Numerical Analysis: Interpolatory Requirement & Coefficient Conditions

In summary, the problem is to find coefficients for a rational function that satisfies the interpolatory requirement R(xi)=yi, i=1,2,3,4. This can be done under the condition that the 4x4 matrix A is invertible, which requires 12 additional equations to be unique.
  • #1
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Homework Statement


take the rational function R(x)=(a+bx)\(c+dx). What does the interpolatory requirment R(xi)=yi, i=1,2,3,4 amount to? under what conditions can you find coefficients? uniquely?


Homework Equations





The Attempt at a Solution


Let y=[y1,y2,y3,y4] and v=[a,b,c,d] and let A be an 4x4 matrix. then I want to try to write this as Av=y and the solution would would exist and be unique when A is invertible. so I write a+bxi=(c+dxi)yi. then I can write a+bxi-cyi-d(xi)(yi)=0 but this won't give me the solution because we could just write a=b=c=d=0 for any x and y. since I know yi and xi I could have a+bxi-cyi = d(xi)(yi) and set up the matrix that way but it still won't give me what I want since I wouldn't be able to find d. Am I on the right track? any suggestions on where to go from here? thanks
 
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  • #2
In A you have 16 matrix elements, but Av = y corresponds to only 4 equations. You need 12 more equations for a unique solution.
 

1. What is the purpose of numerical analysis in interpolatory requirement and coefficient conditions?

Numerical analysis is a branch of mathematics that focuses on developing methods and algorithms for solving mathematical problems using numerical approximations. In the context of interpolatory requirement and coefficient conditions, numerical analysis is used to determine the necessary conditions for a set of data points to be interpolated by a polynomial function, as well as the coefficients of the polynomial.

2. What is an interpolatory requirement in numerical analysis?

An interpolatory requirement in numerical analysis refers to the conditions that a set of data points must satisfy in order to be interpolated by a polynomial function. This involves ensuring that the polynomial passes through all the data points and that there are no oscillations or anomalies in the interpolation.

3. What are coefficient conditions in numerical analysis?

Coefficient conditions in numerical analysis refer to the constraints that must be met by the coefficients of the polynomial function used for interpolation. These conditions are based on mathematical principles and are necessary for the polynomial to accurately represent the data points.

4. How do the interpolatory requirement and coefficient conditions affect the accuracy of the interpolation?

The interpolatory requirement and coefficient conditions play a crucial role in determining the accuracy of the interpolation. If these conditions are not met, the polynomial function used for interpolation may not accurately represent the data points, resulting in errors or inconsistencies in the interpolation.

5. What are some common methods used in numerical analysis for interpolatory requirement and coefficient conditions?

There are several methods used in numerical analysis for determining interpolatory requirement and coefficient conditions, such as Lagrange interpolation, Newton's divided differences, and spline interpolation. These methods involve using different techniques and algorithms to find the most accurate polynomial function for interpolating a given set of data points.

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