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Conjugate gradient solver.

    operator, rhs, preconditioner=None, x=None, tol=1e-05, max_iter=20,

Solves a linear system of equations A*x = rhs for self-adjoint, positive definite matrix A and right-hand side vector rhs, using an iterative, matrix-free algorithm where the action of the matrix A is represented by operator. The iteration terminates when either the number of iterations exceeds max_iter or when the residual norm has been reduced to tol times its initial value, i.e. \(||rhs - A x_k|| <= tol ||rhs||\).


  • operator: A LinearOperator that is self-adjoint and positive definite.
  • rhs: A possibly batched vector of shape [..., N] containing the right-hand size vector.
  • preconditioner: A LinearOperator that approximates the inverse of A. An efficient preconditioner could dramatically improve the rate of convergence. If preconditioner represents matrix M(M approximates A^{-1}), the algorithm uses preconditioner.apply(x) to estimate A^{-1}x. For this to be useful, the cost of applying M should be much lower than computing A^{-1} directly.
  • x: A possibly batched vector of shape [..., N] containing the initial guess for the solution.
  • tol: A float scalar convergence tolerance.
  • max_iter: An integer giving the maximum number of iterations.
  • name: A name scope for the operation.


  • output: A namedtuple representing the final state with fields:
    • i: A scalar int32 Tensor. Number of iterations executed.
    • x: A rank-1 Tensor of shape [..., N] containing the computed solution.
    • r: A rank-1 Tensor of shape [.., M] containing the residual vector.
    • p: A rank-1 Tensor of shape [..., N]. A-conjugate basis vector.
    • gamma: \(r \dot M \dot r\), equivalent to \(||r||_2^2\) when preconditioner=None.