Left preconditioning multiplies both sides of the linear system by the same invertible matrix . It leaves the unknown vector unchanged. The original linear-system residual is ; the transformed linear-system residual is . Applying the transformed linear map to first computes , then solves .
For symmetric positive-definite matrices and , is generally not a symmetric matrix in the Euclidean inner product. Its positive eigenvalues alone do not justify ordinary Euclidean conjugate gradient method recurrences. The remedy is symmetric positive-definite preconditioning, implemented through the Preconditioned conjugate gradient method.
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