Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem

$${\displaystyle Ax=\lambda Bx,}$$

for a given pair ${\displaystyle (A,B)}$ of complex Hermitian or real symmetric matrices, where the matrix ${\displaystyle B}$ is also assumed positive-definite.

Kantorovich in 1948 proposed calculating the smallest eigenvalue ${\displaystyle \lambda _{1}}$ of a symmetric matrix ${\displaystyle A}$ by steepest descent using a direction ${\displaystyle r=Ax-\lambda (x)x}$ of a scaled gradient of a Rayleigh quotient ${\displaystyle \lambda (x)=\langle x,Ax\rangle /\langle x,x\rangle }$ in a scalar product ${\displaystyle \langle x,y\rangle =x^{\mathsf {T}}y}$, with the step size computed by minimizing the Rayleigh quotient in the linear span of the vectors ${\displaystyle x}$ and ${\displaystyle r}$, i.e. in a locally optimal manner. Samokish proposed applying a preconditioner ${\displaystyle T}$ to the residual vector ${\displaystyle r}$ to generate the preconditioned direction ${\displaystyle w=Tr}$ and derived asymptotic, as ${\displaystyle x}$ approaches the eigenvector, convergence rate bounds. D'yakonov suggested spectrally equivalent preconditioning and derived non-asymptotic convergence rate bounds. Block locally optimal multi-step steepest descent for eigenvalue problems was described in. Local minimization of the Rayleigh quotient on the subspace spanned by the current approximation, the current residual and the previous approximation, as well as its block version, appeared in. The preconditioned version was analyzed in and.

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The method performs an iterative maximization (or minimization) of the generalized Rayleigh quotient

$${\displaystyle \rho (x):=\rho (A,B;x):={\frac {x^{\mathsf {T}}Ax}{x^{\mathsf {T}}Bx}},}$$

which results in finding largest (or smallest) eigenpairs of ${\displaystyle Ax=\lambda Bx.}$