Gram-Schmidt Orthonormalization Process

     A set of basis vectors for an inner product space can be made orthogonal and of unit length. This process is called the Gram-Schmidt Orthonomalization process. The formulas for performing theses conversion are:

   Use the Gram-Schmidt orthonormalization process to create a cubic polynomial basis for: P3(x) that can approximate functions over the interval [ 2, 4 ].

   A cubic basis means a set of four linearly independent polynomials that can span the P3(x) space, thus you can start with { 1, x, x2, x3 }.  Then you need to convert this basis into “orthogonal” polynomial vectors using the calculus inner product with respect to the given interval [ 2, 4 ].  Convert this set into unit polynomial vectors to “normalized” them.

Try these Examples:

  • Example 1 -  Construct an orthonormal basis with Gram -Schmidt using a cubic polynomial and the Calculus inner product over a specific interval.
  • Example 2 -  Go to the next section on Inner Product Applications - Example 2, to see another cubic polynomial orthonormal basis, but over a different interval.

 

   Example 1:

Create the calculus inner product function needed for the Gram-Schmidt Process.

  • iprod := (f,g,a,b) -> int(f*g,x=a..b)


Setup the cubic polynomial basis we need to convert into an orthonormal basis.

  • v:=[1,x,x^2,x^3];

      

Start the Gram-Schmidt orthonormalization process.

  • w1:=v[1]

  

  • w2:=v[2]-(iprod(v[2],w1,2,4))/(iprod(w1,w1,2,4))*w1

      

  • w3:=v[3]-(iprod(v[3],w1,2,4)/(iprod(w1,w1,2,4))*w1)
         - ((iprod(v[3],w2,2,4)/(iprod(w2,w2,2,4))*w2));

      

  • w4:=v[4]-(iprod(v[4],w1,2,4)/(iprod(w1,w1,2,4))*w1) -((iprod(v[4],w2,2,4)/(iprod(w2,w2,2,4))*w2)) - ((iprod(v[4],w3,2,4)/(iprod(w3,w3,2,4))*w3))

      

This set of ‘w’  vectors is orthogonal :

Now lets make them unit vectors :

  • r[1]:=w1/sqrt(iprod(w1,w1,2,4)); r[2]:=w2/sqrt(iprod(w2,w2,2,4)); r[3]:=w3/sqrt(iprod(w3,w3,2,4));
    r[4]:=w4/sqrt(iprod(w4,w4,2,4));

          

      

This set is now an orthonormal basis for P3(x):

   Look to the next section where we demonstrate how such a basis can be used to approximate continuous functions over specific intervals.

Orthonormal Basis