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-rwxr-xr-x1670/CH4/EX4.1/4_1.sce19
-rwxr-xr-x1670/CH4/EX4.10/4_10.sce30
-rwxr-xr-x1670/CH4/EX4.11/4_11.sce30
-rwxr-xr-x1670/CH4/EX4.12/4_12.sce36
-rwxr-xr-x1670/CH4/EX4.13/4_13.sce23
-rwxr-xr-x1670/CH4/EX4.14/4_14.sce23
-rwxr-xr-x1670/CH4/EX4.15/4_15.sce25
-rwxr-xr-x1670/CH4/EX4.16/4_16.sce25
-rwxr-xr-x1670/CH4/EX4.17/4_17.sce18
-rwxr-xr-x1670/CH4/EX4.2/4_2.sce20
-rwxr-xr-x1670/CH4/EX4.3/4_3.sce20
-rwxr-xr-x1670/CH4/EX4.4/4_4.sce20
-rwxr-xr-x1670/CH4/EX4.5/4_5.sce37
-rwxr-xr-x1670/CH4/EX4.6/4_6.sce49
-rwxr-xr-x1670/CH4/EX4.7/4_7.sce32
-rwxr-xr-x1670/CH4/EX4.8/4_8.sce30
-rwxr-xr-x1670/CH4/EX4.9/4_9.sce30
17 files changed, 467 insertions, 0 deletions
diff --git a/1670/CH4/EX4.1/4_1.sce b/1670/CH4/EX4.1/4_1.sce
new file mode 100755
index 000000000..4043d5311
--- /dev/null
+++ b/1670/CH4/EX4.1/4_1.sce
@@ -0,0 +1,19 @@
+//Example 4.1
+//Power Method
+//Page no. 89
+clc;close;clear;
+
+A=[1,3,-1;3,2,4;-1,4,10];
+e=0.001;
+q0=[0;0;1];
+for i=1:5
+ q1=A*q0;
+ a=max(q1)
+ for j=1:3
+ q2(j)=q1(j)/a;
+ end
+ printf('\nq(%i) = %.4f a = %.4f Scaled q(%i) = %.3f\n %.3f %.3f\n %.3f %i\n\n',i,q1(1),a,i,q2(1),q1(2),q2(2),q1(3),q2(3))
+ q1=q2;
+ q0=q1;
+end
+printf('Hence the largest eigenvalue is %.2f with the corresponding eigenvector as %.3f\n %.3f\n %i',a,q0(1),q0(2),q0(3)) \ No newline at end of file
diff --git a/1670/CH4/EX4.10/4_10.sce b/1670/CH4/EX4.10/4_10.sce
new file mode 100755
index 000000000..ab713b8ab
--- /dev/null
+++ b/1670/CH4/EX4.10/4_10.sce
@@ -0,0 +1,30 @@
+//Example 4.10
+//Givens Method
+//Page no. 105
+clc;close;clear;
+
+A=[8,-6,2;-6,7,-4;2,-4,3];
+n=3;
+for k=1:1
+ max1=0
+i1=2;j1=3;
+fi=(atan((2*A(i1,j1))/(A(i1,i1)-A(j1,j1)+10^-20)))/2
+disp(fi,'fi = ')
+O1=eye(n,n)
+O1(i1,j1)=-sin(fi)
+O1(j1,i1)=sin(fi)
+O1(i1,i1)=cos(fi)
+O1(j1,j1)=cos(fi)
+disp(O1,'O1 = ')
+A=inv(O1)*A*O1
+disp(A,'B = ')
+end
+printf('\n\n')
+l=poly(0,'lb')
+A=A-l*eye(n,n)
+disp(det(A),'Characteristic Equation = ')
+A=roots(det(A))
+printf('\n\n The eigenvalues are : \n\n')
+for i=1:n
+ printf('\tl%i = %g\t',i,A(i))
+end \ No newline at end of file
diff --git a/1670/CH4/EX4.11/4_11.sce b/1670/CH4/EX4.11/4_11.sce
new file mode 100755
index 000000000..63c49dd08
--- /dev/null
+++ b/1670/CH4/EX4.11/4_11.sce
@@ -0,0 +1,30 @@
+//Example 4.11
+//Givens Method
+//Page no. 106
+clc;close;clear;
+
+A=[1,2,2;2,1,2;2,2,1];
+n=3;
+for k=1:1
+ max1=0
+i1=2;j1=3;
+fi=(atan((2*A(i1,j1))/(A(i1,i1)-A(j1,j1)+10^-20)))/2
+disp(fi,'fi = ')
+O1=eye(n,n)
+O1(i1,j1)=-sin(fi)
+O1(j1,i1)=sin(fi)
+O1(i1,i1)=cos(fi)
+O1(j1,j1)=cos(fi)
+disp(O1,'O1 = ')
+A=inv(O1)*A*O1
+disp(A,'B = ')
+end
+printf('\n\n')
+l=poly(0,'lb')
+A=A-l*eye(n,n)
+disp(-det(A),'Characteristic Equation = ')
+A=roots(det(A))
+printf('\n\n The eigenvalues are : \n\n')
+for i=1:n
+ printf('\tl%i = %g\t',i,A(i))
+end \ No newline at end of file
diff --git a/1670/CH4/EX4.12/4_12.sce b/1670/CH4/EX4.12/4_12.sce
new file mode 100755
index 000000000..56ceeb5e8
--- /dev/null
+++ b/1670/CH4/EX4.12/4_12.sce
@@ -0,0 +1,36 @@
+//Example 4.12
+//Givens Method
+//Page no. 107
+clc;close;clear;
+
+A=[1,2,2,2;2,1,2,2;2,2,1,3;2,2,3,1];
+n=4;
+for k=1:3
+ max1=0
+ if k==1 then
+ i1=2;j1=3;
+ elseif k==2
+ i1=2;j1=4;
+ else
+ i1=3;j1=4;
+ end
+fi=(atan((2*A(i1,j1))/(A(i1,i1)-A(j1,j1)+10^-20)))/2
+disp(fi,'fi = ')
+O1=eye(n,n)
+O1(i1,j1)=-sin(fi)
+O1(j1,i1)=sin(fi)
+O1(i1,i1)=cos(fi)
+O1(j1,j1)=cos(fi)
+disp(O1,'O1 = ')
+A=inv(O1)*A*O1
+disp(A,'B = ')
+end
+printf('\n\n')
+l=poly(0,'lb')
+A=A-l*eye(n,n)
+disp(-det(A),'Characteristic Equation = ')
+A=roots(det(A))
+printf('\n\n The eigenvalues are : \n\n')
+for i=1:n
+ printf('\tl%i = %g\t',i,A(i))
+end \ No newline at end of file
diff --git a/1670/CH4/EX4.13/4_13.sce b/1670/CH4/EX4.13/4_13.sce
new file mode 100755
index 000000000..4d4cb3c6b
--- /dev/null
+++ b/1670/CH4/EX4.13/4_13.sce
@@ -0,0 +1,23 @@
+//Example 4.13
+//House Holder Transformation
+//Page no. 113
+clc;clear;close;
+
+A=[3,2,1;2,3,2;1,2,3]
+disp(A,'A=')
+k=0;
+for j=2:3
+ k=k+A(j,1)^2;
+end
+a=A(2,1)*abs(1/A(2,1))*sqrt(k);
+disp(a,'alpha=')
+U=[0;a+A(2,1);A(3,1)];
+disp(U,'U=')
+U1=U'*U;
+disp(U1,'UT*U=')
+U2=U*U';
+disp(U2,'U*UT=')
+P=eye(3,3)-(2*U2)/U1;
+disp(P,'P=');
+B=P*A*P;
+disp(B,'B='); \ No newline at end of file
diff --git a/1670/CH4/EX4.14/4_14.sce b/1670/CH4/EX4.14/4_14.sce
new file mode 100755
index 000000000..a8f26a1fe
--- /dev/null
+++ b/1670/CH4/EX4.14/4_14.sce
@@ -0,0 +1,23 @@
+//Example 4.14
+//House Holder Transformation
+//Page no. 114
+clc;clear;close;
+
+A=[1,3,4;3,1,2;4,2,1]
+disp(A,'A=')
+k=0;
+for j=2:3
+ k=k+A(j,1)^2;
+end
+a=A(2,1)*abs(1/A(2,1))*sqrt(k);
+disp(a,'alpha=')
+U=[0;a+A(2,1);A(3,1)];
+disp(U,'U=')
+U1=U'*U;
+disp(U1,'UT*U=')
+U2=U*U';
+disp(U2,'U*UT=')
+P=eye(3,3)-(2*U2)/U1;
+disp(P,'P=');
+B=P*A*P;
+disp(B,'B='); \ No newline at end of file
diff --git a/1670/CH4/EX4.15/4_15.sce b/1670/CH4/EX4.15/4_15.sce
new file mode 100755
index 000000000..3c4b2d935
--- /dev/null
+++ b/1670/CH4/EX4.15/4_15.sce
@@ -0,0 +1,25 @@
+//Example 4.15
+//Strum Sequence
+//Page no. 116
+clc;clear;close;
+
+A=[1,2,2;2,1,2;2,2,1]
+disp(A,'A=')
+k=0;
+for j=2:3
+ k=k+A(j,1)^2;
+end
+a=A(2,1)*abs(1/A(2,1))*sqrt(k);
+U=[0;a+A(2,1);A(3,1)];
+U1=U'*U;
+U2=U*U';
+P=eye(3,3)-(2*U2)/U1;
+B=P*A*P;
+disp(B,'Reduced Matrix = ');
+lb=poly(0,"lb")
+f0l=1; //strum sequence
+f1l=(B(1,1)-lb)*f0l;
+f2l=(B(2,2)-lb)*f1l-B(1,2)^2*f0l
+f3l=(B(3,3)-lb)*f2l-B(2,3)^2*f1l
+disp(f3l,"f3(lambda) = ")
+disp(roots(f3l),"Therefore the eigenvalues are : ") \ No newline at end of file
diff --git a/1670/CH4/EX4.16/4_16.sce b/1670/CH4/EX4.16/4_16.sce
new file mode 100755
index 000000000..a507e0337
--- /dev/null
+++ b/1670/CH4/EX4.16/4_16.sce
@@ -0,0 +1,25 @@
+//Example 4.16
+//Strum Sequence
+//Page no. 117
+clc;clear;close;
+
+A=[8,-6,2;-6,7,-4;2,-4,3]
+disp(A,'A=')
+k=0;
+for j=2:3
+ k=k+A(j,1)^2;
+end
+a=A(2,1)*abs(1/A(2,1))*sqrt(k);
+U=[0;a+A(2,1);A(3,1)];
+U1=U'*U;
+U2=U*U';
+P=eye(3,3)-(2*U2)/U1;
+B=P*A*P;
+disp(B,'Reduced Matrix = ');
+lb=poly(0,"lb")
+f0l=1; //strum sequence
+f1l=(B(1,1)-lb)*f0l;
+f2l=(B(2,2)-lb)*f1l-B(1,2)^2*f0l
+f3l=(B(3,3)-lb)*f2l-B(2,3)^2*f1l
+disp(f3l,"f3(lambda) = ")
+disp(roots(f3l),"Therefore the eigenvalues are : ") \ No newline at end of file
diff --git a/1670/CH4/EX4.17/4_17.sce b/1670/CH4/EX4.17/4_17.sce
new file mode 100755
index 000000000..659bda4d1
--- /dev/null
+++ b/1670/CH4/EX4.17/4_17.sce
@@ -0,0 +1,18 @@
+//Example 4.17
+//Gerschgorin Circles
+//Page no. 118
+clc;clear;close;
+
+A=[1,2,3;2,4,6;3,6,1];
+j=2;
+k=3;
+printf('The Gerschgorin Circles are : \n\n A =' )
+for i=1:3
+ printf('\t|z-%i| = |%i| + |%i| = %i\n',A(i,i),A(i,j),A(i,k),A(i,j)+A(i,k))
+ if j~=1 then
+ j=j-1
+ end
+ if i==2 then
+ k=k-1
+ end
+end \ No newline at end of file
diff --git a/1670/CH4/EX4.2/4_2.sce b/1670/CH4/EX4.2/4_2.sce
new file mode 100755
index 000000000..24c7196c1
--- /dev/null
+++ b/1670/CH4/EX4.2/4_2.sce
@@ -0,0 +1,20 @@
+//Example 4.2
+//Power Method
+//Page no. 90
+clc;close;clear;
+
+A=[1,-3,2;4,4,-1;6,3,5];
+e=0.001;
+q0=[1;1;1];
+for i=1:9
+ q1=A*q0;
+ a=max(q1)
+ for j=1:3
+ q2(j)=q1(j)/a;
+ end
+ printf('\nq(%i) = %.4f a = %.4f Scaled q(%i) = %.3f\n %.3f %.3f\n %.3f %i\n\n',i,q1(1),a,i,q2(1),q1(2),q2(2),q1(3),q2(3))
+ q1=q2;
+ q0=q1;
+end
+q0=q0*30
+printf('Hence the largest eigenvalue is %.1g with the corresponding eigenvector as %.1g\n %.1g\n %i',a,q0(1),q0(2),q0(3)) \ No newline at end of file
diff --git a/1670/CH4/EX4.3/4_3.sce b/1670/CH4/EX4.3/4_3.sce
new file mode 100755
index 000000000..659796729
--- /dev/null
+++ b/1670/CH4/EX4.3/4_3.sce
@@ -0,0 +1,20 @@
+//Example 4.3
+//Power Method
+//Page no. 91
+clc;close;clear;
+
+A=[2,-1,0;-1,2,-1;0,-1,2];
+e=0.001;
+q0=[1;1;1];
+for i=1:6
+ q1=A*q0;
+ a=max(q1)
+ for j=1:3
+ q2(j)=q1(j)/a;
+ end
+ printf('\nq(%i) = %.4f a = %.4f Scaled q(%i) = %.3f\n %.3f %.3f\n %.3f %i\n\n',i,q1(1),a,i,q2(1),q1(2),q2(2),q1(3),q2(3))
+ q1=q2;
+ q0=q1;
+end
+q0=-q0/q0(2)
+printf('Hence the largest eigenvalue is %.3f with the corresponding eigenvector as %.1f\n %.1g\n %.1f',a,q0(1),q0(2),q0(3)) \ No newline at end of file
diff --git a/1670/CH4/EX4.4/4_4.sce b/1670/CH4/EX4.4/4_4.sce
new file mode 100755
index 000000000..0452e0c06
--- /dev/null
+++ b/1670/CH4/EX4.4/4_4.sce
@@ -0,0 +1,20 @@
+//Example 4.4
+//Power Method
+//Page no. 93
+clc;close;clear;
+
+A=[3,-1,0;-1,2,-1;0,-1,3];
+e=0.001;
+q0=[1;1;1];
+for i=1:5
+ q1=A*q0;
+ a=max(q1)
+ for j=1:3
+ q2(j)=q1(j)/a;
+ end
+ printf('\nq(%i) = %.4f a = %.4f Scaled q(%i) = %.3f\n %.3f %.3f\n %.3f %i\n\n',i,q1(1),a,i,q2(1),q1(2),q2(2),q1(3),q2(3))
+ q1=q2;
+ q0=q1;
+end
+q0=-q0/q0(2)
+printf('Hence the largest eigenvalue is %.1g with the corresponding eigenvector as %.1g\n %.1g\n %.1g',a,q0(1),q0(2),q0(3)) \ No newline at end of file
diff --git a/1670/CH4/EX4.5/4_5.sce b/1670/CH4/EX4.5/4_5.sce
new file mode 100755
index 000000000..34a049b36
--- /dev/null
+++ b/1670/CH4/EX4.5/4_5.sce
@@ -0,0 +1,37 @@
+//Example 4.5
+//Jacobi Method
+//Page no. 95
+clc;close;clear;
+
+A=[10,7,8,7;7,5,6,5;8,6,10,9;7,5,9,10];
+n=4;
+for k=1:14
+ max1=0
+for i=1:n
+ for j=1:n
+ if A(i,j)>max1 & i~=j then
+ max1=A(i,j)
+ i1=i;j1=j;
+ end
+ end
+end
+fi=(atan((2*A(i1,j1))/(A(i1,i1)-A(j1,j1)+10^-20)))/2
+disp(fi,'fi = ')
+O1=eye(n,n)
+O1(i1,j1)=-sin(fi)
+O1(j1,i1)=sin(fi)
+O1(i1,i1)=cos(fi)
+O1(j1,j1)=cos(fi)
+disp(O1,'O1 = ')
+A=inv(O1)*A*O1
+disp(A,'A1 = ')
+end
+printf('\n\n The eigenvalues are : \n\n')
+for i=1:n
+ printf('\tl%i = %g\t',i,A(i,i))
+end
+printf('\n\n')
+l=poly(0,'lb')
+A=A-l*eye(n,n)
+disp(det(A),'Characteristic Equation = ')
+printf("\n\n\n\n\nNote : Computation Errors in some parts in calculation performed in book") \ No newline at end of file
diff --git a/1670/CH4/EX4.6/4_6.sce b/1670/CH4/EX4.6/4_6.sce
new file mode 100755
index 000000000..bbe85501d
--- /dev/null
+++ b/1670/CH4/EX4.6/4_6.sce
@@ -0,0 +1,49 @@
+//Example 4.6
+//Jacobi Method
+//Page no. 97
+clc;close;clear;
+
+A=[1,sqrt(2),2;sqrt(2),3,sqrt(2);2,sqrt(2),1];
+C=A;
+V=[sqrt(2),0,1/2;sqrt(2),0,1/4;3/(4*sqrt(2)),-1/(4*sqrt(2)),2]
+S=eye(3,3)
+disp(A,"A =")
+VI=0;
+for i=1:3
+ for j=1:3
+ if(i~=j)
+ VI=VI+A(i,j)^2 //initial off diag norm
+ end
+ end
+end
+VI=sqrt(VI);
+VF=VI*10^-7; //final threshold
+V1=VI/3;
+o=poly(0,"o");
+for i=1:3
+for q=2:3
+ for p=q-1:-1:1
+ if(A(p,q)>V1)
+ a=-A(p,q);
+ b=(A(p,p)-A(q,q))/2
+ if(b~=0)
+ w=b*abs(1/b)*(a/sqrt(a^2+b^2));
+ else
+ w=(a/sqrt(a^2+b^2));
+ end
+ sin0=w/sqrt(2*(1+sqrt(1-w^2)));
+ cos0=sqrt(1-sin0^2)
+ end
+ B(p,p)=A(p,p)*cos0^2+A(q,q)*sin0^2-2*A(p,q)*sin0*cos0
+ B(q,q)=A(p,p)*sin0^2+A(q,q)*cos0^2+2*A(p,q)*sin0*cos0
+ B(p,q)=(A(p,p)-A(q,q))*sin0*cos0+A(p,q)*(cos0^2-sin0^2)
+ S(i,i)=S(i,i)
+ S(i,p)=S(i,p)*cos0-S(i,q)*sin0
+ S(i,q)=S(i,p)*sin0+S(i,q)*cos0
+
+ end
+end
+end
+disp(B,"B =")
+disp(S,"S =")
+printf('\n\n\nComputation error in the solution provided by book') \ No newline at end of file
diff --git a/1670/CH4/EX4.7/4_7.sce b/1670/CH4/EX4.7/4_7.sce
new file mode 100755
index 000000000..5211d5285
--- /dev/null
+++ b/1670/CH4/EX4.7/4_7.sce
@@ -0,0 +1,32 @@
+//Example 4.7
+//Jacobi Method
+//Page no. 99
+clc;close;clear;
+
+A=[2,3,1;3,2,2;1,2,1];
+n=3;
+for k=1:10
+ max1=0
+for i=1:n
+ for j=1:n
+ if A(i,j)>max1 & i~=j then
+ max1=A(i,j)
+ i1=i;j1=j;
+ end
+ end
+end
+fi=(atan((2*A(i1,j1))/(A(i1,i1)-A(j1,j1)+10^-20)))/2
+disp(fi,'fi = ')
+O1=eye(n,n)
+O1(i1,j1)=-sin(fi)
+O1(j1,i1)=sin(fi)
+O1(i1,i1)=cos(fi)
+O1(j1,j1)=cos(fi)
+disp(O1,'O1 = ')
+A=inv(O1)*A*O1
+disp(A,'A1 = ')
+end
+printf('\n\n The eigenvalues are : \n\n')
+for i=1:n
+ printf('\tl%i = %g\t',i,A(i,i))
+end \ No newline at end of file
diff --git a/1670/CH4/EX4.8/4_8.sce b/1670/CH4/EX4.8/4_8.sce
new file mode 100755
index 000000000..6f214fa7c
--- /dev/null
+++ b/1670/CH4/EX4.8/4_8.sce
@@ -0,0 +1,30 @@
+//Example 4.8
+//Givens Method
+//Page no. 103
+clc;close;clear;
+
+A=[2,3,1;3,2,2;1,2,1];
+n=3;
+for k=1:1
+ max1=0
+i1=2;j1=3;
+fi=(atan((2*A(i1,j1))/(A(i1,i1)-A(j1,j1)+10^-20)))/2
+disp(fi,'fi = ')
+O1=eye(n,n)
+O1(i1,j1)=-sin(fi)
+O1(j1,i1)=sin(fi)
+O1(i1,i1)=cos(fi)
+O1(j1,j1)=cos(fi)
+disp(O1,'O1 = ')
+A=inv(O1)*A*O1
+disp(A,'B = ')
+end
+printf('\n\n')
+l=poly(0,'lb')
+A=A-l*eye(n,n)
+disp(-det(A),'Characteristic Equation = ')
+A=roots(det(A))
+printf('\n\n The approximate roots of characteristic equation are: \n\n')
+for i=1:n
+ printf('\tl%i = %g\t',i,A(i))
+end \ No newline at end of file
diff --git a/1670/CH4/EX4.9/4_9.sce b/1670/CH4/EX4.9/4_9.sce
new file mode 100755
index 000000000..0a315d3cf
--- /dev/null
+++ b/1670/CH4/EX4.9/4_9.sce
@@ -0,0 +1,30 @@
+//Example 4.9
+//Givens Method
+//Page no. 104
+clc;close;clear;
+
+A=[3,2,1;2,3,2;1,2,3];
+n=3;
+for k=1:1
+ max1=0
+i1=2;j1=3;
+fi=(atan((2*A(i1,j1))/(A(i1,i1)-A(j1,j1)+10^-20)))/2
+disp(fi,'fi = ')
+O1=eye(n,n)
+O1(i1,j1)=-sin(fi)
+O1(j1,i1)=sin(fi)
+O1(i1,i1)=cos(fi)
+O1(j1,j1)=cos(fi)
+disp(O1,'O1 = ')
+A=inv(O1)*A*O1
+disp(A,'B = ')
+end
+printf('\n\n')
+l=poly(0,'lb')
+A=A-l*eye(n,n)
+disp(-det(A),'Characteristic Equation = ')
+A=roots(det(A))
+printf('\n\n The eigenvalues are : \n\n')
+for i=1:n
+ printf('\tl%i = %g\t',i,A(i))
+end \ No newline at end of file
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