Interpolation and Approximation 1 Lagrange Interpolation n The

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Interpolation and Approximation 1

Interpolation and Approximation 1

Lagrange Interpolation n The basic interpolation problem can be posed in one of two

Lagrange Interpolation n The basic interpolation problem can be posed in one of two ways: 2

Lagrange Interpolating Polynomials 3

Lagrange Interpolating Polynomials 3

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exist unique 7

exist unique 7

Example e-1/2 8

Example e-1/2 8

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Discussion n n There are circumstances in which polynomial interpolation as approximation will work

Discussion n n There are circumstances in which polynomial interpolation as approximation will work very well, and other circumstances in which it will not. The Lagrange form of the interpolating polynomial is not well suited for actual computations, and there is an alternative construction that is far superior to it. 10

Newton Interpolation and Divided Differences n The disadvantage of the Lagrange form n n

Newton Interpolation and Divided Differences n The disadvantage of the Lagrange form n n If we decide to add a point to the set of nodes, we have to completely re-compute all of the functions. Here we introduce an alternative form of the polynomial: the Newton form n It can allow us to easily write in terms of 11

General Form of Newton’s Interpolating Polynomials Bracketed function evaluations are finite divided differences 12

General Form of Newton’s Interpolating Polynomials Bracketed function evaluations are finite divided differences 12

Errors of Newton’s Interpolating Polynomials n Structure of interpolating polynomials is similar to the

Errors of Newton’s Interpolating Polynomials n Structure of interpolating polynomials is similar to the Taylor series expansion in the sense that finite divided differences are added sequentially to capture the higher order derivatives. n For an nth-order interpolating polynomial, an analogous relationship for the error is: x Is somewhere containing the unknown and he data n For non differentiable functions, if an additional point f(xn+1) is available, an alternative formula can be used that does not require prior knowledge of the function: 13

Newton Interpolation 14

Newton Interpolation 14

=0 15

=0 15

Example 16

Example 16

Discussion n n The coefficients are called divided differences. We can use divided-difference table

Discussion n n The coefficients are called divided differences. We can use divided-difference table to find them. 17

Example 18

Example 18

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Example (Con. ) 20

Example (Con. ) 20

Table 4. 5 21

Table 4. 5 21