Lagrangian Interpolation Formula

-

Lagrangian Interpolation Formula :

       Let y = f(x) be a real valued function which is defined in an interval [a,b] . Let 
x₀ , x₁ ,x₂ ,............xₙ be n+1 distinct points in that interval at which the respective values y₀ ,y₁,y₂ ............yₙ are tabulated.

     Now our aim is to construct a polynomial Φ(x) of degree ≤ n , which interpolates f(x) such that 

     Φ(xᵢ) = y(xᵢ) , i = 1,2,3,.......,n .........(1)

   Let us suppose that the polynomial Φ(x)
                                  n
be given by Φ(x) = Σ  lᵢ(x) y(xᵢ)  .........(2)
                                 i= 0

     where each lᵢ(x) is a polynomial of degree ≤n in xᵢ , called  Lagrangian function.

     The function given in equation (1) if each lᵢ (x) satisfied lᵢ(xⱼ) = 0 where i ≠ j  ,lᵢ(xⱼ)=1  when i=j  .........(3)
Now ,
  the polynomial lᵢ(x) vanishes at the (n+1) points xₒ , x₁ , ........., xₙ therefore , we can write this polynomial in the following form

lᵢ(x) = cᵢ (x-x₀)(x-x₁)........(x-xᵢ₋₁)(x-xᵢ₊₁)....(x-xₙ)
                                                              ........(4)

where cᵢ's are constant co_efficients

putting x = xᵢ in equation (4) , we get

lᵢ(xᵢ) =cᵢ(xᵢ-x₀)(xᵢ-x₁).....(xᵢ-xᵢ₋₁)(xᵢ-xᵢ₊₁)...(xᵢ-xₙ)

or, 1 = cᵢ(xᵢ-x₀)(xᵢ-x₁).....(xᵢ-xᵢ₋₁)(xᵢ-xᵢ₊₁)
                                       ......(xᵢ-xₙ)     as lᵢ(xᵢ) =1

or,cᵢ = 1/(xᵢ-x₀)(xᵢ-x₁)...(xᵢ-xᵢ₋₁)(xᵢ-xᵢ₊₁)...(xᵢ-xₙ)
                                                         ..........(5)

putting the value of cᵢ from equation (5) in equation (4) , we get

lᵢ(x) = (x-x₀)(x-x₁)...(x-xᵢ₋₁)(x-xᵢ₊₁) /
             (xᵢ-x₀)(xᵢ-x₁)....(xᵢ-xᵢ₋₁)(xᵢ-xᵢ₊₁).....(xᵢ-xₙ)
                                                               .......(6)

putting the values of lᵢ(x) from equation (6) in equation (2) we get ,
         
Φ(x) =
  n
  Σ (x-x₀)(x-x₁)...(x-xᵢ₋₁)(x-xᵢ₊₁)....(x-xₙ)   y(xᵢ)  /
 i=1    (xᵢ-x₀)(xᵢ-x₁)...(xᵢ-xᵢ₋₁)(xᵢ-xᵢ₊₁)...(xᵢ-xₙ)
                                                  .......(8)

Equation (8) is known as Lagrangian Interpolation Formula.

For More :



About Scientist :


Joseph-Louis Lagrange was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics. 

Comments

Post a Comment

If Any Doubt Ask Me

Popular posts from this blog

Accuracy Of Numbers

-
Image
              Accuracy Of Numbers 1.  Approximate Numbers :            There are two types of numbers exact and approximate . Exact numbers are 2,4,9,13,7,2,6.45....etc . But there are numbers such as 4/3 =1.3333....  ,          √(2)= 1.41413 and finite number of digits . These may be approximated by numbers 1.3333, 1.4141 , 3.1416 respectively . Such numbers which represent the given numbers to certain degree of accuracy are called approximate numbers .  2. Significant Figures :                     The digits used to express a number is called significant digits (figures) . Thus each of the numbers 7845, 3.589, 0.4758 contains four significant figures while the numbers 0.00389 , 0.000587 and 0.0000296 contain only three significant figures since zeros only help to fix the position of the decimal poin...
Read more

PFAFFIAN Differential Equations And It's Solutions

-
Image
PFAFFIAN Differential Equations And It's Solutions :-      The relation              n             Σ  Fᵢ(x₁,x₂,......xₙ) dxᵢ = 0            i=1   where Fᵢ's (i=1,2,....n) are functions of all or some of the n independent variables x₁,x₂,.....xₙ is called a Pfaffian differential equation . Here we shall study Pfaffian differential equation in three independent variables which is of the form              P dx + Q dy + R dz = 0    ...........(1)    where P, Q , R are functions of x , y and z . Setting -->                        -->               X = (P,Q,R)  and dr = (dx,dy,dz), equation (1) can be written in the vector notation as            ...
Read more

Derivation Of Composite Trapezoidal Rule

-
Image
Derivation Of Composite Trapezoidal Rule :                                                   b       Let us find the value of ∫ f(x) dx                                                  a numerically by composite trapezoidal rule. Solution :              Let [a,b] be divided into n equal sub intervals of length h each . Let                            a= x₀,x₁,x₂,.......xₙ=b  be the points of sub division .  i.e x₁ = x₀+ih  for i= 0,1,2,......n By Newton's Forward Difference Interpolation  f(x) = f(x₀)+uΔf(x₀) +u(u-1)/2!  Δ²f(x₀)+........       ...
Read more