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MINRES minimizes ``norm(Ax - b)`` for a real symmetric or complex hermitian matrix `A`. Unlike the Conjugate Gradient method, `A` can be indefinite or singular. If ``shift != 0`` then the method solves ``(A - shift*I)x = b``. Parameters ---------- A : {sparse array, ndarray, LinearOperator} The real symmetric or complex hermitian N-by-N matrix of the linear system. Alternatively, `A` can be a linear operator which can produce ``Ax`` using, e.g., ``scipy.sparse.linalg.LinearOperator``. b : ndarray Right hand side of the linear system. Has shape (N,) or (N,1). Returns ------- x : ndarray The converged solution. info : int Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown Other Parameters ---------------- x0 : ndarray Starting guess for the solution. rtol : float Tolerance to achieve. The algorithm terminates when the relative residual is below ``rtol``. shift : float Value to apply to the system ``(A - shift * I)x = b``. Default is 0. maxiter : int Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : {sparse array, ndarray, LinearOperator} Preconditioner for `A`. The preconditioner should approximate the inverse of `A`. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as ``callback(xk)``, where ``xk`` is the current solution vector. show : bool If ``True``, print out a summary and metrics related to the solution during iterations. Default is ``False``. check : bool If ``True``, run additional input validation to check that `A` and `M` (if specified) are symmetric. Default is ``False``. Notes ----- This file is a translation of the MATLAB implementation [2]_. References ---------- .. [1] Solution of sparse indefinite systems of linear equations, C. C. Paige and M. A. Saunders (1975), SIAM J. Numer. Anal. 12(4), pp. 617-629. https://web.stanford.edu/group/SOL/software/minres/ .. [2] https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_array >>> from scipy.sparse.linalg import minres >>> A = csc_array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) >>> A = A + A.T >>> b = np.array([2, 4, -1], dtype=float) >>> x, exitCode = minres(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True zEnter minres. zExit minres. r) z3 beta2 = 0. If M = I, b and x are eigenvectors z/ beta1 = 0. 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