d02uec finds the solution of a linear constant coefficient boundary value problem by using the Chebyshev integration formulation on a Chebyshev Gauss–Lobatto grid.
where $1\le m\le 4$, $B$ is an $m\times (m+1)$ matrix of constant coefficients and ${\beta}_{i}$ are constants. The points ${y}_{i}$ are usually either $a$ or $b$.
The function $f\left(x\right)$ is supplied as an array of Chebyshev coefficients ${c}_{j}$, $j=0,1,\dots ,n$ for the function discretized on $n+1$ Chebyshev Gauss–Lobatto points (as returned by d02ucc); the coefficients are normally obtained by a previous call to d02uac. The solution and its derivatives (up to order $m$) are returned, in the form of their Chebyshev series representation, as arrays of Chebyshev coefficients; subsequent calls to d02ubc will return the corresponding function and derivative values at the Chebyshev Gauss–Lobatto discretization points on $[a,b]$. Function and derivative values can be obtained on any uniform grid over the same range $[a,b]$ by calling the interpolation function d02uwc.
4References
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 Computation65(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 Mathematics234(2) 317–342
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, where the number of grid points is $n+1$.
On entry: the Chebyshev coefficients ${c}_{j}$, $j=0,1,\dots ,n$, for the right-hand side of the boundary value problem. Usually these are obtained by a previous call of d02uac.
Note: the $(i,j)$th element of the matrix is stored in ${\mathbf{bmat}}\left[(j-1)\times {\mathbf{m}}+i-1\right]$.
On entry: ${\mathbf{bmat}}\left[\mathit{j}\times {\mathbf{m}}+\mathit{i}-1\right]$ must contain the coefficients ${B}_{\mathit{i},\mathit{j}+1}$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=0,1,\dots ,m$, in the problem formulation of Section 3.
On exit: the coefficients have been scaled to form an equivalent problem defined on the domain $[\mathrm{-1},1]$.
On entry: the coefficients,
${f}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,m+1$, in the formulation of the linear boundary value problem given in Section 3. The highest order term, ${\mathbf{f}}\left[{\mathbf{m}}\right]$, needs to be nonzero to have a well posed problem.
On exit: the coefficients have been scaled to form an equivalent problem defined on the domain $[\mathrm{-1},1]$.
Note: the $(i,j)$th element of the matrix is stored in ${\mathbf{uc}}\left[(j-1)\times \left({\mathbf{n}}+1\right)+i-1\right]$.
On exit: the Chebyshev coefficients in the Chebyshev series representations of the solution and derivatives of the solution to the boundary value problem. The coefficients of $U$ are stored as the first $n+1$ elements of uc, the first derivative coefficients are stored as the next $n+1$ elements of uc, and so on.
11: $\mathbf{resid}$ – double *Output
On exit: the maximum residual resulting from substituting the solution vectors returned in uc into both linear equations of Section 3 representing the linear boundary value problem and associated boundary conditions. That is
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_CONVERGENCE
During iterative refinement, convergence was achieved, but the residual is more than $100\times \mathit{machineprecision}$. $\text{Residual achieved on convergence}=\u27e8\mathit{\text{value}}\u27e9$.
NE_INT
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $1\le {\mathbf{m}}\le 4$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: n is even.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 8$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Internal error while unpacking matrix during iterative refinement.
Please contact NAG.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_2
On entry, ${\mathbf{a}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{b}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{a}}<{\mathbf{b}}$.
NE_REAL_ARRAY
On entry, ${\mathbf{f}}\left[{\mathbf{m}}\right]=0.0$.
NE_SINGULAR_MATRIX
Singular matrix encountered during iterative refinement.
Please check that your system is well posed.
NE_TOO_MANY_ITER
During iterative refinement, the maximum number of iterations was reached. $\text{Number of iterations}=\u27e8\mathit{\text{value}}\u27e9$ and $\text{residual achieved}=\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
The accuracy should be close to machine precision for well conditioned boundary value problems.
8Parallelism and Performance
d02uec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d02uec makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The number of operations is of the order $n\mathrm{log}\left(n\right)$ and the memory requirements are $\mathit{O}\left(n\right)$; thus the computation remains efficient and practical for very fine discretizations (very large values of $n$). Collocation methods will be faster for small problems, but the method of d02uec should be faster for larger discretizations.
10Example
This example solves the third-order problem $4{U}_{xxx}+3{U}_{xx}+2{U}_{x}+U=2\mathrm{sin}x-2\mathrm{cos}x$ on $[-\pi /2,\pi /2]$ subject to the boundary conditions $U[-\pi /2]=0$, $3{U}_{xx}[-\pi /2]+2{U}_{x}[-\pi /2]+U[-\pi /2]=2$, and $3{U}_{xx}[\pi /2]+2{U}_{x}[\pi /2]+U[\pi /2]=\mathrm{-2}$ using the Chebyshev integration formulation on a Chebyshev Gauss–Lobatto grid of order $16$.