NAG Library Function Document

nag_ode_bvp_ps_lin_grid_vals (d02uwc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_ode_bvp_ps_lin_grid_vals (d02uwc) interpolates from a set of function values on a supplied grid onto a set of values for a uniform grid on the same range. The interpolation is performed using barycentric Lagrange interpolation. nag_ode_bvp_ps_lin_grid_vals (d02uwc) is primarily a utility function to map a set of function values specified on a Chebyshev Gauss–Lobatto grid onto a uniform grid.

2 Specification

#include <nag.h>

#include <nagd02.h>

void	nag_ode_bvp_ps_lin_grid_vals (Integer n, Integer nip, const double x[], const double f[], double xip[], double fip[], NagError *fail)

3 Description

nag_ode_bvp_ps_lin_grid_vals (d02uwc) interpolates from a set of

n + 1

function values,

f (x_{i})

, on a supplied grid,

x_{i}

, for

i = 0, 1, \dots, n

, onto a set of

m

values,

\hat{f} ({\hat{x}}_{j})

, on a uniform grid,

{\hat{x}}_{j}

, for

j = 1, 2, \dots, m

. The image

\hat{x}

has the same range as

x

, so that

{\hat{x}}_{j} = x_{\min} + ((j - 1) / (m - 1)) \times (x_{\max} - x_{\min})

, for

j = 1, 2, \dots, m

. The interpolation is performed using barycentric Lagrange interpolation as described in Berrut and Trefethen (2004).

nag_ode_bvp_ps_lin_grid_vals (d02uwc) is primarily a utility function to map a set of function values specified on a Chebyshev Gauss–Lobatto grid computed by nag_ode_bvp_ps_lin_cgl_grid (d02ucc) onto an evenly-spaced grid with the same range as the original grid.

4 References

Berrut J P and Trefethen L N (2004) Barycentric lagrange interpolation SIAM Rev. 46(3) 501–517

5 Arguments

1: n – IntegerInput: On entry: $n$ , where the number of grid points for the input data is $n + 1$ .
Constraint: $n > 0$ and n is even.
2: nip – IntegerInput: On entry: the number, $m$ , of grid points in the uniform mesh $\hat{x}$ onto which function values are interpolated. If $nip = 1$ then on successful exit from nag_ode_bvp_ps_lin_grid_vals (d02uwc), $fip [0]$ will contain the value $f (x_{n})$ .
Constraint: $nip > 0$ .
3: x[ $n + 1$ ] – const doubleInput: On entry: the grid points, $x_{i}$ , for $i = 0, 1, \dots, n$ , at which the function is specified.
Usually this should be the array of Chebyshev Gauss–Lobatto points returned in nag_ode_bvp_ps_lin_cgl_grid (d02ucc).
4: f[ $n + 1$ ] – const doubleInput: On entry: the function values, $f (x_{i})$ , for $i = 0, 1, \dots, n$ .
5: xip[nip] – doubleOutput: On exit: the evenly-spaced grid points, ${\hat{x}}_{j}$ , for $j = 1, 2, \dots, m$ .
6: fip[nip] – doubleOutput: On exit: the set of interpolated values $\hat{f} ({\hat{x}}_{j})$ , for $j = 1, 2, \dots, m$ . Here $\hat{f} ({\hat{x}}_{j}) \approx f (x = {\hat{x}}_{j})$ .
7: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n > 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: n is even.
On entry, $nip = ⟨value⟩$ .
Constraint: $nip > 0$ .
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.

7 Accuracy

nag_ode_bvp_ps_lin_grid_vals (d02uwc) is intended, primarily, for use with Chebyshev Gauss–Lobatto input grids. For such input grids and for well-behaved functions (no discontinuities, peaks or cusps), the accuracy should be a small multiple of machine precision.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example interpolates the function

x + \cos (5 x)

, as specified on a

65

-point Gauss–Lobatto grid on

[- 1, 1]

, onto a coarse uniform grid.

NAG Library Function Documentnag_ode_bvp_ps_lin_grid_vals (d02uwc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_ode_bvp_ps_lin_grid_vals (d02uwc)