PP form for the reverse function (not inverse function)

Hello friends!

Consider a pp form for a set of (x,y) data points. Imagine now that we wish to find the pp form for the (x,1/y) data

points using the pp form for the (x,y) data. I am wondering whether matlab has a function which can calculate this?

Of course, I can simply calculate it indirectly but I am not sure if this is the best way to perform such a calculation

in terms of both speed and acuracy. For instance, the following do the job for the function cos(x) over [-pi/2 pi/2]:

x=-pi:0.5:pi/2;

y=cos(x);

pp=spline(x,y);

mesh=linspace(x(1),x(end),200);

pp_inv=spline(mesh,1./ppval(pp,mesh));

I seriously doubt whther my approach is efficient? (I feel not)

Any idea?

Thanks in advance!

Babak

Answer 1

[Aditya]

Your approach to finding the piecewise polynomial (pp) form for the reciprocal of a function using MATLAB's `spline` function is generally correct, but it has some potential issues regarding efficiency and accuracy.

The problem with evaluating `1./ppval(pp, mesh)` is that it does not take into account the original spline's structure and may introduce inaccuracies, especially near zeros of the original function `y`. Additionally, the evaluation of the spline at a dense mesh followed by another spline fitting can be computationally intensive.

Unfortunately, MATLAB does not have a built-in function that directly converts a pp form spline for `(x, y)` data into a pp form spline for `(x, 1/y)` data. However, you can achieve a more efficient and potentially more accurate result by working with the pp form directly.

Here's an approach that modifies the coefficients of the original pp spline to create a new pp spline for the reciprocal:

1. Extract the pp form of the original spline.

2. Invert the polynomial coefficients analytically to obtain the coefficients for the reciprocal function.

3. Create a new pp form with the inverted coefficients.

This method can be complex because inverting a polynomial is not straightforward and may not always be possible for higher-order polynomials. However, for simple cases where `y` is a low-order polynomial, you might be able to perform the inversion analytically.

Here's an example of how you might approach this problem using MATLAB's built-in functions to extract and manipulate the pp form:

% Original data points
x = -pi:0.5:pi/2;
y = cos(x);

% Fit a spline to the original data
pp = spline(x, y);

% Evaluate the spline on a mesh
mesh = linspace(x(1), x(end), 200);
y_vals = ppval(pp, mesh);

% Invert the y-values
inv_y_vals = 1 ./ y_vals;

% Fit a new spline to the inverted y-values
pp_inv = spline(mesh, inv_y_vals);

% Evaluate the inverted spline on a new mesh
new_mesh = linspace(x(1), x(end), 200);
inv_y_spline_vals = ppval(pp_inv, new_mesh);

% Plot the original and inverted spline
plot(mesh, y_vals, 'b-', new_mesh, inv_y_spline_vals, 'r--');
legend('Original Spline', 'Inverted Spline');

In this example, the inversion is done numerically by evaluating the original spline on a mesh and then fitting a new spline to the reciprocal of those values. This approach is similar to what you've done, but it's important to note that if `y` approaches zero, the reciprocal will tend to infinity, which could lead to inaccuracies or undefined behavior.