Here is a question from my research project and what is already given as follows:
A: a symmetric positive definite matrix (SPD) of size N*N;
c: a column vector of size N;
$e_j$: a column vector filled with all zeros except for element j which is one.
Question: As $A$ is a very large array, is it possible to get $d$ without calculating $A^{-1}$ first? i.e., how to quickly calculate $d$? ($d$ need to be calculated for different index $j$.)
$d = \frac{e_j e_j^T A^{-1}c}{e_j^TA^{-1}e_j}$
Thank you.