In numerical this method is used for polynomial interpolation. This method is used to find a polynomial by taking certain values at arbitrary points. The set of points are given such that no two points are of the same values, and this function works in such a way that the functions coincide at each point.
Let y = f(x) be a real valued function defined in an interval [a, b] and let be (n + 1) distinct points in the intervals at which the respective values are given.
Assume the polynomial
This is called the Lagrangian function.
It should be in the form:
Thus we have
Using the product notation, we have,
Again, we can write
Differentiating with respect to x we get
Thus, the interpolating formula can be written as:
The values of f(x) and x with the value for which the f(x) is to be calculated
The value of f(x) for the required x
Step 1: [Taking the inputs from the user] Read n [The number of points] for i = 1 to n repeat Read x[i] [The values of x] [End of ‘for’ loop] for i = 1 to n repeat Read y[i] [the values of f(x)] [End of ‘for’ loop] Read x1 [The point of interpolation] Step 2: [Lagrange’s Interpolation] Set u ← 1.0 Set s ← 0.0 for i = 1 to n repeat Set d[i] ← 1.0 for j = 1 to n repeat If j = i then Set a[i][j] ← x1 - x[j] else Set a[i][j] ← x[i] - x[j] [End of ‘if’] Set d[i] ← d[i] × a[i][j] [End of ‘for’ loop] Set s ← s + y[i]/d[i] Set u ← u × a[i][i] [End of ‘for’ loop] for i = 1 to n repeat Print x[i], y[i] [End of ‘for’ loop] Print u × s [End of Lagrange’s Interpolation]
1. This polynomial can be used on functions that are tabulated at equal intervals as well as on the functions which are tabulated at unequal intervals.
2. It has an important theoretical role in the development of numerical differentiation and numerical integration of the function.
1. Computation of this method is difficult.
2. An nth degree Lagrange's interpolation at a point x is to be calculated by performing at least 2(n + 1) multiplications or divisions and (2n + 1) additions or subtractions.
1. It is generally used for arguments that are spaced unequally.