سوال ۱۸ کنکور ارشد ریاضی ۱۴۰۰

PASSAGE 1 (excerpt) — Number of Steps, Work per Step, and Preconditioning

"We shall see that the number of steps required for convergence to a satisfactory precision typically depends on spectral properties of the matrix A, if the word "spectral" is interpreted broadly. For example, the conjugate gradient iteration is guaranteed to solve a hermitian positive definite system $Ax = b$ quickly if the eigenvalues of A are clustered well away from the origin. Similarly, the Lanczos iteration is guaranteed to compute certain eigenvalues of a real hermitian matrix quickly if those eigenvalues are well separated from the rest of the spectrum (and if the initial vector that starts the iteration is suitably generic)."

Starting with a proper vector, if certain eigenvalues of a real hermitian matrix are apart from the spectrum, then Lanczos method ------------.