数值分析书面作业代写 函数代写

Written assignment #3

数值分析书面作业代写 Consider the monomials: mk(x) = xk.                          (1) The monomials m0(x), m1(x), . . . , mn(x) form a basis for the set of degree n real polynomials:

Consider the monomials:

m_k(x) = x^k.                          (1)

The monomials m₀(x), m₁(x), . . . , m_n(x) form a basis for the set of degree n real polynomials:

P_n = α₀ + α₁x + α₂x₂ + · · · + α_nx_n : (α₀, . . . , α_n) ∈ R_n+1 .                (2)

Define an inner product on the set of square-integrable functions defined on [−1, 1] for a pair of functions f, g : [−1, 1] → R by:

 

 

Problem 2.

Plot p₀, . . . , p₅ on [−1, 1] using matplotlib and insert the plot inline in your submission using the LATEX graphicx package.

 

Problem 3.   数值分析书面作业代写

As we will see in the coming weeks, orthogonal polynomials are useful for approximating functions. Let f : [−1, 1] → R be square-integrable, and let p₀, . . . , p_n be a set of orthogonal polynomials on [−1, 1]. For a function g : [−1, 1] → R, define ∥g∥ by:

 

 

Problem 4.  数值分析书面作业代写

Pick three square-integrable functions defined on [−1, 1] which are not polynomials. They do not need to be smooth or even continuous. Now, for n = 5, compute the coefficients c₀, . . . , c₅ for each function f and form the approximation:

 

 

The relative error e_kversus k, for k = 0, . . . , 5. Use a semilogy plot for this one.

You are free to do this problem by hand (except for the graphing, which must be done using Python), or you can do it using Python. If you use Python, please investigate using scipy.integrate.quadrature to compute the integrals—this will make your life easier.

 

 

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