cg#
- scipy.sparse.linalg.cg(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=None, M=None, callback=None)[source]#
- Solve - Ax = bwith the Conjugate Gradient method, for a symmetric, positive-definite A.- Parameters:
- A{sparse array, ndarray, LinearOperator}
- The real or complex N-by-N matrix of the linear system. A must represent a hermitian, positive definite matrix. Alternatively, A can be a linear operator which can produce - Axusing, e.g.,- scipy.sparse.linalg.LinearOperator.
- bndarray
- Right hand side of the linear system. Has shape (N,) or (N,1). 
- x0ndarray
- Starting guess for the solution. 
- rtol, atolfloat, optional
- Parameters for the convergence test. For convergence, - norm(b - A @ x) <= max(rtol*norm(b), atol)should be satisfied. The default is- atol=0.and- rtol=1e-5.
- maxiterinteger
- 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. M must represent a hermitian, positive definite matrix. It should approximate the inverse of A (see Notes). Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. 
- callbackfunction
- User-supplied function to call after each iteration. It is called as - callback(xk), where- xkis the current solution vector.
 
- Returns:
- xndarray
- The converged solution. 
- infointeger
- Provides convergence information:
- 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations 
 
 
 - Notes - The preconditioner M should be a matrix such that - M @ Ahas a smaller condition number than A, see [2].- References [1]- “Conjugate Gradient Method, Wikipedia, https://en.wikipedia.org/wiki/Conjugate_gradient_method [2]- “Preconditioner”, Wikipedia, https://en.wikipedia.org/wiki/Preconditioner - Examples - >>> import numpy as np >>> from scipy.sparse import csc_array >>> from scipy.sparse.linalg import cg >>> P = np.array([[4, 0, 1, 0], ... [0, 5, 0, 0], ... [1, 0, 3, 2], ... [0, 0, 2, 4]]) >>> A = csc_array(P) >>> b = np.array([-1, -0.5, -1, 2]) >>> x, exit_code = cg(A, b, atol=1e-5) >>> print(exit_code) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True