nag_ode_bvp_ps_lin_grid_vals (d02uw) 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 (d02uw) is primarily a utility function to map a set of function values specified on a Chebyshev Gauss–Lobatto grid onto a uniform grid.

Syntax

[xip, fip, ifail] = d02uw(n, nip, x, f)

[xip, fip, ifail] = nag_ode_bvp_ps_lin_grid_vals(n, nip, x, f)

Description

nag_ode_bvp_ps_lin_grid_vals (d02uw) 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 (d02uw) 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 (d02uc) onto an evenly-spaced grid with the same range as the original grid.

References

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

Parameters

Compulsory Input Parameters

1: $n$ – int64int32nag_int scalar: $n$ , where the number of grid points for the input data is $n + 1$ .

Constraint: $n > 0$ and n is even.
2: $nip$ – int64int32nag_int scalar: 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 (d02uw), $fip (1)$ will contain the value $f (x_{n})$ .

Constraint: $nip > 0$ .
3: $x (n + 1)$ – double array: 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 (d02uc).
4: $f (n + 1)$ – double array: The function values, $f (x_{i})$ , for $i = 0, 1, \dots, n$ .

Optional Input Parameters

None.

Output Parameters

1: $xip (nip)$ – double array: The evenly-spaced grid points, ${\hat{x}}_{j}$ , for $j = 1, 2, \dots, m$ .
2: $fip (nip)$ – double array: The set of interpolated values $\hat{f} ({\hat{x}}_{j})$ , for $j = 1, 2, \dots, m$ . Here $\hat{f} ({\hat{x}}_{j}) \approx f (x = {\hat{x}}_{j})$ .
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $n > 0$ .

Constraint: n is even.

$ifail = 2$: Constraint: $nip > 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

nag_ode_bvp_ps_lin_grid_vals (d02uw) 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.

Further Comments

None.

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.

Open in the MATLAB editor: d02uw_example

function d02uw_example


fprintf('d02uw example results\n\n');

n   = int64(64);
a = -1;
b =  1;

% Set up Chebyshev grid
[x, ifail] = d02uc(n, a, b);

% Set up function on grid
f = x + cos(5*x);

% Interpolate onto smaller equally spaced grid
nip = int64(17);
[xip, fip, ifail] = d02uw(n, nip, x, f);

% Display interpolated values
fprintf('\nInterpolated function values\n');
fprintf('      x          F\n');
fprintf('%10.4f %10.4f \n', [xip fip]');

d02uw example results


Interpolated function values
      x          F
   -1.0000    -0.7163 
   -0.8750    -1.2060 
   -0.7500    -1.5706 
   -0.6250    -1.6249 
   -0.5000    -1.3011 
   -0.3750    -0.6745 
   -0.2500     0.0653 
   -0.1250     0.6860 
    0.0000     1.0000 
    0.1250     0.9360 
    0.2500     0.5653 
    0.3750     0.0755 
    0.5000    -0.3011 
    0.6250    -0.3749 
    0.7500    -0.0706 
    0.8750     0.5440 
    1.0000     1.2837

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox