initial commit / add all books

22 files changed, 557 insertions, 0 deletions

diff --git a/1332/CH11/EX11.1/11_1.pdf b/1332/CH11/EX11.1/11_1.pdf
new file mode 100755
index 000000000..ed2819e08
--- /dev/null
+++ b/1332/CH11/EX11.1/11_1.pdf
diff --git a/1332/CH11/EX11.1/11_1.sce b/1332/CH11/EX11.1/11_1.sce
new file mode 100755
index 000000000..8149a5d76
--- /dev/null
+++ b/1332/CH11/EX11.1/11_1.sce
@@ -0,0 +1,17 @@
+//Example 11.1

+//Eigenvalues and Eigenvectors

+//Page no. 333

+clc;clear;close;

+

+A1=[0.6;0.2];A2=[-0.2;0.6];A3=[-0.6;-0.2];A4=[0.2;-0.6];

+T=[1.1,-0.3;-0.3,1.9];

+B1=T*A1;B2=T*A2;B3=T*A3;B4=T*A4;

+disp(B4,B3,B2,B1,'The transformed vectors are :')

+disp('These points lie on the ellipse:')

+printf('     2      2\n(x-3y)+(3x+y)\n------ ------\n  16     4\n\n')

+A5=[0;2/sqrt(10)];

+disp('The vector (0,2/10^(1/2)) lies on the circle:')

+printf(' 2   2\nx + y = 4\n        -\n        10\n\n')

+B5=T*A5;

+disp('Also lies on the same ellipse',B5)

+printf('\n\nWe can see that there is a linear relationship between the first 4 vectors and their respective transformend vectors through the scalars known as eigenvectors and eigenvalues respectively')
\ No newline at end of file
diff --git a/1332/CH11/EX11.10/11_10.pdf b/1332/CH11/EX11.10/11_10.pdf
new file mode 100755
index 000000000..f586d6d7a
--- /dev/null
+++ b/1332/CH11/EX11.10/11_10.pdf
diff --git a/1332/CH11/EX11.10/11_10.sce b/1332/CH11/EX11.10/11_10.sce
new file mode 100755
index 000000000..bb91d6b91
--- /dev/null
+++ b/1332/CH11/EX11.10/11_10.sce
@@ -0,0 +1,49 @@
+//Example 11.10

+//LU Method

+//Page no. 363

+clc;close;clear;

+

+A=[120,80,40,-16;80,120,16,-40;40,16,120,-80;-16,-40,-80,120];

+disp(A,"A =")

+L=eye(4,4);

+for l=1:20

+for j=1:4

+    for i=1:j

+        k=0

+        for p=1:i-1

+            k=k-A(i,p)*A(p,j)

+        end

+        A(i,j)=A(i,j)+k

+    end

+    for i=j+1:4

+        k=0;

+        for p=1:j-1

+            k=k-A(i,p)*A(p,j)

+        end

+        A(i,j)=(A(i,j)+k)/A(j,j) 

+    end

+end

+disp(A,"Modified A = ")

+    for i=1:4

+    for j=1:4

+        if i>j then

+            L(i,j)=A(i,j)

+        else

+            U(i,j)=A(i,j)

+        end

+    end

+end

+disp(U,"U =",L,"L =")

+A=U*L;

+printf('\n\nAfter %i iterations, matrix A =\n\n',l)

+for i=1:4

+    for j=1:4

+        printf('    %.2f\t',A(i,j))

+    end

+    printf('\n')

+end

+end

+printf('\n\nTherefore the eigenvalues are the diagonal elements f the transformed triangular matrix are:\n\n')

+for i=1:4

+    printf(' %.2f,',A(i,i))

+end
\ No newline at end of file
diff --git a/1332/CH11/EX11.11/11_11.pdf b/1332/CH11/EX11.11/11_11.pdf
new file mode 100755
index 000000000..cc1ab5a87
--- /dev/null
+++ b/1332/CH11/EX11.11/11_11.pdf
diff --git a/1332/CH11/EX11.11/11_11.sce b/1332/CH11/EX11.11/11_11.sce
new file mode 100755
index 000000000..745b06f80
--- /dev/null
+++ b/1332/CH11/EX11.11/11_11.sce
@@ -0,0 +1,119 @@
+//Example 11.11

+//Generalized Eigenvalue Problem

+//Page no. 365

+clc;close;clear;

+

+A=[1,1,0.5;1,1,0.25;0.5,0.25,2]

+B=[2,2,2;2,5,5;2,5,11]

+disp(B,"B =",A,"A =")

+for i=1:3

+    G(i,i)=sqrt(B(i,i))

+end

+G=[B;eye(3,3)];

+

+//transformation to frobenius matrix

+for k=3:-1:2

+    g(k)=0;

+    for j=1:k-1

+        if(g(k)<G(k,j))

+            g(k)=G(k,j)

+            p=j;

+        end

+    end

+    if(g(k)~=0)

+        for j=1:3

+            r(1,j)=G(k,j)

+        end

+        for i=1:6

+            G(i,k-1)=G(i,k-1)/g(k)

+        end

+        for j=1:3

+            if(j~=k-1)

+                l=G(k,j)

+                for i=1:6

+                    G(i,j)=G(i,j)-l*G(i,k-1)

+                end

+            end

+        end

+    end

+    for j=1:3

+        for i=1:3

+            c(i,1)=G(i,j)

+        end

+        G(k-1,j)=0

+        for i=1:3

+            G(k-1,j)=G(k-1,j)+r(1,i)*c(i,1)

+        end

+    end

+end

+

+//partition g

+for i=4:6

+    for j=1:3

+        T(i-3,j)=G(i,j)

+    end

+end

+

+//eigenvalues computation

+p=poly(B,'x')

+a=roots(p)

+printf('\n\nDiagonalized Matrix B = \n\n')

+for i=1:3

+    for j=1:3

+        if i~=j then

+            B(i,j)=0

+        else

+            B(i,j)=a(i)

+        end

+    end

+end

+disp(B)

+//eigenvectors computation

+for k=1:3

+    m=2

+    for l=1:3

+        y(l,k)=a(k)^(m)

+        m=m-1;

+    end

+end

+printf('\n\n')

+

+

+for k=1:3

+    for l=1:3

+        y1(l,1)=y(l,1)

+        y2(l,1)=y(l,2)

+        y3(l,1)=y(l,3)

+    end

+    x1=T*y3;

+    x2=T*y2;

+    x3=T*y1;

+end

+printf('\n\nEigenvectors of B are :\n\n')

+for i=1:3

+    printf('|%.5f|\t\t|%.5f|\t\t|%.5f|',x3(i,1),x2(i,1),x1(i,1))

+    printf('\n')

+end

+x=[x3,x2,x1]

+

+

+

+

+

+B=[2,2,2;2,5,5;2,5,11]

+G=0

+for i=1:3

+    for j=1:3

+        if i==j then

+            G(i,j)=sqrt(B(i,j))

+        else

+            G(i,j)=0;

+        end

+    end

+end

+

+B=inv(G)*x'*A*x*inv(G)

+disp(B,"Eigenvectors of A =")

+

+printf('\n\n\nNote : Computation Error in book in caculation of eigenvector of B thus for A')

+

diff --git a/1332/CH11/EX11.2/11_2.pdf b/1332/CH11/EX11.2/11_2.pdf
new file mode 100755
index 000000000..78f0fd729
--- /dev/null
+++ b/1332/CH11/EX11.2/11_2.pdf
diff --git a/1332/CH11/EX11.2/11_2.sce b/1332/CH11/EX11.2/11_2.sce
new file mode 100755
index 000000000..4bb0413f5
--- /dev/null
+++ b/1332/CH11/EX11.2/11_2.sce
@@ -0,0 +1,43 @@
+//Example 11.2

+//Leverrier's Method

+//Page no. 337

+clc;close;clear;

+

+A=[2,2,2;2,5,5;2,5,1];

+A1=A;

+C(1)=0;

+    for j=1:3

+        for k=1:3

+            if(j==k)

+                C(1)=C(1)+A1(j,k)

+            end

+        end

+    end

+    disp(A,'A=')

+        disp(A1,'A1=')

+    printf('\nC1=')

+    disp(C(1));

+for i=2:3

+    A2=A*(A1-C(i-1)*eye(3,3));

+    printf('\n\n\nA%i=',i)

+    disp(A2);

+    C(i)=0;

+    for j=1:3

+        for k=1:3

+            if(j==k)

+                C(i)=C(i)+A2(j,k)/i

+            end

+        end

+    end

+    printf('\nC%i=',i)

+    disp(C(i))

+    A1=A2;

+end

+printf('\n\n\nTherefore the characteristic polynomial is:\n 3    2\nx - %ix - %ix %i = 0',C(1),C(2),C(3))

+

+//verification

+printf('\n\nVerification:')

+s=poly(0,"s");

+p=poly(A,'x');

+A=A-eye(3,3)*%s;

+disp(p,'=',A)

diff --git a/1332/CH11/EX11.3/11_3.pdf b/1332/CH11/EX11.3/11_3.pdf
new file mode 100755
index 000000000..afc59990b
--- /dev/null
+++ b/1332/CH11/EX11.3/11_3.pdf
diff --git a/1332/CH11/EX11.3/11_3.sce b/1332/CH11/EX11.3/11_3.sce
new file mode 100755
index 000000000..f25a0d626
--- /dev/null
+++ b/1332/CH11/EX11.3/11_3.sce
@@ -0,0 +1,90 @@
+//Example 11.3

+//Danilevsky Method

+//Page no. 341

+clc;close;clear;

+

+A=[-1,0,0;1,-2,3;0,2,-3];

+G=[A;eye(3,3)];

+disp(G);

+//transformation to frobenius matrix

+for k=3:-1:2

+    g(k)=0;

+    for j=1:k-1

+        if(g(k)<G(k,j))

+            g(k)=G(k,j)

+            p=j;

+        end

+    end

+    if(g(k)~=0)

+        for j=1:3

+            r(1,j)=G(k,j)

+        end

+        for i=1:6

+            G(i,k-1)=G(i,k-1)/g(k)

+        end

+        disp(G)

+        for j=1:3

+            if(j~=k-1)

+                l=G(k,j)

+                for i=1:6

+                    G(i,j)=G(i,j)-l*G(i,k-1)

+                end

+            end

+        end

+        disp(G)

+    end

+    for j=1:3

+        for i=1:3

+            c(i,1)=G(i,j)

+        end

+        G(k-1,j)=0

+        for i=1:3

+            G(k-1,j)=G(k-1,j)+r(1,i)*c(i,1)

+        end

+    end

+    disp(G)

+end

+

+//partition g

+for i=4:6

+    for j=1:3

+        T(i-3,j)=G(i,j)

+    end

+end

+disp(T,'T=')

+

+//eigenvalues computation

+printf('\n\n\nCharateristic polynomial:')

+p=poly(A,'x')

+disp(p)

+printf('\n\n\nEigenvalues:')

+a=roots(p)

+disp(a')

+//eigenvectors computation

+for k=1:3

+    m=2

+    for l=1:3

+        y(l,k)=a(k,1)^(m)

+        m=m-1;

+    end

+end

+printf('\n\n')

+disp(y,'y=')

+

+//eigenvector computation

+

+for k=1:3

+    for l=1:3

+        y1(l,1)=y(l,1)

+        y2(l,1)=y(l,2)

+        y3(l,1)=y(l,3)

+    end

+    x1=T*y3;

+    x2=T*y2;

+    x3=T*y1;

+end

+printf('\n\nEigenvectors :\n')

+for i=1:3

+    printf('|%.1f|\t\t|%.1f|\t\t|%.1f|',x1(i,1),x2(i,1),x3(i,1))

+    printf('\n')

+end

diff --git a/1332/CH11/EX11.4/11_4.pdf b/1332/CH11/EX11.4/11_4.pdf
new file mode 100755
index 000000000..7afaedb01
--- /dev/null
+++ b/1332/CH11/EX11.4/11_4.pdf
diff --git a/1332/CH11/EX11.4/11_4.sce b/1332/CH11/EX11.4/11_4.sce
new file mode 100755
index 000000000..6e787e59a
--- /dev/null
+++ b/1332/CH11/EX11.4/11_4.sce
@@ -0,0 +1,19 @@
+//Example 11.4

+//Power Method

+//Page no. 345

+clc;close;clear;

+

+A=[1,2;3,4];

+e=0.001;

+q0=[1;1];

+for i=1:5

+    q1=A*q0;

+    a=max(q1)

+    for j=1:2

+        q2(j)=q1(j)/a;

+    end

+    printf('\nq(%i) = %.4f     a = %.4f      Scaled q(%i) = %.4f\n       %.4f                                      %i\n\n',i,q1(1),a,i,q2(1),q1(2),q2(2))

+    q1=q2;

+    q0=q1;

+end

+printf('Hence the largest eigenvalue is %.4f with the corresponding eigenvector as %.4f\n                                                                                %i',a,q0(1),q0(2))
\ No newline at end of file
diff --git a/1332/CH11/EX11.5/11_5.pdf b/1332/CH11/EX11.5/11_5.pdf
new file mode 100755
index 000000000..1db354995
--- /dev/null
+++ b/1332/CH11/EX11.5/11_5.pdf
diff --git a/1332/CH11/EX11.5/11_5.sce b/1332/CH11/EX11.5/11_5.sce
new file mode 100755
index 000000000..8c4c7be78
--- /dev/null
+++ b/1332/CH11/EX11.5/11_5.sce
@@ -0,0 +1,41 @@
+//Example 11.5

+//Inverse Power Method

+//Page no. 347

+clc;close;clear;

+

+A=[7,6,-3;-12,-20,24;-6,-12,16];

+e=10^-6;

+X=[1;1;1];

+B=0;

+Y=[0;0;0]

+a=0;l=0;

+for i=1:2

+    printf('When a=%i\n',a);

+    C=A-a*eye();

+    disp(C,"C=")

+    C_1=inv(C);

+    disp(C_1,"Inverse of C=");

+    printf('\n\nItr       lambda                             X')

+    printf('\n------------------------------------------------------------------\n')

+    for j=1:10

+        printf('\n%i        %f           %f     %f      %f',j-1,l,X(1),X(2),X(3));

+        Y=C_1*X;

+        B=max(Y);

+        e1=abs(l-B);

+        X=Y/B;

+        m=0;

+        for k=1:3

+            m=m+(Y(k)-X(k))^2;

+        end

+        e2=sqrt(m);

+        er=max(e1,e2);

+        if(er<e)

+            break

+        end

+        l=B;

+        

+    end

+    a=-3;

+    printf('\n\n\n\n')

+end

+printf('\n\n\nNote : Computation of Y is wrong given in the book')
\ No newline at end of file
diff --git a/1332/CH11/EX11.6/11_6.pdf b/1332/CH11/EX11.6/11_6.pdf
new file mode 100755
index 000000000..614f45376
--- /dev/null
+++ b/1332/CH11/EX11.6/11_6.pdf
diff --git a/1332/CH11/EX11.6/11_6.sce b/1332/CH11/EX11.6/11_6.sce
new file mode 100755
index 000000000..1ead55877
--- /dev/null
+++ b/1332/CH11/EX11.6/11_6.sce
@@ -0,0 +1,26 @@
+//Example 11.6

+//Rayleigh Quotient

+//Page no. 348

+clc;close;clear;

+

+A=[10,7,8,7;7,5,6,5;8,6,10,9;7,5,9,10];

+q0=[1;1;1;1];

+for i=0:4

+    X=(A^i)*q0;

+    l=(X'*A*X)/(X'*X)

+    printf('\nLambda(%i) = %f\n',i+1,l)

+end

+printf('\n\nDominant Eigenvalue = %f\n\n\n',l)

+

+e=0.001;

+for i=1:5

+    q1=A*q0;

+    a=max(q1)

+    for j=1:4

+        q2(j)=q1(j)/a;

+    end

+

+    q1=q2;

+    q0=q1;

+end

+disp(q2,'Corresponding Eigenvector = ')
\ No newline at end of file
diff --git a/1332/CH11/EX11.7/11_7.pdf b/1332/CH11/EX11.7/11_7.pdf
new file mode 100755
index 000000000..537aae140
--- /dev/null
+++ b/1332/CH11/EX11.7/11_7.pdf
diff --git a/1332/CH11/EX11.7/11_7.sce b/1332/CH11/EX11.7/11_7.sce
new file mode 100755
index 000000000..4aba85754
--- /dev/null
+++ b/1332/CH11/EX11.7/11_7.sce
@@ -0,0 +1,49 @@
+//Example 11.7

+//Jacobi's Method

+//Page no. 355

+clc;close;clear;

+

+A=[1,1,1/2;1,1,1/4;1/2,1/4,2];

+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/1332/CH11/EX11.8/11_8.pdf b/1332/CH11/EX11.8/11_8.pdf
new file mode 100755
index 000000000..03ccb3fcd
--- /dev/null
+++ b/1332/CH11/EX11.8/11_8.pdf
diff --git a/1332/CH11/EX11.8/11_8.sce b/1332/CH11/EX11.8/11_8.sce
new file mode 100755
index 000000000..d366036b9
--- /dev/null
+++ b/1332/CH11/EX11.8/11_8.sce
@@ -0,0 +1,16 @@
+//Example 11.8

+//Recursive Formula

+//Page no. 357

+clc;close;clear;

+

+A=[2,-1,0,0;-1,2,-1,0;0,-1,2,-1;0,0,-1,2];

+l=poly(0,"l");

+p0=1;

+p1=A(1,1)-l;

+for i=2:4

+    p2=(A(i,i)-l)*p1-A(i,i-1)^2*p0;

+    p0=p1;

+    p1=p2;

+    printf('\n\np%i(l) = ',i);

+    disp(p2)

+end

diff --git a/1332/CH11/EX11.9/11_9.pdf b/1332/CH11/EX11.9/11_9.pdf
new file mode 100755
index 000000000..6cd99ab28
--- /dev/null
+++ b/1332/CH11/EX11.9/11_9.pdf
diff --git a/1332/CH11/EX11.9/11_9.sce b/1332/CH11/EX11.9/11_9.sce
new file mode 100755
index 000000000..5c95364bb
--- /dev/null
+++ b/1332/CH11/EX11.9/11_9.sce
@@ -0,0 +1,88 @@
+//Example 11.9

+//QR Method

+//Page no. 360

+clc;close;clear;

+

+A=[2,-1,0;-1,2,-1;0,-1,2];

+deff('y=c(i,j)','y=A(j,j)/sqrt((A(i,j)^2+A(j,j)^2))')

+deff('y=s2(i,j)','y=A(i,j)/sqrt((A(i,j)^2+A(j,j)^2))')

+disp(A,'A=')

+l0=0;f=1;m=0;s=0;w=0;

+for n=1:5

+    for j=1:2

+        for k=1:2

+            V(j,k)=A(j,k)

+        end

+    end

+    disp(V,'V=')

+    p=poly(V,'x');

+    disp('=0',p);

+    a=roots(p);

+    for j=1:2

+        printf('\na(%i) = %f',j,a(j))

+    end

+        if(abs(a(1)-V(1,1))<=abs(a(2)-V(1,1)))

+            a=a(1)

+        else

+            a=a(2)

+        end

+    printf('\na = %f\n',a)

+    s=s+a;

+    A=A-a*eye()

+    R=A;Q=eye(3,3);

+    

+   for j=1:2

+       for i=j+1:3

+           for k=1:3             //C matrix evaluation

+               for l=1:3

+                   if(k==l)

+                       if(k==i | k==j)

+                           C(k,l)=c(i,j)

+                       else

+                           C(k,l)=1

+                       end

+                   end

+                   if(k>l)

+                       if(k==i & l==j)

+                           C(k,l)=-1*s2(i,j)

+                      else

+                           C(k,l)=0

+                       end

+                   end

+                   if(k<l)

+                       if(k==j & l==i)

+                           C(k,l)=s2(i,j)

+                       else

+                           C(k,l)=0

+                       end

+                   end

+               end

+          end

+           

+           R=C*R;

+           Q=Q*C';

+           

+       end

+    end

+disp(Q,'Q=',R,'R=')

+disp(Q*R,'Q*R=')

+A=R*Q;

+disp(A,'A=')

+end

+l1=l0+s;

+for i=2:3

+    for j=2:3

+        V(i-1,j-1)=A(i,j)

+    end

+end

+disp(V,'V=')

+    p=poly(V,'x');

+    disp('=0',p);

+    a=roots(p);

+    for j=1:2

+        printf('\na(%i) = %f',j,a(j))

+    end

+    l2=l1+a(1)

+    l3=l1+a(2)

+    disp(l3,'l3=',l2,'l2=',l1,'l1=')

+printf('\n\n\nNote : Values of V varies in each step resulting in different results due to error in book calculation')
\ No newline at end of file