3.4 Approximation of the Second Derivative
51
The leading truncation error term is first order but vanishes when the spacing between the points is uniform, making the approximation second-order
accurate. However, even when the grid is non-uniform, the argument given
above shows that the truncation error is reduced in a second-order manner
when the grid is refined. When a compound interest grid is used, the error decreases in the same way as for the CDS approximation of the first derivative,
see Eq. (3.25).
Higher-order approximations for the second derivative can be obtained by
including more data points, say xi-2 or xi+^.
Finally, one can use interpolation t o fit a polynomial of degree n through
n + 1 data points. From that interpolation, approximations t o all derivatives
up to the nth can be obtained by differentiation. Using quadratic interpolation on three points leads t o the formulas given above. Approaches like those
described in Sect. 3.3.3 can also be extended to the second derivative.
In general, the truncation error of the approximation to the second derivative is the degree of the interpolating polynomial minus one (first order for
parabolas, second order for cubics, etc.). One order is gained when the spacing
is uniform and even-order polynomials are used. For example, a polynomial
of degree four fit through five points leads to a fourth-order approximation
on uniform grids:
One can also use approximations of the second derivative to increase the
accuracy of approximations to the first derivative. For example, using the
FDS expression for the first derivative, Eq. (3.4), keeping just two terms
on the right-hand side, and using the CDS expression (3.30) for the second
derivative, results in the following expression for the first derivative:
This expression possesses a second-order truncation error on any grid and
reduces to the standard CDS expression for the first derivative on uniform
grids. This approximation is identical to Eq. (3.26). In a similar way, one can
upgrade any approximation by eliminating the derivative in the leading truncation error term. Higher-order approximations always involve more nodes,
yielding more complex equations to solve and more complicated treatment
51
The leading truncation error term is first order but vanishes when the spacing between the points is uniform, making the approximation second-order
accurate. However, even when the grid is non-uniform, the argument given
above shows that the truncation error is reduced in a second-order manner
when the grid is refined. When a compound interest grid is used, the error decreases in the same way as for the CDS approximation of the first derivative,
see Eq. (3.25).
Higher-order approximations for the second derivative can be obtained by
including more data points, say xi-2 or xi+^.
Finally, one can use interpolation t o fit a polynomial of degree n through
n + 1 data points. From that interpolation, approximations t o all derivatives
up to the nth can be obtained by differentiation. Using quadratic interpolation on three points leads t o the formulas given above. Approaches like those
described in Sect. 3.3.3 can also be extended to the second derivative.
In general, the truncation error of the approximation to the second derivative is the degree of the interpolating polynomial minus one (first order for
parabolas, second order for cubics, etc.). One order is gained when the spacing
is uniform and even-order polynomials are used. For example, a polynomial
of degree four fit through five points leads to a fourth-order approximation
on uniform grids:
One can also use approximations of the second derivative to increase the
accuracy of approximations to the first derivative. For example, using the
FDS expression for the first derivative, Eq. (3.4), keeping just two terms
on the right-hand side, and using the CDS expression (3.30) for the second
derivative, results in the following expression for the first derivative:
This expression possesses a second-order truncation error on any grid and
reduces to the standard CDS expression for the first derivative on uniform
grids. This approximation is identical to Eq. (3.26). In a similar way, one can
upgrade any approximation by eliminating the derivative in the leading truncation error term. Higher-order approximations always involve more nodes,
yielding more complex equations to solve and more complicated treatment