d02uwf 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. d02uwf is primarily a utility routine to map a set of function values specified on a Chebyshev Gauss–Lobatto grid onto a uniform grid.

2 Specification

Fortran Interface

Subroutine d02uwf (

n, nip, x, f, xip, fip, ifail)

Integer, Intent (In)	::	n, nip
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(n+1), f(n+1)
Real (Kind=nag_wp), Intent (Out)	::	xip(nip), fip(nip)

C Header Interface

#include <nag.h>

void	d02uwf_ (const Integer n, const Integer nip, const double x[], const double f[], double xip[], double fip[], Integer *ifail)

The routine may be called by the names d02uwf or nagf_ode_bvp_ps_lin_grid_vals.

3 Description

d02uwf 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).

d02uwf is primarily a utility routine to map a set of function values specified on a Chebyshev Gauss–Lobatto grid computed by d02ucf 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: $n$ – Integer Input: On entry: $n$ , where the number of grid points for the input data is $n + 1$ .

Constraint: $n > 0$ and n is even.
2: $nip$ – Integer Input: 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 d02uwf, $fip (1)$ will contain the value $f (x_{n})$ .

Constraint: $nip > 0$ .
3: $x (n + 1)$ – Real (Kind=nag_wp) array Input: 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 d02ucf.
4: $f (n + 1)$ – Real (Kind=nag_wp) array Input: On entry: the function values, $f (x_{i})$ , for $i = 0, 1, \dots, n$ .
5: $xip (nip)$ – Real (Kind=nag_wp) array Output: On exit: the evenly-spaced grid points, ${\hat{x}}_{j}$ , for $j = 1, 2, \dots, m$ .
6: $fip (nip)$ – Real (Kind=nag_wp) array Output: 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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n > 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: n is even.

$ifail = 2$: On entry, $nip = ⟨ value ⟩$ .
Constraint: $nip > 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

d02uwf 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

Background information to multithreading can be found in the Multithreading documentation.

d02uwf is not threaded in any implementation.

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.

d02uw: FL CL CPP AD PY MB

NAG FL Interfaced02uwf (bvp_​ps_​lin_​grid_​vals)

▸▿ 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 FL Interface
d02uwf (bvp_ps_lin_grid_vals)