I need help to understand how to use the result given by an algorithm for constructing an orthonormal polynomial basis over $L^{2}(X)$, where $X\subset\mathbb{R}^2$, with respect to the inner product $(f,g) = \iint_X f(x,y)g(x,y)\,dxdy$. We denote $\mathcal{P}_{n} = \{\text{Polynomials }P(x,y): \text{ total degree of } P \leq n\}\subset L^{2}(X)$. The algorithm is as follows.

(1) Consider the vector corresponding to the constant polynomial $1$. Normalize it. The resulting vector provides an orthonormal basis of $\mathcal{P}_{0}$.

(2) Suppose we have constructed an orthonormal basis of $\mathcal{P}_{k}$, consisting of $K$ vectors. Multiply each of the vectors in this basis by $x$ and $y$, respectively, and put the resulting vectors, together with the vectors in the basis of $\mathcal{P}_{k}$, in a matrix having $3K$ columns. Calculate the SVD of the matrix. The SVD yields an orthonormal basis of the column space of the matrix, which is also an orthonormal basis of $\mathcal{P}_{k+1}$.

(3) Apply the procedure described in step (2) for each $k$ in $\{0, 1, \ldots , n − 1\}$ in order to obtain an orthonormal basis of $\mathcal{P}_n$.

My thoughts: One need to implement an SVD that uses projection with respect to the given inner product. Also, I assume that a quadrature rule is given to compute the inner product.

Questions: (1) What does it mean to multiply with x and y in step (2)? The SVD is not performed symbolically. Should I sample $X$ and multiply by those values?

(2) Executing the algorithm above, how do I use the result to evaluate the resulting orthonormal polynomial basis at a point in $X$? How do I take the derivative of this basis?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.