Q : Simpson rule Matlab code to calculates the numerical approximation to integral x^3 from 0 to 1 with 3 sub-intervals?

Answer :

To calculate the numerical approximation of

\[\int_{0}^{1} x^3 dx \]

with 3 sub-intervals

Simpson 1/3 rule :

\[\int_{a}^{b} f(x) dx =\frac{\delta x}{3}\left(f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+…..+2f(x_{n-2})+4f(x_{n-1})+f(x_{n})\right)\]

clear all
%Simpson 1/3 rule Matlab code to calculates the numerical approximation to a definite integral
f=@(x) x.^3 % function
a=0 % left point of the interval [a, b]
b=1 % right point of the interval [a, b]
n=3 % step size
A=0
h=(b-a)/n
for i = 1:n
    x(i) = a +h* (i-1); 
    y(i) = f(x(i));
end
%Simpson 1/3 rule
for i = 1:n
    if ( i== 1 || i == n)
        A = A+y(i);
    elseif(rem(i,2)==1)
        A = A+4*y(i);
    else(rem(i,2)==2)
        A = A+2*y(i);
    end
end
clear all
%Simpson 1/3 rule Matlab code to calculates the numerical approximation to a definite integral
f=@(x) x.^3 % function
a=0 % left point of the interval [a, b]
b=1 % right point of the interval [a, b]
n=3 % step size
A=0
h=(b-a)/n
for i = 1:n
    x(i) = a +h* (i-1); 
    y(i) = f(x(i));
end
%Simpson 1/3 rule
for i = 1:n
    if ( i== 1 || i == n)
        A = A+y(i);
    elseif(rem(i,2)==1)
        A = A+4*y(i);
    else(rem(i,2)==2)
        A = A+2*y(i);
    end
end


Output:
f =

@(x) x .^ 3

a = 0
b = 1
n = 3
A = 0
h = 0.3333
ans = 0
A = 0.041152
A= 0.0411523

Q: What is a Matrix ? Explain addition, subtraction and multiplication of two matrices.

Answer :

Matrix : A matrix is an arrangement of elements into rows and columns.

Examples

• \(\left(\begin{array}{ccc}3 & -2 & 1 \\4 & -5 & 8\\\end{array}\right)\) is a matrix with two rows and three columns.
• \(\left(\begin{array}{ccc}2 & 3 & 4 \\3 & 2 & 1 \\4 & 6 & 8 \\\end{array}\right)\) is a matrix with three rows and three columns

Addition and Subtraction of Two Matrices : If Two matrices have equal number of rows and columns are equal then they can be added element wise.

Let \(A=\left(\begin{array}{ccc}0 & -1 & 2 \\-3 & 2 & 0 \\4 & 2 & 0 \\\end{array}\right) and B=\left(\begin{array}{ccc}9 & 6 & 3 \\-5 & 0 & 2 \\3 & 3 & 2 \\\end{array}\right)\)

Both A and B have equal number of rows and matrices. So we can find their addition and subtraction as follows

Addition of A and B

\[\left(\begin{array}{ccc}0+9 & -1+6 & 2+3 \\-3-5 & 2+0& 0+2\\ 4+3&2+3&0+2\\\end{array}\right)=\left(\begin{array}{ccc}9 & 5 & 5 \\-8 & 2& 2\\ 7&5&2\\\end{array}\right) \]

Subtraction of A and B

\[\left(\begin{array}{ccc}0-9 & -1-6 & 2-3 \\-3-(-5) & 2-0& 0-2\\ 4-3&2-3&0-2\\\end{array}\right)=\left(\begin{array}{ccc}-9 & -7 & -1 \\2 & 2& -2\\ 1&-1&-2\\\end{array}\right) \]

Scalar Multiplication : When we want to multiply by a number \(k\) to a matrix then it is multiplied with each entry \( a(i, j) \) of matrix \(A\) and it is called scalar multiplication of k with A.

Example : Suppose \(k = 5\) and \(A=\left(\begin{array}{ccc}2 & -1 & 1 \\0 & 4 & 2 \\0 & -3 & -5 \\\end{array}\right)\) then scalar multiplication of k with A is

\[kA=5A = \left(\begin{array}{ccc}2*5 & -1*5 & 1*5 \\0*5 & 4*5 & 2*5 \\0*5 & -3*5 & -5*5 \\\end{array}\right) = \left(\begin{array}{ccc}10 & -5 & 5 \\0 & 20 & 10\\0 & -15 & -25 \\\end{array}\right)\]

Multiplication of Two Matrices :

If number of rows of a matrix A are equal to number of columns of a matrix B then the product AB is find as follows

Let \(A =\left(\begin{array}{cc}2 & 8 \\3 & 4 \end{array}\right), B =\left(\begin{array}{cc}-3 & 1 \\0 & -1 \end{array}\right)\)

Clearly number of rows in A are two equal to number of columns of B equal to 2.

\[AB =\left(\begin{array}{cc}2 & 8 \\3 & 4 \end{array}\right)\left(\begin{array}{cc}-3 & 1 \\0 & -1 \end{array}\right)\]

\[=\left(\begin{array}{cc}2*(-3)+8*0 & 2*1+8*(-1) \\3*(-3)+4*0 & 3*1+4*(-1) \end{array}\right) \]

\[AB=\left(\begin{array}{cc}-6 & -6 \\-9 & -1 \end{array}\right)\]