The Matlab function bvp4c solves boundary value problems. It takes a differential equation, boundary conditions and an initial guess as input, and returns a structure array containing arrays of x, y and yp (which stands for "y prime", or y').
The length of the output arrays should be the same as that of the initial guess, but I found that it isn't always. I have checked the dimensions of the input (the initial guess, always 1x101 double for x and 16x101 double for y) and the output (sometimes 1x101 double for x and 16x101 double for y and yp as it should be, but often different values, such as 1x91 double and 16x91 double or 1x175 double and 16x175 double).
Looking at the output array x when its length is off, some extra values are squeezed in, or some are taken out. For example, the initial guess has 100 positions between x=0 and x=1, and the x array should be [0 0.01 0.02 ... 1], but sometimes a new position like 0.015 shows up.
Question: Why does this happen, and how can this be solved?
"The length of the output arrays should be the same as that of the initial guess ...." This is incorrect.
As described in the bvp4c documentation, sol.x contains a "[mesh] selected by bvp4c" with an "[approximation] to y(x) at the mesh points of sol.x". In order to evaluate bvp4c's solution on your mesh, use deval.
Why does bvp4c choose a mesh? Quoting from the cited paper1, which you can get in full here if you have a MathWorks account:
Because BVPs can have more than one solution, BVP codes require users to supply a guess for the solution desired. The guess includes a guess for an initial mesh that reveals the behavior of the desired solution. The codes then adapt the mesh so as to obtain an accurate numerical solution with a modest number of mesh points.
Because a steady BVP generally has a global behavior strongly dependent on its boundary values, the spatial mesh between the two boundaries may need to be refined in order to properly approximate the desired solution with the locally chosen basis functions for the method. However, there may also be portions of the mesh that do not need to be refined and can even be coarsened in some cases to maintain a reasonably small residual and accurate approximation. Therefore, for general efficiency, the guess mesh is adaptively refined or coarsened depending on some locally chosen metric (since bvp4c is collocation based, the metric is probably point-based or division-integrated based) such that the mesh returned by bvp4c is, in some sense, adequate enough for generic interpolation within the boundaries.
I'll also note that this is different from numerically solving IVPs since their state is not global across the entire time integration locus and only depends on the current state to the next time-step, and possibly previous time steps if using a multi-step method or solving a delay differential equation, which makes the refinement inherently local. This local behavior of IVPs is what allows functions like ode45 to return a solution at pre-selected time values because it can locally refine the solution at the selected point while performing the time march (this is known as dense output).
1 Shampine, L.F., M.W. Reichelt, and J. Kierzenka, "Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c".