Question

Question:(M) Let \({{\rm{P}}_4}\) have the inner product as in Example 5, and let \({p_{0,}}{p_{1,\,}}{p_2}\)be the orthogonal polynomials from that example. Using your matrix program, apply the Gram–Schmidt process to the set\(\left\{ {{p_0},{p_{1\,}},{p_2},{t^3},{t^4}} \right\}\) to create an orthogonal basis for \({{\rm{P}}_4}\).

Short answer

The new orthogonal polynomials are multiple of \( - 17t + 5{t^3}\)and\(72 - 155{t^2} + 35{t^4}\) .

Step-by-step solution

Use the given information

Let matrix A defines the basis \(A = \left\{ {{p_0},{p_1},{p_2},{t^3},{t^4}} \right\}\).

From example (5), polynomial vectors are:

\({p_0} = \left( {\begin{array}{*{20}{c}}1\\1\\1\\1\\1\end{array}} \right)\),\({p_1} = \left( {\begin{array}{*{20}{c}}{ - 2}\\{ - 1}\\0\\1\\2\end{array}} \right)\),\({p_2} = \left( {\begin{array}{*{20}{c}}2\\{ - 1}\\{ - 2}\\{ - 1}\\2\end{array}} \right)\)

Use the following steps to find the associated values for the obtained data in MATLAB.

1. Formulate the matrix A using the polynomials \(\left\{ {{p_0},{p_1},{p_2}} \right\}\) and enter it in the tab,as \(\left( {\left( {\begin{array}{*{20}{c}}1&{ - 2}&4&1&1\\1&{ - 1}&1&1&1\\1&{\,\,\,0}&0&1&1\\1&{\,\,\,1}&1&1&1\\1&{\,\,\,2}&4&1&1\end{array}} \right)} \right)\).
2. Run the command “gramschmidt(A)”.
3. Press ENTER.

By using the matrix program, then new orthogonal polynomials are obtained as multiple of \( - 17t + 5{t^3}\)and\(72 - 155{t^2} + 35{t^4}\) .

Scale the polynomials

To obtain the small values of integers of the polynomials at \( - 2, - 1\), 0, 1 and 2, the polynomials must be scaled.