Chebyshev polynomial approximation to approximate partial differential equations

with n₁ + n₂ = n .

Once the basis functions (approximants) have been chosen and the approximant nodes
defined, the basis coefficients c_i can be obtained. If we define the following Chebyshev-
Vandermode type matrix Τ :

To( 51) Γ1( 51) . ...Γ n-1( 5 S

Γo( 5 2) Γ1( 5 2) . Γ n-1( 5 2)
^Т =
.                                         .                          .                                  .

ʃθ(5n ) Γ1(5n ) .    Γn-1(5n) _

then the coefficients c_i ; c = (c_o,c₁,...,c_n-1)' of V(5) solve Τc = V , with Γ_ij = Γ_i(5_i) being
the j basis function evaluated at the i-th interpolation node. When 5 is allowed to vary
over some other interval, say [t,T] ≠ [-1,1], we rescale the value of 5 to 5^*where
5 ∙ = |((T — t ) 5 + (T + t ))6

As an example of using different basis functions, we anticipate some of the
empirical results presented in the next section and after pricing an European option we
calculate the approximation error. We use two different basis functions (i.e. Chebyshev
basis and spline basis). The approximation error is shown in Figures (1-2).

Insert Figures (1-2) here

As can be seen, when the approximation is calculated using Chebyshev basis functions
the error is of the order of 1×1o^-15 for a polynomial of order 2o. Spline functions do not

⁶ An interesting issue here is the non-singularity of the Vandermode matrix over Chebyshev basis as above.
In theory, there is no guarantee that the matrix is non-singular. However, in practice, in general applications
such as ours, we can conjecture that as long as the number of indeterminates exceeds the sparsity with
respect to Τ , non-singularity should hold. Alternatively, we suggest two ways to overcome the problem:
(a) simply use the singular value decomposition of Τ ; (b) use the generalised Vandermode matrix over
Chebyshev. In fact, for this type of matrix Werther (1993) proves that, as long as the indeterminates take a
value [1, ∞], the generalised Vandermode matrix over Chebyshev basis is non-singular.

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