Engineering Mathematics : Numerical Methods

1.NUMERICAL METHOD AND ANALYTIC METHOD

We use numerical method to find approximate solution of problems by numerical calculations with aid of calculator. For better accuracy we have to minimize the error.

Error = Exact value – Approximate value

Absolute error = Modulus of error

Relative error = Absolute error / (Exact value)

Percentage error = 100 X Relative error

1.1.   Significant Digits

It is defined as the digits to the left of the first non-zero digit to fix the position of decimal point.

1.2.   Truncation Error

The term truncation error is used to denote error, which results from approximating a smooth function by truncating its Taylor series representation to a finite number of terms.

Using truncation till e digits is given by

2.   Bisection Method

 3.   Regula Falsi Method

In this method, we first find a sufficiently small interval [a₀, b₀], such that f(a₀).f(b₀) < 0, by tabulation or graphical method, and which contains only one root α (say) of f(x) = 0, i. e. f’(x) maintains same sign in [a₀,b₀].

This method is based on the assumption that the graph of y = f(x) in the small interval [a₀, b₀] can be represented by the chord joining (a₀, f(a₀)) and (b₀, f(b₀,)). Therefore, at the point x = x₁ = a₀ + h₀, at which the chord meets the x-axis, we obtain two intervals [a₀, x₁] and [x₁, b₀], one of which must contain the root α, depending upon the condition f(a₀)f(x₁) < 0 or f(x_1,)f(b₀) < 0.

Let f(x_1,)f(b₀) < 0, then α lies in the interval [x₁, b₀] which we rename as [a₁, b₁] Again, we consider that the graph of y = f(x) in [a₁,b₁] as the chord joining (a₁,f(a₁)) and (b₁,f(b₁)) . Thus, the point of intersection of the chord with the x-axis (say) x₂ = a₁ + h₁ gives us an approximate value of the root α of the equation f(x) = 0.

 4.   Newton-Raphson Method

When the derivative of f(x) is of the simple form, the real root (non-repeated) of the equation f(x) = 0, can be computed rapidly by a process known as the Newton Raphson method. Usually the problem is to find a recurrence relation which enables us to find out a sequence {x_n} converging to the desired root α.

4. Secant Method

In order to implement the Newton-Raphson method, f’(x) needs to be found analytically and evaluated numerically. In some cases, the analytical (or its numerical) evaluation may not be feasible or desirable.

5. Trapezoidal Rule of integration

Let us approximate integrand f by a line segment in each subinterval. Then coordinate of end points of subintervals are

(x₀, y₀), (x₁, y₁), (x₂, y₂), …., (x_n, y_n)

Then from x=a to x=b the area under curve of y = f(x) is approximately equal to sum of the areas of n trapezoids of each n subintervals.

 6. Simpsons rule of Numerical integration (Simpsons 1/3rd rule)

Where integrand f(x) is a given function and a and b are known which are end points of the interval [a, b]. Either f(x) is given or a table of values of f(x) are given.

Let us approximate integrand f by a line segment in each subinterval. Then coordinate of end points of subintervals are (x₀, y₀), (x₁, y₁), (x₂, y₂),…., (x_n, y_n).

7. Numerical Solution of Ordinary Differential Equations (ODE)