NFFT: FPT - Fast polynomial transform

NFFT 3.5.3alpha

This module implements fast polynomial transforms. More...

Macros
#define	FPT_NO_FAST_ALGORITHM (1U << 2)
	If set, TODO complete comment.

#define	FPT_NO_DIRECT_ALGORITHM (1U << 3)
	If set, TODO complete comment.

#define	FPT_NO_STABILIZATION (1U << 0)
	If set, no stabilization will be used.

#define	FPT_PERSISTENT_DATA (1U << 4)
	If set, TODO complete comment.

#define	FPT_FUNCTION_VALUES (1U << 5)
	If set, the output are function values at Chebyshev nodes rather than Chebyshev coefficients.

#define	FPT_AL_SYMMETRY (1U << 6)
	If set, TODO complete comment.

Detailed Description

This module implements fast polynomial transforms.

In the following, we abbreviate the term "fast polynomial transforms" by FPT.

Let $\alpha_n,\;\beta_n,\;\gamma_n,\;n=0,\dots,N$ be given recursion coefficients of the polynomials $P_n$ defined by $P_{-1}(x) = 0$ , $P_{0}(x) = 1$ and

$P_n(x) = (\alpha_nx+\beta_n) P_{n-1}(x) + \gamma_n P_{n-2}(x) ,\qquad n=1,2,\dots$

for $x\in[-1,1]$ . The Chebyshev polnyomials of the first kind are defined by

$T_n(x) = \cos(n\, \arccos x).$

Let $f\colon [-1,1]\to\mathbb R$ be a polynomial of degree $N\in\mathbb N$ . The FPT transforms the polynomial coefficients $[x_n]_{n=0..N}$ from

$f = \sum_{n=0}^N x_n P_n$

into Chebyshev coefficients $[y_n]_{n=0..N}$ from

$f = \sum_{n=0}^N y_n T_n.$

Macro Definition Documentation

◆ FPT_NO_FAST_ALGORITHM

#define FPT_NO_FAST_ALGORITHM (1U << 2)

If set, TODO complete comment.

Definition at line 644 of file nfft3.h.

◆ FPT_NO_DIRECT_ALGORITHM

#define FPT_NO_DIRECT_ALGORITHM (1U << 3)

If set, TODO complete comment.

Definition at line 645 of file nfft3.h.

◆ FPT_NO_STABILIZATION

#define FPT_NO_STABILIZATION (1U << 0)

If set, no stabilization will be used.

Definition at line 643 of file nfft3.h.

◆ FPT_PERSISTENT_DATA

#define FPT_PERSISTENT_DATA (1U << 4)

If set, TODO complete comment.

Definition at line 646 of file nfft3.h.

◆ FPT_FUNCTION_VALUES

#define FPT_FUNCTION_VALUES (1U << 5)

If set, the output are function values at Chebyshev nodes rather than Chebyshev coefficients.

Definition at line 650 of file nfft3.h.

◆ FPT_AL_SYMMETRY

#define FPT_AL_SYMMETRY (1U << 6)

If set, TODO complete comment.

Definition at line 651 of file nfft3.h.