`scipy.integrate._bvp:solve_newton`

source: /scipy/integrate/_bvp.py :348

Signature

   def     solve_newton (    n    ,    m    ,    h    ,    col_fun    ,    bc    ,    jac    ,    y    ,    p    ,    B    ,    bvp_tol    ,    bc_tol      )

Summary

Solve the nonlinear collocation system by a Newton method.

Extended Summary

This is a simple Newton method with a backtracking line search. As advised in ^[1], an affine-invariant criterion function F = ||J^-1 r||^2 is used, where J is the Jacobian matrix at the current iteration and r is the vector or collocation residuals (values of the system lhs).

The method alters between full Newton iterations and the fixed-Jacobian iterations based

There are other tricks proposed in ^[1], but they are not used as they don't seem to improve anything significantly, and even break the convergence on some test problems I tried.

All important parameters of the algorithm are defined inside the function.

Parameters

n : int: Number of equations in the ODE system.
m : int: Number of nodes in the mesh.
h : ndarray, shape (m-1,): Mesh intervals.
col_fun : callable: Function computing collocation residuals.
bc : callable: Function computing boundary condition residuals.
jac : callable: Function computing the Jacobian of the whole system (including collocation and boundary condition residuals). It is supposed to return csc_matrix.
y : ndarray, shape (n, m): Initial guess for the function values at the mesh nodes.
p : ndarray, shape (k,): Initial guess for the unknown parameters.
B : ndarray with shape (n, n) or None: Matrix to force the S y(a) = 0 condition for a problems with the singular term. If None, the singular term is assumed to be absent.
bvp_tol : float: Tolerance to which we want to solve a BVP.
bc_tol : float: Tolerance to which we want to satisfy the boundary conditions.

Returns

y : ndarray, shape (n, m): Final iterate for the function values at the mesh nodes.
p : ndarray, shape (k,): Final iterate for the unknown parameters.
singular : bool: True, if the LU decomposition failed because Jacobian turned out to be singular.

Aliases

scipy.integrate._bvp.solve_newton