# [Math] Why does Givens rotation avoid iteration and Jacobi rotation doesn’t in case of reducing a symmetric matrix to tridiagonal

eigenvalues-eigenvectorslinear algebramatrices

I am currently implementing symmetric matrix reduction to tridiagonal.
I read that Givens rotation avoids iteration when it is used for reducing a matrix to tridiagonal whereas Jacobi rotation is iterative. Givens rotation try to annihilate the element $a_{i-1j}$ by a rotation in $ij-$plane and Jacobi rotation try to annihilate the element $a_{ij}$ by a rotation in $ij-$plane. But just because of that, how can Givens rotation avoid iteration as both use same equations?

$a^{'}_{rp}=ca_{rp} – sa_{rq}$

$a^{'}_{rq}=ca_{rp} + sa_{rq}$

$a^{'}_{pp}=c^{2}a_{pp}+s^{2}a_{qq}-2sca_{pq}$

$a^{'}_qq=s^{2}a_{pp}+c^{2}a_{qq}+2sca_{pq}$

$a^{'}_{pq}=(c^{2}-s^{2})a_{pq}+sc(a_{pp}-a_{qq})$

Now in Jacobi rotation , they try to zero out $a^{'}_{pq}$ and in Givens rotation they try to zero out $a^{'}_{p-1q}$ i.e.,$a^{'}_{rq}$.