Constructing Polynomials with Specific Value Sequences

When dealing with sequences of values, one common task is to find a polynomial that passes through a specified set of points. This process, known as polynomial interpolation, is a fundamental concept in numerical analysis and applied mathematics. In this article, we will explore how to construct such a polynomial using both a systematic and a more formal approach, ultimately providing a clear algorithm for solving the problem.

Introduction to Polynomial Interpolation

Polynomial interpolation is the process of finding a polynomial that takes on specific values at a given set of points. Given a sequence of values and corresponding points, the goal is to find a polynomial ( P(x) ) such that ( P(x_i) y_i ) for each ( i ).

Systematic Approach

Let's solve an example where we want a polynomial that takes the values 3, 1, 4, 1, 5, and 9 at ( x 0, 1, 2, 3, 4, ) and 5, respectively.

Step 1: Start with the first value

Our polynomial must satisfy ( P(0) 3 ) . Therefore, we start with a polynomial of the form ( P(x) 3 ).

Step 2: Incorporate the second value

We next want ( P(1) 1 ) . To achieve this, we modify our polynomial to include a factor that makes it zero at ( x 1 ). We add a term ( -2x - 1 ), yielding ( P(x) 3x - 2x 1 x - 1 ). This polynomial now satisfies both conditions: ( P(0) 3 ) and ( P(1) 1 ).

Step 3: Add the third value

Next, we want ( P(2) 4 ) . We add a term in the form of a polynomial that is zero at ( x 0 ) and ( x 1 ) but not at ( x 2 ). We compute the value required for this term as follows:

[ P(x) x - 1 E(x-1)(x-2) ]

We choose ( E ) such that

[ E(2-1)(2-2) 4 - (2 - 1) ]

[ E(1)(0) 4 - 1 ]

[ E frac{5}{2} ]

Thus, our polynomial is:

[ P(x) (x - 1) frac{5}{2}(x - 1)(x - 2) ]

This polynomial now satisfies ( P(0) 3 ), ( P(1) 1 ), and ( P(2) 4 ).

Step 4: Continue adding terms

We continue this process, adding terms that are zero at all previously chosen points and contributing to the required value at the next point. The general form of our polynomial becomes:

[ P(x) 3x - 2x - 1 frac{5}{2}x^{(2)} - 1x^{(1)} - 1x^{(0)} frac{5}{2}x^{(2)} - 1x^{(1)} - 2x^{(0)} ]

After simplification, we get:

[ P(x) 3x - frac{5}{2}x^2 frac{5}{2}x - 1 ]

Formal Approach: Lagrange Interpolation

An alternative, more elegant but computationally more complex, method is to use Lagrange interpolation. The Lagrange polynomial is defined as:

[ P(x) sum_{i0}^{n} y_i ell_i(x) ]

where

[ ell_i(x) prod_{substack{0 leq j leq n j eq i}} frac{x - x_j}{x_i - x_j} ]

In our example:

[ P(x) 1 ell_0(x) 3 ell_1(x) 4 ell_2(x) 1 ell_3(x) 5 ell_4(x) 9 ell_5(x) ]

Each (ell_i(x)) is a second-degree polynomial zero at all ( x_j ) except ( x_i ).

Constructing (ell_i(x))

For the third term, ( ell_2(x) ), we compute:

[ ell_2(x) frac{(x - 0)(x - 1)(x - 3)(x - 4)(x - 5)}{(2 - 0)(2 - 1)(2 - 3)(2 - 4)(2 - 5)} ]

[ ell_2(x) frac{(x - 0)(x - 1)(x - 3)(x - 4)(x - 5)}{2 cdot 1 cdot -1 cdot -2 cdot -3} ]

[ ell_2(x) frac{(x - 0)(x - 1)(x - 3)(x - 4)(x - 5)}{-12} ]

[ ell_2(x) -frac{(x)(x-1)(x-3)(x-4)(x-5)}{12} ]

Substituting ( y_2 4 ) into the Lagrange formula, the third term is:

[ 4 ell_2(x) 4 left( -frac{(x)(x-1)(x-3)(x-4)(x-5)}{12} right) ]

[ 4 ell_2(x) -frac{4(x)(x-1)(x-3)(x-4)(x-5)}{12} ]

[ 4 ell_2(x) -frac{4(x)(x-1)(x-3)(x-4)(x-5)}{12} ]

This is the general form for the polynomial interpolation using the Lagrange method.

Conclusion

Polynomial interpolation is a powerful technique for fitting a polynomial to a given sequence of values. The systematic approach and the formal Lagrange interpolation approach each have their merits and computational requirements. The choice of method depends on the specific problem and available computational resources. Understanding both approaches is crucial for solving a wide range of interpolation problems in various fields, including engineering, computer science, and data analysis.

References

1. Burden, R. L., Faires, J. D. (2011). . Cengage Learning.

2. Cheney, E. W., Kincaid, D. (2009). . Brooks/Cole, Cengage Learning.