5
This is my comment expanded into an answer.
MATLAB is a column-major programming language; it is not very hard to find it in the documentation once you know what to look for:
https://www.mathworks.com/help/matlab/matlab_external/matlab-data.html#f22019
The layout of data structures (most commonly row-major or column-major) used in a programming language is ...
2
Gonna answer by points:
NR authors are referring to the fact that $h$ itself is affected by roundoff too. Also, if your $h$ does not have a finite binary representation, like $h = 0.1$, you're sure you have error for every $h$ in your code.
What you really want is that the difference between $x$ and $x+h$ is exactly representable in finite precision ...
2
The inverse of the (1,1) block of
$$
\begin{bmatrix}
A & B\\
C & D
\end{bmatrix}^{-1}
$$
is $A-BD^{-1}C$ (Schur complement). This is what you are trying to compute, if I understand correctly from your explanation ("marginalize" may be standard in your domain, but it is not standard linear algebra language). So at least you can reduce to ...
1
Here is a fast implementation using sparse matrices and sparse Jacobian estimation:
% define square domain [-1,1] x [-1,1]
n = 51;
x=linspace(-1,1,n);
y=x;
[X,Y]=meshgrid(x,x);
% build finite differences operators
dx=x(2)-x(1);
e=ones(n,1);
d0x=ones(n,1);
grad = spdiags([-e 0*e e],-1:1,n,n)/2/dx;
% use Kronecker product to build matrix of d/dx and d/dy
...
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