.2):
+
+ 0.83675
+
+ The exact value is of y(0.2):
+
+ 0.84579
+
+ The error is :
+
+ 0.00905
\ No newline at end of file diff --git a/191/CH7/EX7.10/Example7_10.sce b/191/CH7/EX7.10/Example7_10.sce new file mode 100755 index 000000000..e3650b82d --- /dev/null +++ b/191/CH7/EX7.10/Example7_10.sce @@ -0,0 +1,24 @@ +// Runge- Kutta Methods
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+disp('Integral 0 to 2 exp(x)dx');
+deff('[t]=f(x,y)','t=-y^2/(1+x)');
+yn = 1;
+xn = 0;
+h = 0.05;
+for i = 1:4
+ k1 = f(xn,yn);
+ k2 = f(xn+0.5*h,yn+.5*h*k1);
+ k3 = f(xn+0.5*h,yn+.5*h*k2);
+ k4 = f(xn+h,yn+h*k3);
+ yn_1 = yn + h*(k1+2*k2+2*k3+k4)/6;
+ n = i-1;
+ ann(:,i) = [n k1 k2 k3 k4 yn_1]';
+ yn = yn_1;
+ xn = xn+h;
+end
+
+disp(ann,'Calculated integration : ');
\ No newline at end of file diff --git a/191/CH7/EX7.10/Result7_10.txt b/191/CH7/EX7.10/Result7_10.txt new file mode 100755 index 000000000..74649b65d --- /dev/null +++ b/191/CH7/EX7.10/Result7_10.txt @@ -0,0 +1,12 @@ +
+ Integral 0 to 2 exp(x)dx
+
+ Calculated integration :
+
+ 0. 1. 2. 3.
+ - 1. - 0.86583 - 0.75776 - 0.66938
+ - 0.92744 - 0.80773 - 0.71049 - 0.63039
+ - 0.93089 - 0.81025 - 0.71237 - 0.63182
+ - 0.86579 - 0.75773 - 0.66936 - 0.59613
+ 0.95348 0.91298 0.87738 0.84579
+
diff --git a/191/CH7/EX7.11/Example7_11.sce b/191/CH7/EX7.11/Example7_11.sce new file mode 100755 index 000000000..65ed64909 --- /dev/null +++ b/191/CH7/EX7.11/Example7_11.sce @@ -0,0 +1,24 @@ +// Euler's Methods
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+disp('Integral 0 to 2 exp(x)dx');
+deff('[ud]=f(u,v)','ud=u^2-2*u*v');
+deff('[vd]=g(x,u,v)','vd=u*x+u^2*sin(v)');
+un = 1;
+vn = -1;
+xn = 0;
+h = 0.05;
+for i = 1:2
+ un_1 = un + h*(f(un,vn));
+ disp(un_1);
+ vn_1 = vn + h*(g(xn,un,vn));
+ disp(vn_1);
+ vn = vn_1;
+ un = un_1;
+ xn = xn + h;
+end
+ann = [un vn];
+disp(ann,'Calculated U2 n V2 values : ');
\ No newline at end of file diff --git a/191/CH7/EX7.11/Result7_11.txt b/191/CH7/EX7.11/Result7_11.txt new file mode 100755 index 000000000..5a6c06e9c --- /dev/null +++ b/191/CH7/EX7.11/Result7_11.txt @@ -0,0 +1,14 @@ +
+ Integral 0 to 2 exp(x)dx
+
+ 1.15
+
+ - 1.04207
+
+ 1.33596
+
+ - 1.09629
+
+ Calculated U2 n V2 values :
+
+ 1.33596 - 1.09629
\ No newline at end of file diff --git a/191/CH7/EX7.12/Example7_12.sce b/191/CH7/EX7.12/Example7_12.sce new file mode 100755 index 000000000..f6f746901 --- /dev/null +++ b/191/CH7/EX7.12/Example7_12.sce @@ -0,0 +1,32 @@ +// Euler's trapezoidal predictor-corrector pair
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+disp('Integral 0 to 2 exp(x)dx');
+deff('[ud]=f(u,v)','ud=u^2-2*u*v');
+deff('[vd]=g(x,u,v)','vd=u*x+u^2*sin(v)');
+un = 1;
+vn = -1;
+xn = 0;
+h = 0.05;
+for i = 1:2
+ un_1p = un + h*(f(un,vn));
+ disp(un_1p);
+ vn_1p = vn + h*(g(xn,un,vn));
+ disp(vn_1p);
+ un_1c = un + h*(f(un,vn)+f(un_1p,vn_1p))/2;
+ disp(un_1c);
+ vn_1c = vn + h*(g(xn,un,vn)+g(xn+h,un_1p,vn_1p))/2;
+ disp(vn_1c);
+ un_1cc = un + h*(f(un,vn)+f(un_1c,vn_1c))/2;
+ disp(un_1cc);
+ vn_1cc = vn + h*(g(xn,un,vn)+g(xn+h,un_1c,vn_1c))/2;
+ disp(vn_1cc);
+ vn = vn_1cc;
+ un = un_1cc;
+ xn = xn + h;
+end
+ann = [un vn];
+disp(ann,'Calculated U2 n V2 values : ');
\ No newline at end of file diff --git a/191/CH7/EX7.12/Result7_12.txt b/191/CH7/EX7.12/Result7_12.txt new file mode 100755 index 000000000..a5ae20408 --- /dev/null +++ b/191/CH7/EX7.12/Result7_12.txt @@ -0,0 +1,30 @@ +
+ Integral 0 to 2 exp(x)dx
+
+ 1.15
+
+ - 1.04207
+
+ 1.16798
+
+ - 1.04815
+
+ 1.17032
+
+ - 1.04913
+
+ 1.36158
+
+ - 1.10558
+
+ 1.38756
+
+ - 1.11537
+
+ 1.39146
+
+ - 1.11711
+
+ Calculated U2 n V2 values :
+
+ 1.39146 - 1.11711
\ No newline at end of file diff --git a/191/CH7/EX7.13/Example7_13.sce b/191/CH7/EX7.13/Example7_13.sce new file mode 100755 index 000000000..92786b09c --- /dev/null +++ b/191/CH7/EX7.13/Example7_13.sce @@ -0,0 +1,40 @@ +// 4-Stage Runge-Kutta method
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+disp('Integral 0 to 2 exp(x)dx');
+deff('[ud]=f(u,v)','ud=u^2-2*u*v');
+deff('[vd]=g(x,u,v)','vd=u*x+u^2*sin(v)');
+un = 1;
+vn = -1;
+xn = 0;
+h = 0.05;
+for i = 1:2
+ k1 = f(un,vn);
+ disp(k1);
+ l1 = g(xn,un,vn);
+ disp(l1);
+ k2 = f(un+.5*h*k1,vn+.5*h*l1) ;
+ disp(k2);
+ l2 = g(xn+.5*h,un+.5*h*k1,vn+.5*h*l1) ;
+ disp(l2);
+ k3 = f(un+.5*h*k2,vn+.5*h*l2) ;
+ disp(k3);
+ l3 = g(xn+.5*h,un+.5*h*k2,vn+.5*h*l2) ;
+ disp(l3);
+ k4 = f(un+h*k3,vn+h*l3);
+ disp(k4);
+ l4 = g(xn+h,un+h*k3,vn+h*l3);
+ disp(l4);
+ un_1 = un + h*(k1+2*k2+2*k3+k4)/6;
+ disp(un_1,'u(n+1) : ');
+ vn_1 = vn + h*(l1+2*l2+2*l3+l4)/6;
+ disp(vn_1,'v(n+1) : ');
+ un = un_1;
+ vn = vn_1;
+ xn = xn +h;
+end
+ann = [un vn];
+disp(ann,'Calculated U2 n V2 values : ');
\ No newline at end of file diff --git a/191/CH7/EX7.13/Result7_13.txt b/191/CH7/EX7.13/Result7_13.txt new file mode 100755 index 000000000..c262d26df --- /dev/null +++ b/191/CH7/EX7.13/Result7_13.txt @@ -0,0 +1,55 @@ +
+ Integral 0 to 2 exp(x)dx
+
+ 3.
+
+ - 0.84147
+
+ 3.35085
+
+ - 0.95847
+
+ 3.39404
+
+ - 0.97619
+
+ 3.82179
+
+ - 1.12751
+
+ u(n+1) :
+
+ 1.16926
+
+ v(n+1) :
+
+ - 1.04865
+
+ 3.81948
+
+ - 1.12654
+
+ 4.3234
+
+ - 1.31351
+
+ 4.3945
+
+ - 1.34436
+
+ 5.02915
+
+ - 1.59417
+
+ u(n+1) :
+
+ 1.3883
+
+ v(n+1) :
+
+ - 1.11562
+
+ Calculated U2 n V2 values :
+
+ 1.3883 - 1.11562
+
\ No newline at end of file diff --git a/191/CH7/EX7.2/Example7_2.sce b/191/CH7/EX7.2/Example7_2.sce new file mode 100755 index 000000000..629a7dc9f --- /dev/null +++ b/191/CH7/EX7.2/Example7_2.sce @@ -0,0 +1,27 @@ +//Euler's trapezoidal predictor-corrector pair
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+deff('[g]=f(x,y)','g= -y^2/(1+x)');
+y = 1;
+x = 0;
+h = 0.05;
+i=0;
+while x<0.2
+ y0 = y - 0.05*y^2/(1+x);
+ disp(y0,'The Y0 :')
+ y1 = y - h*(y^2/(1+x)+y0^2/(1+x+h))/2;
+ disp(y1,'The Y1 :')
+ y2 = y - h*(y^2/(1+x)+y1^2/(1+x+h))/2;
+ disp(y2,'The Y2 :')
+ y = y2;
+ x = x + h;
+end
+disp(y2,'The calculated value of y(0.2):');
+x = 0.2;
+act = 1/(1+log(1+x));
+disp(act,'The exact value is of y(0.2): ');
+err = act - y2;
+disp(err,'The error is :');
\ No newline at end of file diff --git a/191/CH7/EX7.2/Result7_2.txt b/191/CH7/EX7.2/Result7_2.txt new file mode 100755 index 000000000..b31b31347 --- /dev/null +++ b/191/CH7/EX7.2/Result7_2.txt @@ -0,0 +1,60 @@ +
+ The Y0 :
+
+ 0.95
+
+ The Y1 :
+
+ 0.95351
+
+ The Y2 :
+
+ 0.95335
+
+ The Y0 :
+
+ 0.91007
+
+ The Y1 :
+
+ 0.91289
+
+ The Y2 :
+
+ 0.91277
+
+ The Y0 :
+
+ 0.87490
+
+ The Y1 :
+
+ 0.87720
+
+ The Y2 :
+
+ 0.87711
+
+ The Y0 :
+
+ 0.84366
+
+ The Y1 :
+
+ 0.84556
+
+ The Y2 :
+
+ 0.84549
+
+ The calculated value of y(0.2):
+
+ 0.84549
+
+ The exact value is of y(0.2):
+
+ 0.84579
+
+ The error is :
+
+ 0.00030
\ No newline at end of file diff --git a/191/CH7/EX7.3/Example7_3.sce b/191/CH7/EX7.3/Example7_3.sce new file mode 100755 index 000000000..3bf51222e --- /dev/null +++ b/191/CH7/EX7.3/Example7_3.sce @@ -0,0 +1,25 @@ +//Mid-point formula
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+deff('[g]=f(x,y)','g= -y^2/(1+x)');
+y0 = 1;
+y1 = 0.95335;
+x = 0.05;
+h = 0.05;
+i=0;
+while x<0.2
+ y2 = y0 - 0.1*y1^2/(1+x);
+ disp(y2,'The Y :')
+ y0 = y1;
+ y1 = y2;
+ x = x + h;
+end
+disp(y2,'The calculated value of y(0.2):');
+x = 0.2;
+act = 1/(1+log(1+x));
+disp(act,'The exact value is of y(0.2): ');
+err = act - y2;
+disp(err,'The error is :');
\ No newline at end of file diff --git a/191/CH7/EX7.3/Result7_3.txt b/191/CH7/EX7.3/Result7_3.txt new file mode 100755 index 000000000..eef26cd1f --- /dev/null +++ b/191/CH7/EX7.3/Result7_3.txt @@ -0,0 +1,24 @@ +
+ The Y :
+
+ 0.91344
+
+ The Y :
+
+ 0.87750
+
+ The Y :
+
+ 0.84648
+
+ The calculated value of y(0.2):
+
+ 0.84648
+
+ The exact value is of y(0.2):
+
+ 0.84579
+
+ The error is :
+
+ - 0.00069
\ No newline at end of file diff --git a/191/CH7/EX7.4/Example7_4.sce b/191/CH7/EX7.4/Example7_4.sce new file mode 100755 index 000000000..36a6dc582 --- /dev/null +++ b/191/CH7/EX7.4/Example7_4.sce @@ -0,0 +1,36 @@ +//Illustraion of Taylor Series for approximation
+//It needs symbolic toolbox
+clc;
+clear;
+close();
+cd ~/Desktop/maxima_symbolic;
+exec symbolic.sce
+y0 = 1;
+x0 = 0;
+y1_0 = -y0^2/(1+x0);
+y2_0 = (2*y0^3+y0^2)/((1+x0)^2);
+y3_0 = -(6*y0^4 + 6*y0^3 + 2*y0^2)/((1+x0)^3);
+//similarly
+y4_0 = 88;
+y5_0 = -694;
+y6_0 = 6578;
+y7_0 = -72792;
+syms r h;
+format('v',10);
+yxr = 1 - r*h + (y2_0*(r*h)^2)/factorial(2) - (y3_0*(r*h)^3)/factorial(3) + (y4_0*(r*h)^4)/factorial(4) - (y5_0*(r*h)^5)/factorial(5) +(y6_0*(r*h)^6)/factorial(6) - (y7_0*(r*h)^7)/factorial(7);
+yxr_5d = 1 - r*h + (y2_0*(r*h)^2)/factorial(2) + (y3_0*(r*h)^3)/factorial(3) + (y4_0*(r*h)^4)/factorial(4);
+h = 0.05;
+r = 1;
+yx1 = eval(yxr_5d);
+format('v',8);
+disp(dbl(yx1), 'Value when r = 1 :');
+
+syms r h;
+format('v',10);
+yxr = 1 - r*h + (y2_0*(r*h)^2)/factorial(2) - (y3_0*(r*h)^3)/factorial(3) + (y4_0*(r*h)^4)/factorial(4) - (y5_0*(r*h)^5)/factorial(5) +(y6_0*(r*h)^6)/factorial(6) - (y7_0*(r*h)^7)/factorial(7);
+yxr_5d = 1 - r*h + (y2_0*(r*h)^2)/factorial(2) + (y3_0*(r*h)^3)/factorial(3) + (y4_0*(r*h)^4)/factorial(4) + (y5_0*(r*h)^5)/factorial(5) ;
+h = 0.05;
+r = 2;
+yx1 = eval(yxr_5d);
+format('v',8);
+disp(dbl(yx1), 'Value when r = 2 :')
\ No newline at end of file diff --git a/191/CH7/EX7.4/Result7_4.txt b/191/CH7/EX7.4/Result7_4.txt new file mode 100755 index 000000000..d816e47e9 --- /dev/null +++ b/191/CH7/EX7.4/Result7_4.txt @@ -0,0 +1,6 @@ + Value when r = 1 :
+
+ 0.95348
+ Value when r = 1 :
+
+ 0.95348
\ No newline at end of file diff --git a/191/CH7/EX7.5/Example7_5.sce b/191/CH7/EX7.5/Example7_5.sce new file mode 100755 index 000000000..a94f1635f --- /dev/null +++ b/191/CH7/EX7.5/Example7_5.sce @@ -0,0 +1,32 @@ +// 3-Step Adams - Bashforth and 2- step Adam-Moulton formula
+clc;
+clear;
+close();
+format('v',8);
+funcprot(0);
+disp('Integral 0 to 2 exp(x)dx');
+deff('[yd]=f(x,y)','yd = -y^2/(1+x)');
+
+y0 = 1;
+x0 = 0;
+h = 0.05;
+x1 = x0+h;
+x2 = x1+h;
+y2 = 0.91298;
+y1 = 0.95348;
+for i = 1:2
+ yn = y2 + h*(23*f(x2,y2)-16*f(x1,y1)+5*f(x0,y0))/12;
+ disp(yn,'yn(0) = ');
+ yn_i = yn;
+ yn_i = y2 + h*(5*f(x2+h,yn_i)+8*f(x2,y2)-f(x1,y1))/12;
+ disp(yn_i , 'yn(i)');
+ yn_i = y2 + h*(5*f(x2+h,yn_i)+8*f(x2,y2)-f(x1,y1))/12;
+ disp(yn_i , 'yn(i)');
+ y0 = y1;y1 = y2;y2 = yn_i;
+ x0 = x1;x1 = x2;x2 = x2+h;
+end
+x = 0.2 ;
+act = 1/(1+log(1+x));
+disp(act,'The exact value is of y(0.2): ');
+err = act - y2;
+disp(err,'The error is :');
\ No newline at end of file diff --git a/191/CH7/EX7.5/Result7_5.txt b/191/CH7/EX7.5/Result7_5.txt new file mode 100755 index 000000000..6b69e7282 --- /dev/null +++ b/191/CH7/EX7.5/Result7_5.txt @@ -0,0 +1,34 @@ +
+ Integral 0 to 2 exp(x)dx
+
+ yn(0) =
+
+ 0.87725
+
+ yn(i)
+
+ 0.87739
+
+ yn(i)
+
+ 0.87738
+
+ yn(0) =
+
+ 0.84571
+
+ yn(i)
+
+ 0.84581
+
+ yn(i)
+
+ 0.84581
+
+ The exact value is of y(0.2):
+
+ 0.84579
+
+ The error is :
+
+ - 0.00001
\ No newline at end of file diff --git a/191/CH8/EX8.1/Example8_1.sce b/191/CH8/EX8.1/Example8_1.sce new file mode 100755 index 000000000..6943479a9 --- /dev/null +++ b/191/CH8/EX8.1/Example8_1.sce @@ -0,0 +1,19 @@ +//The finite difference method
+clc;
+clear;
+close();
+format('v',7);
+funcprot(0);
+disp('Integral 0 to 2 exp(x)dx');
+deff('[pp]=p(x)','pp=x');
+deff('[qq]=q(x)','qq=-3');
+deff('[rr]=r(x)','rr=exp(x)');
+y0 = 1;
+yn = 2;
+x = [.2 .4 .6 .8 1];
+h = 0.2;
+A = [-2-h^2*q(x(1)) 1-h*p(x(1))/2 0 0;1+h*p(x(2))/2 -2-h^2*q(x(2)) 1-h*p(x(2))/2 0;0 1+h*p(x(3))/2 -2-h^2*q(x(3)) 1-h*p(x(3))/2;0 0 1+h*p(x(4))/2 -2-h^2*q(x(4))];
+disp(A,'A');
+c = [h^2*r(x(1))-(1+h*p(x(1))/2)*y0;h^2*r(x(2));h^2*r(x(3));h^2*r(x(4))-(1-h*p(x(4))/2)*yn];
+Y = inv(A)*c;
+disp(Y','The respective values of y1,y2,y3,y4 : ');
\ No newline at end of file diff --git a/191/CH8/EX8.1/Result8_1.txt b/191/CH8/EX8.1/Result8_1.txt new file mode 100755 index 000000000..e9b4c8313 --- /dev/null +++ b/191/CH8/EX8.1/Result8_1.txt @@ -0,0 +1,13 @@ +
+ Integral 0 to 2 exp(x)dx
+
+ A
+
+ - 1.88 0.98 0. 0.
+ 1.04 - 1.88 0.96 0.
+ 0. 1.06 - 1.88 0.94
+ 0. 0. 1.08 - 1.88
+
+ The respective values of y1,y2,y3,y4 :
+
+ 1.4651 1.8196 2.0383 2.1023
\ No newline at end of file |
