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Lagrange Polynomial of Order 4

Lagrange Polynomial of Order 4
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2022-01-01 06:29
FunctionCoordinate GeometryGeometric Construction
This example demonstrates the construction of a fourth-order Lagrange polynomial. It involves the concept of polynomial functions and constructs a polynomial given five points. The construction process uses GeoGebra commands such as point creation, line drawing, and polynomial generation. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Point A(1.0, 1.0)
  2. Point B(4.28, 2.22)
  3. Point C(7.58, 0.58)
  4. Point D(10.28, 1.48)
  5. x(A)
  6. x(B)
  7. x(C)
  8. x(D)
  9. y(A)
  10. y(B)
  11. y(C)
  12. y(D)
  13. L1(x) = ((((x - x_2) * (x - x_3)) * (x - x_4))) / ((((x_1 - x_2) * (x_1 - x_3)) * (x_1 - x_4)))
  14. L2(x) = ((((x - x_1) * (x - x_3)) * (x - x_4))) / ((((x_2 - x_1) * (x_2 - x_3)) * (x_2 - x_4)))
  15. L3(x) = ((((x - x_1) * (x - x_2)) * (x - x_4))) / ((((x_3 - x_1) * (x_3 - x_2)) * (x_3 - x_4)))
  16. L4(x) = ((((x - x_1) * (x - x_2)) * (x - x_3))) / ((((x_4 - x_1) * (x_4 - x_2)) * (x_4 - x_3)))
  17. p(x) = (L1(x) * y_1) + (L2(x) * y_2) + (L3(x) * y_3) + (L4(x) * y_4)

AI Drawing Prompt Reference

Draw a fourth-order Lagrange interpolation polynomial with five points A(1.0, 1.0), B(4.28, 2.22), C(7.58, 0.58), D(10.28, 1.48), and an unknown point E. First, create these five points, then construct the Lagrange basis polynomials L1(x), L2(x), L3(x), and L4(x) based on these points, and finally calculate the interpolation polynomial p(x) using these basis polynomials and the y-coordinates of the points.
The prompt above was compiled from the original drawing commands and is for reference only.