In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method.

In linear algebra and numerical analysis, a preconditioner ${\displaystyle P}$ of a matrix ${\displaystyle A}$ is a matrix such that ${\displaystyle P^{-1}A}$ has a smaller condition number than ${\displaystyle A}$. It is also common to call ${\displaystyle T=P^{-1}}$ the preconditioner, rather than ${\displaystyle P}$, since ${\displaystyle P}$ itself is rarely explicitly available. In modern preconditioning, the application of ${\displaystyle T=P^{-1}}$, i.e., multiplication of a column vector, or a block of column vectors, by ${\displaystyle T=P^{-1}}$, is commonly performed in a matrix-free fashion, i.e., where neither ${\displaystyle P}$, nor ${\displaystyle T=P^{-1}}$ (and often not even ${\displaystyle A}$) are explicitly available in a matrix form.

Preconditioners are useful in iterative methods to solve a linear system ${\displaystyle Ax=b}$ for ${\displaystyle x}$ since the rate of convergence for most iterative linear solvers increases because the condition number of a matrix decreases as a result of preconditioning. Preconditioned iterative solvers typically outperform direct solvers, e.g., Gaussian elimination, for large, especially for sparse, matrices. Iterative solvers can be used as matrix-free methods, i.e. become the only choice if the coefficient matrix ${\displaystyle A}$ is not stored explicitly, but is accessed by evaluating matrix-vector products.

Instead of solving the original linear system ${\displaystyle Ax=b}$ for ${\displaystyle x}$, one may consider the right preconditioned system $${\displaystyle AP^{-1}(Px)=b}$$ and solve $${\displaystyle AP^{-1}y=b}$$ for ${\displaystyle y}$ and $${\displaystyle Px=y}$$ for ${\displaystyle x}$.

Alternatively, one may solve the left preconditioned system $${\displaystyle P^{-1}(Ax-b)=0.}$$

Both systems give the same solution as the original system as long as the preconditioner matrix ${\displaystyle P}$ is nonsingular. The left preconditioning is more traditional.

The two-sided preconditioned system $${\displaystyle QAP^{-1}(Px)=Qb}$$ may be beneficial, e.g., to preserve the matrix symmetry: if the original matrix ${\displaystyle A}$ is real symmetric and real preconditioners ${\displaystyle Q}$ and ${\displaystyle P}$ satisfy ${\displaystyle Q^{T}=P^{-1}}$ then the preconditioned matrix ${\displaystyle QAP^{-1}}$ is also symmetric. The two-sided preconditioning is common for diagonal scaling where the preconditioners ${\displaystyle Q}$ and ${\displaystyle P}$ are diagonal and scaling is applied both to columns and rows of the original matrix ${\displaystyle A}$, e.g., in order to decrease the dynamic range of entries of the matrix.

The goal of preconditioning is reducing the condition number, e.g., of the left or right preconditioned system matrix ${\displaystyle P^{-1}A}$ or ${\displaystyle AP^{-1}}$. Small condition numbers benefit fast convergence of iterative solvers and improve stability of the solution with respect to perturbations in the system matrix and the right-hand side, e.g., allowing for more aggressive quantization of the matrix entries using lower computer precision.