naginterfaces.library.ode.bvp_ps_lin_solve(a, b, c, bmat, y, bvec, f)[source]

bvp_ps_lin_solve finds the solution of a linear constant coefficient boundary value problem by using the Chebyshev integration formulation on a Chebyshev Gauss–Lobatto grid.

For full information please refer to the NAG Library document for d02ue


, the lower bound of domain .


, the upper bound of domain .

cfloat, array-like, shape

The Chebyshev coefficients , , for the right-hand side of the boundary value problem. Usually these are obtained by a previous call of bvp_ps_lin_coeffs().

bmatfloat, array-like, shape

must contain the coefficients , for , for , in the problem formulation of Notes.

yfloat, array-like, shape

The points, , for , where the boundary conditions are discretized.

bvecfloat, array-like, shape

The values, , for , in the formulation of the boundary conditions given in Notes.

ffloat, array-like, shape

The coefficients, , for , in the formulation of the linear boundary value problem given in Notes. The highest order term, , needs to be nonzero to have a well posed problem.

bmatfloat, ndarray, shape

The coefficients have been scaled to form an equivalent problem defined on the domain .

ffloat, ndarray, shape

The coefficients have been scaled to form an equivalent problem defined on the domain .

ucfloat, ndarray, shape

The Chebyshev coefficients in the Chebyshev series representations of the solution and derivatives of the solution to the boundary value problem. The elements contain the coefficients representing the solution , for . contains the coefficients representing the th derivative of , for .


The maximum residual resulting from substituting the solution vectors returned in into the equations representing the linear boundary value problem and associated boundary conditions.

(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: is even.

(errno )

On entry, and .

Constraint: .

(errno )

On entry, .

(errno )

On entry, .

Constraint: .

(errno )

Internal error while unpacking matrix during iterative refinement.

Please contact NAG.

(errno )

Singular matrix encountered during iterative refinement.

Please check that your system is well posed.

(errno )

During iterative refinement, the maximum number of iterations was reached.

and .

(errno )

During iterative refinement, convergence was achieved, but the residual is more than . .


bvp_ps_lin_solve solves the constant linear coefficient ordinary differential problem

subject to a set of linear constraints at points , for :

where , is an matrix of constant coefficients and are constants. The points are usually either or .

The function is supplied as an array of Chebyshev coefficients , for the function discretized on Chebyshev Gauss–Lobatto points (as returned by bvp_ps_lin_cgl_grid()); the coefficients are normally obtained by a previous call to bvp_ps_lin_coeffs(). The solution and its derivatives (up to order ) are returned, in the form of their Chebyshev series representation, as arrays of Chebyshev coefficients; subsequent calls to bvp_ps_lin_cgl_vals() will return the corresponding function and derivative values at the Chebyshev Gauss–Lobatto discretization points on . Function and derivative values can be obtained on any uniform grid over the same range by calling the interpolation function bvp_ps_lin_grid_vals().


Clenshaw, C W, 1957, The numerical solution of linear differential equations in Chebyshev series, Proc. Camb. Phil. Soc. (53), 134–149

Coutsias, E A, Hagstrom, T and Torres, D, 1996, An efficient spectral method for ordinary differential equations with rational function coefficients, Mathematics of Computation (65(214)), 611–635

Greengard, L, 1991, Spectral integration and two-point boundary value problems, SIAM J. Numer. Anal. (28(4)), 1071–80

Lundbladh, A, Hennigson, D S and Johannson, A V, 1992, An efficient spectral integration method for the solution of the Navier–Stokes equations, Technical report FFA–TN, 1992–28, Aeronautical Research Institute of Sweden

Muite, B K, 2010, A numerical comparison of Chebyshev methods for solving fourth-order semilinear initial boundary value problems, Journal of Computational and Applied Mathematics (234(2)), 317–342