Interpolation

Interpolation is a concept that is used in numerical analysis. It means finding an intermediate value from the given data. That is, for a set of function values, sometimes a situation arises to know what the value of the function is, for some intermediate value of the variable.

Let us explain the concept with a simple example. Let the ordered pairs of the data of a function be (0, -1), (1, 7), (2, 22), (3, 40), (4, 69), (5, 98) and we need to find the function value of 3.8 of the variable. The attempt to find that value is called interpolation and there are several methods. Let us describe those one by one.

The first method is to round the given value of the variable to the nearest value in the data and take the corresponding value of the function. So for the given data, the value 3.8 of the variable can be rounded to 4 and assume the value of the function approximately as 40. But as one can easily see it is a crude method and far from accurate. This method can only be used just for guidance.

The next method is called as linear interpolation. That is, the function is assumed to be linear in the interval that contains the required value of the variable and accordingly the value of the function is determined. In the given example 3.8 falls in the interval [3, 4]. Considering the function to be linear in this interval, the slope of the function in this interval is (69 – 40)/(1) = 29. So the value of the function at x = 3.8 is, 29(0.8) + 40 = 63.2. Though this method also is not very accurate still the accuracy is much better than that in method 1. This method, even at the cost of sacrificing some accuracy, is preferred because it is easy to work with. In general the linear interpolation y at a point x  is given by the formula y = [(y_b – y_a)/(x_b – x_a)](x – x_a) + y_a, where [a, b] is the interval in which the required point occurs. This is the algorithm used in any linear interpolation calculator.

It is always possible a curve rather than a straight line could better cover the points plotted from a data. In other words a polynomial function can give a better interpolation. Finding a suitable polynomial function for a given data is called as polynomial regression.  Suppose we consider the same data, the three degree polynomial function f(x) = 3x² + 5x – 1 will be very close to the desired results. In such a case evaluating the function for x = 3.8, the value of the function is will be a very accurate interpolation. The accuracy can further be improved when a polynomial function of degree same as the number of data points is determined.

A high level interpolation polynomial can be derived by extensive methods and such an interpolation is called as Lagrange interpolation introduces by the famous Italian mathematician.

Analytical Geometry

Analytical Geometry:

Another term for analytical geometry is Cartesian geometry or co ordinate geometry. It refers to the study of relationships between points, lines, planes etc against a back drop of the co ordinate system, be it in three dimensions or two dimensions. In our lower grade normal geometry, we knew of geometric shapes such as triangles, angles, squares, rectangles etc. The same shapes in co ordinate geometry are described by the co ordinates of their vertices, or by the equations and slopes of the lines joining these vertices.

The basic of analytical geometry deals with the concept of lines. The topics covered are parallel lines, perpendicular lines and inclination of lines with respect to the coordinate systems or with respect to each other. A line is described by its slope or gradient. This slope can be found using the co ordinates of two points that lie on the line. If two lines have the same gradient then they are said to be parallel to each other. If the product of the slopes of two lines is -1, then the two lines are said to be perpendicular to each other. The slope can also be defined by the tangent of the angle that the line makes with the positive x axis direction. Thus if we know the slope we can use the arctan function to find the inclination of the line with respect to the positive x direction.

All the other geometric shapes are studies with the help of this concept of straight lines. For example consider a quadrilateral. It is a closed shape formed by four line segments that are parts of four lines. If both the pairs of opposite sides are parallel and congruent to each other, (that means if both the pairs of opposite sides have the same slope and same length) then such a quadrilateral is called a parallelogram. In a parallelogram if all the adjacent sides are perpendicular to each other, (in other words, if the product of the slopes of adjacent sides is -1), then the parallelogram becomes a rectangle. Similarly other geometric shapes can also be studied in this way.

Analytical geometry is most useful in studying three dimensional objects. This is sometimes also called analytical solid geometry. In three dimnesional space, there are three coordinate systems that can be used. Besides the Cartesian co ordinate system, there can also be the cylindrical coordinate system and the spherical co ordinate system. All these three systems can be interchangeably used to study various types of curves, surfaces or solids in space. Just like how a point in the Cartesian co ordinate system is defined by its x y and z co ordinate, in the cylindrical co ordinate system it is defined by three parameters. They are, (a) its distance from the origin (r), (b) the angle of the line joining the point and the origin with the positive x axis (θ) and (c) its perpendicular distance from the x-y plane (z co ordinate).

Also check out the video streaming of Analytical Geometry

Variables

                                                                                                               
What are Variables?

A variable is something which always varies. That is it does not have a fixed value. In any situation if we are not sure of a value for a quantity then we represent such things using alphabets like x, y, z, a, b, c etc known as variables.

Let’s assume that Peter is working in a hotel and earns $12 per hour and on top of his hourly wage he also get tips for each hour. This expression $(12 + x) would give how much peter earn in a given hour. Here x denotes the tips earned. Now, you might also realize that number of tips or the amount of tips Peter make per might change dramatically from hour to hour. It varies consistently. Since, we are not sure of the tips Peter make each hour exactly, this value is called a variable.

Random Variables

In this section i will introduce you to the concept of a random variable. For me this is something I had lot of trouble for getting my head around. That’s really because it called variable. Generally, variable are unknowns used in algebraic equations/ expressions.

For example: x + 2 = 15. We can find the value of x, by subtracting 2 from both sides.
Or if we have an equation in two unknowns that is y = 2x + 5, here x is an independent quantity and y is a dependent quantity. So, we can assume any x value and find respective value for y. In this case we will have infinite combinations of x and y satisfying the equation.

A random variable is kind of same thing that it can take multiple values but it is not something which you really ever solve for. A random variable is usually denoted by a capital letters say X. It can take bunch of different values but we are never solving for it. In fact it a function that maps from you from the world of random processes to the actual numbers. Let us see one example.

Let the random process be it is going to rain tomorrow or not. Now, how are we going to quantify this, let say X = 1 if it rains tomorrow and X = 0 if does not rain. It is not compulsory that we have to use 1 and 0. Here we can assign to any number. So, we need to keep in mind is a random variable is not a traditional variable.

It is defined as a numerical value to each outcome of a particular experiment. For every element of an experiment’s sample space, it can take only one value. When a random variable can take finite (countable) finite set of outcomes then it is known as discrete random variable. Individual outcomes for a random variable are denoted by lower case letters. A continuous random variable would take on any of the countless number of values in the given line interval.

Using Matrices

In this article we shall study about one of the methods used to solve system of linear equations using matrices. Before we study about the method let us first see few definitions

Consider the following system of simultaneous non – homogeneous linear equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂

Expressing the above equations in matrices, we get

These equations can be represented as a matrix equation as AX = D, where







Here A is called the coefficient matrix.X is called the variable matrix.
D is called the constant matrix.

Augmented Matrices

The coefficient matrix augmented with constant column matrix, is called the augmented matrix, generally denoted by [AD]. Hence the augmented matrix of the above system of simultaneous linear equations is 

Sub Matrices

A matrix obtained by deleting some rows or columns (or both) of a matrix is called a sub matrix.

Definition (Rank of a matrix)

Let A be a non – zero matrices. The rank of A is defined as the maximum of the orders of the non – singular square matrices of A. The rank of a null matrix is defined as zero. The rank of A is denoted by rank (A).

It is to be noted that :-

If A is a non zero matrix of order 3 then rank of A is
(i)    1 if every 2 x 2 sub matrices is singular
(ii)    2 if A is singular and atleast one of its 2 x 2 sub matrices is non – singular.
(iii)    3 if A is non – singular.

Consistent and Inconsistent systems

A system of linear equations is said to be
(i)    Consistent if it has a solution
(ii)    Inconsistent if it has no solution.

A system of three simultaneous equations in three unknowns whose matrix form is AX = D has
(i)    A unique solution if rank (A) = rank ([AD]) = 3
(ii)    Infinitely many solutions if rank (A) = rank([AD]) < 3
(iii)    No solution if rank (A) is not equal to rank ([AD])

It is to be noted that the system is consistent if and only if rank (A) = rank ([AD])

The different ways of solving non homogenous systems of equations are
(i)    Cramer’s Rule
(ii)    Matrix inversion method
(iii)    Gauss – Jordan method

In Gauss – Jordan method we try to transform the augmented matrix by using elementary row transformations. So that the solution is completely visible that is x = α, y = β and z = γ. We may get infinitely many solutions or no solution also.

For solving a system of three equations in three unknowns by Gauss – Jordan method, elementary row operations are performed on the augmented matrix as indicated below.

(i)    Transform the first element of 1^st row and 1^st column to 1 and transform the other non zero elements if any in of 1^st row and 1^st column to zero.
(ii)    Transform the second element of 2^nd row and 2^nd column to 1 and transform the other non zero elements if any in of 2^nd row and 2^nd column to zero.
(iii)    Transform the third element of 3^rd row and 3^rd column to 1 and transform the other non zero elements if any in of 3^rd row and 3^rd column to zero. 

Matrices Calculator
We shall study about Matrices Calculator in some other article.

Right Circular Cylinder

Right Circular Cylinder :- If the number of circular sheets is arranged in a stack, if the stack is kept vertically up then it is called the right circular cyl. If the base is circular and the other sheets are kept at ninety degrees at the base, then it is called the right circular cyl.  The right circular cyl can be made by a rectangular sheet by folding it in round shape. The area of the sheet gives us the curved surface area of the cyl because the length of the sheet is equal to the circumference of the circular base which is equal to 2 π r . Therefore the curved surface area of the cyl is equal to the area of the rectangular sheet. The area of the rectangular sheet is equal to the product of the length and the breadth. The curved surface area of the cylinders is also equal to the product of the perimeter of the base and the height of the cylinders.  If the top and bottom of the cylinders is also consider because it is needed to make the circular tin, If the area of the bottom and top of the cylinders is included in the curved surface area we get the total surface area. It means total surface area of the right circular cylinder is equal to the area of the base plus the area of the top of the cylinder plus the curved surface area of the cylinder. The cylinder has two circular base at the bottom and at the top.The diameter of the base of the cylinder can be measured directly with the help of the scale. 

What is the Formula for Volume of a Cylinder :- As we know that the volume of cuboids is equal to the product of the length, breadth and the height. The volume is three dimensional. The cuboids are built up with the rectangle of the same size sheets. In the same way the right circular cylinders can be made by using the same principle. So, by using the same argument as for cuboids, we can see that the volume of a cylinders can be obtained by the product of the base and height of the cylinders.  Therefore, the volume of a cylinder = the base area of the cylinder X height of the cylinder. As we know that the base of the cylinder is the circle. The area of the circle is equal toπ r². The height of the cylinder can be assumed as h. Therefore the volume of the cylinder = π r²h, where r is the base radius and h is the height of the cylinder.

Example :- Let us find the volume of the cylinder which has the radius of the base is twenty one centimeter and its height is thirty five centimeter.

Solution :- The formula for the volume of the cylinder, V = π r²h
Radius r = 21 centimeter, height h = 35 centimeter
Therefore the volume of the cylinder V = π ×21² ×35 = 48490.48 cubic centimeters.