1. 4 Interpolation and Polynomial Approximation
Pade Approximation
Numerical Methods
Dr. Lai Shengjian
Research Building 710#, UESTC

2. Pade Approximations
The notion of rational approximations for 𝒇(𝒙) that will be approximated over a small portion of its domain.
A rational approximation to 𝒇(𝒙) on [𝒂,𝒃] is the quotient of two polynomial
𝑷 𝑵 (𝒙) and 𝑸 𝑴 𝒙
Goal is to make the maximum error as small as possible.
The method of pade requires that 𝒇(𝒙) and its derivative be continuous at 𝒙= 𝟎.
The rational function 𝑹 𝑵,𝑴 (𝒙) has N+M+1 unknown coefficients. Assume 𝒇(𝒙) is analytic and has the Maclaurin expansiong
Form the difference 𝒇 𝒙 𝑸 𝑴 𝒙 − 𝑷 𝑵 𝒙 =𝒁(𝒙)

3. Pade Approximations
The lower index j=M+N+1 in the summation on the right side of Eq. because the first N+M derivatives of 𝒇(𝒙) and 𝑹 𝑵,𝑴 (𝒙) are to agree at 𝒙=𝟎. Set equal to zero for k=0,1,2,…,N+M to obtain N+M+1 linear equations.

4. Pade Approximations
Ex Establish the Pade approximation 𝑹 𝟒,𝟒 (𝒙) over [−𝟓,𝟓] Simplify the computations if we start with 𝒇 𝒙 =𝒄𝒐𝒔( 𝒙 𝟏/𝟐 )

5. Continued Fraction Form

6. Continued Fraction Form
Ex The pade requires a minimum of 12 arithmetic operations to perform an evaluation. How to reduce this number to seven by the use of continued fractions.

7. Continued Fraction Form
Compare 𝑹 𝟒,𝟒 (𝒙) with the Taylor polynomial 𝑷 𝟔 (𝒙) of degree 𝑵=𝟔

8. Excercise
Homework Exercises P174 4 Program (Report4: 插值与多项式逼近的数值计算方法实验) Algorithm4.1.3 P139 1; Algorithm4.3.5 P154 2; Algorithm P170 6,7; Algorithm4.6.3 P175 1,3 Report File Format: 学号_姓名_report4.doc Upload the Online Class(互动教学空间网络学堂物理学 院数值计算方法).