Final answer:
To derive the interpolation functions for the 1D problem, we need to consider the specific interpolation method mentioned in each part of the question. The interpolation functions for quadratic, linear, cubic, and spline interpolation are provided.
Step-by-step explanation:
To derive the interpolation functions for the 1D problem, we need to consider the specific interpolation method mentioned in each part of the question. Let's go through each method one by one:
a) Quadratic interpolation:
In quadratic interpolation, we use three data points to fit a quadratic polynomial. The interpolation functions are:
n₁(x) = (x - x₂)(x - x₃) / (x₁ - x₂)(x₁ - x₃)
n₂(x) = (x - x₁)(x - x₃) / (x₂ - x₁)(x₂ - x₃)
n₃(x) = (x - x₁)(x - x₂) / (x₃ - x₁)(x₃ - x₂)
b) Linear interpolation:
In linear interpolation, we use two data points to fit a straight line. The interpolation functions are:
n₁(x) = (x - x₂) / (x₁ - x₂)
n₂(x) = (x - x₁) / (x₂ - x₁)
c) Cubic interpolation:
In cubic interpolation, we use four data points to fit a cubic polynomial. The interpolation functions are:
n₁(x) = (x - x₂)(x - x₃)(x - x₄) / (x₁ - x₂)(x₁ - x₃)(x₁ - x₄)
n₂(x) = (x - x₁)(x - x₃)(x - x₄) / (x₂ - x₁)(x₂ - x₃)(x₂ - x₄)
n₃(x) = (x - x₁)(x - x₂)(x - x₄) / (x₃ - x₁)(x₃ - x₂)(x₃ - x₄)
n₄(x) = (x - x₁)(x - x₂)(x - x₃) / (x₄ - x₁)(x₄ - x₂)(x₄ - x₃)
d) Spline interpolation:
Spline interpolation uses multiple piecewise-defined polynomials. The interpolation functions depend on the specific spline method used, such as natural, clamped, or not-a-knot spline.