(a) Consider a piecewise quadratic polynomial that is a differentible interpolant to the n+1 datapoints (xo, yo),.. (xn, Yn). Is this unique? If not, which conditions might be imposed forthe uniqueness?(b) Suppose that a quintic spline S(x) is fitted to a function ƒ(x) = C4 at nodes {xo, ..., xn}with function values {f(xo),..., f(xn)}. What conditions must S satisfy? What conditionsmight be imposed to ensure a unique spline?

Answered: (a) Consider a piecewise quadratic…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(a) Consider a piecewise quadratic polynomial that is a differentible interpolant to the n+1 data
points (xo, yo),.. (xn, Yn). Is this unique? If not, which conditions might be imposed for
the uniqueness?
(b) Suppose that a quintic spline S(x) is fitted to a function ƒ(x) = C4 at nodes {xo, ..., xn}
with function values {f(xo),..., f(xn)}. What conditions must S satisfy? What conditions
might be imposed to ensure a unique spline?

Expert Solution

Step 1: For Question a:

a) A piecewise quadratic polynomial interpolant to n+1 data points can be unique under certain conditions, but it is not always unique. The conditions for uniqueness depend on how you define the piecewise quadratic interpolant.

In general, a piecewise quadratic interpolant is defined by specifying quadratic polynomials for different subintervals $left square bracket x subscript i comma x subscript i plus 1 end subscript right square bracket$ , where i ranges from 0 to n-1. If you want to ensure uniqueness, you can impose one or more of the following conditions:

Unique Interpolation Points: Make sure that all data points $(x_{i}, y_{i})$ are distinct. If two or more data points coincide, the interpolant may not be unique.
Interpolation Conditions: Specify interpolation conditions at each data point $(x_{i}, y_{i})$ . For example, you can require that the interpolant passes through all data points $(y_{i} = f (x_{i}))$ , which will uniquely determine the quadratic polynomials for each subinterval.
Continuity Conditions: Enforce continuity between adjacent quadratic polynomials. For example, require that the interpolant and its first derivative are continuous at the junction points $left parenthesis x subscript i comma x subscript i plus 1 end subscript right parenthesis$ . This can help ensure uniqueness by reducing the degrees of freedom in the interpolant.
Smoothness Conditions: You can also impose smoothness conditions, such as requiring the interpolant to be twice continuously differentiable $(C^{2})$ or more, which will further constrain the interpolant.

The choice of conditions depends on your specific problem and the behavior you want from the interpolant. Depending on the data and conditions, you may need to solve a system of equations to find the unique piecewise quadratic interpolant.